Quantum Gibbs Sampling Thermal States

Short answer: Quantum Gibbs sampling is the task of preparing thermal states of quantum systems on a quantum computer, a quantum counterpart of classical Markov chain Monte Carlo (MCMC). Recent theory papers, including one in Nature, propose algorithms with provable costs, and a review covers methods aimed at near-term devices. These are algorithm proposals, and none of these papers reports a useful result on a real quantum computer.

Why thermal states matter

A 2023 arXiv paper says preparing ground states and thermal states is essential for simulating quantum systems on quantum computers. It notes that, despite hopes for practical quantum advantage in simulation, popular state preparation approaches have been challenged. A Nature paper says that although many quantum algorithms exist for simulating quantum dynamics, a general-purpose method for simulating low-temperature quantum phenomena remains unknown. In classical settings, it says, sampling from thermal distributions has been largely addressed by MCMC methods.

The Nature proposal

The Nature paper proposes an efficient quantum algorithm for thermal simulation that, like MCMC, exhibits detailed balance, respects locality and serves as a toy model for thermalization in open quantum systems. Its text describes the work as settling an open problem: an efficient algorithm that satisfies quantum detailed balance exactly while inheriting the locality of the Hamiltonian, up to a range depending on the temperature. It says the algorithm resembles effective descriptions of open-system dynamics when a system couples to a heat bath. The authors speculate that it might be among the first useful quantum simulation algorithms for low-temperature properties in the forthcoming fault-tolerant era. That is a stated expectation, not a result.

The 2023 Lindbladian approach

The arXiv paper says Monte Carlo-style quantum Gibbs samplers had emerged as an alternative, but earlier proposals were unsatisfactory due to technical obstacles rooted in energy-time uncertainty. It introduces simple continuous-time quantum Gibbs samplers that overcome these obstacles by efficiently simulating quantum master equations known as Lindbladians. It also constructs what it calls the first provably accurate and efficient algorithm for preparing certain purified Gibbs states, called thermal field double states in high-energy physics, for rapidly thermalizing systems, with a quantum walk speedup. The costs depend provably on temperature, accuracy and the mixing time, or spectral gap, of the relevant Lindbladian. The authors say they complete the first rigorous proof of finite-time thermalization for physically derived Lindbladians.

Near-term approaches

A 2023 overview paper focuses on preparing Gibbs states on NISQ devices. It says variational quantum algorithms show the most promise there and surveys methods including joint Hamiltonian evolution of a system-environment coupling, quantum imaginary time evolution, and variational algorithms using the Helmholtz free energy as a cost function. It benchmarks one recent variational algorithm on the spin-1/2 one-dimensional XY model. It also recalls the quantum Metropolis algorithm of Temme and co-authors, which prepares Gibbs states through random walks and gets a quadratic speedup over direct Metropolis-Hastings by using Grover-type techniques.

What this does not tell you

These abstracts give no resource estimates for a problem of practical interest. The efficiency claims are proven for the stated algorithms and assumptions, such as rapid thermalization, and may not apply to every system.

FAQ

What is Gibbs sampling here?

Preparing thermal states of a quantum system, the quantum analogue of classical MCMC sampling.

What did the Nature paper add?

An efficient quantum algorithm for thermal simulation with detailed balance and locality, per its abstract.

What are Lindbladians?

Quantum master equations describing open-system dynamics, which the 2023 arXiv paper's samplers simulate.

Does it work on current hardware?

The overview paper discusses near-term variational methods and benchmarks one on a small model. The provable algorithms are aimed at the fault-tolerant era.

What limits the cost?

The 2023 arXiv paper says cost depends on temperature, accuracy and mixing time.