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arXiv:2203.08882 · Quantum

Quantum algorithms from fluctuation theorems: Thermal-state preparation

Zoë Holmes, Gopikrishnan Muraleedharan, Rolando D. Somma, Yiğit Subaşı, Burak Şahinoğlu

Published Oct 6, 2022• 32 citations
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Abstract

Fluctuation theorems provide a correspondence between properties of quantum systems in thermal equilibrium and a work distribution arising in a non-equilibrium process that connects two quantum systems with Hamiltonians H 0 and H 1 = H 0 + V . Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of H 1 at inverse temperature β ≥ 0 starting from a purification of the thermal state of H 0 . The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is O ~ ( e β ( Δ A − w l ) / 2 ) , where Δ A is the free-energy difference between H 1 and H 0 , and w l is a work cutoff that depends on the properties of the work distribution and the approximation error ϵ > 0 . If the non-equilibrium process is trivial, this complexity is exponential in β ‖ V ‖ , where ‖ V ‖ is the spectral norm of V . This represents a significant improvement of prior quantum algorithms that have complexity exponential in β ‖ H 1 ‖ in the regime where ‖ V ‖ ≪ ‖ H 1 ‖ . The dependence of the complexity in ϵ varies according to the structure of the quantum systems. It can be exponential in 1 / ϵ in general, but we show it to be sublinear in 1 / ϵ if H 0 and H 1 commute, or polynomial in 1 / ϵ if H 0

DOI
10.22331/q-2022-10-06-825

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Fluctuation theorems provide a correspondence between properties of quantum systems in thermal equilibrium and a work distribution arising in a non-equilibrium process that connects two quantum systems with Hamiltonians H 0 and H 1 = H 0 + V . Building upon these theorems, we present a quantum algorithm to prepare a purification of the thermal state of H 1 at inverse temperature β ≥ 0 starting from a purification of the thermal state of H 0 . The complexity of the quantum algorithm, given by the number of uses of certain unitaries, is O ~ ( e β ( Δ A − w l ) / 2 ) , where Δ A is the free-energy difference between H 1 and H 0 , and w l is a work cutoff that depends on the properties of the work distribution and the approximation error ϵ > 0 . If the non-equilibrium process is trivial, this complexity is exponential in β ‖ V ‖ , where ‖ V ‖ is the spectral norm of V . This represents a significant improvement of prior quantum algorithms that have complexity exponential in β ‖ H 1 ‖ in the regime where ‖ V ‖ ≪ ‖ H 1 ‖ . The dependence of the complexity in ϵ varies according to the structure of the quantum systems. It can be exponential in 1 / ϵ in general, but we show it to be sublinear in 1 / ϵ if H 0 and H 1 commute, or polynomial in 1 / ϵ if H 0
We present quantum algorithms for solving two problems regarding stochastic processes. The first algorithm prepares the thermal Gibbs state of a quantum system and runs in time almost linear in p Nβ/Z and polynomial in log(1/epsilon), where N is the Hilbert space dimension, β is the inverse temperature, Z is the partition function, and epsilon is the desired precision of the output state. Our quantum algorithm exponentially improves the complexity dependence on 1/epsilon and polynomially improves the dependence on β of known quantum algorithms for this problem. The second algorithm estimates the hitting time of a Markov chain. For a sparse stochastic matrix P, it runs in time almost linear in 1/(epsilon ∆3/2 ), where epsilon is the absolute precision in the estimation and ∆ is a parameter determined by P, and whose inverse is an upper bound of the hitting time. Our quantum algorithm quadratically improves the complexity dependence on 1/epsilon and 1/∆ of the analog classical algorithm for hitting-time estimation. Both algorithms use tools recently developed in the context of Hamiltonian simulation, spectral gap amplification, and solving linear systems of equations.
We derive the fluctuation theorem for quantum-state statistics that can be obtained when we initially measure the total energy of a quantum system at thermal equilibrium, let the system evolve unitarily, and record the quantum-state data reconstructed at the end of the process. The obtained theorem shows that the quantum-state statistics for the forward and backward processes is related to the equilibrium free-energy difference through an infinite series of independent relations, which gives the quantum work fluctuation theorem as a special case, and reproduces the out-of-time-order fluctuation-dissipation theorem near thermal equilibrium. The quantum-state statistics exhibits a system-size scaling behavior that differs between integrable and non-integrable (quantum chaotic) systems as demonstrated numerically for one-dimensional quantum lattice models.
Nature is governed by precise physical laws, which can inspire the discovery of new computer-run simulation algorithms. Thermal states are the most ubiquitous for they are the equilibrium states of matter. Simulating thermal states of quantum matter has applications ranging from quantum machine learning to better understanding of high-temperature superconductivity and quantum chemistry. The computational complexity of this task is hopelessly hard for classical computers. The existing quantum algorithms come with caveats: most either require quantum phase estimation rendering them impractical for current noisy hardware, or are variational which face obstacles such as initialization, barren plateaus, and a general lack of provable guarantee. We provide two quantum algorithms with provable guarantees to prepare thermal states on (near-term) quantum computers that avoid these drawbacks. The first algorithm is inspired by the natural thermalization process where the ancilla qubits act as the infinite thermal bath. This algorithm can potentially run in polynomial time to sample thermal distributions of ergodic systems -- the vast class of physical systems that equilibrate in isolation with respect to local observables. The second algorithm works for any system and in general runs in exponential time. However, it requires significantly smaller quantum resources than previous such algorithms. In addition, we provide an error mitigation technique for both algorithms to fight back decoherence, which enables us to run our algorithms on the near-term quantum devices. To illustration, we simulate the thermal state of the hardcore Bose-Hubbard model on the latest generation of available quantum computers.
The simulation of quantum physical systems is expected to be an important application for quantum computers. The work presented in this dissertation aims to improve the resource requirements of quantum computers for solving simulation problems, by providing both novel quantum algorithms and improved implementations of existing ones. I present three main results that cover diverse aspects of simulation including equilibrium physics, the preparation of useful quantum states, and simulations based on classical stochastic processes. The results rely on established quantum algorithms and other recent techniques which I review. My first original contribution is a new quantum algorithm to sample from the thermal Gibbs state of quantum Hamiltonians. The complexity of this algorithm has an exponentially improved dependence on precision compared to an earlier quantum algorithm, and also provides polynomial improvements in other parameters. My second contribution is an improved algorithm to implement reflections about the eigenstates of unitary operators and Hamiltonians, an operation that can be used to prepare ground states of many-body Hamiltonians by quantum amplitude amplification. The improved algorithm for implementing reflections significantly reduces the number of ancilla qubits required over earlier implementations. The improvement relies in part on a novel quantum algorithm for efficiently preparing states with Gaussian-like amplitudes, which may have broad applications. I also prove a lower bound for implementing reflections, which shows that our algorithm is almost optimal. My third and final contribution regards quantum algorithms for classical stochastic processes, which are of importance in statistical physics. I present a quantum algorithm that estimates the average running time of a stochastic process, and provides a quadratic speed-up over a comparable classical algorithm.
In a recent work [10], Poulin and one of us presented a quantum algorithm for preparing thermal Gibbs states of interacting quantum systems. This algorithm is based on Grovers's technique for quantum state engineering, and its running time is dominated by the factor D/Z(β), where D and Z(β) denote the dimension of the quantum system and its partition function at inverse temperature β, respectively. We present here a modified algorithm and a more detailed analysis of the errors that arise due to imperfect simulation of Hamiltonian time evolutions and limited performance of phase estimation (finite accuracy and nonzero probability of failure). This modfication together with the tighter analysis allows us to prove a better running time by the effect of these sources of error on the overall complexity. We think that the ideas underlying of our new analysis could also be used to prove a better performance of quantum Metropolis sampling by Temme et al. [12].
Dissipative algorithms for preparing thermal and ground states typically require a total Hamiltonian simulation time that grows at least linearly with system size NNN, even when their mixing time grows only polylogarithmically. We achieve sublinear simulation time for free fermions by engineering local interactions with a bath whose size grows in proportion to the system size. Starting from arbitrary initial states, our thermal state preparation algorithm achieves trace-norm error εεε in total simulation time O~(N/ε)\widetilde{O}(\sqrt{N}/ε)O(N​/ε) at fixed inverse temperature and bounded single-particle bandwidth, while ground state preparation of gapped systems requires only polylog(N/ε)\mathrm{polylog}(N/ε)polylog(N/ε) time. We further establish lower bounds for thermal state preparation within the bath protocols considered, identifying two obstacles to faster preparation, namely fourth-order deviations from the target thermal state and fluctuations induced by a random frequency shared by all bath qubits. Our results demonstrate that the linear simulation time barrier in dissipative state preparation can be overcome and clarify the bath and filter-design constraints that govern further complexity improvements.