Hardness and Ease of Curing the Sign Problem for Two-Local Qubit Hamiltonians

Abstract

We examine the problem of determining whether a multi-qubit 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 performs this task in a number of arithmetic operations over R which is polynomial in the number of qubits.

Determining QMC simulability with geometric phases

“…First and foremost, it clearly illustrates that 'curing' the sign problem of a model, i.e., finding a unitary transformation that produces an SPF representation for it, is markedly different from curing non-stoquasticity (finding unitary transformations that make the Hamiltonian stoquastic). In fact, our result shows that the latter approach, which is the current standard practice [12][13][14][15][16], should not be used toward rendering a Hamiltonian simulable. In addition, our result demonstrates that 'stoquastization' of sign-problematic Hamiltonians, the method normally used for assigning positive weights to QMC configurations is, in general, a sub-optimal choice; a superior alternative can be given in terms of the geometric phases of the Hamiltonian.…”

“…In the field of quantum Monte Carlo (QMC) simulations [5,6], the partition function of stoquastic Hamiltonians can always be written as a sum of efficiently computable strictly positive weights [7][8][9]. As a consequence, such Hamiltonians do not suffer from a 'sign problem' [10,11], i.e., from the existence of negative summands, which greatly impede the convergence of QMC algorithms [10][11][12][13].…”

“…That stoquasticity leads to sign-problem-free (SPF) representations of quantum many-body physical models has served as the main motivation in numerous recent studies that examine the conditions under which various different classes of non-stoquastic Hamiltonians can be unitarily transformed to equivalent stoquastic [12][13][14][15], or 'minimally non-stoquastic' [16] representations.…”

Phase transitions in the frustrated Ising ladder with stoquastic and nonstoquastic catalysts

“…Since the general problem of deciding whether a local curing transformation exists is NP-complete even for singlequbit Clifford transformations [51,52], we do not consider here the non-stoquasticity of Ĥ in more general cases than the conditions given by Eq. ( 3).…”

“…Otherwise the Hamiltonian is called nonstoquastic, and the inevitable positive or complex offdiagonal matrix elements of the Hamiltonian lead to the infamous sign problem [49,50]. We note that even when the Hamiltonian is stoquastic, but it is presented in a form in which this stoquasticity is unapparent, the problem of deciding whether there exists a local transformation to a basis that "cures the sign problem" (makes it stoquastic) by making all of the Hamiltonian matrix elements real and nonpositive, is NP-complete [51,52].…”