Efficient Quantum Walk Circuits for Metropolis-Hastings Algorithm

Abstract

We present a detailed circuit implementation of Szegedy's quantization of the Metropolis-Hastings walk. This quantum walk is usually defined with respect to an oracle. We find that a direct implementation of this oracle requires costly arithmetic operations. We thus reformulate the quantum walk, circumventing its implementation altogether by closely following the classical Metropolis-Hastings walk. We also present heuristic quantum algorithms that use the quantum walk in the context of discrete optimization problems and numerically study their performances. Our numerical results indicate polynomial quantum speedups in heuristic settings.

Faster Coherent Quantum Algorithms for Phase, Energy, and Amplitude Estimation

“…However, for some applications of phase estimation maintaining coherence remains crucial. The original strategy for quantum matrix inversion [HHL08], quantum Metropolis sampling [Temme&09,YA11,Lemi&19], and a protocol for partition function estimation [Mon15], thermal state preparation, and Bayesian inference [HW19,AHNTW20] all violate the assumptions above. The eigenvalues must be estimated while preserving the superposition, and there is no guarantee that the input state is an eigenstate.…”

Prospects of quantum computing for molecular sciences

“…FF and DFT methods work reliably in applications such as binding free energy predictions for drug-protein docking, although they are generally prohibitively expensive for high-throughput screening studies. On the quantum algorithm side, the quantum-Metropolis-Hastings algorithm can provide a quadratic speedup for Monte Carlo simulations (Szegedy 2004;Lemieux et al 2020). However, the algorithm needs separated bases for electronic and vibrational degrees of freedom.…”

“…Heuristics like genetic algorithms (Deaven and Ho 1995) or simulated annealing (Mundim and Tsallis 1996) are often used to search for global minimum geometry. In this case, quantum walks (Szegedy 2004;Lemieux et al 2020) may replace classical random walks in simulated annealing to provide a quadratic speed up. However, due to the faster clock speed of classical computers, quadratic speedups are unlikely to be practical in the foreseeable future.…”

Antibody loop modelling on a quantum computer

“…( 3). Following [64], a quantum walk operator W equivalent to W can be implemented by four quantum operators, acting on System (S), Move (M) and Coin (C) quantum registers, defined as:…”

“…As mentioned in Section III B, the main tasks are to propose a state space Ω and set of rules M corresponding to a Markov chain that allows the biologically relevant conformations of antibody loops to be efficiently explored; and to design quantum circuits that can implement the V, B, F, R operators defined in Section IV E. For the former, our approach is conceptually similar to that used in [65] as well as the Rosetta software package, modified to be more convenient for implementing on a quantum computer. For the latter, at a high level the approach we take follows [60,64], but the process is complicated by the need to coherently convert from dihedral to Cartesian coordinates.…”