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arXiv:2207.14313 · Physical Review X

Quantum Many-Body Jarzynski Equality and Dissipative Noise on a Digital Quantum Computer

Dominik Hahn, Maxime Dupont, Markus Schmitt, David J. Luitz, Marin Bukov

Published Nov 2, 2023• 12 citations
arXivHTMLPDF

Abstract

The quantum Jarzynski equality and the Crooks relation are fundamental laws connecting equilibrium processes with nonequilibrium fluctuations.They are promising tools to benchmark quantum devices and measure free energy differences.While they are well established theoretically and also experimental realizations for few-body systems already exist, their experimental validity in the quantum many-body regime has not been observed so far.Here, we present results for nonequilibrium protocols in systems with up to 16 interacting degrees of freedom obtained on trapped ion and superconducting qubit quantum computers, which test the quantum Jarzynski equality and the Crooks relation in the many-body regime.To achieve this, we overcome present-day limitations in the preparation of thermal ensembles and in the measurement of work distributions on noisy intermediate-scale quantum devices.We discuss the accuracy to which the Jarzynski equality holds on different quantum computing platforms subject to platformspecific errors.The analysis reveals the validity of Jarzynski's equality in a regime with energy dissipation, compensated for by a fast unitary drive.This provides new insights for analyzing errors in many-body quantum simulators.

DOI
10.1103/physrevx.13.041023

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The quantum Jarzynski equality and the Crooks relation are fundamental laws connecting equilibrium processes with nonequilibrium fluctuations.They are promising tools to benchmark quantum devices and measure free energy differences.While they are well established theoretically and also experimental realizations for few-body systems already exist, their experimental validity in the quantum many-body regime has not been observed so far.Here, we present results for nonequilibrium protocols in systems with up to 16 interacting degrees of freedom obtained on trapped ion and superconducting qubit quantum computers, which test the quantum Jarzynski equality and the Crooks relation in the many-body regime.To achieve this, we overcome present-day limitations in the preparation of thermal ensembles and in the measurement of work distributions on noisy intermediate-scale quantum devices.We discuss the accuracy to which the Jarzynski equality holds on different quantum computing platforms subject to platformspecific errors.The analysis reveals the validity of Jarzynski's equality in a regime with energy dissipation, compensated for by a fast unitary drive.This provides new insights for analyzing errors in many-body quantum simulators.
The non-equilibrium dynamics of quantum many-body systems is one of the most fascinating problems in physics. Open fundamental and practical questions range from how they relax to equilibrium, to how to extract useful work from them. Here, we derive a set of exact results that relate out-of-equilibrium fluctuations in the energy and other observables of a quantum system to its equilibrium properties for a very general family of initial conditions. These quantum fluctuation relations generalise the Jarzynski and Crooks relations to quantum systems with conserved quantities, and can be applied to protocols driving the system between integrable and chaotic regimes, or coupling it to different reservoirs. We illustrate our results with simulations of an integrable model subject to quenches realisable with current technology. Our findings will help guiding research on the interplay of quantum and thermal fluctuations in quantum simulation, and their exploitation in the design of new quantum devices.
Thermodynamics constrains changes to the energy of a system, both deliberate and random, via its first and second laws. When the system is not in equilibrium, fluctuation theorems such as the Jarzynski equality further restrict the distributions of deliberate work done. Such fluctuation theorems have been experimentally verified in small, nonequilibrium quantum systems undergoing unitary or decohering dynamics. Yet, their validity in systems governed by a non-Hermitian Hamiltonian has long been contentious due to the false premise of the Hamiltonian's dual and equivalent roles in dynamics and energetics. Here we show that work fluctuations in a non-Hermitian qubit obey the Jarzynski equality even if its Hamiltonian has complex or purely imaginary eigenvalues. With postselection on a dissipative superconducting circuit undergoing a cyclic parameter sweep, we experimentally quantify the work distribution using projective energy measurements and show that the fate of the Jarzynski equality is determined by the parity-time symmetry of, and the energetics that result from, the corresponding non-Hermitian, Floquet Hamiltonian. By distinguishing the energetics from non-Hermitian dynamics, our results provide the recipe for investigating the nonequilibrium quantum thermodynamics of such open systems. Published by the American Physical Society 2024
The classical Jarzynski equality establishes an exact relation between the stochastic work performed on a system driven out of thermal equilibrium and the free energy difference in a corresponding quasistatic process. This fluctuation theorem bears experimental relevance, as it enables the determination of the free energy difference through the measurement of externally applied work in a nonequilibrium process. In the quantum case, the Jarzynski equality only holds if the measurement procedure of the stochastic work is drastically changed: it is replaced by a so-called two-point measurement scheme that requires the knowledge of the initial and final Hamiltonian and therefore lacks the predictive power for the free energy difference that the classical Jarzynski equation is known for. Here, we propose a quantum fluctuation theorem that is valid for externally measurable quantum work determined during the driving protocol. In contrast to the two-point measurement case, the theorem also applies to open quantum systems and the scenario can be realized without knowing the system Hamiltonian. Our fluctuation theorem comes in the form of an inequality and therefore only yields bounds to the true free energy difference. The inequality is saturated in the quasiclassical case of vanishing energy coherences at the beginning and at the end of the protocol. Thus, there is a clear quantum disadvantage.
Based on the observation that the thermodynamic equilibrium free energy of an open quantum system in contact with a thermal environment is the difference between the free energy of the total system and that of the bare environment, the validity of the Crooks theorem and of the Jarzynski equality is extended to open quantum systems. No restrictions on the nature of the environment or on the strength of the coupling between system and environment need to be imposed. This free energy entering the Crooks theorem and the Jarzynski equality is closely related to the Hamiltonian of mean force that generalizes the classical statistical mechanical concept of the potential of mean force.
The central quantity in the celebrated quantum Jarzynski equality is ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, where WWW is work and $\ensuremath{\beta}$ is the inverse temperature. The impact of quantum randomness on the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ and hence on the predictive power of the Jarzynski estimator is an important problem. Working on a single nitrogen-vacancy center in diamond and riding on an implementation of the two-point measurement of nonequilibrium work with single-shot readouts, we have conducted a direct experimental investigation of the relationship between the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ and adiabaticity of nonequilibrium work protocols. It is observed that adiabatic processes minimize the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, thus verifying an early theoretical concept, the so-called principle of minimal work fluctuations. Furthermore, it is experimentally demonstrated that shortcuts-to-adiabaticity control can be exploited to minimize the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ in fast work protocols. Our work should stimulate further experimental studies of quantum effects on the bias and error in the estimates of free-energy differences based on the Jarzynski equality.
We consider the quantum mechanical generalization of Crooks Fluctuation Theorem and Jarzynski Equality for an open quantum system. The explicit expression for microscopic work for an arbitrary prescribed protocol is obtained, and the relation between quantum Crooks Fluctuation Theorem, quantum Jarzynski Equality and their classical counterparts are clarified. Numerical simulations based on a two-level toy model are used to demonstrate the validity of the quantum version of the two theorems beyond linear response theory regime.