Abstract: A locally testable code is an error-correcting code that admits very efficient probabilistic tests of membership. Tensor codes provide a simple family of combinatorial constructions of locally testable codes that generalize the family of Reed-Muller codes. The natural test for tensor codes, the axis-parallel line vs. point test, plays an essential role in constructions of probabilistically checkable proofs. We analyze the axis-parallel line vs. point test as a two-prover game and show that the test is sound against quantum provers sharing entanglement. Our result implies the quantum-soundness of the low individual degree test, which is an essential component of the MIP* = RE theorem. Our proof also generalizes to the infinite-dimensional commuting-operator model of quantum provers.

Publication: arXiv
ID: CaltechAUTHORS:20220202-191902193

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Abstract: The study of quantum correlation sets initiated by Tsirelson in the 1980s and originally motivated by questions in the foundations of quantum mechanics has more recently been tied to questions in quantum cryptography, complexity theory, operator space theory, group theory, and more. Synchronous correlation sets introduced by Paulsen et al. [J. Funct. Anal. 270, 2188–2222 (2016)] are a subclass of correlations that has proven particularly useful to study and arises naturally in applications. We show that any correlation that is almost synchronous, in a natural ℓ1 sense, arises from a state and measurement operators that are well-approximated by a convex combination of projective measurements on a maximally entangled state. This extends a result of Paulsen et al. [J. Funct. Anal. 270, 2188–2222 (2016)] that applies to exactly synchronous correlations. Crucially, the quality of approximation is independent of the dimension of the Hilbert spaces or of the size of the correlation. Our result allows one to reduce the analysis of many classes of nonlocal games, including rigidity properties, to the case of strategies using maximally entangled states that are generally easier to manipulate.

Publication: Journal of Mathematical Physics Vol.: 63 No.: 2 ISSN: 0022-2488

ID: CaltechAUTHORS:20211006-163212999

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Abstract: Achieving quantum computational advantage requires solving a classically intractable problem on a quantum device. Natural proposals rely upon the intrinsic hardness of classically simulating quantum mechanics; however, verifying the output is itself classically intractable. On the other hand, certain quantum algorithms (e.g. prime factorization via Shor's algorithm) are efficiently verifiable, but require more resources than what is available on near-term devices. One way to bridge the gap between verifiability and implementation is to use "interactions" between a prover and a verifier. By leveraging cryptographic functions, such protocols enable the classical verifier to enforce consistency in a quantum prover's responses across multiple rounds of interaction. In this work, we demonstrate the first implementation of an interactive quantum advantage protocol, using an ion trap quantum computer. We execute two complementary protocols -- one based upon the learning with errors problem and another where the cryptographic construction implements a computational Bell test. To perform multiple rounds of interaction, we implement mid-circuit measurements on a subset of trapped ion qubits, with subsequent coherent evolution. For both protocols, the performance exceeds the asymptotic bound for classical behavior; maintaining this fidelity at scale would conclusively demonstrate verifiable quantum advantage.

Publication: arXiv
ID: CaltechAUTHORS:20220202-191905591

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Abstract: The complexity class NP characterizes the collection of computational problems that have efficiently verifiable solutions. With the goal of classifying computational problems that seem to lie beyond NP, starting in the 1980s complexity theorists have considered extensions of the notion of efficient verification that allow for the use of randomness (the class MA), interaction (the class IP), and the possibility to interact with multiple proofs, or provers (the class MIP). The study of these extensions led to the celebrated PCP theorem and its applications to hardness of approximation and the design of cryptographic protocols. In this work, we study a fourth modification to the notion of efficient verification that originates in the study of quantum entanglement. We prove the surprising result that every problem that is recursively enumerable, including the Halting problem, can be efficiently verified by a classical probabilistic polynomial-time verifier interacting with two all-powerful but noncommunicating provers sharing entanglement. The result resolves long-standing open problems in the foundations of quantum mechanics (Tsirelson's problem) and operator algebras (Connes' embedding problem).

Publication: Communications of the ACM Vol.: 64 No.: 11 ISSN: 0001-0782

ID: CaltechAUTHORS:20200417-131646685

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Abstract: Self-testing is a method to characterise an arbitrary quantum system based only on its classical input-output correlations, and plays an important role in device-independent quantum information processing as well as quantum complexity theory. Prior works on self-testing require the assumption that the system's state is shared among multiple parties that only perform local measurements and cannot communicate. Here, we replace the setting of multiple non-communicating parties, which is difficult to enforce in practice, by a single computationally bounded party. Specifically, we construct a protocol that allows a classical verifier to robustly certify that a single computationally bounded quantum device must have prepared a Bell pair and performed single-qubit measurements on it, up to a change of basis applied to both the device's state and measurements. This means that under computational assumptions, the verifier is able to certify the presence of entanglement, a property usually closely associated with two separated subsystems, inside a single quantum device. To achieve this, we build on techniques first introduced by Brakerski et al. (2018) and Mahadev (2018) which allow a classical verifier to constrain the actions of a quantum device assuming the device does not break post-quantum cryptography.

Publication: Quantum Vol.: 5ISSN: 2521-327X

ID: CaltechAUTHORS:20200417-132557882

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Abstract: We consider a new model for the testing of untrusted quantum devices, consisting of a single polynomial time bounded quantum device interacting with a classical polynomial time verifier. In this model, we propose solutions to two tasks—a protocol for efficient classical verification that the untrusted device is “truly quantum” and a protocol for producing certifiable randomness from a single untrusted quantum device. Our solution relies on the existence of a new cryptographic primitive for constraining the power of an untrusted quantum device: post-quantum secure trapdoor claw-free functions that must satisfy an adaptive hardcore bit property. We show how to construct this primitive based on the hardness of the learning with errors (LWE) problem.

Publication: Journal of the ACM Vol.: 68 No.: 5 ISSN: 0004-5411

ID: CaltechAUTHORS:20210921-144712064

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Abstract: We establish a strong monogamy-of-entanglement property for subspace coset states, which are uniform superpositions of vectors in a linear subspace of F^n₂ to which has been applied a quantum one-time pad. This property was conjectured recently by [Coladangelo, Liu, Liu, and Zhandry, Crypto'21] and shown to have applications to unclonable decryption and copy-protection of pseudorandom functions. We present two proofs, one which directly follows the method of the original paper and the other which uses an observation from [Vidick and Zhang, Eurocrypt'20] to reduce the analysis to a simpler monogamy game based on BB'84 states. Both proofs ultimately rely on the same proof technique, introduced in [Tomamichel, Fehr, Kaniewski and Wehner, New Journal of Physics '13].

Publication: arXiv
ID: CaltechAUTHORS:20211006-152638528

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Abstract: The generation of certifiable randomness is the most fundamental information-theoretic task that meaningfully separates quantum devices from their classical counterparts. We propose a protocol for exponential certified randomness expansion using a single quantum device. The protocol calls for the device to implement a simple quantum circuit of constant depth on a 2D lattice of qubits. The output of the circuit can be verified classically in linear time, and is guaranteed to contain a polynomial number of certified random bits assuming that the device used to generate the output operated using a (classical or quantum) circuit of sub-logarithmic depth. This assumption contrasts with the locality assumption used for randomness certification based on Bell inequality violation and more recent proposals for randomness certification based on computational assumptions. Furthermore, to demonstrate randomness generation it is sufficient for a device to sample from the ideal output distribution within constant statistical distance. Our procedure is inspired by recent work of Bravyi et al. (Science 362(6412):308–311, 2018), who introduced a relational problem that can be solved by a constant-depth quantum circuit, but provably cannot be solved by any classical circuit of sub-logarithmic depth. We develop the discovery of Bravyi et al. into a framework for robust randomness expansion. Our results lead to a new proposal for a demonstrated quantum advantage that has some advantages compared to existing proposals. First, our proposal does not rest on any complexity-theoretic conjectures, but relies on the physical assumption that the adversarial device being tested implements a circuit of sub-logarithmic depth. Second, success on our task can be easily verified in classical linear time. Finally, our task is more noise-tolerant than most other existing proposals that can only tolerate multiplicative error, or require additional conjectures from complexity theory; in contrast, we are able to allow a small constant additive error in total variation distance between the sampled and ideal distributions.

Publication: Communications in Mathematical Physics Vol.: 382 No.: 1 ISSN: 0010-3616

ID: CaltechAUTHORS:20190320-100502117

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Abstract: We introduce a three-player nonlocal game, with a finite number of classical questions and answers, such that the optimal success probability of 1 in the game can only be achieved in the limit of strategies using arbitrarily high-dimensional entangled states. Precisely, there exists a constant 0 < c ≤ 1 such that to succeed with probability 1 − ε in the game it is necessary to use an entangled state of at leastΩ(ε^(−c)) qubits, and it is sufficient to use a state of at most O(ε⁻¹) qubits. The game is based on the coherent state exchange game of Leung et al. (CJTCS 2013). In our game, the task of the quantum verifier is delegated to a third player by a classical referee. Our results complement those of Slofstra (arXiv:1703.08618) and Dykema et al. (arXiv:1709.05032), who obtained two-player games with similar (though quantitatively weaker) properties based on the representation theory of finitely presented groups and C∗-algebras respectively.

Publication: Quantum Vol.: 4ISSN: 2521-327X

ID: CaltechAUTHORS:20190204-154622144

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Abstract: Low degree tests play an important role in classical complexity theory, serving as basic ingredients in foundational results such as MIP = NEXP [BFL91] and the PCP theorem [AS98,ALM+98]. Over the last ten years, versions of these tests which are sound against quantum provers have found increasing applications to the study of nonlocal games and the complexity class MIP^*. The culmination of this line of work is the result MIP^* = RE [arXiv:2001.04383]. One of the key ingredients in the first reported proof of MIP^* = RE is a two-prover variant of the low degree test, initially shown to be sound against multiple quantum provers in [arXiv:1302.1242]. Unfortunately a mistake was recently discovered in the latter result, invalidating the main result of [arXiv:1302.1242] as well as its use in subsequent works, including [arXiv:2001.04383]. We analyze a variant of the low degree test called the low individual degree test. Our main result is that the two-player version of this test is sound against quantum provers. This soundness result is sufficient to re-derive several bounds on MIP^* that relied on [arXiv:1302.1242], including MIP^* = RE.

Publication: arXiv
ID: CaltechAUTHORS:20211004-222652076

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Abstract: For all n ≥ 1, we give an explicit construction of m × m matrices A_1,…,A_n with m = 2^([n/2]) such that for any d and d × d matrices A′_1,…,A′_n that satisfy ∥A_′i−A′_j∥S_1 ≤ ∥A_i−A_j∥S_1 ≤ (1+δ)∥A′_i−A′_j∥S_1 for all i,j∈{1,…,n} and small enough δ = O(n^(−c)), where c > 0 is a universal constant, it must be the case that d ≥ 2^([n/2]−1). This stands in contrast to the metric theory of commutative ℓ_p spaces, as it is known that for any p ≥ 1, any n points in ℓ_p embed exactly in ℓ^d_p for d = n(n−1)/2. Our proof is based on matrices derived from a representation of the Clifford algebra generated by n anti-commuting Hermitian matrices that square to identity, and borrows ideas from the analysis of nonlocal games in quantum information theory.

No.: 2266
ID: CaltechAUTHORS:20190320-095834301

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Abstract: A proof of quantumness is a method for provably demonstrating (to a classical verifier) that a quantum device can perform computational tasks that a classical device with comparable resources cannot. Providing a proof of quantumness is the first step towards constructing a useful quantum computer. There are currently three approaches for exhibiting proofs of quantumness: (i) Inverting a classically-hard one-way function (e.g. using Shor’s algorithm). This seems technologically out of reach. (ii) Sampling from a classically-hard-to-sample distribution (e.g. BosonSampling). This may be within reach of near-term experiments, but for all such tasks known verification requires exponential time. (iii) Interactive protocols based on cryptographic assumptions. The use of a trapdoor scheme allows for efficient verification, and implementation seems to require much less resources than (i), yet still more than (ii). In this work we propose a significant simplification to approach (iii) by employing the random oracle heuristic. (We note that we do not apply the Fiat-Shamir paradigm.) We give a two-message (challenge-response) proof of quantumness based on any trapdoor claw-free function. In contrast to earlier proposals we do not need an adaptive hard-core bit property. This allows the use of smaller security parameters and more diverse computational assumptions (such as Ring Learning with Errors), significantly reducing the quantum computational effort required for a successful demonstration.

Publication: arXiv
ID: CaltechAUTHORS:20200728-144326318

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Abstract: We show that every language in QMA admits a classical-verifier, quantum-prover zero-knowledge argument system which is sound against quantum polynomial-time provers and zero-knowledge for classical (and quantum) polynomial-time verifiers. The protocol builds upon two recent results: a computational zero-knowledge proof system for languages in QMA, with a quantum verifier, introduced by Broadbent et al. (FOCS 2016), and an argument system for languages in QMA, with a classical verifier, introduced by Mahadev (FOCS 2018).

Publication: Quantum Vol.: 4ISSN: 2521-327X

ID: CaltechAUTHORS:20190320-095213331

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Abstract: We define the notion of a proof of knowledge in the setting where the verifier is classical, but the prover is quantum, and where the witness that the prover holds is in general a quantum state. We establish simple properties of our definition, including that nondestructive classical proofs of quantum knowledge are impossible for nontrivial states, and that, under certain conditions on the parameters in our definition, a proof of knowledge protocol for a hard-to-clone state can be used as a (destructive) quantum money verification protocol. In addition, we provide two examples of protocols (both inspired by private-key classical verification protocols for quantum money schemes) which we can show to be proofs of quantum knowledge under our definition. In so doing, we introduce new techniques for the analysis of such protocols which build on results from the literature on nonlocal games. Finally, we show that, under our definition, the verification protocol introduced by Mahadev (FOCS 2018) is a classical argument of quantum knowledge for QMA relations.

Publication: arXiv
ID: CaltechAUTHORS:20200728-145122122

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Abstract: Rapid technological advances point to a near future where engineered devices based on the laws of quantum mechanics are able to implement computations that can no longer be emulated on a classical computer. Once that stage is reached, will it be possible to verify the results of the quantum device? Recently, Mahadev introduced a solution to the following problem: Is it possible to delegate a quantum computation to a quantum device in a way that the final outcome of the computation can be verified on a classical computer, given that the device may be faulty or adversarial and given only the ability to generate classical instructions and obtain classical readout information in return? Mahadev's solution combines the framework of interactive proof systems from complexity theory with an ingenious use of classical cryptographic techniques to tie a "cryptographic leash'' around the quantum device. In these notes I give a self-contained introduction to her elegant solution, explaining the required concepts from complexity, quantum computing, and cryptography, and how they are brought together in Mahadev's protocol for classical verification of quantum computations.

Publication: Bulletin of the American Mathematical Society Vol.: 57 No.: 1 ISSN: 0273-0979

ID: CaltechAUTHORS:20200316-150528835

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Abstract: A non-interactive zero-knowledge (NIZK) proof system for a language L∈NP allows a prover (who is provided with an instance x∈L, and a witness w for x) to compute a classical certificate π for the claim that x∈L such that π has the following properties: 1) π can be verified efficiently, and 2) π does not reveal any information about w, besides the fact that it exists (i.e. that x∈L). NIZK proof systems have recently been shown to exist for all languages in NP in the common reference string (CRS) model and under the learning with errors (LWE) assumption. We initiate the study of NIZK arguments for languages in QMA. Our first main result is the following: if LWE is hard for quantum computers, then any language in QMA has an NIZK argument with preprocessing. The preprocessing in our argument system consists of (i) the generation of a CRS and (ii) a single (instance-independent) quantum message from verifier to prover. The instance-dependent phase of our argument system involves only a single classical message from prover to verifier. Importantly, verification in our protocol is entirely classical, and the verifier needs not have quantum memory; its only quantum actions are in the preprocessing phase. Our second contribution is to extend the notion of a classical proof of knowledge to the quantum setting. We introduce the notions of arguments and proofs of quantum knowledge (AoQK/PoQK), and we show that our non-interactive argument system satisfies the definition of an AoQK. In particular, we explicitly construct an extractor which can recover a quantum witness from any prover who is successful in our protocol. We also show that any language in QMA has an (interactive) proof of quantum knowledge.

Publication: arXiv
ID: CaltechAUTHORS:20200110-140701565

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Abstract: We introduce a protocol between a classical polynomial-time verifier and a quantum polynomial-time prover that allows the verifier to securely delegate to the prover the preparation of certain single-qubit quantum states The prover is unaware of which state he received and moreover, the verifier can check with high confidence whether the preparation was successful. The delegated preparation of single-qubit states is an elementary building block in many quantum cryptographic protocols. We expect our implementation of "random remote state preparation with verification", a functionality first defined in (Dunjko and Kashefi 2014), to be useful for removing the need for quantum communication in such protocols while keeping functionality. The main application that we detail is to a protocol for blind and verifiable delegated quantum computation (DQC) that builds on the work of (Fitzsimons and Kashefi 2018), who provided such a protocol with quantum communication. Recently, both blind an verifiable DQC were shown to be possible, under computational assumptions, with a classical polynomial-time client (Mahadev 2017, Mahadev 2018). Compared to the work of Mahadev, our protocol is more modular, applies to the measurement-based model of computation (instead of the Hamiltonian model) and is composable. Our proof of security builds on ideas introduced in (Brakerski et al. 2018).

ID: CaltechAUTHORS:20200109-143243905

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Abstract: Quantum mechanics and the theory of operator algebras have been intertwined since their origin. In the 1930s [20] von Neumann laid the foundations for the theory of (what are now known as) von Neumann algebras with the explicit goal of establishing Heisenberg’s matrix mechanics on a rigorous footing (quoting from the preface, in the translation by Beyer: “The object of this book is to present the new quantum mechanics in a unified representation which, so far as it is possible and useful, is mathematically rigorous”). Following the initial explorations of Murray and von Neumann, the new theory took on a life of its own, eventually leading to multiple applications unrelated to quantum mechanics, such as to free probability or noncommutative geometry.

Publication: Notices of the American Mathematical Society Vol.: 66 No.: 10 ISSN: 0002-9920

ID: CaltechAUTHORS:20200728-152043230

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Abstract: We show that any language solvable in nondeterministic time exp( exp(⋯exp(n))), where the number of iterated exponentials is an arbitrary function R(n), can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness 1 and soundness 1 − exp(−Cexp(⋯exp(n))), where the number of iterated exponentials is R(n)−1 and C>0 is a universal constant. The result was previously known for R=1 and R=2; we obtain it for any time-constructible function R. The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC’17). As a separate consequence of this technique we obtain a different proof of Slofstra’s recent result on the uncomputability of the entangled value of multiprover games (Forum of Mathematics, Pi 2019). Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP* contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelson’s problem on the relation between the commuting operator and tensor product models for quantum correlations.

Publication: arXiv
ID: CaltechAUTHORS:20190204-112657116

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Abstract: In privacy amplification, two mutually trusted parties aim to amplify the secrecy of an initial shared secret X in order to establish a shared private key K by exchanging messages over an insecure communication channel. If the channel is authenticated the task can be solved in a single round of communication using a strong randomness extractor; choosing a quantum-proof extractor allows one to establish security against quantum adversaries. In the case that the channel is not authenticated, this simple solution is no longer secure. Nevertheless, Dodis and Wichs (STOC’09) showed that the problem can be solved in two rounds of communication using a non-malleable extractor, a stronger pseudo-random construction than a strong extractor. We give the first construction of a non-malleable extractor that is secure against quantum adversaries. The extractor is based on a construction by Li (FOCS’12), and is able to extract from source of min-entropy rates larger than 1 / 2. Combining this construction with a quantum-proof variant of the reduction of Dodis and Wichs, due to Cohen and Vidick (unpublished) we obtain the first privacy amplification protocol secure against active quantum adversaries.

Publication: arXiv No.: 11477
ID: CaltechAUTHORS:20190320-102401828

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Abstract: The problem of reliably certifying the outcome of a computation performed by a quantum device is rapidly gaining relevance. We present two protocols for a classical verifier to verifiably delegate a quantum computation to two non-communicating but entangled quantum provers. Our protocols have near-optimal complexity in terms of the total resources employed by the verifier and the honest provers, with the total number of operations of each party, including the number of entangled pairs of qubits required of the honest provers, scaling as O(g\log g) for delegating a circuit of size g. This is in contrast to previous protocols, whose overhead in terms of resources employed, while polynomial, is far beyond what is feasible in practice. Our first protocol requires a number of rounds that is linear in the depth of the circuit being delegated, and is blind, meaning neither prover can learn the circuit or its input. The second protocol is not blind, but requires only a constant number of rounds of interaction. Our main technical innovation is an efficient rigidity theorem which allows a verifier to test that two entangled provers perform measurements specified by an arbitrary m-qubit tensor product of single-qubit Clifford observables on their respective halves of m shared EPR pairs, with a robustness that is independent of m. Our two-prover classical-verifier delegation protocols are obtained by combining this rigidity theorem with a single-prover quantum-verifier protocol for the verifiable delegation of a quantum computation, introduced by Broadbent.

Publication: arXiv No.: 11478
ID: CaltechAUTHORS:20190320-123759874

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Abstract: Quantum cryptography promises levels of security that are impossible to attain in a classical world. Can this security be guaranteed to classical users of a quantum protocol, who may not even trust the quantum devices used to implement the protocol? This central question dates back to the early 1990s when the challenge of achieving Device-Independent Quantum Key Distribution (DIQKD) was first formulated. We answer the challenge by rigorously proving the device-independent security of an entanglement-based protocol building on Ekert's original proposal for quantum key distribution. The proof of security builds on techniques from the classical theory of pseudo-randomness to achieve a new quantitative understanding of the non-local nature of quantum correlations.

Publication: Communications of the ACM Vol.: 62 No.: 4 ISSN: 0001-0782

ID: CaltechAUTHORS:20190321-152633091

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Abstract: Device-independent security is the gold standard for quantum cryptography: not only is security based entirely on the laws of quantum mechanics, but it holds irrespective of any a priori assumptions on the quantum devices used in a protocol, making it particularly applicable in a quantum-wary environment. While the existence of device-independent protocols for tasks such as randomness expansion and quantum key distribution has recently been established, the underlying proofs of security remain very challenging, yield rather poor key rates, and demand very high quality quantum devices, thus making them all but impossible to implement in practice. We introduce a technique for the analysis of device-independent cryptographic protocols. We provide a flexible protocol and give a security proof that provides quantitative bounds that are asymptotically tight, even in the presence of general quantum adversaries. At a high level our approach amounts to establishing a reduction to the scenario in which the untrusted device operates in an identical and independent way in each round of the protocol. This is achieved by leveraging the sequential nature of the protocol and makes use of a newly developed tool, the “entropy accumulation theorem” of Dupuis, Fawzi, and Renner [Entropy Accumulation, preprint, 2016]. As concrete applications we give simple and modular security proofs for device-independent quantum key distribution and randomness expansion protocols based on the CHSH inequality. For both tasks, we establish essentially optimal asymptotic key rates and noise tolerance. In view of recent experimental progress, which has culminated in loophole-free Bell tests, it is likely that these protocols can be practically implemented in the near future.

Publication: SIAM Journal on Computing Vol.: 48 No.: 1 ISSN: 0097-5397

ID: CaltechAUTHORS:20190206-150209557

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Abstract: We give a protocol for producing certifiable randomness from a single untrusted quantum device that is polynomial-time bounded. The randomness is certified to be statistically close to uniform from the point of view of any computationally unbounded quantum adversary, that may share entanglement with the quantum device. The protocol relies on the existence of post-quantum secure trapdoor claw-free functions, and introduces a new primitive for constraining the power of an untrusted quantum device. We then show how to construct this primitive based on the hardness of the learning with errors (LWE) problem. The randomness protocol can also be used as the basis for an efficiently verifiable "quantum supremacy" proposal, thus answering an outstanding challenge in the field.

ID: CaltechAUTHORS:20190201-143229032

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Abstract: We relate the amount of entanglement required to play linear system non-local games near-optimally to the hyperlinear profile of finitely presented groups. By calculating the hyperlinear profile of a certain group, we give an example of a finite non-local game for which the amount of entanglement required to play ϵ-optimally is at least Ω(1/ϵ^k), f or some k > 0. Since this function approaches infinity as ϵ approaches zero, this provides a quantitative version of a theorem of the first author.

Publication: Annales Henri Poincaré Vol.: 19 No.: 10 ISSN: 1424-0637

ID: CaltechAUTHORS:20180926-132554192

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Abstract: We show that given an explicit description of a multiplayer game, with a classical verifier and a constant number of players, it is QMA-hard, under randomized reductions, to distinguish between the cases when the players have a strategy using entanglement that succeeds with probability 1 in the game, or when no such strategy succeeds with probability larger than 1/2. This proves the “games quantum PCP conjecture” of Fitzsimons and the second author (ITCS'15), albeit under randomized reductions. The core component in our reduction is a construction of a family of two-player games for testing n-qubit maximally entangled states. For any integer n ≥ 2, we give such a game in which questions from the verifier are O(log n) bits long, and answers are poly(loglogn) bits long. We show that for any constant ε ≥ 0, any strategy that succeeds with probability at least 1 - ε in the test must use a state that is within distance δ(ε) = O(ε c ) from a state that is locally equivalent to a maximally entangled state on n qubits, for some universal constant c > 0. The construction is based on the classical plane-vs-point test for multivariate low-degree polynomials of Raz and Safra (STOC'97). We extend the classical test to the quantum regime by executing independent copies of the test in the generalized Pauli X and Z bases over Fq, where q is a sufficiently large prime power, and combine the two through a test for the Pauli twisted commutation relations. Our main complexity-theoretic result is obtained by combining this family of games with techniques from the classical PCP literature. More specifically, we use constructions of PCPs of proximity introduced by Ben-Sasson et al. (CCC'05), and crucially rely on a linear property of such PCPs. Another consequence of our results is a deterministic reduction from the games quantum PCP conjecture to a suitable formulation of the constraint satisfaction quantum PCP conjecture.

ID: CaltechAUTHORS:20190201-143229217

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Abstract: Bell-inequality violations establish that two systems share some quantum entanglement. We give a simple test to certify that two systems share an asymptotically large amount of entanglement, n EPR states. The test is efficient: unlike earlier tests that play many games, in sequence or in parallel, our test requires only one or two CHSH games. One system is directed to play a CHSH game on a random specified qubit i, and the other is told to play games on qubits {i,j}, without knowing which index is i. The test is robust: a success probability within delta of optimal guarantees distance O(n^{5/2} sqrt{delta}) from n EPR states. However, the test does not tolerate constant delta; it breaks down for delta = Omega~(1/sqrt{n}). We give an adversarial strategy that succeeds within delta of the optimum probability using only O~(delta^{-2}) EPR states.

Publication: Quantum Vol.: 2ISSN: 2521-327X

ID: CaltechAUTHORS:20171108-142443122

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Abstract: We show that for any ε > 0 there is an XOR game G = G(ε) with Θ(ε^(−1/5)) inputs for one player and Θ(ε^(−2/5)) inputs for the other player such that Ω(ε^(−1/5)) ebits are required for any strategy achieving bias that is at least a multiplicative factor (1−ε) from optimal. This gives an exponential improvement in both the number of inputs or outputs and the noise tolerance of any previously-known self-test for highly entangled states. Up to the exponent −1/5 the scaling of our bound with ε is tight: for any XOR game there is an ε-optimal strategy using ⌈ε^(−1)⌉ ebits, irrespective of the number of questions in the game.

Publication: Quantum Information and Computation Vol.: 18 No.: 7-8 ISSN: 1533-7146

ID: CaltechAUTHORS:20180926-101512002

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Abstract: We show that it is NP-hard to approximate, to within an additive constant, the maximum success probability of players sharing quantum entanglement in a two-player game with classical questions of logarithmic length and classical answers of constant length. As a corollary, the inclusion NEXP subseteq MIP^*, first shown by Ito and Vidick (FOCS'12) with three provers, holds with two provers only. The proof is based on a simpler, improved analysis of the low-degree test of Raz and Safra (STOC'97) against two entangled provers.

Publication: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH, Wadern/Saarbruecken, Germany
ID: CaltechAUTHORS:20180822-141142977

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Abstract: Device-independent cryptography goes beyond conventional quantum cryptography by providing security that holds independently of the quality of the underlying physical devices. Device-independent protocols are based on the quantum phenomena of non-locality and the violation of Bell inequalities. This high level of security could so far only be established under conditions which are not achievable experimentally. Here we present a property of entropy, termed “entropy accumulation”, which asserts that the total amount of entropy of a large system is the sum of its parts. We use this property to prove the security of cryptographic protocols, including device-independent quantum key distribution, while achieving essentially optimal parameters. Recent experimental progress, which enabled loophole-free Bell tests, suggests that the achieved parameters are technologically accessible. Our work hence provides the theoretical groundwork for experimental demonstrations of device-independent cryptography.

Publication: Nature Communications Vol.: 9ISSN: 2041-1723

ID: CaltechAUTHORS:20180130-110708768

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Abstract: The success of polynomial-time tensor network methods for computing ground states of certain quantum local Hamiltonians has recently been given a sound theoretical basis by Arad et al. [Math. Phys. 356, 65 (2017)]. The convergence proof, however, relies on “rigorous renormalization group” (RRG) techniques which differ fundamentally from existing algorithms. We introduce a practical adaptation of the RRG procedure which, while no longer theoretically guaranteed to converge, finds matrix product state ansatz approximations to the ground spaces and low-lying excited spectra of local Hamiltonians in realistic situations. In contrast to other schemes, RRG does not utilize variational methods on tensor networks. Rather, it operates on subsets of the system Hilbert space by constructing approximations to the global ground space in a treelike manner. We evaluate the algorithm numerically, finding similar performance to density matrix renormalization group (DMRG) in the case of a gapped nondegenerate Hamiltonian. Even in challenging situations of criticality, large ground-state degeneracy, or long-range entanglement, RRG remains able to identify candidate states having large overlap with ground and low-energy eigenstates, outperforming DMRG in some cases.

Publication: Physical Review B Vol.: 96 No.: 21 ISSN: 2469-9950

ID: CaltechAUTHORS:20170627-090122309

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Abstract: In a recent work, Moshkovitz [FOCS'14] presented a transformation n two-player games called "fortification", and gave an elementary proof of an (exponential decay) parallel repetition theorem for fortified two-player projection games. In this paper, we give an analytic reformulation of Moshkovitz's fortification framework, which was originally cast in combinatorial terms. This reformulation allows us to expand the scope of the fortification method to new settings. First, we show any game (not just projection games) can be fortified, and give a simple proof of parallel repetition for general fortified games. Then, we prove parallel repetition and fortification theorems for games with players sharing quantum entanglement, as well as games with more than two players. This gives a new gap amplification method for general games in the quantum and multiplayer settings, which has recently received much interest. An important component of our work is a variant of the fortification transformation, called "ordered fortification", that preserves the entangled value of a game. The original fortification of Moshkovitz does not in general preserve the entangled value of a game, and this was a barrier to extending the fortification framework to the quantum setting.

No.: 67
ID: CaltechAUTHORS:20160321-071142064

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Abstract: One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance by Landau et al. gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unresolved, including whether there exist rigorous efficient algorithms when the ground space is degenerate (and poly(n) dimensional), or for the poly(n) lowest energy states for 1D systems, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm for finding low energy states for 1D systems, based on a rigorously justified renormalization group (RG)-type transformation. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n^(O(log n)) algorithm for the poly(n) lowest energy states for 1D systems (under a mild density condition). We note that for these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is O(nM(n)), where M(n) is the time required to multiply two n by n matrices.

Publication: Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH, Wadern/Saarbruecken, Germany
ID: CaltechAUTHORS:20200804-100730896

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Abstract: One of the central challenges in the study of quantum many-body systems is the complexity of simulating them on a classical computer. A recent advance (Landau et al. in Nat Phys, 2015) gave a polynomial time algorithm to compute a succinct classical description for unique ground states of gapped 1D quantum systems. Despite this progress many questions remained unsolved, including whether there exist efficient algorithms when the ground space is degenerate (and of polynomial dimension in the system size), or for the polynomially many lowest energy states, or even whether such states admit succinct classical descriptions or area laws. In this paper we give a new algorithm, based on a rigorously justified RG type transformation, for finding low energy states for 1D Hamiltonians acting on a chain of nparticles. In the process we resolve some of the aforementioned open questions, including giving a polynomial time algorithm for poly(n) degenerate ground spaces and an n^(O(log n)) algorithm for the poly(n) lowest energy states (under a mild density condition). For these classes of systems the existence of a succinct classical description and area laws were not rigorously proved before this work. The algorithms are natural and efficient, and for the case of finding unique ground states for frustration-free Hamiltonians the running time is Õ(nM(n)), where M(n) is the time required to multiply two n × n matrices.

Publication: Communications in Mathematical Physics Vol.: 356 No.: 1 ISSN: 0010-3616

ID: CaltechAUTHORS:20160321-072746620

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Abstract: In this work we consider the ground space connectivity problem for commuting local Hamiltonians. The ground space connectivity problem asks whether it is possible to go from one (efficiently preparable) state to another by applying a polynomial length sequence of 2-qubit unitaries while remaining at all times in a state with low energy for a given Hamiltonian H. It was shown in [Gharibian and Sikora, ICALP15] that this problem is QCMA-complete for general local Hamiltonians, where QCMA is defined as QMA with a classical witness and BQP verifier. Here we show that the commuting version of the problem is also QCMA-complete. This provides one of the first examples where commuting local Hamiltonians exhibit complexity theoretic hardness equivalent to general local Hamiltonians.

Publication: Quantum Vol.: 1ISSN: 2521-327X

ID: CaltechAUTHORS:20171011-112512941

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Abstract: We introduce a simple two-player test which certifies that the players apply tensor products of Pauli σ_X and σ_Z observables on the tensor product of n EPR pairs. The test has constant robustness: any strategy achieving success probability within an additive of the optimal must be poly(ε)-close, in the appropriate distance measure, to the honest n-qubit strategy. The test involves 2n-bit questions and 2-bit answers. The key technical ingredient is a quantum version of the classical linearity test of Blum, Luby, and Rubinfeld. As applications of our result we give (i) the first robust self-test for n EPR pairs; (ii) a quantum multiprover interactive proof system for the local Hamiltonian problem with a constant number of provers and classical questions and answers, and a constant completeness-soundness gap independent of system size; (iii) a robust protocol for verifiable delegated quantum computation with a constant number of quantum polynomial-time provers sharing entanglement.

ID: CaltechAUTHORS:20170710-154654821

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Abstract: We study the parallel repetition of one-round games involving players that can use quantum entanglement. A major open question in this area is whether parallel repetition reduces the entangled value of a game at an exponential rate - in other words, does an analogue of Raz's parallel repetition theorem hold for games with players sharing quantum entanglement? Previous results only apply to special classes of games. We introduce a class of games we call anchored. We then introduce a simple transformation on games called anchoring, inspired in part by the Feige-Kilian transformation, that turns any (multiplayer) game into an anchored game. Unlike the Feige-Kilian transformation, our anchoring transformation is completeness preserving. We prove an exponential-decay parallel repetition theorem for anchored games that involve any number of entangled players. We also prove a threshold version of our parallel repetition theorem for anchored games. Together, our parallel repetition theorems and anchoring transformation provide the first hardness amplification techniques for general entangled games. We give an application to the games version of the Quantum PCP Conjecture.

ID: CaltechAUTHORS:20170710-152910604

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Abstract: We give an arguably simpler and more direct proof of a recent result by Miller, Jain and Shi, who proved device-independent security of a protocol for quantum key distribution in which the devices can be used in parallel. Our proof combines existing results on immunization (Kempe et al., SICOMP 2011) and parallel repetition (Bavarian et al., STOC 2017) of entangled games.

ID: CaltechAUTHORS:20190320-102806367

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Abstract: An ideal system of n qubits has 2^n dimensions. This exponential grants power, but also hinders characterizing the system's state and dynamics. We study a new problem: the qubits in a physical system might not be independent. They can "overlap," in the sense that an operation on one qubit slightly affects the others. We show that allowing for slight overlaps, n qubits can fit in just polynomially many dimensions. (Defined in a natural way, all pairwise overlaps can be ≤ ϵ in n^(O(1/ϵ^2)) dimensions.) Thus, even before considering issues like noise, a real system of n qubits might inherently lack any potential for exponential power. On the other hand, we also provide an efficient test to certify exponential dimensionality. Unfortunately, the test is sensitive to noise. It is important to devise more robust tests on the arrangements of qubits in quantum devices.

Publication: 8th Innovations in Theoretical Computer Science Conference (ITCS 2017) No.: 67
ID: CaltechAUTHORS:20171011-113818136

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Abstract: The field of quantum information is born out of a sequence of surprising discoveries in the 1980s, all building on the same deep insight: the counter-intuitive quantum properties of particles such as photons or electrons can be put to task in order to accomplish certain computational, cryptographic, and information-theoretic tasks impossible to realize by purely classical means. A famous example is the cryptographic problem of key distribution, for which Bennett and Brassard devised the first quantum protocol in 1984 [6] and whose security relies on the no-cloning principle of quantum mechanics. Another example is the computational problem of factoring large numbers, for which Shor devised the first efficient quantum algorithm in 1994 [32] by exploiting the possibility for quantum systems to evolve in superpositions of exponentially many different states.

Publication: New Journal of Physics Vol.: 18 No.: 10 ISSN: 1367-2630

ID: CaltechAUTHORS:20161205-151744898

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Abstract: We give a simple proof of the exponential de Finetti theorem due to Renner. Like Renner's proof, ours combines the post-selection de Finetti theorem, the Gentle Measurement lemma, and the Chernoff bound, but avoids virtually all calculations, including any use of the theory of types.

ID: CaltechAUTHORS:20190320-103022957

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Abstract: We show that for any Є > 0 the problem of finding a factor (2 - Є) approximation to the entangled value of a three-player XOR game is NP-hard. Equivalently, the problem of approximating the largest possible quantum violation of a tripartite Bell correlation inequality to within any multiplicative constant is NP-hard. These results are the first constant-factor hardness of approximation results for entangled games or quantum violations of Bell inequalities shown under the sole assumption that P≠NP. They can be thought of as an extension of Håstad's optimal hardness of approximation results for MAX-E3-LIN2 [J. ACM, 48 (2001), pp. 798--859] to the entangled-player setting. The key technical component of our work is a soundness analysis of a plane-vs-point low-degree test against entangled players. This extends and simplifies the analysis of the multilinearity test by Ito and Vidick [Proceedings of the 53rd FOCS, IEEE, Piscataway, NJ, 2012, pp. 243-252]. Our results demonstrate the possibility of efficient reductions between entangled-player games and our techniques may lead to further hardness of approximation results.

Publication: SIAM Journal on Computing Vol.: 45 No.: 3 ISSN: 0097-5397

ID: CaltechAUTHORS:20161103-145636436

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Abstract: The detectability lemma is a useful tool for probing the structure of gapped ground states of frustration-free Hamiltonians of lattice spin models. The lemma provides an estimate on the error incurred by approximating the ground space projector with a product of local projectors. We provide a simpler proof for the detectability lemma which applies to an arbitrary ordering of the local projectors, and show that it is tight up to a constant factor. As an application, we show how the lemma can be combined with a strong converse by Gao to obtain local spectral gap amplification: We show that by coarse graining a local frustration-free Hamiltonian with a spectral gap γ>0 to a length scale O(γ^(−1/2)), one gets a Hamiltonian with an Ω(1) spectral gap.

Publication: Physical Review B Vol.: 93 No.: 20 ISSN: 1098-0121

ID: CaltechAUTHORS:20160318-153303794

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Abstract: We give the first construction of a family of quantum-proof extractors that has optimal seed length dependence O(log(n/ǫ)) on the input length n and error ǫ. Our extractors support any min-entropy k = Ω(log n + log1+α (1/ǫ)) and extract m = (1 − α)k bits that are ǫ-close to uniform, for any desired constant α > 0. Previous constructions had a quadratically worse seed length or were restricted to very large input min-entropy or very few output bits. Our result is based on a generic reduction showing that any strong classical condenser is automatically quantum-proof, with comparable parameters. The existence of such a reduction for extractors is a long-standing open question; here we give an affirmative answer for condensers. Once this reduction is established, to obtain our quantum-proof extractors one only needs to consider high entropy sources. We construct quantum-proof extractors with the desired parameters for such sources by extending a classical approach to extractor construction, based on the use of block-sources and sampling, to the quantum setting. Our extractors can be used to obtain improved protocols for device-independent randomness expansion and for privacy amplification.

Publication: arXiv
ID: CaltechAUTHORS:20160517-182619760

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Abstract: Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class NP, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class QSZK, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.

Publication: Foundations and Trends in Theoretical Computer Science Vol.: 11 No.: 1-2
ID: CaltechAUTHORS:20160622-144016671

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Abstract: We prove a moment majorization principle for matrix-valued functions with domain {−1,1}^m, m∈N. The principle is an inequality between higher-order moments of a non-commutative multilinear polynomial with different random matrix ensemble inputs, where each variable has small influence and the variables are instantiated independently. This technical result can be interpreted as a noncommutative generalization of one of the two inequalities of the seminal invariance principle of Mossel, O'Donnell and Oleszkiewicz. Our main application is sharp Unique Games hardness for two versions of the noncommutative Grothendieck inequality. This generalizes a result of Raghavendra and Steurer who established hardness of approximation for the commutative Grothendieck inequality. A similar application was proven recently by Briët, Regev and Saket using different techniques.

ID: CaltechAUTHORS:20190320-103637958

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Abstract: In the context of multiplayer games, the parallel repetition problem can be phrased as follows: given a game G with optimal winning probability 1 - α and its repeated version G^n (in which n games are played together, in parallel), can the players use strategies that are substantially better than ones in which each game is played independently? This question is relevant in physics for the study of correlations and plays an important role in computer science in the context of complexity and cryptography. In this paper, the case of multiplayer non-signaling games is considered, i.e., the only restriction on the players is that they are not allowed to communicate during the game. For complete-support games (games where all possible combinations of questions have non-zero probability to be asked) with any number of players, we prove a threshold theorem stating that the probability that non-signaling players win more than a fraction 1-α+β of the n games is exponentially small in nβ^2 for every 0 ≤ β ≤ α. For games with incomplete support, we derive a similar statement for a slightly modified form of repetition. The result is proved using a new technique based on a recent de Finetti theorem, which allows us to avoid central technical difficulties that arise in standard proofs of parallel repetition theorems.

Publication: IEEE Transactions on Information Theory Vol.: 62 No.: 3 ISSN: 0018-9448

ID: CaltechAUTHORS:20160318-101440389

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Abstract: This review article is concerned with a recently uncovered connection between operator spaces, a noncommutative extension of Banach spaces, and quantum nonlocality, a striking phenomenon which underlies many of the applications of quantum mechanics to information theory, cryptography, and algorithms. Using the framework of nonlocal games, we relate measures of the nonlocality of quantum mechanics to certain norms in the Banach and operator space categories. We survey recent results that exploit this connection to derive large violations of Bell inequalities, study the complexity of the classical and quantum values of games and their relation to Grothendieck inequalities, and quantify the nonlocality of different classes of entangled states.

Publication: Journal of Mathematical Physics Vol.: 57 No.: 1 ISSN: 0022-2488

ID: CaltechAUTHORS:20160225-142342994

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Abstract: We give a quantum multiprover interactive proof system for the local Hamiltonian problem in which there is a constant number of provers, questions are classical of length polynomial in the number of qubits, and answers are of constant length. The main novelty of our protocol is that the gap between completeness and soundness is directly proportional to the promise gap on the (normalized) ground state energy of the Hamiltonian. This result can be interpreted as a concrete step towards a quantum PCP theorem giving entangled-prover interactive proof systems for QMA-complete problems. The key ingredient is a quantum version of the classical linearity test of Blum, Luby, and Rubinfeld, where the function f : {0,1}^n → {0,1} is replaced by a pair of functions X,Z : {0,1}^n → Obs_d(C), the set of d-dimensional Hermitian matrices that square to identity. The test enforces that (i) each function is exactly linear, X(a)X(b) = X(a+b) and Z(a)Z(b) = Z(a+b), and (ii) the two functions are approximately complementary, X(a)Z(b) ≈ (−1)^(a⋅b)Z(b)X(a).

ID: CaltechAUTHORS:20160318-160143988

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Abstract: Quantum entanglement is known to provide a strong advantage in many two-party distributed tasks. We investigate the question of how much entanglement is needed to reach optimal performance. For the first time we show that there exists a purely classical scenario for which no finite amount of entanglement suffices. To this end we introduce a simple two-party nonlocal game H, inspired by Lucien Hardy’s paradox. In our game each player has only two possible questions and can provide bit strings of any finite length as answer. We exhibit a sequence of strategies which use entangled states in increasing dimension d and succeed with probability 1 - O(d^(-c)) for some c ≥ 0.13. On the other hand, we show that any strategy using an entangled state of local dimension d has success probability at most 1 - Ω (d^(-2)). In addition, we show that any strategy restricted to producing answers in a set of cardinality at most d has success probability at most 1 - Ω (d^(-2)). Finally, we generalize our construction to derive similar results starting from any game G with two questions per player and finite answers sets in which quantum strategies have an advantage.

Publication: Quantum Information and Computation Vol.: 15 No.: 15-16 ISSN: 1533-7146

ID: CaltechAUTHORS:20160818-080941623

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Abstract: Two major open problems regarding the parallel repetition of games are whether an analogue of Raz's parallel-repetition theorem holds for (a) games with more than two players, and (b) games with quantum players using entanglement. We make progress on both problems: we introduce a class of games we call anchored, and prove exponential-decay parallel repetition theorems for anchored games in the multiplayer and entangled-player settings. We introduce a simple transformation on games called anchoring and show that this transformation turns any game into an anchored game. Together, our parallel repetition theorem and our anchoring transformation provide a simple and efficient hardness-amplification technique in both the classical multiplayer and quantum settings.

ID: CaltechAUTHORS:20160318-152740730

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Abstract: We introduce quantum XOR games, a model of two-player, one-round games that extends the model of XOR games by allowing the referee’s questions to the players to be quantum states. We give examples showing that quantum XOR games exhibit a wide range of behaviors that are known not to exist for standard XOR games, such as cases in which the use of entanglement leads to an arbitrarily large advantage over the use of no entanglement. By invoking two deep extensions of Grothendieck’s inequality, we present an efficient algorithm that gives a constant-factor approximation to the best performance that players can obtain in a given game, both in the case that they have no shared entanglement and that they share unlimited entanglement. As a byproduct of the algorithm, we prove some additional interesting properties of quantum XOR games, such as the fact that sharing a maximally entangled state of arbitrary dimension gives only a small advantage over having no entanglement at all.

Publication: ACM Transactions on Computation Theory Vol.: 7 No.: 4 ISSN: 1942-3454

ID: CaltechAUTHORS:20160321-083901879

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Abstract: The density matrix renormalization group method has been extensively used to study the ground state of 1D many-body systems since its introduction two decades ago. In spite of its wide use, this heuristic method is known to fail in certain cases and no certifiably correct implementation is known, leaving researchers faced with an ever-growing toolbox of heuristics, none of which is guaranteed to succeed. Here we develop a polynomial time algorithm that provably finds the ground state of any 1D quantum system described by a gapped local Hamiltonian with constant ground-state energy. The algorithm is based on a framework that combines recently discovered structural features of gapped 1D systems with an efficient construction of a class of operators called approximate ground-state projections (AGSPs). The combination of these tools yields a method that is guaranteed to succeed in all 1D gapped systems. An AGSP-centric approach may help guide the search for algorithms for more general quantum systems, including for the central challenge of 2D systems, where even heuristic methods have had more limited success.

Publication: Nature Physics Vol.: 11 No.: 7 ISSN: 1745-2473

ID: CaltechAUTHORS:20150422-093309397

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Abstract: The class MIP∗ of promise problems that can be decided through an interactive proof system with multiple entangled provers provides a complexity-theoretic framework for the exploration of the nonlocal properties of entanglement. Very little is known in terms of the power of this class. The only proposed approach for establishing upper bounds is based on a hierarchy of semidefinite programs introduced independently by Pironio et al. and Doherty et al. in 2006. This hierarchy converges to a value, the field-theoretic value, that is only known to coincide with the provers’ maximum success probability in a given proof system under a plausible but difficult mathematical conjecture, Connes’ embedding conjecture. No bounds on the rate of convergence are known. We introduce a rounding scheme for the hierarchy, establishing that any solution to its N -th level can be mapped to a strategy for the provers in which measurement operators associated with distinct provers have pairwise commutator bounded by O(ℓ^2/√N) in operator norm, where ℓ is the number of possible answers per prover. Our rounding scheme motivates the introduction of a variant of quantum multiprover interactive proof systems, called MIP∗_δ in which the soundness property is required to hold against provers allowed to operate on the same Hilbert space as long as the commutator of operations performed by distinct provers has norm at most δ. Our rounding scheme implies the upper bound MIP∗_δ ⊆ DTIME(exp(exp(poly)/δ^2)). In terms of lower bounds we establish that MIP∗_(2−poly) contains NEXP with completeness 1 and soundness 1−2^(−poly). We discuss connections with the mathematical literature on approximate commutation and applications to device-independent cryptography.

No.: 9134
ID: CaltechAUTHORS:20151207-141218015

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Abstract: We study the behavior of the entangled value of two-player one-round projection games under parallel repetition. We show that for any projection game G of entangled value 1−ϵ<1, the value of the k-fold repetition of G goes to zero as O((1−ϵ^c)^k), for some universal constant c≥1 furthermore the constraint graph of G is expanding, we obtain the optimal c = 1. Previously exponential decay of the entangled value under parallel repetition was only known for the case of XOR and unique games. To prove the theorem, we extend an analytical framework introduced by Dinur and Steurer for the study of the classical value of projection games under parallel repetition. Our proof, as theirs, relies on the introduction of a simple relaxation of the entangled value that is perfectly multiplicative. The main technical component of the proof consists in showing that the relaxed value remains tightly connected to the entangled value, thereby establishing the parallel repetition theorem. More generally, we obtain results on the behavior of the entangled value under products of arbitrary (not necessarily identical) projection games. Relating our relaxed value to the entangled value is done by giving an algorithm for converting a relaxed variant of quantum strategies that we call “vector quantum strategy” to a quantum strategy. The algorithm is considerably simpler in case the bipartite distribution of questions in the game has good expansion properties. When this is not the case, the algorithm relies on a quantum analogue of Holenstein’s correlated sampling lemma which may be of independent interest. Our “quantum correlated sampling lemma” generalizes results of van Dam and Hayden on universal embezzlement to the following approximate scenario: two non-communicating parties, given classical descriptions of bipartite states |ψ⟩,|φ⟩, respectively, such that |ψ⟩≈|φ⟩, are able to locally generate a joint entangled state |Ψ⟩≈|ψ⟩≈|φ⟩ using an initial entangled state that is independent of their inputs.

Publication: Computational Complexity Vol.: 24 No.: 2 ISSN: 1016-3328

ID: CaltechAUTHORS:20150615-140934465

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Abstract: We give a quantum interactive proof system for the local Hamiltonian problem on n qubits in which (i) the verifier has a single round of interaction with five entangled provers, (ii) the verifier sends a classical message on O(log n) bits to each prover, who replies with a constant number of qubits, and (iii) completeness and soundness are separated by an inverse polynomial in $n$. As the same class of proof systems, without entanglement between the provers, is included in QCMA, our result provides the first indication that quantum multiprover interactive proof systems with entangled provers may be strictly more powerful than unentangled-prover interactive proof systems. A distinguishing feature of our protocol is that the completeness property requires honest provers to share a large entangled state, obtained as the encoding of the ground state of the local Hamiltonian via an error-correcting code. Our result can be interpreted as a first step towards a multiprover variant of the quantum PCP conjecture.

ID: CaltechAUTHORS:20150218-115725417

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Abstract: Quantum cryptography promises levels of security that are impossible to replicate in a classical world. Can this security be guaranteed even when the quantum devices on which the protocol relies are untrusted? This central question dates back to the early 1990s when the challenge of achieving device-independent quantum key distribution was first formulated. We answer this challenge by rigorously proving the device-independent security of a slight variant of Ekert's original entanglement-based protocol against the most general (coherent) attacks. The resulting protocol is robust: While assuming only that the devices can be modeled by the laws of quantum mechanics and are spatially isolated from each other and from any adversary's laboratory, it achieves a linear key rate and tolerates a constant noise rate in the devices. In particular, the devices may have quantum memory and share arbitrary quantum correlations with the eavesdropper. The proof of security is based on a new quantitative understanding of the monogamous nature of quantum correlations in the context of a multiparty protocol.

Publication: Physical Review Letters Vol.: 113 No.: 14 ISSN: 0031-9007

ID: CaltechAUTHORS:20150108-142044094

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Abstract: The classical Grothendieck inequality has applications to the design of approximation algorithms for NP-hard optimization problems. We show that an algorithmic interpretation may also be given for a noncommutative generalization of the Grothendieck inequality due to Pisier and Haagerup. Our main result, an efficient rounding procedure for this inequality, leads to a polynomial-time constant-factor approximation algorithm for an optimization problem which generalizes the Cut Norm problem of Frieze and Kannan, and is shown here to have additional applications to robust principal component analysis and the orthogonal Procrustes problem.

Publication: Theory of Computing Vol.: 10 No.: 1 ISSN: 1557-2862

ID: CaltechAUTHORS:20200731-152129927

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Abstract: We provide alternative proofs of two recent Grothendieck theorems for jointly completely bounded bilinear forms, originally due to Pisier and Shlyakhtenko (Grothendieck's theorem for operator spaces, Invent. Math. 150(2002), 185-217) and Haagerup and Musat (The Effros-Ruan conjecture for bilinear forms on C*-algebras, Invent. Math. 174(2008), 139-163). Our proofs are elementary and are inspired by the so-called embezzlement states in quantum information theory. Moreover, our proofs lead to quantitative estimates.

Publication: Journal of Operator Theory Vol.: 71 No.: 2 ISSN: 1841-7744

ID: CaltechAUTHORS:20160318-152323237

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Abstract: We study the behavior of the entangled value of two-player one-round projection games under parallel repetition. We show that for any projection game G of entangled value 1 - ε <; 1, the value of the k-fold repetition of G goes to zero as O((1 - ε^c)^k), for some universal constant c ≥ 1. Previously parallel repetition with an exponential decay in k was only known for the case of XOR and unique games. To prove the theorem we extend an analytical framework recently introduced by Dinur and Steurer for the study of the classical value of projection games under parallel repetition. Our proof, as theirs, relies on the introduction of a simple relaxation of the entangled value that is perfectly multiplicative. The main technical component of the proof consists in showing that the relaxed value remains tightly connected to the entangled value, thereby establishing the parallel repetition theorem. More generally, we obtain results on the behavior of the entangled value under products of arbitrary (not necessarily identical) projection games. Relating our relaxed value to the entangled value is done by giving an algorithm for converting a relaxed variant of quantum strategies that we call “vector quantum strategy” to a quantum strategy. The algorithm is considerably simpler in case the bipartite distribution of questions in the game has good expansion properties. When this is not the case, rounding relies on a quantum analogue of Holenstein's correlated sampling lemma which may be of independent interest. Our “quantum correlated sampling lemma” generalizes results of van Dam and Hayden on universal embezzlement to the following approximate scenario: two isolated parties, given classical descriptions of arbitrary bipartite states |ψ〉, |φ〉 respectively such that |ψ〉 ≈ |φ〉, are able to locally generate a joint entangled state|- Ψ〉 ≈ |ψ〉 ≈ |φ〉 using an initial entangled state that is independent of their inputs.

ID: CaltechAUTHORS:20140910-132001940

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Abstract: Computing ground states of local Hamiltonians is a fundamental problem in condensed matter physics. The problem is known to be QMA-complete, even for one-dimensional Hamiltonians. This means that we do not even expect that there is a sub-exponential size description of the ground state that allows efficient computation of local observables such as the energy. In sharp contrast, the heuristic density matrix renormalization group (DMRG) algorithm invented two decades ago has been remarkably successful in practice on one-dimensional problems. The situation is reminiscent of the unexplained success of the simplex algorithm before the advent of ellipsoid and interior-point methods. Is there a principled explanation for this, in the form of a large class of one-dimensional Hamiltonians whose ground states can be provably efficiently approximated? Here we give such an algorithm for gapped one-dimensional Hamiltonians: our algorithm outputs an (inverse-polynomial) approximation to the ground state, expressed as a matrix product state (MPS) of polynomial bond dimension. The running time of the algorithm is polynomial in the number of qudits n and the approximation quality δ, for a fixed local dimension d and gap Δ > 0. A key ingredient of our algorithm is a new construction of an operator called an approximate ground state projector (AGSP), a concept first introduced in to derive an improved area law for gapped one-dimensional systems. For this purpose the AGSP has to be efficiently constructed; the particular AGSP we construct relies on matrix-valued Chernoff bounds. Other ingredients of the algorithm include the use of convex programming, recently discovered structural features of gapped 1D quantum systems, and new techniques for manipulating and bounding the complexity of matrix product states.

ID: CaltechAUTHORS:20140909-142344205

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Abstract: Quantum cryptography is based on the discovery that the laws of quantum mechanics allow levels of security that are impossible to replicate in a classical world. Can such levels of security be guaranteed even when the quantum devices on which the protocol relies are untrusted? This fundamental question in quantum cryptography dates back to the early nineties when the challenge of achieving device independent quantum key distribution, or DIQKD, was first formulated. We answer this challenge affirmatively by exhibiting a robust protocol for DIQKD and rigorously proving its security. The protocol achieves a linear key rate while tolerating a constant noise rate in the devices. The security proof assumes only that the devices can be modeled by the laws of quantum mechanics and are spatially isolated from each other and any adversary's laboratory. In particular, we emphasize that the devices may have quantum memory. All previous proofs of security relied either on the use of many independent pairs of devices, or on the absence of noise. To prove security for a DIQKD protocol it is necessary to establish at least that the generated key is truly random even in the presence of a quantum adversary. This is already a challenge, one that was recently resolved. DIQKD is substantially harder, since now the protocol must also guarantee that the key is completely secret from the quantum adversary's point of view, and the entire protocol is robust against noise; this in spite of the substantial amounts of classical information leaked to the adversary throughout the protocol, as part of the error estimation and information reconciliation procedures. Our proof of security builds upon a number of techniques, including randomness extractors that are secure against quantum storage as well as ideas originating in the coding strategy used in the proof of the Holevo-Schumacher-Westmoreland theorem which we apply to bound correlations across multiple rounds in a way not unrelated to information-theoretic proofs of the parallel repetition property for multiplayer games. Our main result can be understood as a new bound on monogamy of entanglement in the type of complex scenario that arises in a key distribution protocol.

ID: CaltechAUTHORS:20140909-145320191

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Abstract: Quantum entanglement is known to provide a strong advantage in many two-party distributed tasks. We investigate the question of how much entanglement is needed to reach optimal performance. For the first time we show that there exists a purely classical scenario for which no finite amount of entanglement suffices. To this end we introduce a simple two-party nonlocal game H, inspired by a paradox of Lucien Hardy. In our game each player has only two possible questions and can provide answers in a countable set. We exhibit a sequence of strategies which use entangled states in increasing dimension d and succeed with probability 1 − O(d^(−c)) for some c ≥ 0.13. On the other hand, we show that any strategy using an entangled state of local dimension d has success probability at most 1 − Ω(d⁻²). In addition, we show that any strategy restricted to producing answers in a set of cardinality at most d has success probability at most 1 − Ω(d⁻²).

Publication: Lecture Notes in Computer Science No.: 8572 ISSN: 0302-9743

ID: CaltechAUTHORS:20200805-153500961

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Abstract: We show that for any ε > 0 the problem of finding a factor (2 - ε) approximation to the entangled value of a three-player XOR game is NP-hard. Equivalently, the problem of approximating the largest possible quantum violation of a tripartite Bell correlation inequality to within any multiplicative constant is NP-hard. These results are the first constant-factor hardness of approximation results for entangled games or quantum violations of Bell inequalities shown under the sole assumption that P≠NP. They can be thought of as an extension of Hástad's optimal hardness of approximation results for MAX-E3-LIN2 (JACM'01) to the entangled-player setting. The key technical component of our work is a soundness analysis of a point-vs-plane low-degree test against entangled players. This extends and simplifies the analysis of the multilinearity test by Ito and Vidick (FOCS'12). Our results demonstrate the possibility for efficient reductions between entangled-player games and our techniques may lead to further hardness of approximation results.

ID: CaltechAUTHORS:20140910-100149541

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Abstract: The study of quantum-mechanical violations of Bell inequalities is motivated by the investigation, and the eventual demonstration, of the nonlocal properties of entanglement. In recent years, Bell inequalities have found a fruitful re-formulation using the language of multiplayer games originating from Computer Science. This paper studies the nonlocal properties of entanglement in the context of the simplest such games, called XOR games. When there are two players, it is well known that the maximum bias—the advantage over random play—of players using entanglement can be at most a constant times greater than that of classical players. Recently, Pérez-García et al. (Commun. Mathe. Phys. 279:455, 2008) showed that no such bound holds when there are three or more players: the use of entanglement can provide an unbounded advantage, and scale with the number of questions in the game. Their proof relies on non-trivial results from operator space theory, and gives a non-explicit existence proof, leading to a game with a very large number of questions and only a loose control over the local dimension of the players’ shared entanglement. We give a new, simple and explicit (though still probabilistic) construction of a family of three-player XOR games which achieve a large quantum-classical gap (QC-gap). This QC-gap is exponentially larger than the one given by Pérez-García et. al. in terms of the size of the game, achieving a QC-gap of order √N with N^2 questions per player. In terms of the dimension of the entangled state required, we achieve the same (optimal) QC-gap of √N for a state of local dimension N per player. Moreover, the optimal entangled strategy is very simple, involving observables defined by tensor products of the Pauli matrices. Additionally, we give the first upper bound on the maximal QC-gap in terms of the number of questions per player, showing that our construction is only quadratically off in that respect. Our results rely on probabilistic estimates on the norm of random matrices and higher-order tensors which may be of independent interest.

Publication: Communications in Mathematical Physics Vol.: 321 No.: 1 ISSN: 0010-3616

ID: CaltechAUTHORS:20160318-154623344

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Abstract: The classical Grothendieck inequality has applications to the design of approximation algorithms for NP-hard optimization problems. We show that an algorithmic interpretation may also be given for a noncommutative generalization of the Grothendieck inequality due to Pisier and Haagerup. Our main result, an efficient rounding procedure for this inequality, leads to a constant-factor polynomial time approximation algorithm for an optimization problem which generalizes the Cut Norm problem of Frieze and Kannan, and is shown here to have additional applications to robust principle component analysis and the orthogonal Procrustes problem.

ID: CaltechAUTHORS:20140910-115031387

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Abstract: The classical PCP theorem is arguably the most important achievement of classical complexity theory in the past quarter century. In recent years, researchers in quantum computational complexity have tried to identify approaches and develop tools that address the question: does a quantum version of the PCP theorem hold? The story of this study starts with classical complexity and takes unexpected turns providing fascinating vistas on the foundations of quantum mechanics and multipartite entanglement, topology and the so-called phenomenon of topological order, quantum error correction, information theory, and much more; it raises questions that touch upon some of the most fundamental issues at the heart of our understanding of quantum mechanics. At this point, the jury is still out as to whether or not such a theorem holds. This survey aims to provide a snapshot of the status in this ongoing story, tailored to a general theory-of-CS audience.

Publication: ACM SIGACT News Vol.: 44 No.: 2 ISSN: 0163-5700

ID: CaltechAUTHORS:20140910-135821275

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Abstract: We introduce quantum XOR games, a model of two-player one-round games that extends the model of XOR games by allowing the referee's questions to the players to be quantum states. We give examples showing that quantum XOR games exhibit a wide range of behaviors that are known not to exist for standard XOR games, such as cases in which the use of entanglement leads to an arbitrarily large advantage over the use of no entanglement. By invoking two deep extensions of Grothendieck's inequality, we present an efficient algorithm that gives a constant-factor approximation to the best performance players can obtain in a given game, both in case they have no shared entanglement and in case they share unlimited entanglement. As a byproduct of the algorithm we prove some additional interesting properties of quantum XOR games, such as the fact that sharing a maximally entangled state of arbitrary dimension gives only a small advantage over having no entanglement at all.

ID: CaltechAUTHORS:20160329-155246836

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Abstract: We study multipartite entanglement in the context of XOR games. In particular, we study the ratio of the entangled and classical biases, which measure the maximum advantage of a quantum or classical strategy over a uniformly random strategy. For the case of two-player XOR games, Tsirelson proved that this ratio is upper bounded by the celebrated Grothendieck constant. In contrast, Pérez-García et al. proved the existence of entangled states that give quantum players an unbounded advantage over classical players in a three-player XOR game. We show that the multipartite entangled states that are most often seen in today’s literature can only lead to a bias that is a constant factor larger than the classical bias. These states include GHZ states, any state local-unitarily equivalent to combinations of GHZ and maximally entangled states shared between different subsets of the players (e.g., stabilizer states), as well as generalizations of GHZ states of the form ∑iɑi|i〉...|i〉 for arbitrary amplitudes ɑi. Our results have the following surprising consequence: classical three-player XOR games do not follow an XOR parallel repetition theorem, even a very weak one. Besides this, we discuss implications of our results for communication complexity and hardness of approximation. Our proofs are based on novel applications of extensions of Grothendieck’s inequality, due to Blei and Tonge, and Carne, generalizing Tsirelson’s use of Grothendieck’s inequality to bound the bias of two-player XOR games.

Publication: Quantum Information and Computation Vol.: 13 No.: 3-4 ISSN: 1533-7146

ID: CaltechAUTHORS:20140909-144447941

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Abstract: We present an analysis of Wiesner’s quantum money scheme, as well as some natural generalizations of it, based on semidefinite programming. For Wiesner’s original scheme, it is determined that the optimal probability for a counterfeiter to create two copies of a bank note from one, where both copies pass the bank’s test for validity, is (3/4)^n for n being the number of qubits used for each note. Generalizations in which other ensembles of states are substituted for the one considered by Wiesner are also discussed, including a scheme recently proposed by Pastawski, Yao, Jiang, Lukin, and Cirac, as well as schemes based on higher dimensional quantum systems. In addition, we introduce a variant of Wiesner’s quantum money in which the verification protocol for bank notes involves only classical communication with the bank. We show that the optimal probability with which a counterfeiter can succeed in two independent verification attempts, given access to a single valid n-qubit bank note, is (3/4+√2/8)^n. We also analyze extensions of this variant to higher-dimensional schemes.

No.: 7582
ID: CaltechAUTHORS:20160318-155200133

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Abstract: A recent sequence of works, initially motivated by the study of the nonlocal properties of entanglement, demonstrate that a source of information-theoretically certified randomness can be constructed based only on two simple assumptions: the prior existence of a short random seed and the ability to ensure that two black-box devices do not communicate (i.e. are non-signaling). We call protocols achieving such certified amplification of a short random seed randomness amplifiers. We introduce a simple framework in which we initiate the systematic study of the possibilities and limitations of randomness amplifiers. Our main results include a new, improved analysis of a robust randomness amplifier with exponential expansion, as well as the first upper bounds on the maximum expansion achievable by a broad class of randomness amplifiers. In particular, we show that non-adaptive randomness amplifiers that are robust to noise cannot achieve more than doubly exponential expansion. Finally, we show that a wide class of protocols based on the use of the CHSH game can only lead to (singly) exponential expansion if adversarial devices are allowed the full power of non-signaling strategies. Our upper bound results apply to all known non-adaptive randomness amplifier constructions to date.

No.: 8096
ID: CaltechAUTHORS:20160318-153752227

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Abstract: We prove a strong limitation on the ability of entangled provers to collude in a multiplayer game. Our main result is the first nontrivial lower bound on the class MIP* of languages having multi-prover interactive proofs with entangled provers, namely MIP* contains NEXP, the class of languages decidable in non-deterministic exponential time. While Babai, Fort now, and Lund (Computational Complexity 1991) proved the celebrated equality MIP = NEXP in the absence of entanglement, ever since the introduction of the class MIP* it was open whether shared entanglement between the provers could weaken or strengthen the computational power of multi-prover interactive proofs. Our result shows that it does not weaken their computational power: MIP* contains MIP. At the heart of our result is a proof that Babai, Fort now, and Lund's multilinearity test is sound even in the presence of entanglement between the provers, and our analysis of this test could be of independent interest. As a byproduct we show that the correlations produced by any entangled strategy which succeeds in the multilinearity test with high probability can always be closely approximated using shared randomness alone.

ID: CaltechAUTHORS:20140910-100733064

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Abstract: Randomness extraction involves the processing of purely classical information and is therefore usually studied with in the framework of classical probability theory. However, such a classical treatment is generally too restrictive for applications where side information about the values taken by classical random variables may be represented by the state of a quantum system. This is particularly relevant in the context of cryptography, where an adversary may make use of quantum devices. Here, we show that the well-known construction paradigm for extractors proposed by Trevisan is sound in the presence of quantum side information. We exploit the modularity of this paradigm to give several concrete extractor constructions, which, e.g., extract all the conditional (smooth) min-entropy of the source using a seed of length polylogarithmic in the input, or only require the seed to be weakly random.

Publication: SIAM Journal on Computing Vol.: 41 No.: 4 ISSN: 0097-5397

ID: CaltechAUTHORS:20160322-084353163

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Abstract: We introduce a protocol through which a pair of quantum mechanical devices may be used to generate n random bits that are ε-close in statistical distance from n uniformly distributed bits, starting from a seed of O(log n log 1/ ϵ) uniform bits. The bits generated are certifiably random, based only on a simple statistical test that can be performed by the user, and on the assumption that the devices obey the no-signalling principle. No other assumptions are placed on the devices' inner workings: it is not necessary to even assume the validity of quantum mechanics.

Publication: Philosophical Transactions A: Mathematical, Physical and Engineering Sciences Vol.: 370 No.: 1971 ISSN: 1364-503X

ID: CaltechAUTHORS:20200804-084834826

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Abstract: Given two sets A, B ⊆ R_n, a measure of their correlation is given by the expected squared inner product between random x ϵ A and y ϵ B. We prove an inequality showing that no two sets of large enough Gaussian measure (at least e^(-δn) for some constant δ > 0) can have correlation substantially lower than would two random sets of the same size. Our proof is based on a concentration inequality for the overlap of a random Gaussian vector on a large set. As an application, we show how our result can be combined with the partition bound of Jain and Klauck to give a simpler proof of a recent linear lower bound on the randomized communication complexity of the Gap-Hamming-Distance problem due to Chakrabarti and Regev.

Publication: Chicago Journal of Theoretical Computer Science Vol.: 18 No.: 1 ISSN: 1073-0486

ID: CaltechAUTHORS:20200804-133447851

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Abstract: We introduce a protocol through which a pair of quantum mechanical devices may be used to generate n bits that are ε-close in statistical distance from n uniformly distributed bits, starting from a seed of O(log n log 1/ε) uniform bits. The bits generated are certifiably random based only on a simple statistical test that can be performed by the user, and on the assumption that the devices do not communicate in the middle of each phase of the protocol. No other assumptions are placed on the devices' inner workings. A modified protocol uses a seed of O(log^3 n) uniformly random bits to generate n bits that are poly^(-1)(n)-indistinguishable from uniform even from the point of view of a quantum adversary who may have had prior access to the devices, and may be entangled with them.

ID: CaltechAUTHORS:20140910-141425231

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Abstract: We present an analysis of Wiesner's quantum money scheme, as well as some natural generalizations of it, based on semidefinite programming. For Wiesner's original scheme, it is determined that the optimal probability for a counterfeiter to create two copies of a bank note from one, where both copies pass the bank's test for validity, is (3/4)^n for n being the number of qubits used for each note. Generalizations in which other ensembles of states are substituted for the one considered by Wiesner are also discussed, including a scheme recently proposed by Pastawski, Yao, Jiang, Lukin, and Cirac, as well as schemes based on higher dimensional quantum systems. In addition, we introduce a variant of Wiesner's quantum money in which the verification protocol for bank notes involves only classical communication with the bank. We show that the optimal probability with which a counterfeiter can succeed in two independent verification attempts, given access to a single valid n-qubit bank note, is (3/4+√2/8)^n. We also analyze extensions of this variant to higher-dimensional schemes.

ID: CaltechAUTHORS:20190320-104220786

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Abstract: We prove that the Banach algebra formed by the space of compact operators on a Hilbert space endowed with the Schur product is a quotient of a uniform algebra (also known as a Q-algebra). Together with a similar result of Pérez-García for the trace class, this completes the answer to a long-standing question of Varopoulos.

Publication: Journal of Functional Analysis Vol.: 262 No.: 1 ISSN: 0022-1236

ID: CaltechAUTHORS:20200728-153958382

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Abstract: We introduce a protocol through which a pair of quantum mechanical devices may be used to generate n bits of true randomness from a seed of O(log n) uniform bits. The bits generated are certifiably random based only on a simple statistical test that can be performed by the user, and on the assumption that the devices obey the no-signaling principle. No other assumptions are placed on the devices' inner workings. A modified protocol uses a seed of O(log^3 n) uniformly random bits to generate n bits of true randomness even conditioned on the state of a quantum adversary who may have had prior access to the devices, and may be entangled with them.

ID: CaltechAUTHORS:20190320-104707093

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Abstract: A central question in our understanding of the physical world is how our knowledge of the whole relates to our knowledge of the individual parts. One aspect of this question is the following: to what extent does ignorance about a whole preclude knowledge of at least one of its parts? Relying purely on classical intuition, one would certainly be inclined to conjecture that a strong ignorance of the whole cannot come without significant ignorance of at least one of its parts. Indeed, we show that this reasoning holds in any noncontextual (NC) hidden-variable model (HV). Curiously, however, such a conjecture is false in quantum theory: we provide an explicit example where a large ignorance about the whole can coexist with an almost perfect knowledge of each of its parts. More specifically, we provide a simple information-theoretic inequality satisfied in any NC HV, but which can be arbitrarily violated by quantum mechanics.

Publication: Physical Review Letters Vol.: 107 No.: 3 ISSN: 0031-9007

ID: CaltechAUTHORS:20160318-151328788

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Abstract: We establish the first hardness results for the problem of computing the value of one-round games played by a verifier and a team of provers who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) a quantum verifier and two entangled provers or (ii) a classical verifier and three entangled provers. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation of entangled-prover games to within a constant. Using our techniques we also show that every language in PSPACE has a two-prover one-round interactive proof system with perfect completeness and soundness 1-1/poly even against entangled provers. We start our proof by describing two ways to modify classical multiprover games to make them resistant to entangled provers. We then show that a strategy for the modified game that uses entanglement can be “rounded” to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P=NP, the values of entangled-prover games cannot be computed by semidefinite programs that are polynomial in the size of the verifier's system, a method that has been successful for more restricted quantum games.

Publication: SIAM Journal on Computing Vol.: 40 No.: 3 ISSN: 0097-5397

ID: CaltechAUTHORS:20110713-155400829

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Abstract: We consider one-round games between a classical referee and two players. One of the main questions in this area is the parallel repetition question: Is there a way to decrease the maximum winning probability of a game without increasing the number of rounds or the number of players? Classically, efforts to resolve this question, open for many years, have culminated in Raz’s celebrated parallel repetition theorem on one hand, and in efficient product testers for PCPs on the other. In the case where players share entanglement, the only previously known results are for special cases of games, and are based on techniques that seem inherently limited. Here we show for the first time that the maximum success probability of entangled games can be reduced through parallel repetition, provided it was not initially 1. Our proof is inspired by a seminal result of Feige and Kilian in the context of classical two-prover one-round interactive proofs. One of the main components in our proof is an orthogonalization lemma for operators, which might be of independent interest.

ID: CaltechAUTHORS:20140910-135518364

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Abstract: Recent numerical investigations [K. Pál and T. Vértesi, Phys. Rev. A 82, 022116 (2010)] suggest that the I3322 inequality, arguably the simplest extremal Bell inequality after the CHSH inequality, has a very rich structure in terms of the entangled states and measurements that maximally violate it. Here we show that for this inequality the maximally entangled state of any dimension achieves the same violation than just a single EPR pair. In contrast, stronger violations can be achieved using higher dimensional states which are less entangled. This shows that the maximally entangled state is not the most nonlocal resource, even when one restricts attention to the most simple extremal Bell inequalities.

Publication: Physical Review A Vol.: 83 No.: 5 ISSN: 1050-2947

ID: CaltechAUTHORS:20160318-115742532

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Abstract: We show that Trevisan's extractor and its variants] are secure against bounded quantum storage adversaries. One instantiation gives the first such extractor to achieve an output length Θ(K-b), where K is the source's entropy and b the adversary's storage, together with a poly-logarithmic seed length. Another instantiation achieves a logarithmic key length, with a slightly smaller output length Θ((K-b)/K^γ) for any γ>0. In contrast, the previous best construction could only extract (K/b)^(1/15) bits. Some of our constructions have the additional advantage that every bit of the output is a function of only a polylogarithmic number of bits from the source, which is crucial for some cryptographic applications. Our argument is based on bounds for a generalization of quantum random access codes, which we call quantum functional access codes. This is crucial as it lets us avoid the local list-decoding algorithm central to the approach in, which was the source of the multiplicative overhead.

ID: CaltechAUTHORS:20140910-145258006

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Abstract: The idea to put computing machines on a physical footing and to use the laws of physics as the basis of a computer already dates back several decades. In the 1980s, Feynman [24,25] was the first to consider quantum mechanics from a computational point of view by observing that the simulation of quantum mechanical systems on a classical computer seemed to require an increase in complexity exponential in the size of the system. He asked whether this exponential overhead was inevitable, and if it was possible to design a universal quantum computer, which could simulate any quantum system without the exponential overhead. In 1985 Deutsch [17] defined the model of the quantum Turing machine, generalizing the classical Turing machine to follow the laws of quantum mechanics. Yao later showed that it was equivalent to the quantum circuit model, also defined by Deutsch.

Publication: Lecture Notes in Physics No.: 808 ISSN: 0075-8450

ID: CaltechAUTHORS:20200804-153453452

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Abstract: Gap Hamming Distance is a well-studied problem in communication complexity, in which Alice and Bob have to decide whether the Hamming distance between their respective n-bit inputs is less than n/2 - √n or greater than n/2 + √n. We show that every k-round bounded-error communication protocol for this problem sends a message of at least Ω(n/(k²logk)) bits. This lower bound has an exponentially better dependence on the number of rounds than the previous best bound, due to Brody and Chakrabarti. Our communication lower bound implies strong space lower bounds on algorithms for a number of data stream computations, such as approximating the number of distinct elements in a stream.

Publication: arXiv No.: 6302
ID: CaltechAUTHORS:20190320-105826650

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Abstract: XOR games are a simple computational model with connections to many areas of complexity theory. Perhaps the earliest use of XOR games was in the study of quantum correlations. XOR games also have an interesting connection to Grothendieck's inequality, a fundamental theorem of analysis, which shows that two players sharing entanglement can achieve at most a constant factor advantage over players following classical strategies in an XOR game. Perez-Garcia et al. show that when the players share GHZ states, this advantage is bounded by a constant. We use a multilinear generalization of Grothendieck's inequality due to Blei and Tonge to simplify the proof of the second result and extend it to the case of so-called Schmidt states, answering an open problem of Perez-Garcia et al. Via a reduction given in that paper, this answers a 35-year-old problem in operator algebras due to Varopoulos, showing that the space of compact operators on a Hilbert space is a Q-algebra under Schur product. A further generalization of Grothendieck's inequality due to Carne lets us show that the gap between the entangled and classical value is at most a constant in any multiplayer XOR game in which the players are allowed to share combinations of GHZ states and EPR pairs of any dimension. As an application of our results, we show that the discrepancy method in communication complexity remains a lower bound in the multiparty model where the players have quantum communication and the kinds of entanglement discussed above. This answers an open question of Lee, Schechtman, and Shraibman.

Publication: arXiv
ID: CaltechAUTHORS:20190320-111726313

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Abstract: The central question in quantum multi-prover interactive proof systems is whether or not entanglement shared among provers affects the verification power of the proof system. We study for the first time positive aspects of prior entanglement and show how it can be used to parallelize any multi-prover quantum interactive proof system to a one-round system with perfect completeness, soundness bounded away from one by an inverse-polynomial in the input size, and one extra prover. Alternatively, we can also parallelize to a three-turn system with the same number of provers, where the verifier only broadcasts the outcome of a coin flip. This “public-coin” property is somewhat surprising, since in the classical case public-coin multi-prover interactive proofs are equivalent to single-prover ones.

Publication: Computational Complexity Vol.: 18 No.: 2 ISSN: 1016-3328

ID: CaltechAUTHORS:20200805-150138118

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Abstract: Geometric intuition suggests that the Néron–Tate height of Heegner points on a rational elliptic curve E should be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross–Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin–Selberg convolution of E with a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twisted L-function of E by a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with these L-series and L-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves.

Publication: Canadian Journal of Mathematics Vol.: 60 No.: 6 ISSN: 0008-414X

ID: CaltechAUTHORS:20190320-142201374

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Abstract: We establish the first hardness results for the problem of computing the value of one-round games played by a referee and a team of players who can share quantum entanglement. In particular, we show that it is NP-hard to approximate within an inverse polynomial the value of a one-round game with (i) quantum referee and two entangled players or (ii) classical referee and three entangled players. Previously it was not even known if computing the value exactly is NP-hard. We also describe a mathematical conjecture, which, if true, would imply hardness of approximation to within a constant.We start our proof by describing two ways to modify classical multi-player games to make them resistant to entangled players. We then show that a strategy for the modified game that uses entanglement can be "rounded'' to one that does not. The results then follow from classical inapproximability bounds. Our work implies that, unless P = NP, the values of entangled-player games cannot be computed by semidefinite programs that are polynomial in the size of the referee's system, a method that has been successful for more restricted quantum games.

ID: CaltechAUTHORS:20140910-085511133

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Abstract: The most famous lattice problem is the Shortest Vector Problem (SVP), which has many applications in cryptology. The best approximation algorithms known for SVP in high dimension rely on a subroutine for exact SVP in low dimension. In this paper, we assess the practicality of the best (theoretical) algorithm known for exact SVP in low dimension: the sieve algorithm proposed by Ajtai, Kumar and Sivakumar (AKS) in 2001. AKS is a randomized algorithm of time and space complexity 2^(O(n)), which is theoretically much lower than the super-exponential complexity of all alternative SVP algorithms. Surprisingly, no implementation and no practical analysis of AKS has ever been reported. It was in fact widely believed that AKS was impractical: for instance, Schnorr claimed in 2003 that the constant hidden in the 2^(O(n)) complexity was at least 30. In this paper, we show that AKS can actually be made practical: we present a heuristic variant of AKS whose running time is (4/3+ϵ)^n polynomial-time operations, and whose space requirement is (4/3+ ϵ)^(n/2) polynomially many bits. Our implementation can experimentally find shortest lattice vectors up to dimension 50, but is slower than classical alternative SVP algorithms in these dimensions.

Publication: Journal of Mathematical Cryptology Vol.: 2 No.: 2 ISSN: 1862-2976

ID: CaltechAUTHORS:20200804-103250325

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Abstract: The central question in quantum multi-prover interactive proof systems is whether or not entanglement shared among provers affects the verification power of the proof system. We study for the first time positive aspects of prior entanglement and show how it can be used to parallelize any multi- prover quantum interactive proof system to a one-round system with perfect completeness, soundness bounded away from 1 by an inverse polynomial in the input size, and one extra proven Alternatively, we can also parallelize to a three-turn system with the same number of provers, where the verifier only broadcasts the outcome of a coin flip. This "public-coin" property is somewhat surprising, since in the classical case public-coin multi-prover interactive proofs are equivalent to single prover ones.

ID: CaltechAUTHORS:20140910-083116800

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Abstract: We show that the value of a general two-prover quantum game cannot be computed by a semidefinite program of polynomial size (unless P=NP), a method that has been successful in more restricted quantum games. More precisely, we show that proof of membership in the NP-complete problem GAP-3D-MATCHING can be obtained by a 2-prover, 1-round quantum interactive proof system where the provers share entanglement, with perfect completeness and soundness s = 1 − 2^(−O(n)), and such that the space of the verifier and the size of the messages are O(log n). This implies that QMIP*_(log n,1,1−2−O(n))⊈ P unless P = NP and provides the first non-trivial lower bound on the power of entangled quantum provers, albeit with an exponentially small gap. The gap achievable by our proof system might in fact be larger, provided a certain conjecture on almost commuting versus nearly commuting projector matrices is true.

ID: CaltechAUTHORS:20160322-085312434

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