On complexity of the quantum Ising model

We study the computational complexity of estimating the ground state energy and simulating the adiabatic evolution for the transverse field Ising model (TIM). It is shown that the ground state energy problem for TIM on a degree-3 graph is complete for the complexity class StoqMA. This is an extension of the classical class MA where the verifier can accept quantum states as a proof and apply classical reversible gates in a coherent fashion. As a corollary, we complete the complexity classification of 2-local Hamiltonians with a fixed set of interactions proposed recently by Cubitt and Montanaro. In the special case of the ferromagnetic TIM we show that the ground state energy problem is contained in BPP. Finally, we prove that the adiabatic quantum computing (AQC) with TIM has the same computational power as AQC with general 2-local stoquastic Hamiltonians. Calculating the ground state energy and thermal equilibrium properties of interacting quantum manybody systems is one of the central problems in quantum chemistry and condensed matter physics. It was realized early on that the complexity of this problem depends on the statistics of the constituent particles. For systems composed of bosons and for certain special classes of spin Hamiltonians the quantum partition function can be mapped to the one of a classical system occupying one extra spatial dimension [1] which often enables efficient Monte Carlo simulation [2, 3, 4, 5, 6, 7]. On the other hand, for systems composed of fermions and for the vast majority of spin Hamiltonians the quantum-to-classical mapping produces a partition function with unphysical Boltzmann weights taking both positive and negative (or complex) values — a phenomenon known as the “sign problem”. Here we focus on quantum spin Hamiltonians avoiding the sign problem also known as stoquastic. The defining property of stoquastic Hamiltonians is that their off-diagonal matrix elements in the standard product basis must be real and non-positive. The stoquastic class encompasses many interesting models such as TIM, the Heisenberg ferromagnetic and antiferromagnetic models (the latter is stoquastic on any bipartite graph), quantum simulated annealing Hamiltonians [8], the toric code Hamiltonian [9], and Hamiltonians derived from reversible Markov chains [10, 11, 12]. Identifying “easy” and “hard” instances of stoquastic Hamiltonians is therefore important as it could give insights on the power and limitations of classical Monte Carlo simulation algorithms [13] and contribute to our understanding of speedups in quantum annealing algorithms [14, 15]. To state our results let us define several classes of stoquastic Hamiltonians. Let TIM(n, J) be the set of all n-qubit transverse field Ising Hamiltonians