Quantum Metropolis Sampling
A quantum analog of the Metropolis algorithm that samples from the thermal (Gibbs) distribution of a quantum Hamiltonian.
The thermal sampling problem
Many physical questions ask for properties of a system in thermal equilibrium at inverse temperature beta, described by the Gibbs state rho = exp(-beta H)/Z, where Z is the partition function. Classical Markov chain Monte Carlo samples such distributions for classical systems, but for a quantum Hamiltonian the eigenstates are unknown and non-commuting terms make direct sampling hard. Quantum Metropolis sampling generalizes the Metropolis-Hastings method to this quantum setting.
The quantum Metropolis step
The algorithm performs a random walk over the eigenstates of H. From a current energy eigenstate, it proposes a move using a random unitary, uses phase estimation to measure the proposed state's energy, and accepts or rejects the move with a Metropolis rule based on the energy change and beta. The central difficulty is undoing a rejected move without having measured (and thereby collapsed) the original state; a reversal procedure restores the previous eigenstate when a move is rejected.
Ingredients
- Phase estimation to read energies of proposed states.
- A proposal unitary that connects eigenstates.
- A Metropolis acceptance probability min(1, exp(-beta*Delta E)).
- A coherent rejection-reversal step to recover the prior state.
What it produces
After enough steps, the walk's stationary distribution is the Boltzmann distribution over eigenstates, so measured observables approximate thermal averages. This gives access to free energies, specific heats, and correlation functions of quantum systems at finite temperature, quantities central to condensed-matter and chemistry studies.
Costs and alternatives
The method relies on accurate phase estimation, making it a fault-tolerant-era algorithm rather than a near-term one; mixing time governs how many steps are needed and can be long near phase transitions. Near-term alternatives prepare thermal states variationally or via imaginary-time evolution, trading rigor for shallower circuits. Quantum Metropolis sampling remains the conceptually cleanest route to provable thermal sampling of quantum systems.