Abstract: Out-of-time-order correlators (OTOCs) are a standard measure of quantum chaos. Of the four operators involved, one pair may be regarded as a source and the other as a probe. A usual approach, applicable to large-N systems such as the SYK model, is to replace the actual source with some mean-field perturbation and solve for the probe correlation function on the double Keldysh contour. We show how to obtain the OTOC by combining two such solutions for perturbations propagating forward and backward in time. These dynamical perturbations, or scrambling modes, are considered on the thermofield double background and decomposed into a coherent and an incoherent part. For the large-q SYK, we obtain the OTOC in a closed form. We also prove a previously conjectured relation between the Lyapunov exponent and high-frequency behavior of the spectral function.

Publication: Journal of High Energy Physics Vol.: 2022 No.: 3 ISSN: 1029-8479

ID: CaltechAUTHORS:20220323-545297000

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Abstract: Interactions in quantum systems can spread initially localized quantum information into the exponentially many degrees of freedom of the entire system. Understanding this process, known as quantum scrambling, is key to resolving several open questions in physics. Here, by measuring the time-dependent evolution and fluctuation of out-of-time-order correlators, we experimentally investigate the dynamics of quantum scrambling on a 53-qubit quantum processor. We engineer quantum circuits that distinguish operator spreading and operator entanglement and experimentally observe their respective signatures. We show that whereas operator spreading is captured by an efficient classical model, operator entanglement in idealized circuits requires exponentially scaled computational resources to simulate. These results open the path to studying complex and practically relevant physical observables with near-term quantum processors.

Publication: Science Vol.: 374 No.: 6574 ISSN: 0036-8075

ID: CaltechAUTHORS:20211028-210102101

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Abstract: The discovery of topological order has revised the understanding of quantum matter and provided the theoretical foundation for many quantum error–correcting codes. Realizing topologically ordered states has proven to be challenging in both condensed matter and synthetic quantum systems. We prepared the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measured a topological entanglement entropy near the expected value of –ln2 and simulated anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigated key aspects of the surface code, including logical state injection and the decay of the nonlocal order parameter. Our results demonstrate the potential for quantum processors to provide insights into topological quantum matter and quantum error correction.

Publication: Science Vol.: 374 No.: 6572 ISSN: 0036-8075

ID: CaltechAUTHORS:20211203-174950058

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Abstract: Realizing the potential of quantum computing requires sufficiently low logical error rates(1). Many applications call for error rates as low as 10⁻¹⁵ (refs. 2,3,4,5,6,7,8,9), but state-of-the-art quantum platforms typically have physical error rates near 10⁻³ (refs. 10,11,12,13,14). Quantum error correction(15,16,17) promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device(18,19) and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.

Publication: Nature Vol.: 595 No.: 7867 ISSN: 0028-0836

ID: CaltechAUTHORS:20210728-191748877

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Abstract: We argue that “stringy” effects in a putative gravity-dual picture for SYK-like models are related to the branching time, a kinetic coefficient defined in terms of the retarded kernel. A bound on the branching time is established assuming that the leading diagrams are ladders with thin rungs. Thus, such models are unlikely candidates for sub-AdS holography. In the weak coupling limit, we derive a relation between the branching time, the Lyapunov exponent, and the quasiparticle lifetime using two different approximations.

Publication: Journal of High Energy Physics Vol.: 2021 No.: 3 ISSN: 1029-8479

ID: CaltechAUTHORS:20210311-125528811

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Abstract: This paper is an attempt to extend the recent understanding of the Page curve for evaporating black holes to more general systems coupled to a heat bath. Although calculating the von Neumann entropy by the replica trick is usually a challenge, we have identified two solvable cases. For the initial section of the Page curve, we sum up the perturbation series in the system-bath coupling κ; the most interesting contribution is of order 2s, where s is the number of replicas. For the saturated regime, we consider the effect of an external impulse on the entropy at a later time and relate it to OTOCs. A significant simplification occurs in the maximal chaos case such that the effect may be interpreted in terms of an intermediate object, analogous to the branching surface of a replica wormhole.

Publication: Journal of High Energy Physics Vol.: 2021 No.: 3 ISSN: 1029-8479

ID: CaltechAUTHORS:20210324-090318969

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Abstract: We study the Sachdev-Ye-Kitaev (SYK₄) model with a weak SYK₂ term of magnitude Γ beyond the simplest perturbative limit considered previously. For intermediate values of the perturbation strength, J/N ≪ Γ ≪ J/√N, fluctuations of the Schwarzian mode are suppressed, and the SYK₄ mean-field solution remains valid beyond the timescale t₀ ∼ N/J up to t∗∼J/Γ². The out-of-time-order correlation function displays at short time intervals exponential growth with maximal Lyapunov exponent 2πT, but its prefactor scales as T at low temperatures T ≤ Γ.

Publication: Physical Review Letters Vol.: 125 No.: 19 ISSN: 0031-9007

ID: CaltechAUTHORS:20201104-130324443

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Abstract: We describe numerous properties of the Sachdev-Ye-Kitaev model for complex fermions with N ≫ 1 flavors and a global U(1) charge. We provide a general definition of the charge in the (G, Σ) formalism, and compute its universal relation to the infrared asymmetry of the Green function. The same relation is obtained by a renormalization theory. The conserved charge contributes a compact scalar field to the effective action, from which we derive the many-body density of states and extract the charge compressibility. We compute the latter via three distinct numerical methods and obtain consistent results. Finally, we present a two dimensional bulk picture with free Dirac fermions for the zero temperature entropy.

Publication: Journal of High Energy Physics Vol.: 2020 No.: 2 ISSN: 1029-8479

ID: CaltechAUTHORS:20200226-133542550

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Abstract: The dynamics of a nearly-AdS_2 spacetime with boundaries is reduced to that of two particles in the anti-de Sitter space. We determine the class of physically meaningful wavefunctions, and prescribe the statistical mechanics of a black hole. We demonstrate how wavefunctions for a two-sided black hole and a regularized notion of trace can be used to construct thermal partition functions, and more generally, arbitrary density matrices. We also obtain correlation functions of external operators.

Publication: Journal of High Energy Physics Vol.: 2019 No.: 5 ISSN: 1126-6708

ID: CaltechAUTHORS:20190529-161258314

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Abstract: We derive an identity relating the growth exponent of early-time OTOCs, the pre-exponential factor, and a third number called “branching time”. The latter is defined within the dynamical mean-field framework, namely, in terms of the retarded kernel. This identity can be used to calculate stringy effects in the SYK and similar models; we also explicitly define “strings” in this context. As another application, we consider an SYK chain. If the coupling strength βJ is above a certain threshold and nonlinear (in the magnitude of OTOCs) effects are ignored, the exponent in the butterfly wavefront is exactly 2π/β.

Publication: Journal of High Energy Physics Vol.: 2019 No.: 2 ISSN: 1126-6708

ID: CaltechAUTHORS:20190220-075207560

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Abstract: We present two techniques that can greatly reduce the number of gates required to realize an energy measurement, with application to ground state preparation in quantum simulations. The first technique realizes that to prepare the ground state of some Hamiltonian, it is not necessary to implement the time-evolution operator: any unitary operator which is a function of the Hamiltonian will do. We propose one such unitary operator which can be implemented exactly, circumventing any Taylor or Trotter approximation errors. The second technique is tailored to lattice models, and is targeted at reducing the use of generic single-qubit rotations, which are very expensive to produce by standard fault tolerant techniques. In particular, the number of generic single-qubit rotations used by our method scales with the number of parameters in the Hamiltonian, which contrasts with a growth proportional to the lattice size required by other techniques.

Publication: Physical Review Letters Vol.: 121 No.: 1 ISSN: 0031-9007

ID: CaltechAUTHORS:20180705-150616687

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Abstract: We give an exposition of the SYK model with several new results. A non-local correction to the Schwarzian effective action is found. The same action is obtained by integrating out the bulk degrees of freedom in a certain variant of dilaton gravity. We also discuss general properties of out-of-time-order correlators.

Publication: Journal of High Energy Physics Vol.: 2018 No.: 5 ISSN: 1126-6708

ID: CaltechAUTHORS:20180530-110303978

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Abstract: The boundary of a fractionalized topological phase can be gapped by condensing a proper set of bosonic quasiparticles. Interestingly, in the presence of a global symmetry, such a boundary can have different symmetry transformation properties. Here we present an explicit example of this kind, in the double semion state with time reversal symmetry. We find two distinct cases where the semionic excitations on the boundary can transform either as time reversal singlets or as time reversal (Kramers) doublets, depending on the coherent phase factor of the Bose condensate. The existence of these two possibilities are demonstrated using both field-theory argument and exactly solvable lattice models. Furthermore, we study the domain walls between these two types of gapped boundaries and find that the application of time reversal symmetry tunnels a semion between them.

Publication: Physical Review B Vol.: 93 No.: 23 ISSN: 2469-9950

ID: CaltechAUTHORS:20160623-125613186

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Abstract: Computing the group of units in a field of algebraic numbers is one of the central tasks of computational algebraic number theory. It is believed to be hard classically, which is of interest for cryptography. In the quantum setting, efficient algorithms were previously known for fields of constant degree. We give a quantum algorithm that is polynomial in the degree of the field and the logarithm of its discriminant. This is achieved by combining three new results. The first is a classical algorithm for computing a basis for certain ideal lattices with doubly exponentially large generators. The second shows that a Gaussian-weighted superposition of lattice points, with an appropriate encoding, can be used to provide a unique representation of a real-valued lattice. The third is an extension of the hidden subgroup problem to continuous groups and a quantum algorithm for solving the HSP over the group ℝ^n.

ID: CaltechAUTHORS:20161010-172823440

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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 give a new proof for the area law for general 1D gapped systems, which exponentially improves Hastings' famous result [1]. Specifically, we show that for a chain of d-dimensional spins, governed by a 1D local Hamiltonian with a spectral gap ε > 0, the entanglement entropy of the ground state with respect to any cut in the chain is upper bounded by O(log^3 d/ε ). Our approach uses the framework of Refs. [2, 3] to construct a Chebyshev-based AGSP (Approximate Ground Space Projection) with favorable factors. However, our construction uses the Hamiltonian directly, instead of using the Detectability lemma, which allows us to work with general (frustrated) Hamiltonians, as well as slightly improving the 1/ε dependence of the bound in Ref. [3]. To achieve that, we establish a new, “random-walk like”, bound on the entanglement rank of an arbitrary power of a 1D Hamiltonian, which might be of independent interest: ER(H^ℓ) ≤ (ℓd)O(√ℓ). Finally, treating d as a constant, our AGSP shows that the ground state is well approximated by a matrix product state with a sublinear bond dimension B = ε ^O(log^(3/4) n/ε^(1/4)). Using this in conjunction with known dynamical programing algorithms, yields an algorithm for a 1=poly(n) approximation of the ground energy with a subexponential running time T ≤ exp (εO(log^(3/4) n/ε^(1/4))).

ID: CaltechAUTHORS:20140130-142058060

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Abstract: We develop a technique to compute the high-frequency asymptotics of spin correlators in weakly interacting disordered spin systems. We show that the dynamical spin correlator decreases exponentially at high frequencies ⟨SS⟩_ω ∼ exp(−τ∗ω) and compute the characteristic time τ∗ of this dependence. In a typical random configuration, some fraction of spins form strongly coupled pairs, which behave as two-level systems. Their switching dynamics is driven by the high-frequency noise from the surrounding spins, resulting in low-frequency 1/f noise in the magnetic susceptibility and other physical quantities. We discuss application of these results to the problem of susceptibility and flux noise in superconducting circuits at mK temperatures.

Publication: Physical Review B Vol.: 86 No.: 13 ISSN: 1098-0121

ID: CaltechAUTHORS:20120713-092458311

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Abstract: We report on the realization of a superinductor, a dissipationless element whose microwave impedance greatly exceeds the resistance quantum R_Q. The design of the superinductor, implemented as a ladder of nanoscale Josephson junctions, enables tuning of the inductance and its nonlinearity by a weak magnetic field. The Rabi decay time of the superinductor-based qubit exceeds 1 μs. The high kinetic inductance and strong nonlinearity offer new types of functionality, including the development of qubits protected from both flux and charge noises, fault tolerant quantum computing, and high-impedance isolation for electrical current standards based on Bloch oscillations.

Publication: Physical Review Letters Vol.: 109 No.: 13 ISSN: 0031-9007

ID: CaltechAUTHORS:20121026-145953727

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Abstract: We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category C as in the Levin-Wen model, whereas the boundary is associated with amodule category over C.We also consider domainwalls (or defect lines) between different bulk phases.Adomainwall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions.

Publication: Communications in Mathematical Physics Vol.: 313 No.: 2 ISSN: 0010-3616

ID: CaltechAUTHORS:20120720-094651607

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Abstract: Inspired by quantum information theory, we look for representations of the braid groups B_n on V^(⊗(n+m−2)) for some fixed vector space V such that each braid generator σ_i, i = 1, ..., n−1, acts on m consecutive tensor factors from i through i +m−1. The braid relation for m = 2 is essentially the Yang-Baxter equation, and the cases for m > 2 are called generalized Yang-Baxter equations. We observe that certain objects in ribbon fusion categories naturally give rise to such representations for the case m = 3. Examples are given from the Ising theory (or the closely related SU(2)_2), SO(N)_2 for N odd, and SU(3)_3. The solution from the Jones-Kauffman theory at a 6th root of unity, which is closely related to SO(3)_2 or SU(2)_4, is explicitly described in the end.

ID: CaltechAUTHORS:20120713-102318475

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Abstract: In this paper we show how the classification of topological phases in insulators and superconductors is changed by interactions, in the case of one-dimensional systems. We focus on the time-reversal-invariant Majorana chain (BDI symmetry class).While the band classification yields an integer topological index k, it is known that phases characterized by values of k in the same equivalence class modulo 8 can be adiabatically transformed one to another by adding suitable interaction terms. Here we show that the eight equivalence classes are distinct and exhaustive, and provide a physical interpretation for the interacting invariant modulo 8. The different phases realize different Altland-Zirnbauer classes of the reduced density matrix for an entanglement bipartition into two half chains. We generalize these results to the classification of all one-dimensional gapped phases of fermionic systems with possible antiunitary symmetries, utilizing the algebraic framework of central extensions. We use matrix product state methods to prove our results.

Publication: Physical Review B Vol.: 83 No.: 7 ISSN: 1098-0121

ID: CaltechAUTHORS:20110308-123056073

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Abstract: We consider the spin- 12 model on the honeycomb lattice in the presence of a weak magnetic field h_α ≪ 1. Such a perturbation destroys the exact integrability of the model in terms of gapless fermions and static Z_2 fluxes. We show that it results in the appearance of a long-range tail in the irreducible dynamic spin correlation function: ≪s^z(t,r)s^z(0,0)≫ α h^2_z f(tr), where f(t,r) α [max(t,r]^(-4) is proportional to the density polarization function of fermions.

Publication: Physical Review Letters Vol.: 106 No.: 6 ISSN: 0031-9007

ID: CaltechAUTHORS:20110316-161521086

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Abstract: We construct a mapping between the two-dimensional toric code model in external magnetic fields, h_z and h_x, and the three-dimensional classical Ising system with plaquette interactions, which is equivalent to the three-dimensional Z_2 gauge Higgs model with anisotropy between the imaginary time and spatial directions. The isotropic limit of the latter model was studied using Monte Carlo simulations on large (up to 60^3) lattices in order to determine the stability of the topological phase against generic magnetic field perturbations and to resolve fine details of the phase diagram. We find that the topological phase is bounded by second-order transition lines, which merge into a first-order line at what appears to be a multicritical point arising from the competition between the Higgs and confinement transitions in the Z_2 gauge system. An effective field theory for this type of multicritical point (if one actually exists) is not known. Our results have potential applications to frustrated magnets, quantum computation, lattice gauge models in particle physics, and critical phenomena.

Publication: Physical Review B Vol.: 82 No.: 8 ISSN: 1098-0121

ID: CaltechAUTHORS:20100907-152559860

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Abstract: We describe in detail a counterexample to the topological classification of free fermion systems. We deal with a one-dimensional chain of Majorana fermions with an unusual T symmetry. The topological invariant for the free fermion classification lies in Z, but with the introduction of interactions the Z is broken to Z_8. We illustrate this in the microscopic model of the Majorana chain by constructing an explicit path between two distinct phases whose topological invariants are equal modulo 8, along which the system remains gapped. The path goes through a strongly interacting region. We also find the field-theory interpretation of this phenomenon. There is a second-order phase transition between the two phases in the free theory, which can be avoided by going through the strongly interacting region. We show that this transition is in the two-dimensional Ising universality class, where a first-order phase transition line, terminating at a second-order transition, can be avoided by going through the analog of a high-temperature paramagnetic phase. In fact, we construct the full phase diagram of the system as a function of the thermal operator (i.e., the mass term that tunes between the two phases in the free theory) and two quartic operators, obtaining a first-order Peierls transition region, a second-order transition region, and a region with no transition.

Publication: Physical Review B Vol.: 81 No.: 13 ISSN: 1098-0121

ID: CaltechAUTHORS:20100526-114410284

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Abstract: Indistinguishable particles in two dimensions can be characterized by anyonic quantum statistics, which is more general than that of bosons or fermions. Anyons emerge as quasiparticles in fractional quantum Hall states and in certain frustrated quantum magnets. Quantum liquids of anyons show degenerate ground states, where the degeneracy depends on the topology of the underlying surface. Here, we present a new type of continuous quantum phase transition in such anyonic quantum liquids, which is driven by quantum fluctuations of the topology. The critical state connecting two anyonic liquids on surfaces with different topologies is reminiscent of the notion of a 'quantum foam' with fluctuations on all length scales. This exotic quantum phase transition arises in a microscopic model of interacting anyons for which we present an exact solution in a linear geometry. We introduce an intuitive physical picture of this model that unifies string nets and loop gases, and provide a simple description of topological quantum phases and their phase transitions.

Publication: Nature Physics Vol.: 5 No.: 11 ISSN: 1745-2473

ID: CaltechAUTHORS:20091208-111910640

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Abstract: Gapped phases of noninteracting fermions, with and without charge conservation and time-reversal symmetry, are classified using Bott periodicity. The symmetry and spatial dimension determines a general universality class, which corresponds to one of the 2 types of complex and 8 types of real Clifford algebras. The phases within a given class are further characterized by a topological invariant, an element of some Abelian group that can be 0, Z, or Z_2. The interface between two infinite phases with different topological numbers must carry some gapless mode. Topological properties of finite systems are described in terms of K-homology. This classification is robust with respect to disorder, provided electron states near the Fermi energy are absent or localized. In some cases (e.g., integer quantum Hall systems) the K-theoretic classification is stable to interactions, but a counterexample is also given.

No.: 1134
ID: CaltechAUTHORS:20100510-100944960

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Abstract: We introduce a toy model that allows us to study the physical properties of a spin impurity coupled to the electrons in the superconducting island. We show that, when the coupling of the spin is of the order of the superconducting gap Delta, two almost degenerate subgap states are formed. By computing the Berry phase that is associated with the superconducting phase rotations in this model, we prove that these subgap states are characterized by a different charge and demonstrate that the switching between these states has the same effect as quasiparticle poisoning (unpoisoning) of the island. We also show that an impurity coupled to both the island and the lead generates Josepshon current fluctuations.

Publication: Physical Review Letters Vol.: 101 No.: 24 ISSN: 0031-9007

ID: CaltechAUTHORS:FAOprl08

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Abstract: The basic building block of quantum computation is the qubit, a system with two (nearly) degenerate states that can be used to encode quantum information. Real systems typically have a full spectrum of excitations that are considered illegal from the point of view of a computation, and lead to decoherence if they couple too strongly into the qubit states during some process (see Fig. 4.1). The essential problem, then, is to preserve the quantum state of the qubit as long as possible to allow time for computations to take place.

No.: 89
ID: CaltechAUTHORS:20110207-134954371

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Abstract: We show how shot noise in an electronic Mach-Zehnder interferometer in the fractional quantum Hall regime probes the charge and statistics of quantum Hall quasiparticles. The dependence of the noise on the magnetic flux through the interferometer allows for a simple way to distinguish Abelian from non-Abelian quasiparticle statistics. In the Abelian case, the Fano factor (in units of the electron charge) is always lower than unity. In the non-Abelian case, the maximal Fano factor as a function of the magnetic flux exceeds 1.

Publication: Physical Review B Vol.: 76 No.: 8 ISSN: 1098-0121

ID: CaltechAUTHORS:FELprb07

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Abstract: We discuss generalizations of quantum spin Hamiltonians using anyonic degrees of freedom. The simplest model for interacting anyons energetically favors neighboring anyons to fuse into the trivial (“identity”) channel, similar to the quantum Heisenberg model favoring neighboring spins to form spin singlets. Numerical simulations of a chain of Fibonacci anyons show that the model is critical with a dynamical critical exponent z=1, and described by a two-dimensional (2D) conformal field theory with central charge c=(7/10). An exact mapping of the anyonic chain onto the 2D tricritical Ising model is given using the restricted-solid-on-solid representation of the Temperley-Lieb algebra. The gaplessness of the chain is shown to have topological origin.

Publication: Physical Review Letters Vol.: 98 No.: 16 ISSN: 0031-9007

ID: CaltechAUTHORS:FEIprl07

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Abstract: Fractionally charged quasiparticles in the quantum Hall state with a filling factor nu=5/2 are expected to obey non-Abelian statistics. We demonstrate that their statistics can be probed by transport measurements in an electronic Mach-Zehnder interferometer. The tunneling current through the interferometer exhibits a characteristic dependence on the magnetic flux and a nonanalytic dependence on the tunneling amplitudes which can be controlled by gate voltages.

Publication: Physical Review Letters Vol.: 97 No.: 18 ISSN: 0031-9007

ID: CaltechAUTHORS:FELprl06

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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: The k-LOCAL Hamiltonian problem is a natural complete problem for the complexity class QMA, the quantum analogue of NP. It is similar in spirit to MAX-k-SAT, which is NP-complete for k >= 2. It was known that the problem is QMA-complete for any k >= 3. On the other hand, 1-LOCAL Hamiltonian is in P and hence not believed to be QMA-complete. The complexity of the 2-LOCAL Hamiltonian problem has long been outstanding. Here we settle the question and show that it is QMA-complete. We provide two independent proofs; our first proof uses only elementary linear algebra. Our second proof uses a powerful technique for analyzing the sum of two Hamiltonians; this technique is based on perturbation theory and we believe that it might prove useful elsewhere. Using our techniques we also show that adiabatic computation with 2-local interactions on qubits is equivalent to standard quantum computation.

Publication: SIAM Journal on Computing Vol.: 35 No.: 5 ISSN: 0097-5397

ID: CaltechAUTHORS:KEMsiamjc06

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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: In this Letter we propose an interferometric experiment to detect non-Abelian quasiparticle statistics—one of the hallmark characteristics of the Moore-Read state expected to describe the observed fractional quantum Hall effect plateau at nu=5/2. The implications for using this state for constructing a topologically protected qubit as has been recently proposed by Das Sarma et al. are also addressed.

Publication: Physical Review Letters Vol.: 96 No.: 1 ISSN: 0031-9007

ID: CaltechAUTHORS:BONprl06.799

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Abstract: A spin-1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 source gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are non-Abelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and non-Abelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.

Publication: Annals of Physics Vol.: 321 No.: 1 ISSN: 0003-4916

ID: CaltechAUTHORS:KITaop06

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Abstract: Gluing two manifolds M_1 and M_2 with a common boundary S yields a closed manifold M. Extending to formal linear combinations x=Sum_i(a_i M_i) yields a sesquilinear pairing p=<,> with values in (formal linear combinations of) closed manifolds. Topological quantum field theory (TQFT) represents this universal pairing p onto a finite dimensional quotient pairing q with values in C which in physically motivated cases is positive definite. To see if such a "unitary" TQFT can potentially detect any nontrivial x, we ask if is non-zero whenever x is non-zero. If this is the case, we call the pairing p positive. The question arises for each dimension d=0,1,2,.... We find p(d) positive for d=0,1, and 2 and not positive for d=4. We conjecture that p(3) is also positive. Similar questions may be phrased for (manifold, submanifold) pairs and manifolds with other additional structure. The results in dimension 4 imply that unitary TQFTs cannot distinguish homotopy equivalent simply connected 4-manifolds, nor can they distinguish smoothly s-cobordant 4-manifolds. This may illuminate the difficulties that have been met by several authors in their attempts to formulate unitary TQFTs for d=3+1. There is a further physical implication of this paper. Whereas 3-dimensional Chern-Simons theory appears to be well-encoded within 2-dimensional quantum physics, eg in the fractional quantum Hall effect, Donaldson-Seiberg-Witten theory cannot be captured by a 3-dimensional quantum system. The positivity of the physical Hilbert spaces means they cannot see null vectors of the universal pairing; such vectors must map to zero.

Publication: Geometry and Topology Vol.: 9 No.: 53 ISSN: 1465-3060

ID: CaltechAUTHORS:FREgt05

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Abstract: We consider a model of quantum computation in which the set of elementary operations is limited to Clifford unitaries, the creation of the state |0>, and qubit measurement in the computational basis. In addition, we allow the creation of a one-qubit ancilla in a mixed state rho, which should be regarded as a parameter of the model. Our goal is to determine for which rho universal quantum computation (UQC) can be efficiently simulated. To answer this question, we construct purification protocols that consume several copies of rho and produce a single output qubit with higher polarization. The protocols allow one to increase the polarization only along certain "magic" directions. If the polarization of rho along a magic direction exceeds a threshold value (about 65%), the purification asymptotically yields a pure state, which we call a magic state. We show that the Clifford group operations combined with magic states preparation are sufficient for UQC. The connection of our results with the Gottesman-Knill theorem is discussed.

Publication: Physical Review A Vol.: 71 No.: Art no ISSN: 1050-2947

ID: CaltechAUTHORS:BRApra05

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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: The k-Local Hamiltonian problem is a natural complete problem for the complexity class QMA, the quantum analog of NP. It is similar in spirit to MAX- k -SAT, which is NP-complete for k ≥ 2. It was known that the problem is QMA-complete for any k ≥ 3. On the other hand 1-Local Hamiltonian is in P, and hence not believed to be QMA-complete. The complexity of the 2-Local Hamiltonian problem has long been outstanding. Here we settle the question and show that it is QMA-complete. We provide two independent proofs; our first proof uses a powerful technique for analyzing the sum of two Hamiltonians; this technique is based on perturbation theory and we believe that it might prove useful elsewhere. The second proof uses elementary linear algebra only. Using our techniques we also show that adiabatic computation with two-local interactions on qubits is equivalent to standard quantum computation.

No.: 3328
ID: CaltechAUTHORS:20191011-072647725

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Abstract: Entanglement, one of the most intriguing features of quantum theory and a main resource in quantum information science, is expected to play a crucial role also in the study of quantum phase transitions, where it is responsible for the appearance of long-range correlations. We investigate, through a microscopic calculation, the scaling properties of entanglement in spin chain systems, both near and at a quantum critical point. Our results establish a precise connection between concepts of quantum information, condensed matter physics, and quantum field theory, by showing that the behavior of critical entanglement in spin systems is analogous to that of entropy in conformal field theories. We explore some of the implications of this connection.

Publication: Physical Review Letters Vol.: 90 No.: 22 ISSN: 0031-9007

ID: CaltechAUTHORS:VIDprl03

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Abstract: A two-dimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is fault-tolerant by its physical nature.

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

ID: CaltechAUTHORS:20111005-144725727

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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: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned “topological models” having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H ≡ 0. These are called topological quantum field theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model “BQP”. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.

Publication: Communications in Mathematical Physics Vol.: 227 No.: 3 ISSN: 0010-3616

ID: CaltechAUTHORS:20111007-112002181

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Abstract: We define a model of quantum computation with local fermionic modes (LFMs)—sites which can be either empty or occupied by a fermion. With the standard correspondence between the Foch space of m LFMs and the Hilbert space of m qubits, simulation of one fermionic gate takes O(m) qubit gates and vice versa. We show that using different encodings, the simulation cost can be reduced to O(log m) and a constant, respectively. Nearest neighbors fermionic gates on a graph of bounded degree can be simulated at a constant cost. A universal set of fermionic gates is found. We also study computation with Majorana fermions which are basically halves of LFMs. Some connection to qubit quantum codes is made.

Publication: Annals of Physics Vol.: 298 No.: 1 ISSN: 0003-4916

ID: CaltechAUTHORS:20111007-111528069

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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: 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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