Quantum algorithm for approximating partition functions

Abstract

We achieve a quantum speed-up of fully polynomial randomized approximation schemes (FPRAS) for estimating partition functions that combine simulated annealing with the Monte-Carlo Markov Chain method and use non-adaptive cooling schedules. The improvement in time complexity is twofold: a quadratic reduction with respect to the spectral gap of the underlying Markov chains and a quadratic reduction with respect to the parameter characterizing the desired accuracy of the estimate output by the FPRAS. Both reductions are intimately related and cannot be achieved separately.First, we use Grover's fixed point search, quantum walks and phase estimation to efficiently prepare approximate coherent encodings of stationary distributions of the Markov chains. The speed-up we obtain in this way is due to the quadratic relation between the spectral and phase gaps of classical and quantum walks. Second, we generalize the method of quantum counting, showing how to estimate expected values of quantum observables. Using this method instead of classical sampling, we obtain the speed-up with respect to accuracy.

Necessary condition for the quantum adiabatic approximation

“…In this article we give a rigorous lower bound for the evolution time (or cost) of adiabatic processes that prepare the final state. This bound is also valid for more general quantum evolutions [5,6,11,13].…”

“…This assertion is quantified via the adiabatic approximation [2], which provides a relation between the rate of change of the perturbation and the fidelity of the evolved state with the final eigenstate. The adiabatic approximation is a key part of quantum computing as it determines the complexity of several quantum algorithms [3][4][5][6]. In fact, adiabatic quantum computation [3], in which the result of a problem is encoded in the ground state of a (final) Hamiltonian, is equivalent to standard quantum computation [7].…”

“…A more efficient method to traverse the eigenpath for this case was introduced in Ref. [6]. The method in Ref.…”

“…The method in Ref. [6] uses Grover's fixed point search and the schedule r(t) is nonmonotonic. It results in a costÕ(L(log L) 2 / ).…”

Low depth quantum circuits for Ising models

“…Whether efficient quantum algorithms exist for preparing ferromagnetic thermal states is as far as we know an open problem but if so than corner magnetisation measurement could prove a useful diagnostic for such algorithms since classical FPRAS is available. Finally, we add that recently quantum algorithms for FPRAS were found which exhibit a quadratic speed up over the classical counterparts [40]. These algorithms are rather different in spirit from measuring corner magnetisation as instead of using mixed states they use a combination of Grover search and phase estimation to prepare pure states of many qubit systems which coherently encode probability distributions of various classical spin configurations.…”