We examine the problem of determining whether a multiqubit two-local Hamiltonian can be made stoquastic by single-qubit unitary transformations. We prove that when such a Hamiltonian contains one-local terms, then this task can be NP-hard. This is shown by constructing a class of Hamiltonians for which performing this task is equivalent to deciding 3-SAT. In contrast, we show that when such a Hamiltonian contains no one-local terms then this task is easy; namely, we present an algorithm which decides, in a number of arithmetic operations over $\mathbb{R}$ which is polynomial in the number of qubits, whether the sign problem of the Hamiltonian can be cured by single-qubit rotations.
We examine the problem of determining whether a multiqubit two-local Hamiltonian can be made stoquastic by single-qubit unitary transformations. We prove that when such a Hamiltonian contains one-local terms, then this task can be NP-hard. This is shown by constructing a class of Hamiltonians for which performing this task is equivalent to deciding 3-SAT. In contrast, we show that when such a Hamiltonian contains no one-local terms then this task is easy; namely, we present an algorithm which decides, in a number of arithmetic operations over R which is polynomial in the number of qubits, whether the sign problem of the Hamiltonian can be cured by single-qubit rotations.
We elucidate the distinction between global and termwise stoquasticity for local Hamiltonians and prove several complexity results. We prove coNP-hardness of deciding global stoquasticity in a fixed basis and Σ2p-hardness of deciding global stoquasticity under single-qubit transformations. We expand the class of sign-curing transformations by showing how Clifford transformations can sign-cure a class of disordered 1D XYZ Hamiltonians.
We examine the problem of determining if a 2-local Hamiltonian is stoquastic by local basis changes. We analyze this problem for two-qubit Hamiltonians, presenting some basic tools and giving a concrete example where using unitaries beyond Clifford rotations is required in order to decide stoquasticity. We report on simple results for n -qubit Hamiltonians with identical 2-local terms on bipartite graphs. Our most significant result is that we give an efficient algorithm to determine whether an arbitrary n -qubit XYZ Heisenberg Hamiltonian is stoquastic by local basis changes.
We elucidate the distinction between global and termwise stoquasticity for local Hamiltonians and prove several complexity results. We show that the stoquastic local Hamiltonian problem is StoqMA-complete even for globally stoquastic Hamiltonians. We study the complexity of deciding whether a local Hamiltonian is globally stoquastic or not. In particular, we prove coNP-hardness of deciding global stoquasticity in a fixed basis and Σ2p-hardness of deciding global stoquasticity under single-qubit transformations. As a last result, we expand the class of sign-curing transformations by showing how Clifford transformations can sign-cure a class of disordered 1D XYZ Hamiltonians.
Quantum many-body systems whose Hamiltonians are non-stoquastic, i.e., have positive off-diagonal matrix elements in a given basis, are known to pose severe limitations on the efficiency of Quantum Monte Carlo algorithms designed to simulate them, due to the infamous sign problem. We study the computational complexity associated with 'curing' non-stoquastic Hamiltonians, i.e., transforming them into sign-problem-free ones. We prove that if such transformations are limited to single-qubit Clifford group elements or general single-qubit orthogonal matrices, finding the curing transformation is NP-complete. We discuss the implications of this result.
A well-known result about satisfiability theory is that the 2-SAT problem can be solved in linear time, despite the NP-hardness of the 3-SAT problem. In the quantum 2-SAT problem, we are given a family of 2-qubit projectors Πij on a system of n qubits, and the task is to decide whether the Hamiltonian H=∑Πij has a 0-eigenvalue, or all eigenvalues are greater than 1/nα for some α=O(1). The problem is not only a natural extension of the classical 2-SAT problem to the quantum case, but is also equivalent to the problem of finding a ground state of 2-local frustration-free Hamiltonians of spin 21, a well-studied model believed to capture certain key properties in modern condensed matter physics. Bravyi has shown that the quantum 2-SAT problem has a deterministic algorithm of complexity O(n4) in the algebraic model of computation where every arithmetic operation on complex numbers can be performed in unit time, and n is the number of variables. In this paper we give a deterministic algorithm in the algebraic model with running time O(n+m), where m is the number of local projectors, therefore achieving the best possible complexity in that model. We also show that if in the input every number has a constant size representation then the bit complexity of our algorithm is O((n+m)M(n)), where M(n) denotes the complexity of multiplying two n-bit integers. A conference version of this paper appeared in the Proceedings of the 43rd International Colloquium on Automata, Languages and Programming (ICALP'16), 2016.
Local-unitary (LU) equivalence asks whether two quantum states differ only by independent changes of basis on their qubits. For graph states, whether this relation can be decided in polynomial time has remained open for over a decade. We give a deterministic algorithm that decides LU equivalence for graphs on n labelled vertices in O(n6.38) bit operations and constructs exact single-qubit unitaries whenever the states are equivalent. Building on Claudet and Perdrix's quasipolynomial algorithm, we replace the enumeration of vertex subsets by a compact system of constraints generated from pairs and triples. The remaining graph transformation is found by solving linear equations over the binary field. These new steps cost O(n5) bit operations; the inherited graph preprocessing sets the overall bound. We also count the local-Clifford (LC) classes of graph states within any LU class: their number is a power of two, computable within the same bound. For any given graph state, this decides whether single-qubit Clifford gates reach every graph state in its LU class, and supplies a counterexample when they do not. The method also decides LU equivalence of stabilizer codes encoding one logical qubit.