In statistical mechanics, a small system exchanges conserved charges---heat, particles, electric charge, etc.---with a bath. The small system thermalizes to the canonical ensemble, or the grand canonical ensemble, etc., depending on the charges. The charges are usually represented by operators assumed to commute with each other. This assumption was removed within quantum-information-theoretic (QI-theoretic) thermodynamics recently. The small system's long-time state was dubbed "the non-Abelian thermal state (NATS)." We propose an experimental protocol for observing a system thermalize to the NATS. We illustrate with a chain of spins, a subset of which form the system of interest. The conserved charges manifest as spin components. Heisenberg interactions push the charges between the system and the effective bath, the rest of the chain. We predict long-time expectation values, extending the NATS theory from abstract idealization to finite systems that thermalize with finite couplings for finite times. Numerical simulations support the analytics: The system thermalizes to the NATS, rather than to the canonical prediction. Our proposal can be implemented with ultracold atoms, nitrogen-vacancy centers, trapped ions, quantum dots, and perhaps nuclear magnetic resonance. This work introduces noncommuting charges from QI-theoretic thermodynamics into quantum many-body physics: atomic, molecular, and optical physics and condensed matter.

%8 6/21/2019 %G eng %U https://arxiv.org/abs/1906.09227 %0 Journal Article %J EPTCS %D 2019 %T Parallel Self-Testing of the GHZ State with a Proof by Diagrams %A Spencer Breiner %A Amir Kalev %A Carl Miller %XQuantum self-testing addresses the following question: is it possible to verify the existence of a multipartite state even when one's measurement devices are completely untrusted? This problem has seen abundant activity in the last few years, particularly with the advent of parallel self-testing (i.e., testing several copies of a state at once), which has applications not only to quantum cryptography but also quantum computing. In this work we give the first error-tolerant parallel self-test in a three-party (rather than two-party) scenario, by showing that an arbitrary number of copies of the GHZ state can be self-tested. In order to handle the additional complexity of a three-party setting, we use a diagrammatic proof based on categorical quantum mechanics, rather than a typical symbolic proof. The diagrammatic approach allows for manipulations of the complicated tensor networks that arise in the proof, and gives a demonstration of the importance of picture-languages in quantum information.

%B EPTCS %V 287 %P 43-66 %8 01/29/2019 %G eng %U https://arxiv.org/abs/1806.04744 %R https://doi.org/10.4204/EPTCS.287.3 %0 Journal Article %D 2018 %T Implicit regularization and solution uniqueness in over-parameterized matrix sensing %A Anastasios Kyrillidis %A Amir Kalev %XWe consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-r positive semi-definite (PSD) matrices in Rn×n, with a sensing mechanism that satisfies the restricted isometry property (RIP). The algorithm we study is that of \emph{factored gradient descent}, where we model the low-rankness and PSD constraints with the factorization UU⊤, where U∈Rn×r. Surprisingly, recent work argues that the choice of r≤n is not pivotal: even setting U∈Rn×n is sufficient for factored gradient descent to find the rank-r solution, which suggests that operating over the factors leads to an implicit regularization. In this note, we provide a different perspective. We show that, in the noiseless case, under certain conditions, the PSD constraint by itself is sufficient to lead to a unique rank-r matrix recovery, without implicit or explicit low-rank regularization. \emph{I.e.}, under assumptions, the set of PSD matrices, that are consistent with the observed data, is a singleton, irrespective of the algorithm used.

%G eng %U https://arxiv.org/abs/1806.02046 %0 Journal Article %J Phys. Rev. Lett. %D 2018 %T Optimal Pure-State Qubit Tomography via Sequential Weak Measurements %A Ezad Shojaee %A Christopher S. Jackson %A Carlos A. Riofrio %A Amir Kalev %A Ivan H. Deutsch %XThe spin-coherent-state positive-operator-valued-measure (POVM) is a fundamental measurement in quantum science, with applications including tomography, metrology, teleportation, benchmarking, and measurement of Husimi phase space probabilities. We prove that this POVM is achieved by collectively measuring the spin projection of an ensemble of qubits weakly and isotropically. We apply this in the context of optimal tomography of pure qubits. We show numerically that through a sequence of weak measurements of random directions of the collective spin component, sampled discretely or in a continuous measurement with random controls, one can approach the optimal bound.

%B Phys. Rev. Lett. %V 121 %G eng %U https://arxiv.org/abs/1805.01012 %N 130404 %R https://doi.org/10.1103/PhysRevLett.121.130404 %0 Journal Article %D 2018 %T Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning %A Fernando G. S. L. Brandão %A Amir Kalev %A Tongyang Li %A Cedric Yen-Yu Lin %A Krysta M. Svore %A Xiaodi Wu %XWe give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank r, and sparsity s. The first algorithm assumes an input model where one is given access to entries of the matrices at unit cost. We show that it has run time O~(s2(m−−√ε−10+n−−√ε−12)), where ε is the error. This gives an optimal dependence in terms of m,n and quadratic improvement over previous quantum algorithms when m≈n. The second algorithm assumes a fully quantum input model in which the matrices are given as quantum states. We show that its run time is O~(m−−√+poly(r))⋅poly(logm,logn,B,ε−1), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only poly-logarithmically in n and polynomially in r. We apply the second SDP solver to the problem of learning a good description of a quantum state with respect to a set of measurements: Given m measurements and copies of an unknown state ρ, we show we can find in time m−−√⋅poly(logm,logn,r,ε−1) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as ρ on the m measurements, up to error ε. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension, which could be of independent interest.

%G eng %U https://arxiv.org/abs/1710.02581 %0 Journal Article %D 2018 %T Validating and Certifying Stabilizer States %A Amir Kalev %A Anastasios Kyrillidis %XWe propose a measurement scheme that validates the preparation of a target n-qubit stabilizer state. The scheme involves a measurement of n Pauli observables, a priori determined from the target stabilizer and which can be realized using single-qubit gates. Based on the proposed validation scheme, we derive an explicit expression for the worse-case fidelity, i.e., the minimum fidelity between the target stabilizer state and any other state consistent with the measured data. We also show that the worse-case fidelity can be certified, with high probability, using O(n) copies of the state of the system per measured observable.

%G eng %U https://arxiv.org/abs/1808.10786 %0 Journal Article %D 2017 %T Exponential Quantum Speed-ups for Semidefinite Programming with Applications to Quantum Learning %A Fernando G. S. L. Brandão %A Amir Kalev %A Tongyang Li %A Cedric Yen-Yu Lin %A Krysta M. Svore %A Xiaodi Wu %XWe give semidefinite program (SDP) quantum solvers with an exponential speed-up over classical ones. Specifically, we consider SDP instances with m constraint matrices of dimension n, each of rank at most r, and assume that the input matrices of the SDP are given as quantum states (after a suitable normalization). Then we show there is a quantum algorithm that solves the SDP feasibility problem with accuracy ǫ by using √ m log m · poly(log n,r, ǫ −1 ) quantum gates. The dependence on n provides an exponential improvement over the work of Brand ˜ao and Svore [6] and the work of van Apeldoorn et al. [23], and demonstrates an exponential quantum speed-up when m and r are small. We apply the SDP solver to the problem of learning a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state ρ, we show we can find in time √ m log m · poly(log n,r, ǫ −1 ) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as ρ on the m measurements up to error ǫ. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes’ principle. As in previous work, we obtain our algorithm by “quantizing” classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension based on the techniques developed in quantum principal component analysis, which could be of independent interest. Our quantum SDP solver is different from previous ones in the following two aspects: (1) it follows from a zero-sum game approach of Hazan [11] of solving SDPs rather than the primal-dual approach by Arora and Kale [5]; and (2) it does not rely on any sparsity assumption of the input matrices.

%8 2017/10/06 %G eng %U https://arxiv.org/abs/1710.02581 %0 Journal Article %D 2017 %T Provable quantum state tomography via non-convex methods %A Anastasios Kyrillidis %A Amir Kalev %A Dohuyng Park %A Srinadh Bhojanapalli %A Constantine Caramanis %A Sujay Sanghavi %XWith nowadays steadily growing quantum processors, it is required to develop new quantum tomography tools that are tailored for high-dimensional systems. In this work, we describe such a computational tool, based on recent ideas from non-convex optimization. The algorithm excels in the compressed-sensing-like setting, where only a few data points are measured from a lowrank or highly-pure quantum state of a high-dimensional system. We show that the algorithm can practically be used in quantum tomography problems that are beyond the reach of convex solvers, and, moreover, is faster than other state-of-the-art non-convex approaches. Crucially, we prove that, despite being a non-convex program, under mild conditions, the algorithm is guaranteed to converge to the global minimum of the problem; thus, it constitutes a provable quantum state tomography protocol.

%8 2017/11/19 %G eng %U https://arxiv.org/abs/1711.02524 %0 Journal Article %J Quantum Science and Technology %D 2017 %T Rigidity of the magic pentagram game %A Amir Kalev %A Carl Miller %XA game is rigid if a near-optimal score guarantees, under the sole assumption of the validity of quantum mechanics, that the players are using an approximately unique quantum strategy. Rigidity has a vital role in quantum cryptography as it permits a strictly classical user to trust behavior in the quantum realm. This property can be traced back as far as 1998 (Mayers and Yao) and has been proved for multiple classes of games. In this paper we prove ridigity for the magic pentagram game, a simple binary constraint satisfaction game involving two players, five clauses and ten variables. We show that all near-optimal strategies for the pentagram game are approximately equivalent to a unique strategy involving real Pauli measurements on three maximally-entangled qubit pairs.

%B Quantum Science and Technology %V 3 %P 015002 %8 2017/11/02 %G eng %U http://iopscience.iop.org/article/10.1088/2058-9565/aa931d/meta %N 1