TY - JOUR T1 - Experimental Observation of Thermalization with Noncommuting Charges JF - PRX Quantum Y1 - 2023 A1 - Florian Kranzl A1 - Aleksander Lasek A1 - Manoj K. Joshi A1 - Amir Kalev A1 - Rainer Blatt A1 - Christian F. Roos A1 - Nicole Yunger Halpern AB -

Quantum simulators have recently enabled experimental observations of quantum many-body systems' internal thermalization. Often, the global energy and particle number are conserved, and the system is prepared with a well-defined particle number - in a microcanonical subspace. However, quantum evolution can also conserve quantities, or charges, that fail to commute with each other. Noncommuting charges have recently emerged as a subfield at the intersection of quantum thermodynamics and quantum information. Until now, this subfield has remained theoretical. We initiate the experimental testing of its predictions, with a trapped-ion simulator. We prepare 6-21 spins in an approximate microcanonical subspace, a generalization of the microcanonical subspace for accommodating noncommuting charges, which cannot necessarily have well-defined nontrivial values simultaneously. We simulate a Heisenberg evolution using laser-induced entangling interactions and collective spin rotations. The noncommuting charges are the three spin components. We find that small subsystems equilibrate to near a recently predicted non-Abelian thermal state. This work bridges quantum many-body simulators to the quantum thermodynamics of noncommuting charges, whose predictions can now be tested.

VL - 4 UR - https://arxiv.org/abs/2202.04652 U5 - 10.1103/prxquantum.4.020318 ER - TY - JOUR T1 - Noncommuting conserved charges in quantum thermodynamics and beyond JF - Nature Reviews Physics Y1 - 2023 A1 - Shayan Majidy A1 - William F. Braasch A1 - Aleksander Lasek A1 - Twesh Upadhyaya A1 - Amir Kalev A1 - Nicole Yunger Halpern AB -

Thermodynamic systems typically conserve quantities ("charges") such as energy and particle number. The charges are often assumed implicitly to commute with each other. Yet quantum phenomena such as uncertainty relations rely on observables' failure to commute. How do noncommuting charges affect thermodynamic phenomena? This question, upon arising at the intersection of quantum information theory and thermodynamics, spread recently across many-body physics. Charges' noncommutation has been found to invalidate derivations of the thermal state's form, decrease entropy production, conflict with the eigenstate thermalization hypothesis, and more. This Perspective surveys key results in, opportunities for, and work adjacent to the quantum thermodynamics of noncommuting charges. Open problems include a conceptual puzzle: Evidence suggests that noncommuting charges may hinder thermalization in some ways while enhancing thermalization in others.

UR - https://arxiv.org/abs/2306.00054 U5 - 10.1038/s42254-023-00641-9 ER - TY - JOUR T1 - Noncommuting conserved charges in quantum many-body thermalization JF - Phys. Rev. E Y1 - 2020 A1 - Nicole Yunger Halpern A1 - Michael E. Beverland A1 - Amir Kalev AB -

In statistical mechanics, a small system exchanges conserved quantities—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 quantities. The conserved quantities are represented by operators usually 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 forms the system of interest. The conserved quantities manifest as spin components. Heisenberg interactions push the conserved quantities 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 near 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 conserved quantities from QI-theoretic thermodynamics into quantum many-body physics: atomic, molecular, and optical physics and condensed matter.

VL - 101 UR - https://journals.aps.org/pre/abstract/10.1103/PhysRevE.101.042117 CP - 042117 U5 - https://doi.org/10.1103/PhysRevE.101.042117 ER - TY - JOUR T1 - Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning JF - Quantum Y1 - 2020 A1 - Zhang Jiang A1 - Amir Kalev A1 - Wojciech Mruczkiewicz A1 - Hartmut Neven AB -

We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an n-mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on ⌈log3(2n+1)⌉ qubits. The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less than log3(2n) qubits on average. We apply it to the problem of learning k-fermion reduced density matrix (RDM), a problem relevant in various quantum simulation applications. We show that using the ternary-tree mapping one can determine the elements of all k-fermion RDMs, to precision ϵ, by repeating a single quantum circuit for ≲(2n+1)kϵ−2 times. This result is based on a method we develop here that allows one to determine the elements of all k-qubit RDMs, to precision ϵ, by repeating a single quantum circuit for ≲3kϵ−2 times, independent of the system size. This improves over existing schemes for determining qubit RDMs.

VL - 4 UR - https://arxiv.org/abs/1910.10746 CP - 276 U5 - https://doi.org/10.22331/q-2020-06-04-276 ER - TY - JOUR T1 - Simulating Hamiltonian Dynamics with an Off-diagonal Series Expansion Y1 - 2020 A1 - Amir Kalev A1 - Itay Hen AB -

We propose an efficient quantum algorithm for simulating the dynamics of general Hamiltonian systems. Our technique is based on a power series expansion of the time-evolution operator in its off-diagonal terms. The expansion decouples the dynamics due to the diagonal component of the Hamiltonian from the dynamics generated by its off-diagonal part, which we encode using the linear combination of unitaries technique. Our method has an optimal dependence on the desired precision and, as we illustrate, generally requires considerably fewer resources than the current state-of-the-art. We provide an analysis of resource costs for several sample models.

UR - https://arxiv.org/abs/2006.02539 ER - TY - JOUR T1 - An approximate description of quantum states Y1 - 2019 A1 - Marco Paini A1 - Amir Kalev AB -

We introduce an approximate description of an N-qubit state, which contains sufficient information to estimate the expectation value of any observable with precision independent of N. We show, in fact, that the error in the estimation of the observables' expectation values decreases as the inverse of the square root of the number of the system's identical preparations and increases, at most, linearly in a suitably defined, N-independent, seminorm of the observables. Building the approximate description of the N-qubit state only requires repetitions of single-qubit rotations followed by single-qubit measurements and can be considered for implementation on today's Noisy Intermediate-Scale Quantum (NISQ) computers. The access to the expectation values of all observables for a given state leads to an efficient variational method for the determination of the minimum eigenvalue of an observable. The method represents one example of the practical significance of the approximate description of the N-qubit state. We conclude by briefly discussing extensions to generative modelling and with fermionic operators

UR - https://arxiv.org/abs/1910.10543 ER - TY - JOUR T1 - Equilibration to the non-Abelian thermal state in quantum many-body physics Y1 - 2019 A1 - Nicole Yunger Halpern A1 - Michael E. Beverland A1 - Amir Kalev AB -

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. 

UR - https://arxiv.org/abs/1906.09227 ER - TY - JOUR T1 - Parallel Self-Testing of the GHZ State with a Proof by Diagrams JF - EPTCS Y1 - 2019 A1 - Spencer Breiner A1 - Amir Kalev A1 - Carl Miller AB -

Quantum 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.

VL - 287 U4 - 43-66 UR - https://arxiv.org/abs/1806.04744 U5 - https://doi.org/10.4204/EPTCS.287.3 ER - TY - JOUR T1 - Validating and Certifying Stabilizer States JF - Phys. Rev. A Y1 - 2019 A1 - Amir Kalev A1 - Anastasios Kyrillidis AB -

We 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.

VL - 99 UR - https://arxiv.org/abs/1808.10786 CP - 042337 U5 - https://doi.org/10.1103/PhysRevA.99.042337 ER - TY - JOUR T1 - Implicit regularization and solution uniqueness in over-parameterized matrix sensing Y1 - 2018 A1 - Anastasios Kyrillidis A1 - Amir Kalev AB -

We 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.

UR - https://arxiv.org/abs/1806.02046 ER - TY - JOUR T1 - Optimal Pure-State Qubit Tomography via Sequential Weak Measurements JF - Phys. Rev. Lett. Y1 - 2018 A1 - Ezad Shojaee A1 - Christopher S. Jackson A1 - Carlos A. Riofrio A1 - Amir Kalev A1 - Ivan H. Deutsch AB -

The 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.

VL - 121 UR - https://arxiv.org/abs/1805.01012 CP - 130404 U5 - https://doi.org/10.1103/PhysRevLett.121.130404 ER - TY - JOUR T1 - Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning JF - To appear at the 46th International Colloquium on Automata, Languages and Programming (ICALP 2019) Y1 - 2018 A1 - Fernando G. S. L. Brandão A1 - Amir Kalev A1 - Tongyang Li A1 - Cedric Yen-Yu Lin A1 - Krysta M. Svore A1 - Xiaodi Wu AB -

We 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.

UR - https://arxiv.org/abs/1710.02581 ER - TY - JOUR T1 - Exponential Quantum Speed-ups for Semidefinite Programming with Applications to Quantum Learning Y1 - 2017 A1 - Fernando G. S. L. Brandão A1 - Amir Kalev A1 - Tongyang Li A1 - Cedric Yen-Yu Lin A1 - Krysta M. Svore A1 - Xiaodi Wu AB -

We 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.

UR - https://arxiv.org/abs/1710.02581 ER - TY - JOUR T1 - Provable quantum state tomography via non-convex methods Y1 - 2017 A1 - Anastasios Kyrillidis A1 - Amir Kalev A1 - Dohuyng Park A1 - Srinadh Bhojanapalli A1 - Constantine Caramanis A1 - Sujay Sanghavi AB -

With 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.

UR - https://arxiv.org/abs/1711.02524 ER - TY - JOUR T1 - Rigidity of the magic pentagram game JF - Quantum Science and Technology Y1 - 2017 A1 - Amir Kalev A1 - Carl Miller AB -

A 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.

VL - 3 U4 - 015002 UR - http://iopscience.iop.org/article/10.1088/2058-9565/aa931d/meta CP - 1 ER -