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Quantum Information and Computation

Monte Carlo simulation of stoquastic Hamiltonians

Sergey B. Bravyi

Published Oct 1, 2015• 36 citations
Publisher

Abstract

Stoquastic Hamiltonians are characterized by the property that their off-diagonal matrix elements in the standard product basis are real and non-positive. Many interesting quantum models fall into this class including the Transverse field Ising Model (TIM), the Heisenberg model on bipartite graphs, and the bosonic Hubbard model. Here we consider the problem of estimating the ground state energy of a local stoquastic Hamiltonian $H$ with a promise that the ground state of $H$ has a non-negligible correlation with some `guiding' state that admits a concise classical description. A formalized version of this problem called Guided Stoquastic Hamiltonian is shown to be complete for the complexity class $\MA$ (a probabilistic analogue of $\NP$). To prove this result we employ the Projection Monte Carlo algorithm with a variable number of walkers. Secondly, we show that the ground state and thermal equilibrium properties of the ferromagnetic TIM can be simulated in polynomial time on a classical probabilistic computer. This result is based on the approximation algorithm for the classical ferromagnetic Ising model due to Jerrum and Sinclair (1993).

DOI
10.26421/qic15.13-14-3

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Stoquastic Hamiltonians are characterized by the property that their off-diagonal matrix elements in the standard product basis are real and non-positive. Many interesting quantum models fall into this class including the Transverse field Ising Model (TIM), the Heisenberg model on bipartite graphs, and the bosonic Hubbard model. Here we consider the problem of estimating the ground state energy of a local stoquastic Hamiltonian HHH with a promise that the ground state of HHH has a non-negligible correlation with some `guiding' state that admits a concise classical description. A formalized version of this problem called Guided Stoquastic Hamiltonian is shown to be complete for the complexity class \MA\MA\MA (a probabilistic analogue of \NP\NP\NP). To prove this result we employ the Projection Monte Carlo algorithm with a variable number of walkers. Secondly, we show that the ground state and thermal equilibrium properties of the ferromagnetic TIM can be simulated in polynomial time on a classical probabilistic computer. This result is based on the approximation algorithm for the classical ferromagnetic Ising model due to Jerrum and Sinclair (1993).
We study complexity of several problems related to the Transverse field Ising Model (TIM). First, we consider the problem of estimating the ground state energy known as the Local Hamiltonian Problem (LHP). It is shown that the LHP for TIM on degree-3 graphs is equivalent modulo polynomial reductions to the LHP for general k-local `stoquastic' Hamiltonians with any constant k≥2k\ge 2k≥2. This result implies that estimating the ground state energy of TIM on degree-3 graphs is a complete problem for the complexity class StoqMA - an extension of the classical class MA. As a corollary, we complete the complexity classification of 2-local Hamiltonians with a fixed set of interactions proposed recently by Cubitt and Montanaro. Secondly, we study quantum annealing algorithms for finding ground states of classical spin Hamiltonians associated with hard optimization problems. We prove that the quantum annealing with TIM Hamiltonians is equivalent modulo polynomial reductions to the quantum annealing with a certain subclass of k-local stoquastic Hamiltonians. This subclass includes all Hamiltonians representable as a sum of a k-local diagonal Hamiltonian and a 2-local stoquastic Hamiltonian.
We study several problems related to properties of non-negative matrices that arise at the boundary between quantum and classical probabilistic computation. Our results are twofold. First, we identify a large class of quantum Hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer. These are stoquastic local Hamiltonians with a "frustration free" ground-state. A Hamiltonian belongs to this class iff it can be represented as H=∑aHaH=\sum_a H_aH=∑a​Ha​ where (1) every term HaH_aHa​ acts non-trivially on a constant number of qubits, (2) every term HaH_aHa​ has real non-positive off-diagonal matrix elements in the standard basis, and (3) the ground-state of HHH is a ground-state of every term HaH_aHa​. Secondly, we generalize the Cook-Levin theorem proving NP-completeness of the satisfiability problem to the complexity class MA -- a probabilistic analogue of NP. Specifically, we construct a quantum version of the k-SAT problem which we call "stoquastic k-SAT" such that stoquastic k-SAT is contained in MA for any constant kkk, and any promise problem in MA is Karp-reducible to stoquastic 6-SAT. This result provides the first non-trivial example of a MA-complete promise problem.
We study several problems related to properties of nonnegative matrices that arise at the boundary between quantum and classical probabilistic computation. Our results are twofold. First, we identify a large class of quantum Hamiltonians describing systems of qubits for which the adiabatic evolution can be efficiently simulated on a classical probabilistic computer. These are stoquastic local Hamiltonians with a “frustration-free” ground-state. A Hamiltonian belongs to this class iff it can be represented as H=∑aHaH=\sum_{a}H_{a}H=∑a​Ha​ where (1) every term HaH_{a}Ha​ acts nontrivially on a constant number of qubits, (2) every term HaH_{a}Ha​ has real nonpositive off-diagonal matrix elements in the standard basis, and (3) the ground-state of H is a ground-state of every term HaH_{a}Ha​. Second, we generalize the Cook–Levin theorem proving NP-completeness of the satisfiability problem to the complexity class MA (Merlin–Arthur games)—a probabilistic analogue of NP. Specifically, we construct a quantum version of the k-SAT problem which we call “stoquastic k-SAT” such that stoquastic k-SAT is contained in MA for any constant k, and any promise problem in MA is Karp-reducible to stoquastic 6-SAT. This result provides the first nontrivial example of a MA-complete promise problem.
We study the computational complexity of the Guided Local Hamiltonian problem: given a local Hamiltonian HHH together with a classical description of a guiding state that has non-negligible overlap with the ground state of HHH, estimate the ground-state energy within inverse-polynomial precision. This setting captures real-world scenarios in quantum chemistry and many-body physics, where trial states derived from classical heuristics can be used to guide quantum algorithms. We identify families of physically-motivated guiding states for which the computational hardness of ground-state energy estimation persists in the guided setting. Extending prior results for semi-classical subset states, we prove BQP-hardness for classes including fixed-weight states, matrix product states, Gaussian states, and Fendley states. Our hardness results are obtained via refined Feynman-Kitaev circuit-to-Hamiltonian constructions that explicitly expose the structural role of the guiding state in the reduction. Complementing these results, we give a constructive proof of BQP containment when the guiding state admits a polynomial-size classical description, establishing BQP-completeness for the canonical formulation of the problem. Our results show that quantum advantage persists for the newly introduced state classes, and classical methods also remain viable when said guiding states admit appropriate descriptions. Together, our results identify a Goldilocks zone of guiding states that are efficiently preparable, succinctly described, and sample-query accessible, within which quantum advantage for ground-state estimation can be meaningfully assessed. We additionally formalise the Guided Fermi-Hubbard Hamiltonian problem and prove BQP-completeness on 2D square and triangular lattices, both with and without magnetic fields, when provided with an appropriate fermionic guiding state.
We construct a probability distribution, induced by the Perron--Frobenius eigenvector of an exponentially large graph, which cannot be efficiently sampled by any classical algorithm, even when provided with the best-possible warm-start distribution. In the quantum setting, this problem can be viewed as preparing the ground state of a stoquastic Hamiltonian given a guiding state as input, and is known to be efficiently solvable on a quantum computer. Our result suggests that no efficient classical algorithm can solve a broad class of stoquastic ground-state problems. Our graph is constructed from a class of high-degree, high-girth spectral expanders to which self-similar trees are attached. This builds on and extends prior work of Gilyén, Hastings, and Vazirani [Quantum 2021, STOC 2021], which ruled out dequantization for a specific stoquastic adiabatic path algorithm. We strengthen their result by ruling out any classical algorithm for guided ground-state preparation.