A quantum algorithm to count weighted ground states of classical spin Hamiltonians

Abstract

Ground state counting plays an important role in several applications in science and engineering, from estimating residual entropy in physical systems, to bounding engineering reliability and solving combinatorial counting problems. While quantum algorithms such as adiabatic quantum optimization (AQO) and quantum approximate optimization (QAOA) can minimize Hamiltonians, they are inadequate for counting ground states. We modify AQO and QAOA to count the ground states of arbitrary classical spin Hamiltonians, including counting ground states with arbitrary nonnegative weights attached to them. As a concrete example, we show how our method can be used to count the weighted fraction of edge covers on graphs, with user-specified confidence on the relative error of the weighted count, in the asymptotic limit of large graphs. We find the asymptotic computational time complexity of our algorithms, via analytical predictions for AQO and numerical calculations for QAOA, and compare with the classical optimal Monte Carlo algorithm (OMCS), as well as a modified Grover's algorithm. We show that for large problem instances with small weights on the ground states, AQO does not have a quantum speedup over OMCS for a fixed error and confidence, but QAOA has a sub-quadratic speedup on a broad class of numerically simulated problems. Our work is an important step in approaching general ground-state counting problems beyond those that can be solved with Grover's algorithm. It offers algorithms that can employ noisy intermediate-scale quantum devices for solving ground state counting problems on small instances, which can help in identifying more problem classes with quantum speedups.

Quantifying the efficiency of state preparation via quantum variational eigensolvers

“…In this section we present our main findings on the correlation between the success of QAOA state preparation and the interaction distance of the target state. As a toy model, we consider the one-dimensional quantum Ising model in the presence of both transverse and longitudinal fields, (10) where X i and Z i are the standard Pauli matrices on site i, and we assume periodic boundary conditions (PBCs) by identifying sites j + N ≡ j. The model is either ferromagnetic (FM) or antiferromagnetic (AFM) depending on whether the coupling of the first term is chosen to be +1 or −1, respectively.…”

“…The Ising models in Eqs. (10) serve as a useful laboratory for studying a number of phenomena in condensed matter physics. The properties of the ground state of the model in Eq.…”

“…We conclude that there is a practical limitation to the "natural" QAOA protocol proposed in Sec. IV, which was obtained as a Trotter splitting of the Hamiltonian (10) into its translation invariant components: the protocol is unable to prepare ground states that are far from being Gaussian (as measured by interaction distance). This limitation is fundamentally related to the probability 11)- (13).…”

“…4,5 QAOA was originally proposed to tackle classical optimization problems, such as the MaxCut problem, 6 and several others. [7][8][9][10] More recently, it has been pointed out that QAOA could also serve as a tool for exactly preparing quantum many-body states, such as the GHZ state, the ground state of the Ising model at the critical point, for both short-range 11 and long-range interactions, 12,13 the ground state of the toric code, 11 the ground state of the two-dimensional Hubbard model, 14 and the thermofield double states. 15,16 In this paper we focus on the latter type of applications of QAOA in the context of preparing ground states of non-integrable quantum Hamiltonians.…”

“…Figure 7. Distribution of angles θ3,j associated to Hamiltonian(13) in QAOA evolution across phase diagrams of Hamiltonians in Eq (10). and Eq.…”