Abstract: Classical machine learning (ML) provides a potentially powerful approach to solving challenging quantum many-body problems in physics and chemistry. However, the advantages of ML over traditional methods have not been firmly established. In this work, we prove that classical ML algorithms can efficiently predict ground-state properties of gapped Hamiltonians after learning from other Hamiltonians in the same quantum phase of matter. By contrast, under a widely accepted conjecture, classical algorithms that do not learn from data cannot achieve the same guarantee. We also prove that classical ML algorithms can efficiently classify a wide range of quantum phases. Extensive numerical experiments corroborate our theoretical results in a variety of scenarios, including Rydberg atom systems, two-dimensional random Heisenberg models, symmetry-protected topological phases, and topologically ordered phases.

Publication: Science Vol.: 377 No.: 6613 ISSN: 0036-8075

ID: CaltechAUTHORS:20221207-387978400.2

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Abstract: Quantum many-body systems involving bosonic modes or gauge fields have infinite-dimensional local Hilbert spaces which must be truncated to perform simulations of real-time dynamics on classical or quantum computers. To analyze the truncation error, we develop methods for bounding the rate of growth of local quantum numbers such as the occupation number of a mode at a lattice site, or the electric field at a lattice link. Our approach applies to various models of bosons interacting with spins or fermions, and also to both abelian and non-abelian gauge theories. We show that if states in these models are truncated by imposing an upper limit Λ on each local quantum number, and if the initial state has low local quantum numbers, then an error at most ϵ can be achieved by choosing Λ to scale polylogarithmically with ϵ⁻¹, an exponential improvement over previous bounds based on energy conservation. For the Hubbard-Holstein model, we numerically compute a bound on Λ that achieves accuracy ϵ, obtaining significantly improved estimates in various parameter regimes. We also establish a criterion for truncating the Hamiltonian with a provable guarantee on the accuracy of time evolution. Building on that result, we formulate quantum algorithms for dynamical simulation of lattice gauge theories and of models with bosonic modes; the gate complexity depends almost linearly on spacetime volume in the former case, and almost quadratically on time in the latter case. We establish a lower bound showing that there are systems involving bosons for which this quadratic scaling with time cannot be improved. By applying our result on the truncation error in time evolution, we also prove that spectrally isolated energy eigenstates can be approximated with accuracy ϵ by truncating local quantum numbers at Λ = polylog(ϵ⁻¹).

Publication: Quantum Vol.: 6ISSN: 2521-327X

ID: CaltechAUTHORS:20221024-125854800.25

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Abstract: Quantum technology promises to revolutionize how we learn about the physical world. An experiment that processes quantum data with a quantum computer could have substantial advantages over conventional experiments in which quantum states are measured and outcomes are processed with a classical computer. We proved that quantum machines could learn from exponentially fewer experiments than the number required by conventional experiments. This exponential advantage is shown for predicting properties of physical systems, performing quantum principal component analysis, and learning about physical dynamics. Furthermore, the quantum resources needed for achieving an exponential advantage are quite modest in some cases. Conducting experiments with 40 superconducting qubits and 1300 quantum gates, we demonstrated that a substantial quantum advantage is possible with today’s quantum processors.

Publication: Science Vol.: 376 No.: 6598 ISSN: 0036-8075

ID: CaltechAUTHORS:20220113-234532429

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Abstract: We simulate, using nonperturbative methods, the real-time dynamics of small bubbles of “false vacuum” in a quantum spin chain near criticality, where the low-energy physics is described by a relativistic (1+1)-dimensional quantum field theory. We consider bubbles whose walls are kink and antikink quasiparticle excitations, so that wall collisions are kink-antikink scattering events. To construct these bubbles in the presence of strong correlations, we extend a recently proposed matrix product state (MPS) ansatz for quasiparticle wavepackets. We simulate dynamics within a window of about 1000 spins embedded in an infinite chain at energies of up to about 5 times the mass gap. By choosing the wavepacket width and the bubble size appropriately, we avoid strong lattice effects and observe relativistic kink-antikink collisions. We use the MPS quasiparticle ansatz to detect scattering outcomes. (i) In the Ising model, with transverse and longitudinal fields, we do not observe particle production despite nonintegrability (supporting recent observations of nonthermalizing states in this model). (ii) Switching on an additional interaction, we see production of confined and unconfined particle pairs. We characterize the amount of entanglement generated as a function of energy and time and conclude that our classical simulation methods will ultimately fail as these increase. We anticipate that kink-antikink scattering in 1+1 dimensions will be an instructive benchmark problem for future quantum computers and analog quantum simulators.

Publication: PRX Quantum Vol.: 3 No.: 2 ISSN: 2691-3399

ID: CaltechAUTHORS:20210512-104051553

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Abstract: We present a comprehensive architectural analysis for a proposed fault-tolerant quantum computer based on cat codes concatenated with outer quantum error-correcting codes. For the physical hardware, we propose a system of acoustic resonators coupled to superconducting circuits with a two-dimensional layout. Using estimated physical parameters for the hardware, we perform a detailed error analysis of measurements and gates, including cnot and Toffoli gates. Having built a realistic noise model, we numerically simulate quantum error correction when the outer code is either a repetition code or a thin rectangular surface code. Our next step toward universal fault-tolerant quantum computation is a protocol for fault-tolerant Toffoli magic state preparation that significantly improves upon the fidelity of physical Toffoli gates at very low qubit cost. To achieve even lower overheads, we devise a new magic state distillation protocol for Toffoli states. Combining these results together, we obtain realistic full-resource estimates of the physical error rates and overheads needed to run useful fault-tolerant quantum algorithms. We find that with around 1000 superconducting circuit components, one could construct a fault-tolerant quantum computer that can run circuits, which are currently intractable for classical computers. Hardware with 18 000 superconducting circuit components, in turn, could simulate the Hubbard model in a regime beyond the reach of classical computing.

Publication: PRX Quantum Vol.: 3 No.: 1 ISSN: 2691-3399

ID: CaltechAUTHORS:20201209-172305164

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Abstract: Steven Weinberg, widely regarded as the preeminent theoretical particle physicist of his era, passed away on 23 July at age 88. Steve took a pivotal step toward establishing what came to be known as the standard model of the fundamental particles and their interactions, for which he shared the 1979 Nobel Prize in Physics with Sheldon Glashow and Abdus Salam. That contribution was just one highlight in a career studded with major accomplishments. In later years, Steve authored a series of highly influential physics textbooks, as well as eloquent books and essays for the general public expounding on societal and scientific issues. He remained scientifically active up to his final days.

Publication: Science Vol.: 373 No.: 6559 ISSN: 0036-8075

ID: CaltechAUTHORS:20210914-191005909

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Abstract: We consider the problem of jointly estimating expectation values of many Pauli observables, a crucial subroutine in variational quantum algorithms. Starting with randomized measurements, we propose an efficient derandomization procedure that iteratively replaces random single-qubit measurements by fixed Pauli measurements; the resulting deterministic measurement procedure is guaranteed to perform at least as well as the randomized one. In particular, for estimating any L low-weight Pauli observables, a deterministic measurement on only of order log(L) copies of a quantum state suffices. In some cases, for example, when some of the Pauli observables have high weight, the derandomized procedure is substantially better than the randomized one. Specifically, numerical experiments highlight the advantages of our derandomized protocol over various previous methods for estimating the ground-state energies of small molecules.

Publication: Physical Review Letters Vol.: 127 No.: 3 ISSN: 0031-9007

ID: CaltechAUTHORS:20210512-104041014

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Abstract: We study the performance of classical and quantum machine learning (ML) models in predicting outcomes of physical experiments. The experiments depend on an input parameter x and involve execution of a (possibly unknown) quantum process E. Our figure of merit is the number of runs of E required to achieve a desired prediction performance. We consider classical ML models that perform a measurement and record the classical outcome after each run of E, and quantum ML models that can access E coherently to acquire quantum data; the classical or quantum data are then used to predict the outcomes of future experiments. We prove that for any input distribution D(x), a classical ML model can provide accurate predictions on average by accessing E a number of times comparable to the optimal quantum ML model. In contrast, for achieving an accurate prediction on all inputs, we prove that the exponential quantum advantage is possible. For example, to predict the expectations of all Pauli observables in an n-qubit system ρ, classical ML models require 2^(Ω(n)) copies of ρ, but we present a quantum ML model using only O(n) copies. Our results clarify where the quantum advantage is possible and highlight the potential for classical ML models to address challenging quantum problems in physics and chemistry.

Publication: Physical Review Letters Vol.: 126 No.: 19 ISSN: 0031-9007

ID: CaltechAUTHORS:20210512-104048123

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Abstract: The great promise of quantum computers comes with the dual challenges of building them and finding their useful applications. We argue that these two challenges should be considered together, by codesigning full-stack quantum computer systems along with their applications in order to hasten their development and potential for scientific discovery. In this context, we identify scientific and community needs, opportunities, a sampling of a few use case studies, and significant challenges for the development of quantum computers for science over the next 2–10 years. This document is written by a community of university, national laboratory, and industrial researchers in the field of Quantum Information Science and Technology, and is based on a summary from a U.S. National Science Foundation workshop on Quantum Computing held on October 21–22, 2019 in Alexandria, VA.

Publication: PRX Quantum Vol.: 2 No.: 1 ISSN: 2691-3399

ID: CaltechAUTHORS:20210514-140222766

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Abstract: We propose a method for detecting bipartite entanglement in a many-body mixed state based on estimating moments of the partially transposed density matrix. The estimates are obtained by performing local random measurements on the state, followed by postprocessing using the classical shadows framework. Our method can be applied to any quantum system with single-qubit control. We provide a detailed analysis of the required number of experimental runs, and demonstrate the protocol using existing experimental data [Brydges et al., Science 364, 260 (2019)].

Publication: Physical Review Letters Vol.: 125 No.: 20 ISSN: 0031-9007

ID: CaltechAUTHORS:20201111-102025217

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Abstract: Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here, we study the compatibility of these two important principles. If a logical quantum system is encoded into n physical subsystems, we say that the code is covariant with respect to a symmetry group G if a G transformation on the logical system can be realized by performing transformations on the individual subsystems. For a G-covariant code with G a continuous group, we derive a lower bound on the error-correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems n or the dimension d of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with n or d as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem for quantum computation: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large d (using random codes) or n (using codes based on W states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.

Publication: Physical Review X Vol.: 10 No.: 4 ISSN: 2160-3308

ID: CaltechAUTHORS:20201027-095348367

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Abstract: Predicting the properties of complex, large-scale quantum systems is essential for developing quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a ‘classical shadow’, can be used to predict many different properties; order log(M) measurements suffice to accurately predict M different functions of the state with high success probability. The number of measurements is independent of the system size and saturates information-theoretic lower bounds. Moreover, target properties to predict can be selected after the measurements are completed. We support our theoretical findings with extensive numerical experiments. We apply classical shadows to predict quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables and the energy variance of many-body local Hamiltonians. The numerical results highlight the advantages of classical shadows relative to previously known methods.

Publication: Nature Physics Vol.: 16 No.: 10 ISSN: 1745-2473

ID: CaltechAUTHORS:20200427-084340790

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Abstract: We study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In particular, we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment.

Publication: New Journal of Physics Vol.: 22 No.: 7 ISSN: 1367-2630

ID: CaltechAUTHORS:20200827-141821017

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Abstract: We construct quantum error-correcting codes that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of rotational states of a rigid body. These codes, which protect against both drift in the body’s orientation and small changes in its angular momentum, may be well suited for robust storage and coherent processing of quantum information using rotational states of a polyatomic molecule. Extensions of such codes to rigid bodies with a symmetry axis are compatible with rotational states of diatomic molecules as well as nuclear states of molecules and atoms. We also describe codes associated with general non-Abelian groups and develop orthogonality relations for coset spaces, laying the groundwork for quantum information processing with exotic configuration spaces.

Publication: Physical Review X Vol.: 10 No.: 3 ISSN: 2160-3308

ID: CaltechAUTHORS:20200413-094120710

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Abstract: We reconsider the black hole firewall puzzle, emphasizing that quantum error- correction, computational complexity, and pseudorandomness are crucial concepts for understanding the black hole interior. We assume that the Hawking radiation emitted by an old black hole is pseudorandom, meaning that it cannot be distinguished from a perfectly thermal state by any efficient quantum computation acting on the radiation alone. We then infer the existence of a subspace of the radiation system which we interpret as an encoding of the black hole interior. This encoded interior is entangled with the late outgoing Hawking quanta emitted by the old black hole, and is inaccessible to computationally bounded observers who are outside the black hole. Specifically, efficient operations acting on the radiation, those with quantum computational complexity polynomial in the entropy of the remaining black hole, commute with a complete set of logical operators acting on the encoded interior, up to corrections which are exponentially small in the entropy. Thus, under our pseudorandomness assumption, the black hole interior is well protected from exterior observers as long as the remaining black hole is macroscopic. On the other hand, if the radiation is not pseudorandom, an exterior observer may be able to create a firewall by applying a polynomial-time quantum computation to the radiation.

Publication: Journal of High Energy Physics Vol.: 2020 No.: 6 ISSN: 1029-8479

ID: CaltechAUTHORS:20200608-102711610

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Abstract: A distributed sensing protocol uses a network of local sensing nodes to estimate a global feature of the network, such as a weighted average of locally detectable parameters. In the noiseless case, continuous-variable (CV) multipartite entanglement shared by the nodes can improve the precision of parameter estimation relative to the precision attainable by a network without shared entanglement; for an entangled protocol, the root mean square estimation error scales like 1/M with the number M of sensing nodes, the so-called Heisenberg scaling, while for protocols without entanglement, the error scales like 1√M. However, in the presence of loss and other noise sources, although multipartite entanglement still has some advantages for sensing displacements and phases, the scaling of the precision with M is less favorable. In this paper, we show that using CV error correction codes can enhance the robustness of sensing protocols against imperfections and reinstate Heisenberg scaling up to moderate values of M. Furthermore, while previous distributed sensing protocols could measure only a single quadrature, we construct a protocol in which both quadratures can be sensed simultaneously. Our work demonstrates the value of CV error correction codes in realistic sensing scenarios.

Publication: New Journal of Physics Vol.: 22 No.: 2 ISSN: 1367-2630

ID: CaltechAUTHORS:20200430-121016107

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Abstract: We propose a new cellular automaton (CA), the sweep rule, which generalizes Toom’s rule to any locally Euclidean lattice. We use the sweep rule to design a local decoder for the toric code in d ≥ 3 dimensions, the sweep decoder, and rigorously establish a lower bound on its performance. We also numerically estimate the sweep decoder threshold for the three-dimensional toric code on the cubic and body-centered cubic lattices for phenomenological phase-flip noise. Our results lead to new CA decoders with provable error-correction thresholds for other topological quantum codes including the color code.

Publication: Physical Review Letters Vol.: 123 No.: 2 ISSN: 0031-9007

ID: CaltechAUTHORS:20190201-155942835

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Abstract: Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50-100 qubits may be able to perform tasks which surpass the capabilities of today's classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably. NISQ devices will be useful tools for exploring many-body quantum physics, and may have other useful applications, but the 100-qubit quantum computer will not change the world right away - we should regard it as a significant step toward the more powerful quantum technologies of the future. Quantum technologists should continue to strive for more accurate quantum gates and, eventually, fully fault-tolerant quantum computing.

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

ID: CaltechAUTHORS:20180521-094354257

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Abstract: Three-dimensional (3D) color codes have advantages for fault-tolerant quantum computing, such as protected quantum gates with relatively low overhead and robustness against imperfect measurement of error syndromes. Here we investigate the storage threshold error rates for bit-flip and phase-flip noise in the 3D color code (3DCC) on the body-centered cubic lattice, assuming perfect syndrome measurements. In particular, by exploiting a connection between error correction and statistical mechanics, we estimate the threshold for 1D stringlike and 2D sheetlike logical operators to be p^((1))_(3DCC) ≃ 1.9% and p^((2))_(3DCC) ≃ 27.6%. We obtain these results by using parallel tempering Monte Carlo simulations to study the disorder-temperature phase diagrams of two new 3D statistical-mechanical models: the four- and six-body random coupling Ising models.

Publication: Physical Review Letters Vol.: 120 No.: 18 ISSN: 0031-9007

ID: CaltechAUTHORS:20171004-145219476

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Abstract: Stephen William Hawking died on 14 March (Albert Einstein's birthday) at the age of 76 after decades of battling the incurable disease amyotrophic lateral sclerosis (ALS). His early scientific work transformed our understanding of general relativity, Einstein's theory of gravitation. Later in life, Stephen became an immensely successful popularizer of science; his courage and high spirits in the face of his disability inspired millions. Stephen Hawking's achievements as a scientist, communicator, and public figure were commensurate with his great fame.

Publication: Science Vol.: 360 No.: 6385 ISSN: 0036-8075

ID: CaltechAUTHORS:20180412-141957828

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Abstract: Quantum metrology has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on measurement precision, called the Heisenberg limit, which can be achieved for noiseless quantum systems, but is not achievable in general for systems subject to noise. Here we study how measurement precision can be enhanced through quantum error correction, a general method for protecting a quantum system from the damaging effects of noise. We find a necessary and sufficient condition for achieving the Heisenberg limit using quantum probes subject to Markovian noise, assuming that noiseless ancilla systems are available, and that fast, accurate quantum processing can be performed. When the sufficient condition is satisfied, a quantum error-correcting code can be constructed that suppresses the noise without obscuring the signal; the optimal code, achieving the best possible precision, can be found by solving a semidefinite program.

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

ID: CaltechAUTHORS:20170720-172112488

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Abstract: Recent work has shown that quantum computers can compute scattering probabilities in massive quantum field theories, with a run time that is polynomial in the number of particles, their energy, and the desired precision. Here we study a closely related quantum field-theoretical problem: estimating the vacuum-to-vacuum transition amplitude, in the presence of spacetime-dependent classical sources, for a massive scalar field theory in (1+1) dimensions. We show that this problem is BQP-hard; in other words, its solution enables one to solve any problem that is solvable in polynomial time by a quantum computer. Hence, the vacuum-to-vacuum amplitude cannot be accurately estimated by any efficient classical algorithm, even if the field theory is very weakly coupled, unless BQP=BPP. Furthermore, the corresponding decision problem can be solved by a quantum computer in a time scaling polynomially with the number of bits needed to specify the classical source fields, and this problem is therefore BQP-complete. Our construction can be regarded as an idealized architecture for a universal quantum computer in a laboratory system described by massive phi^4 theory coupled to classical spacetime-dependent sources.

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

ID: CaltechAUTHORS:20170720-172919513

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Abstract: Almheiri, Dong, and Harlow [J. High Energy Phys. 04 (2015) 163.] proposed a highly illuminating connection between the AdS/CFT holographic correspondence and operator algebra quantum error correction (OAQEC). Here, we explore this connection further. We derive some general results about OAQEC, as well as results that apply specifically to quantum codes that admit a holographic interpretation. We introduce a new quantity called price, which characterizes the support of a protected logical system, and find constraints on the price and the distance for logical subalgebras of quantum codes. We show that holographic codes defined on bulk manifolds with asymptotically negative curvature exhibit uberholography, meaning that a bulk logical algebra can be supported on a boundary region with a fractal structure. We argue that, for holographic codes defined on bulk manifolds with asymptotically flat or positive curvature, the boundary physics must be highly nonlocal, an observation with potential implications for black holes and for quantum gravity in AdS space at distance scales that are small compared to the AdS curvature radius.

Publication: Physical Review X Vol.: 7 No.: 2 ISSN: 2160-3308

ID: CaltechAUTHORS:20170517-114206968

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Abstract: Recently, W. Lechner, P. Hauke, and P. Zoller [Sci. Adv. 1, e1500838 (2015)] have proposed a quantum annealing architecture, in which a classical spin glass with all-to-all pairwise connectivity is simulated by a spin glass with geometrically local interactions. We interpret this architecture as a classical error-correcting code, which is highly robust against weakly correlated bit-flip noise, and we analyze the code's performance using a belief-propagation decoding algorithm. Our observations may also apply to more general encoding schemes and noise models.

Publication: Physical Review A Vol.: 93 No.: 5 ISSN: 2469-9926

ID: CaltechAUTHORS:20160315-111407944

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Abstract: We study restrictions on locality-preserving unitary logical gates for topological quantum codes in two spatial dimensions. A locality-preserving operation is one which maps local operators to local operators — for example, a constant-depth quantum circuit of geometrically local gates, or evolution for a constant time governed by a geometrically local bounded-strength Hamiltonian. Locality-preserving logical gates of topological codes are intrinsically fault tolerant because spatially localized errors remain localized, and hence sufficiently dilute errors remain correctable. By invoking general properties of two-dimensional topological field theories, we find that the locality-preserving logical gates are severely limited for codes which admit non-abelian anyons, in particular, there are no locality-preserving logical gates on the torus or the sphere with M punctures if the braiding of anyons is computationally universal. Furthermore, for Ising anyons on the M-punctured sphere, locality-preserving gates must be elements of the logical Pauli group. We derive these results by relating logical gates of a topological code to automorphisms of the Verlinde algebra of the corresponding anyon model, and by requiring the logical gates to be compatible with basis changes in the logical Hilbert space arising from local F-moves and the mapping class group.

Publication: Journal of Mathematical Physics Vol.: 57 No.: 2 ISSN: 0022-2488

ID: CaltechAUTHORS:20141209-131847639

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Abstract: We propose a universal form for quark and lepton mass matrices, which applies in a “leading order” approximation where CP-violating phases are ignored. Down-quark mass ratios are successfully predicted in our scheme using the measured Cabibbo-Kobayashi-Maskawa mixing angles as input. Assuming an additional discrete symmetry in the neutrino sector, we obtain the “golden ratio” pattern in the leading-order Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix; in addition we predict an inverted neutrino mass hierarchy with m_1-≃-m_2 ≃ 74 meV, m_3 ≃ 55 meV, and neutrinoless double beta decay mass parameter m_(0νββ) ≃ 33 meV. When CP-violating phases are included, our scheme suggests a residual ℤ_2 antiunitary symmetry of the neutrino mass matrix, in which the interchange of μ and τ neutrinos is accompanied by a time reversal transformation, thus predicting that the CP-violating angle in the neutrino sector is close to the maximal value δ = ± π/2, and that the diagonal phases in the PMNS matrix are α_ 1 ≃ 0, α_2 ≃ π.

Publication: Physical Review D Vol.: 92 No.: 11 ISSN: 2470-0010

ID: CaltechAUTHORS:20160104-165102169

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Abstract: We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].

Publication: Journal of High Energy Physics Vol.: 2015 No.: 6 ISSN: 1029-8479

ID: CaltechAUTHORS:20150717-122900464

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Abstract: A two-dimensional topologically ordered quantum memory is well protected against error if the energy gap is large compared to the temperature, but this protection does not improve as the system size increases. We review and critique some recent proposals for improving the memory time by introducing long-range interactions among anyons, noting that instability with respect to small local perturbations of the Hamiltonian is a generic problem for such proposals. We also discuss some broader issues regarding the prospects for scalable quantum memory in two-dimensional systems.

Publication: Physical Review A Vol.: 91 No.: 3 ISSN: 1050-2947

ID: CaltechAUTHORS:20150420-105848086

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Abstract: Quantum field theory provides the framework for the most fundamental physical theories to be confirmed experimentally, and has enabled predictions of unprecedented precision. However, calculations of physical observables often require great computational complexity and can generally be performed only when the interaction strength is weak. A full understanding of the foundations and rich consequences of quantum field theory remains an outstanding challenge. We develop a quantum algorithm to compute relativistic scattering amplitudes in massive phi-fourth theory in spacetime of four and fewer dimensions. The algorithm runs in a time that is polynomial in the number of particles, their energy, and the desired precision, and applies at both weak and strong coupling. Thus, it offers exponential speedup over existing classical methods at high precision or strong coupling.

Publication: Quantum Information and Computation Vol.: 14 No.: 11-12 ISSN: 1533-7146

ID: CaltechAUTHORS:20120712-151413773

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Abstract: Almheiri et al. have emphasized that otherwise reasonable beliefs about black hole evaporation are incompatible with the monogamy of quantum entanglement, a general property of quantum mechanics. We investigate the final-state projection model of black hole evaporation proposed by Horowitz and Maldacena, pointing out that this model admits cloning of quantum states and polygamous entanglement, allowing unitarity of the evaporation process to be reconciled with smoothness of the black hole event horizon. Though the model seems to require carefully tuned dynamics to ensure exact unitarity of the black hole S-matrix, for a generic final-state boundary condition the deviations from unitarity are exponentially small in the black hole entropy; furthermore observers inside black holes need not detect any deviations from standard quantum mechanics. Though measurements performed inside old black holes could potentially produce causality-violating phenomena, the computational complexity of decoding the Hawking radiation may render the causality violation unobservable. Final-state projection models illustrate how inviolable principles of standard quantum mechanics might be circumvented in a theory of quantum gravity.

Publication: Journal of High Energy Physics Vol.: 2014 No.: 8 ISSN: 1126-6708

ID: CaltechAUTHORS:20140611-133437673

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Abstract: Quantum theory is over a century old, yet physicists continue to be perplexed and delighted by the weirdness of the quantum world. Whereas the laws of classical physics successfully explain the phenomena we experience every day, atoms and other tiny objects obey quantum laws that sometimes seem to defy common sense, baffling our feeble human minds. In the 21st century, we hope to put this weirdness to work by building quantum computers capable of performing amazing tasks.

Publication: Physics World Vol.: 26 No.: 10 ISSN: 0953-8585

ID: CaltechAUTHORS:20131202-091656426

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Abstract: We analyze the accuracy of quantum phase gates acting on “0-π qubits” in superconducting circuits, where the gates are protected against thermal and Hamiltonian noise by continuous-variable quantum error-correcting codes. The gates are executed by turning on and off a tunable Josephson coupling between an LC oscillator and a qubit or pair of qubits; assuming perfect qubits, we show that the gate errors are exponentially small when the oscillator's impedance √L/C is large compared to ℏ/4e^2≈1 kΩ. The protected gates are not computationally universal by themselves, but a scheme for universal fault-tolerant quantum computation can be constructed by combining them with unprotected noisy operations. We validate our analytic arguments with numerical simulations.

Publication: Physical Review A Vol.: 87 No.: 5 ISSN: 1050-2947

ID: CaltechAUTHORS:20130619-094717394

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Abstract: We study the performance of Bacon-Shor codes, quantum subsystem codes which are well suited for applications to fault-tolerant quantum memory because the error syndrome can be extracted by performing two-qubit measurements. Assuming independent noise, we find the optimal block size in terms of the bit-flip error probability pX and the phase error probability pZ, and determine how the probability of a logical error depends on pX and pZ. We show that a single Bacon-Shor code block, used by itself without concatenation, can provide very effective protection against logical errors if the noise is highly biased (pZ/pX ≫1) and the physical error rate pZ is a few percent or below. We also derive an upper bound on the logical error rate for the case where the syndrome data is noisy.

Publication: Quantum Information and Computation Vol.: 13 No.: 5-6 ISSN: 1533-7146

ID: CaltechAUTHORS:20130325-085516990

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Abstract: We develop a scheme for fault-tolerant quantum computation based on asymmetric Bacon-Shor codes, which works effectively against highly biased noise dominated by dephasing. We find the optimal Bacon-Shor block size as a function of the noise strength and the noise bias, and estimate the logical error rate and overhead cost achieved by this optimal code. Our fault-tolerant gadgets, based on gate teleportation, are well suited for hardware platforms with geometrically local gates in two dimensions.

Publication: Physical Review A Vol.: 87 No.: 3 ISSN: 1050-2947

ID: CaltechAUTHORS:20130404-091531009

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Abstract: I study the effectiveness of fault-tolerant quantum computation against correlated Hamiltonian noise, and derive a sufficient condition for scalability. Arbitrarily long quantum computations can be executed reliably provided that noise terms acting collectively on k system qubits are sufficiently weak, and decay sufficiently rapidly with increasing k and with increasing spatial separation of the qubits.

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

ID: CaltechAUTHORS:20130321-160602027

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Abstract: We study the structure of logical operators in local D-dimensional quantum codes, considering both subsystem codes with geometrically local gauge generators and codes defined by geometrically local commuting projectors. We show that if the code distance is d, then any logical operator can be supported on a set of specified geometry containing ˜d qubits, where ˜dd^(1/(D−1)) = O(n) and n is the code length. Our results place limitations on partially self-correcting quantum memories, in which at least some logical operators are protected by energy barriers that grow with system size. We also show that for any two-dimensional local commuting projector code there is a nontrivial logical “string” operator supported on a narrow strip, where the operator is only slightly entangling across any cut through the strip.

Publication: Physical Review A Vol.: 86 No.: 3 ISSN: 1050-2947

ID: CaltechAUTHORS:20121012-142001671

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Abstract: Quantum field theory reconciles quantum mechanics and special relativity, and plays a central role in many areas of physics. We develop a quantum algorithm to compute relativistic scattering probabilities in a massive quantum field theory with quartic self-interactions (φ^4 theory) in spacetime of four and fewer dimensions. Its run time is polynomial in the number of particles, their energy, and the desired precision, and applies at both weak and strong coupling. In the strong-coupling and high-precision regimes, our quantum algorithm achieves exponential speedup over the fastest known classical algorithm.

Publication: Science Vol.: 336 No.: 6085 ISSN: 0036-8075

ID: CaltechAUTHORS:20120522-122303309

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Abstract: We study how dynamical decoupling (DD) pulse sequences can improve the reliability of quantum computers. We prove upper bounds on the accuracy of DD-protected quantum gates and derive sufficient conditions for DD-protected gates to outperform unprotected gates. Under suitable conditions, fault-tolerant quantum circuits constructed from DD-protected gates can tolerate stronger noise and have a lower overhead cost than fault-tolerant circuits constructed from unprotected gates. Our accuracy estimates depend on the dynamics of the bath that couples to the quantum computer and can be expressed either in terms of the operator norm of the bath’s Hamiltonian or in terms of the power spectrum of bath correlations; we explain in particular how the performance of recursively generated concatenated pulse sequences can be analyzed from either viewpoint. Our results apply to Hamiltonian noise models with limited spatial correlations.

Publication: Physical Review A Vol.: 84 No.: 1 ISSN: 1050-2947

ID: CaltechAUTHORS:20110715-131156639

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Abstract: We propose and analyze an interface between a topological qubit and a superconducting flux qubit. In our scheme, the interaction between Majorana fermions in a topological insulator is coherently controlled by a superconducting phase that depends on the quantum state of the flux qubit. A controlled-phase gate, achieved by pulsing this interaction on and off, can transfer quantum information between the topological qubit and the superconducting qubit.

Publication: Physical Review Letters Vol.: 106 No.: 13 ISSN: 0031-9007

ID: CaltechAUTHORS:20110419-095245511

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Abstract: We study the robustness of a fault-tolerant quantum computer subject to Gaussian non-Markovian quantum noise, and we show that scalable quantum computation is possible if the noise power spectrum satisfies an appropriate “threshold condition.” Our condition is less sensitive to very-high-frequency noise than previously derived threshold conditions for non-Markovian noise.

Publication: Physical Review A Vol.: 79 No.: 3 ISSN: 1050-2947

ID: CaltechAUTHORS:20090728-112832875

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Abstract: We present a universal scheme of pulsed operations suitable for the IBM oscillator-stabilized flux qubit comprising the controlled-sigma(z) (CPHASE) gate, single-qubit preparations and measurements. Based on numerical simulations, we argue that the error rates for these operations can be as low as about 0.5% and that noise is highly biased, with phase errors being stronger than all other types of errors by a factor of nearly 10^3. In contrast, the design of a controlled σ(x) (CNOT) gate for this system with an error rate of less than about 1.2% seems extremely challenging. We propose a special encoding that exploits the noise bias allowing us to implement a logical CNOT gate where phase errors and all other types of errors have nearly balanced rates of about 0.4%. Our results illustrate how the design of an encoding scheme can be adjusted and optimized according to the available physical operations and the particular noise characteristics of experimental devices.

Publication: New Journal of Physics Vol.: 11ISSN: 1367-2630

ID: CaltechAUTHORS:ALInjp09

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Abstract: We rigorously analyze Knill's Fibonacci scheme for fault-tolerant quantum computation, which is based on the recursive preparation of Bell states protected by a concatenated error-detecting code. We prove lower bounds on the threshold fault rate of 0.67×10^−3 for adversarial local stochastic noise, and 1.25×10^−3 for independent depolarizing noise. In contrast to other schemes with comparable proved accuracy thresholds, the Fibonacci scheme has a significantly reduced overhead cost because it uses postselection far more sparingly.

Publication: Physical Review A Vol.: 79 No.: 1 ISSN: 1050-2947

ID: CaltechAUTHORS:ALIpra09

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Abstract: We formulate a scheme for fault-tolerant quantum computation that works effectively against highly biased noise, where dephasing is far stronger than all other types of noise. In our scheme, the fundamental operations performed by the quantum computer are single-qubit preparations, single-qubit measurements, and conditional-phase (CPHASE) gates, where the noise in the CPHASE gates is biased. We show that the accuracy threshold for quantum computation can be improved by exploiting this noise asymmetry; e.g., if dephasing dominates all other types of noise in the CPHASE gates by four orders of magnitude, we find a rigorous lower bound on the accuracy threshold higher by a factor of 5 than for the case of unbiased noise.

Publication: Physical Review A Vol.: 78 No.: 5 ISSN: 1050-2947

ID: CaltechAUTHORS:ALIpra08

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Abstract: We study information retrieval from evaporating black holes, assuming that the internal dynamics of a black hole is unitary and rapidly mixing, and assuming that the retriever has unlimited control over the emitted Hawking radiation. If the evaporation of the black hole has already proceeded past the ``half-way'' point, where half of the initial entropy has been radiated away, then additional quantum information deposited in the black hole is revealed in the Hawking radiation very rapidly. Information deposited prior to the half-way point remains concealed until the half-way point, and then emerges quickly. These conclusions hold because typical local quantum circuits are efficient encoders for quantum error-correcting codes that nearly achieve the capacity of the quantum erasure channel. Our estimate of a black hole's information retention time, based on speculative dynamical assumptions, is just barely compatible with the black hole complementarity hypothesis.

Publication: Journal of High Energy Physics Vol.: 2007 No.: 09 ISSN: 1126-6708

ID: CaltechAUTHORS:HAYjhep07

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Abstract: We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal density operator rho for the degrees of freedom in the interior. The von Neumann entropy of rho, a measure of the entanglement of the interior and exterior variables, has the form S(rho)=alphaL-gamma+[centered ellipsis], where the ellipsis represents terms that vanish in the limit L-->[infinity]. We show that -gamma is a universal constant characterizing a global feature of the entanglement in the ground state. Using topological quantum field theory methods, we derive a formula for gamma in terms of properties of the superselection sectors of the medium.

Publication: Physical Review Letters Vol.: 96 No.: 11 ISSN: 0031-9007

ID: CaltechAUTHORS:KITprl06

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Abstract: We prove a new version of the quantum accuracy threshold theorem that applies to non-Markovian noise with algebraically decaying spatial correlations. We consider noise in a quantum computer arising from a perturbation that acts collectively on pairs of qubits and on the environment, and we show that an arbitrarily long quantum computation can be executed with high reliability in D spatial dimensions, if the perturbation is sufficiently weak and decays with the distance r between the qubits faster than 1/r^D.

Publication: Physical Review Letters Vol.: 96 No.: 5 ISSN: 0031-9007

ID: CaltechAUTHORS:AHAprl06

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Abstract: We show that superselection rules do not enhance the information-theoretic security of quantum cryptographic protocols. Our analysis employs two quite different methods. The first method uses the concept of a reference system—in a world subject to a superselection rule, unrestricted operations can be simulated by parties who share access to a reference system with suitable properties. By this method, we prove that if an n-party protocol is secure in a world subject to a superselection rule, then the security is maintained even if the superselection rule is relaxed. However, the proof applies only to a limited class of superselection rules, those in which the superselection sectors are labeled by unitary irreducible representations of a compact symmetry group. The second method uses the concept of the format of a message sent between parties—by verifying the format, the recipient of a message can check whether the message could have been sent by a party who performed charge-conserving operations. By this method, we prove that protocols subject to general superselection rules (including those pertaining to non-Abelian anyons in two dimensions) are no more secure than protocols in the unrestricted world. However, the proof applies only to two-party protocols. Our results show in particular that, if no assumptions are made about the computational power of the cheater, then secure quantum bit commitment and strong quantum coin flipping with arbitrarily small bias are impossible in a world subject to superselection rules.

Publication: Physical Review A Vol.: 69 No.: 5 ISSN: 1050-2947

ID: CaltechAUTHORS:KITpra04

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Abstract: Horowitz and Maldacena have suggested that the unitarity of the black hole S-matrix can be reconciled with Hawking's semiclassical arguments if a final-state boundary condition is imposed at the spacelike singularity inside the black hole. We point out that, in this scenario, departures from unitarity can arise due to interactions between the collapsing body and the infalling Hawking radiation inside the event horizon. The amount of information lost when a black hole evaporates depends on the extent to which these interactions are entangling.

Publication: Journal of High Energy Physics Vol.: 2004 No.: 3 ISSN: 1126-6708

ID: CaltechAUTHORS:GOTjhep04

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Abstract: We prove the security of the Bennett-Brassard (BB84) quantum key distribution protocol for an arbitrary source whose averaged states are basis independent, a condition that is automatically satisfied if the source is suitably designed. The proof is based on the observation that, to an adversary, the key extraction process is equivalent to a measurement in the sigma-hatx basis performed on a pure sigma-hatz-basis eigenstate. The dependence of the achievable key length on the bit error rate is the same as that established by Shor and Preskill [Phys. Rev. Lett. 85, 441 (2000)] for a perfect source, indicating that the defects in the source are efficiently detected by the protocol.

Publication: Physical Review Letters Vol.: 90 No.: 5 ISSN: 0031-9007

ID: CaltechAUTHORS:KOAprl03

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Abstract: We study the ±J random-plaquette Z_2 gauge model (RPGM) in three spatial dimensions, a three-dimensional analog of the two-dimensional ±J random-bond Ising model (RBIM). The model is a pure Z_2 gauge theory in which randomly chosen plaquettes (occurring with concentration p) have couplings with the “wrong sign” so that magnetic flux is energetically favored on these plaquettes. Excitations of the model are one-dimensional “flux tubes” that terminate at “magnetic monopoles” located inside lattice cubes that contain an odd number of wrong-sign plaquettes. Electric confinement can be driven by thermal fluctuations of the flux tubes, by the quenched background of magnetic monopoles, or by a combination of the two. Like the RBIM, the RPGM has enhanced symmetry along a “Nishimori line” in the p–T plane (where T is the temperature). The critical concentration p_c of wrong-sign plaquettes at the confinement-Higgs phase transition along the Nishimori line can be identified with the accuracy threshold for robust storage of quantum information using topological error-correcting codes: if qubit phase errors, qubit bit-flip errors, and errors in the measurement of local check operators all occur at rates below p_c, then encoded quantum information can be protected perfectly from damage in the limit of a large code block. Through Monte-Carlo simulations, we measure p_(c0), the critical concentration along the T=0 axis (a lower bound on p_c), finding p_(c0)=.0293±.0002. We also measure the critical concentration of antiferromagnetic bonds in the two-dimensional RBIM on the T=0 axis, finding p_(c0)=.1031±.0001. Our value of p_(c0) is incompatible with the value of p_c=.1093±.0002 found in earlier numerical studies of the RBIM, in disagreement with the conjecture that the phase boundary of the RBIM is vertical (parallel to the T axis) below the Nishimori line. The model can be generalized to a rank-r antisymmetric tensor field in d dimensions, in the presence of quenched disorder.

Publication: Annals of Physics Vol.: 303 No.: 1 ISSN: 0003-4916

ID: CaltechAUTHORS:20111017-135406894

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Abstract: We analyze surface codes, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An order-disorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the accuracy threshold), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a three-dimensional Z(2) lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are local, that qubits can be measured rapidly, and that polynomial-size classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however, for this procedure the quantum gates are local only if the qubits are arranged in four or more spatial dimensions. We discuss procedures for encoding, measurement, and performing fault-tolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.

Publication: Journal of Mathematical Physics Vol.: 43 No.: 9 ISSN: 0022-2488

ID: CaltechAUTHORS:DENjmp02.842

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Abstract: We show that the nondemolition measurement of a spacelike Wilson loop operator W(C) is impossible in a relativistic non-Abelian gauge theory. In particular, if two spacelike-separated magnetic flux tubes both link with the loop C, then a nondemolition measurement of W(C) would cause electric charge to be transferred from one flux tube to the other, a violation of relativistic causality. A destructive measurement of W(C) is possible in a non-Abelian gauge theory with suitable matter content. In an Abelian gauge theory, many cooperating parties distributed along the loop C can perform a nondemolition measurement of the Wilson loop operator if they are equipped with a shared entangled ancilla that has been prepared in advance. We also note that Abelian electric charge (but not non-Abelian charge) can be transported superluminally, without any accompanying transmission of information.

Publication: Physical Review D Vol.: 65 No.: 6 ISSN: 2470-0010

ID: CaltechAUTHORS:BECprd02

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Abstract: We study the fault tolerance of quantum computation by adiabatic evolution, a quantum algorithm for solving various combinatorial search problems. We describe an inherent robustness of adiabatic computation against two kinds of errors, unitary control errors and decoherence, and we study this robustness using numerical simulations of the algorithm.

Publication: Physical Review A Vol.: 65 No.: 1 ISSN: 1050-2947

ID: CaltechAUTHORS:CHIpra02

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Abstract: We study the properties of quantum stabilizer codes that embed a finite-dimensional protected code space in an infinite-dimensional Hilbert space. The stabilizer group of such a code is associated with a symplectically integral lattice in the phase space of 2N canonical variables. From the existence of symplectically integral lattices with suitable properties, we infer a lower bound on the quantum capacity of the Gaussian quantum channel that matches the one-shot coherent information optimized over Gaussian input states.

Publication: Physical Review A Vol.: 64 No.: 6 ISSN: 1050-2947

ID: CaltechAUTHORS:HARpra01

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Abstract: We examine constraints on quantum operations imposed by relativistic causality. A bipartite superoperator is said to be localizable if it can be implemented by two parties (Alice and Bob) who share entanglement but do not communicate; it is causal if the superoperator does not convey information from Alice to Bob or from Bob to Alice. We characterize the general structure of causal complete-measurement superoperators, and exhibit examples that are causal but not localizable. We construct another class of causal bipartite superoperators that are not localizable by invoking bounds on the strength of correlations among the parts of a quantum system. A bipartite superoperator is said to be semilocalizable if it can be implemented with one-way quantum communication from Alice to Bob, and it is semicausal if it conveys no information from Bob to Alice. We show that all semicausal complete-measurement superoperators are semilocalizable, and we establish a general criterion for semicausality. In the multipartite case, we observe that a measurement superoperator that projects onto the eigenspaces of a stabilizer code is localizable.

Publication: Physical Review A Vol.: 64 No.: 5 ISSN: 1050-2947

ID: CaltechAUTHORS:BECpra01

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Abstract: Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes exploit the noncommutative geometry of phase space to protect against errors that shift the values of the canonical variables q and p. In the setting of quantum optics, fault-tolerant universal quantum computation can be executed on the protected code subspace using linear optical operations, squeezing, homodyne detection, and photon counting; however, nonlinear mode coupling is required for the preparation of the encoded states. Finite-dimensional versions of these codes can be constructed that protect encoded quantum information against shifts in the amplitude or phase of a d-state system. Continuous-variable codes can be invoked to establish lower bounds on the quantum capacity of Gaussian quantum channels.

Publication: Physical Review A Vol.: 64 No.: 1 ISSN: 1050-2947

ID: CaltechAUTHORS:GOTpra01b

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Abstract: We prove the security of a quantum key distribution scheme based on transmission of squeezed quantum states of a harmonic oscillator. Our proof employs quantum error-correcting codes that encode a finite-dimensional quantum system in the infinite-dimensional Hilbert space of an oscillator, and protect against errors that shift the canonical variables p and q. If the noise in the quantum channel is weak, squeezing signal states by 2.51 dB (a squeeze factor er=1.34) is sufficient in principle to ensure the security of a protocol that is suitably enhanced by classical error correction and privacy amplification. Secure key distribution can be achieved over distances comparable to the attenuation length of the quantum channel.

Publication: Physical Review A Vol.: 63 No.: 2 ISSN: 1050-2947

ID: CaltechAUTHORS:GOTpra01a

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Abstract: We prove that the 1984 protocol of Bennett and Brassard (BB84) for quantum key distribution is secure. We first give a key distribution protocol based on entanglement purification, which can be proven secure using methods from Lo and Chau's proof of security for a similar protocol. We then show that the security of this protocol implies the security of BB84. The entanglement purification based protocol uses Calderbank-Shor-Steane codes, and properties of these codes are used to remove the use of quantum computation from the Lo-Chau protocol.

Publication: Physical Review Letters Vol.: 85 No.: 2 ISSN: 0031-9007

ID: CaltechAUTHORS:SHOprl00

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Abstract: I consider some promising future directions for quantum information theory that could influence the development of 21st century physics. Advances in the theory of the distinguishability of superoperators may lead to new strategies for improving the precision of quantum-limited measurements. A better grasp of the properties of multi-partite quantum entanglement may lead to deeper understanding of strongly-coupled dynamics in quantum many-body systems, quantum field theory, and quantum gravity.

Publication: Journal of Modern Optics Vol.: 47 No.: 2-3 ISSN: 0950-0340

ID: CaltechAUTHORS:20111129-141152912

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Abstract: Some quantum states are hard to create and maintain, but are a valuable resource for computing. Twenty-first century entrepreneurs could make a fortune selling disposable quantum states.

Publication: Nature Vol.: 402 No.: 6760 ISSN: 0028-0836

ID: CaltechAUTHORS:20150608-104757494

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Abstract: During the second half of this century we have witnessed staggering progress in the development of information technology. At present, the pace of progress shows no sign of slowing. But in the early twenty-first century, conventional integrated-circuit technology will approach the fundamental limitations imposed by the atomic size scale. At that stage, continued improvement in computing performance, and the continued expansion of the world economy, may hinge on the development of radically new methods for processing information.

Publication: Nature Vol.: 398 No.: 6723 ISSN: 0028-0836

ID: CaltechAUTHORS:20150608-103533216

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Abstract: Twenty-first century computers could achieve astonishing speed by exploiting the principles of quantum mechanics. New techniques of quantum error correction will be essential to prevent those machines from crashing.

Publication: Nature Vol.: 391 No.: 6668 ISSN: 0028-0836

ID: CaltechAUTHORS:20150605-102149860

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Abstract: I assess the potential of quantum computation. Broad and important applications must be found to justify construction of a quantum computer; I review some of the known quantum algorithms and consider the prospects for finding new ones. Quantum computers are notoriously susceptible to making errors; I discuss recently developed fault–tolerant procedures that enable a quantum computer with noisy gates to perform reliably. Quantum computing hardware is still in its infancy; I comment on the specifications that should be met by future hardware. Over the past few years, work on quantum computation has erected a new classification of computational complexity, has generated profound insights into the nature of decoherence, and has stimulated the formulation of new techniques in high–precision experimental physics. A broad interdisciplinary effort will be needed if quantum computers are to fulfil their destiny as the world's fastest computing devices. This paper is an expanded version of remarks that were prepared for a panel discussion at the ITP Conference on Quantum Coherence and Decoherence, December 1996.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 454 No.: 1969 ISSN: 1364-5021

ID: CaltechAUTHORS:20200916-090616410

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Abstract: The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolled interactions with the environment. Recovery from errors can work effectively even if occasional mistakes occur during the recovery procedure. Furthermore, encoded quantum information can be processed without serious propagation of errors. Hence, an arbitrarily long quantum computation can be performed reliably, provided that the average probability of error per quantum gate is less than a certain critical value, the accuracy threshold. A quantum computer storing about 10⁶ qubits, with a probability of error per quantum gate of order 10⁻⁶, would be a formidable factoring engine. Even a smaller less–accurate quantum computer would be able to perform many useful tasks. This paper is based on a talk presented at the ITP Conference on Quantum Coherence and Decoherence, 15 to 18 December 1996.

Publication: Proceedings of the Royal Society A: Mathematical, physical, and engineering sciences Vol.: 454 No.: 1969 ISSN: 1364-5021

ID: CaltechAUTHORS:20200929-143507282

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Abstract: We consider how to optimize memory use and computation time in operating a quantum computer. In particular, we estimate the number of memory quantum bits (qubits) and the number of operations required to perform factorization, using the algorithm suggested by Shor [in Proceedings of the 35th Annual Symposium on Foundations of Computer Science, edited by S. Goldwasser (IEEE Computer Society, Los Alamitos, CA, 1994), p. 124]. A K-bit number can be factored in time of order K3 using a machine capable of storing 5K+1 qubits. Evaluation of the modular exponential function (the bottleneck of Shor’s algorithm) could be achieved with about 72K3 elementary quantum gates; implementation using a linear ion trap would require about 396K3 laser pulses. A proof-of-principle demonstration of quantum factoring (factorization of 15) could be performed with only 6 trapped ions and 38 laser pulses. Though the ion trap may never be a useful computer, it will be a powerful device for exploring experimentally the properties of entangled quantum states.

Publication: Physical Review A Vol.: 54 No.: 2 ISSN: 1050-2947

ID: CaltechAUTHORS:BECpra96

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Abstract: Black hole evaporation is investigated in a (1+1)-dimensional model of quantum gravity. Quantum corrections to the black hole entropy are computed, and the fine-grained entropy of the Hawking radiation is studied. A generalized second law of thermodynamics is formulated, and shown to be valid under suitable conditions. It is also shown that, in this model, a black hole can consume an arbitrarily large amount of information.

Publication: Physical Review D Vol.: 50 No.: 6 ISSN: 2470-0010

ID: CaltechAUTHORS:FIOprd94

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Abstract: The electric charge of a wormhole mouth and the magnetic flux "linked" by the wormhole are non-commuting observables, and so cannot be simultaneously diagonalized. We use this observation to resolve some puzzles in wormhole electrodynamics and chromodynamics. Specifically, we analyze the color electric field that results when a colored object traverses a wormhole, and we discuss the measurement of the wormhole charge and flux using Aharonov-Bohm interference effects.

Publication: Physics Letters B Vol.: 318 No.: 2 ISSN: 0370-2693

ID: CaltechAUTHORS:20120229-152505223

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Abstract: We study the interactions of non-Abelian vortices in two spatial dimensions. These interactions have novel features, because the Aharonov-Bohm effect enables a pair of vortices to exchange quantum numbers. The cross section for vortex-vortex scattering is typically a multivalued function of the scattering angle. There can be an exchange contribution to the vortex-vortex scattering amplitude that adds coherently with the direct amplitude, even if the two vortices have distinct quantum numbers. Thus two vortices can be "indistinguishable" even though they are not the same.

Publication: Physical Review D Vol.: 48 No.: 10 ISSN: 2470-0010

ID: CaltechAUTHORS:LOHprd93

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Abstract: We systematically analyze the decay of metastable topological defects that arise from the spontaneous breakdown of gauge or global symmetries. Quantum-mechanical tunneling rates are estimated for a variety of decay processes. The decay rate for a global string, vortex, domain wall, or kink is typically suppressed compared to the decay rate for its gauged counterpart. We also discuss the decay of global texture, and of semilocal and electroweak strings.

Publication: Physical Review D Vol.: 47 No.: 6 ISSN: 2470-0010

ID: CaltechAUTHORS:PREprd93

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Abstract: I analyze the interplay of gauge and global symmetries in the theory of topological defects. In a two-dimensional model in which both gauge symmetries and exact global symmetries are spontaneously broken, stable vortices may fail to exist even though magnetic flux is topologically conserved. Following Vachaspati and Achúcarro, I formulate the condition that must be satisfied by the pattern of symmetry breakdown for finite-energy configurations to exist in which the conserved magnetic flux is spread out instead of confined to a localized vortex. If this condition is met, vortices are always unstable at sufficiently weak gauge coupling. I also describe the properties of defects in models with an "accidental" symmetry that is partially broken by gauge-boson exchange. In some cases, the spontaneously broken accidental symmetry is not restored inside the core of the defect. Then the structure of the defect can be analyzed using an effective field theory; the details of the physics responsible for the spontaneous symmetry breakdown need not be considered. Examples include domain walls and vortices that are classically unstable, but are stabilized by loop corrections, and magnetic monopoles that have an unusual core structure. Finally, I examine the general theory of the "electroweak strings" that were recently discussed by Vachaspati. These arise only in models with gauge-boson "mixing," and can always end on magnetic monopoles. Cosmological implications are briefly discussed.

Publication: Physical Review D Vol.: 46 No.: 10 ISSN: 2470-0010

ID: CaltechAUTHORS:PREprd92

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Abstract: We analyze the charges carried by loops of string in models with non-abelian local discrete symmetry. The charge on a loop has no localized source, but can be detected by means of the Aharonov-Bohm interaction of the loop with another string. We describe the process of charge detection, and the transfer of charge between point particles and string loops, in terms of gauge-invariant correlation functions.

Publication: Nuclear Physics B Vol.: 386 No.: 1 ISSN: 0550-3213

ID: CaltechAUTHORS:20120314-135655591

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Abstract: We analyze the unlocalized “Cheshire charge” carried by “Alice strings.” The magnetic charge on a string loop is carefully defined, and the transfer of magnetic charge from a monopole to a string loop is analyzed using global topological methods. A semiclassical theory of electric charge transfer is also described.

Publication: Nuclear Physics B Vol.: 386 No.: 1 ISSN: 0550-3213

ID: CaltechAUTHORS:20120314-135309738

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Abstract: We develop an operator formalism for investigating the properties of non-abelian cosmic strings (and vortices) in quantum field theory. Operators are constructed that introduce classical string sources and that create dynamical string loops. The operator construction in lattice gauge theory is explicitly described, and correlation functions are computed in the strong-coupling and weak-coupling limits. These correlation functions are used to study the long-range interactions of non-abelian strings, taking account of charge-screening effects due to virtual particles. Among the phenomena investigated are the Aharonov-Bohm interactions of strings with charged particles, holonomy interactions between string loops, string entanglement, the transfer of “Cheshire charge” to a string loop, and domain-wall decay via spontaneous string nucleation. We also anayze the Aharonov-Bohm interactions of magnetic monopoles with electric flux tubes in a confining gauge theory. We propose that the Aharonov-Bohm effect can be invoked to distinguish among various phases of a non-abelian gauge theory coupled to matter.

Publication: Nuclear Physics B Vol.: 384 No.: 1-2 ISSN: 0550-3213

ID: CaltechAUTHORS:20120314-135053542

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Abstract: A black hole may carry quantum numbers that are not associated with massless gauge fields, contrary to the spirit of the “no-hair” theorems. We describe in detail two different types of black-hole hair that decay exponentially at long range. The first type is associated with discrete gauge charge and the screening is due to the Higgs mechanism. The second type is associated with color magnetic charge, and the screening is due to color confinement. In both cases, we perform semiclassical calculations of the effect of the hair on local observables outside the horizon, and on black-hole thermodynamics. These effects are generated by virtual cosmic strings, or virtual electric flux tubes, that sweep around the event horizon. The effects of discrete gauge charge are nonperturbative in ħ, but the effects of color magnetic charge become ħ-independent in a suitable limit. We present an alternative treatment of discrete gauge charge using dual variables, and examine the possibility of black-hole hair associated with discrete global symmetry. We draw the distinction between primary hair, which endows a black hole with new quantum numbers, and secondary hair, which does not, and we point out some varieties of secondary hair that occur in the standard model of particle physics.

Publication: Nuclear Physics B Vol.: 378 No.: 1-2 ISSN: 0550-3213

ID: CaltechAUTHORS:20120314-134754496

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Abstract: It is shown that the breakdown of a global symmetry group to a discrete subgroup can lead to analogs of the Aharonov-Bohm effect. At sufficiently low momentum transfer, the cross section for scattering of a particle with nontrivial Z2 charge off a global vortex is almost equal to (but definitely different from) maximal Aharonov-Bohm scattering; the effect goes away at large momentum transfer. The scattering of a spin-1/2 particle off a magnetic vortex provides an amusing experimentally realizable example.

Publication: Physical Review Letters Vol.: 68 No.: 17 ISSN: 0031-9007

ID: CaltechAUTHORS:MARprl92

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Abstract: A black hole can carry quantum numbers that are not associated with massless gauge fields, contrary to the spirit of the "no-hair" theorems. In the Higgs phase of a gauge theory, electric charge on a black hole generates a nonzero electric field outside the event horizon. This field is nonperturbative in ħ and is exponentially screened far from the hole. It arises from the cloud of virtual cosmic strings that surround the black hole. In the confinement phase, a magnetic charge on a black hole generates a classical field that is screened at long range by nonperturbative effects. Despite the sharp difference in their formal descriptions, the electric and magnetic cases are closely similar physically.

Publication: Physical Review Letters Vol.: 67 No.: 15 ISSN: 0031-9007

ID: CaltechAUTHORS:COLprl91

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Abstract: We calculate in chiral perturbation theory the dependence of Newton's gravitational constant G on the θ parameter of quantum chromodynamics, and we find that G, as a function of θ, is minimized at θ≌π. This calculation suggests that quantum fluctuations in the topology of spacetime would cause θ to assume a value very near π, contrary to the phenomenological evidence indicating that θ is actually near 0.

Publication: Physics Letters B Vol.: 223 No.: 1 ISSN: 0370-2693

ID: CaltechAUTHORS:20170810-153648292

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Abstract: The statistical significance of voids in the spatial distribution of galaxies or clusters of galaxies is studied. The probability per unit volume of finding a large void is expressed in terms of correlation functions. Numerical simulations are used to estimate the likelihood of observing a large void in the distribution of rich clusters of galaxies, assuming that rich clusters arose wherever suitably averaged primordial density fluctuations were unusually large.

Publication: Astrophysical Journal Vol.: 304 No.: 1 ISSN: 0004-637X

ID: CaltechAUTHORS:20161005-151942807

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Abstract: Expressions are derived for the expected abundance of clusters and voids in a sample of randomly distributed objects.

Publication: Physical Review Letters Vol.: 56 No.: 2 ISSN: 0031-9007

ID: CaltechAUTHORS:POLprl86

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Abstract: How is it possible to justify a lengthy review of the physics of the magnetic monopole when nobody has ever seen one? In spite of the unfortunate lack of favorable experimental evidence, there are sound theoretical reasons for believing that the magnetic monopole must exist. The case for its existence is surely as strong as the case for any other undiscovered particle. Moreover, as of this writing (early 1984), it is not certain that nobody has ever seen one. What seems certain is that nobody has ever seen two.

Publication: Annual Review of Nuclear and Particle Science Vol.: 34ISSN: 0163-8998

ID: CaltechAUTHORS:20120706-134558232

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Abstract: It is pointed out that the use of the "decoupling" constraints on the spectrum of composite massless particles is not justified without further assumptions. There is an alternative condition, whose use would not be subject to the same criticisms, which would lead to the same constraints as the decoupling condition, and which would lead to other results as well, for instance that the nonchiral global symmetries in quantum chromodynamics (QCD) with n massless flavors can not be spontaneously broken. However, this condition is found to be violated in a specific model. It is still an open possibility that the chiral symmetries of QCD are unbroken for n not a multiple of 3.

Publication: Physical Review D Vol.: 24 No.: 4 ISSN: 2470-0010

ID: CaltechAUTHORS:PREprd81

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Abstract: To analyze the physical consequences of a dynamically broken theory of the weak interactions, we must know how the weak gauge group is aligned in an approximate flavor-symmetry group. For a large class of models, spectral-function sum rules enables us to determine this alignment explicitly. We work out the pattern of the electroweak symmetry breakdown for several sample models. Critical values of weak mixing angles are found at which the breakdown pattern changes discontinously. We compute pseudo-Goldstone boson masses, and find that some models contain unusually light charged or colored pseudo-Goldstone bosons.

Publication: Nuclear Physics B Vol.: 177 No.: 1 ISSN: 0550-3213

ID: CaltechAUTHORS:20140724-133643360

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Abstract: Dynamically broken gauge theories of electroweak interactions provide a natural mechanism for generating CP nonconservation. Even if all vacuum angles are unobservable, strong CP nonconservation is not automatically avoided. In the absence of strong CP nonconservation, the neutron electric dipole moment is expected to be of order 10^-24e·cm.

Publication: Physical Review Letters Vol.: 45 No.: 4 ISSN: 0031-9007

ID: CaltechAUTHORS:EICprl80

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Abstract: Grand unified models of elementary particle interactions contain stable superheavy magnetic monopoles. The density of such monopoles in the early universe is estimated to be unacceptably large. Cosmological monopole production may be suppressed if the phase transition at the grand unification mass scale is strongly first order.

Publication: Physical Review Letters Vol.: 43 No.: 19 ISSN: 0031-9007

ID: CaltechAUTHORS:PREprl79

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