Two-local qubit Hamiltonians: when are they stoquastic?

Abstract

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 2local 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.Accepted in Quantum 2019-03-29, click title to verify arXiv:1806.05405v3 [quant-ph] 29 Apr 2019 1 It is worthwhile noting that the entrywise non-negativity of the matrix exp(−H/kT ) for a given temperature T does not imply that H is a symmetric Z-matrix. It is simple to generate numerical 4 × 4 cases where G = A T A is a non-negative matrix by taking A as a non-negative matrix, but H = − log G is not a symmetric Z-matrix. Thus when H is a symmetric Z-matrix, the non-negativity of exp(−H/kT ) is guaranteed for all values of kT , while in other cases exp(−H/kT ) might be non-negative only for sufficiently small values of kT .2 [1] points out that for 2-local Hamiltonians one can prove that termwise stoquastic is the same as stoquastic. This is however not true for k > 2-local Hamiltonians generally.

Effective Hamiltonians for interacting superconducting qubits: local basis reduction and the Schrieffer–Wolff transformation

“…According to reference [40], a two-local two-qubit Hamiltonian of the form H = h xI σ σ σ x,1 + h Ix σ σ σ x,2 + h zI σ σ σ z,1 + h Iz σ σ σ z,2 + +h xx σ σ σ x,1 σ σ σ x,2 + h yy σ σ σ y,1 σ σ σ y,2 + h zz σ σ σ z,1 σ σ σ z,2…”

Demonstration of a Nonstoquastic Hamiltonian in Coupled Superconducting Flux Qubits

“…Otherwise, representation of the basis vectors requires exponential resources. A Hamiltonian is stoquastic if there exists a local basis such that all its off-diagonal elements are real and nonpositive [14][15][16]. A positive off-diagonal element would cause negative transition probabilities, which cannot be simulated by stochastic processes.…”

Superconducting qubit circuit emulation of a vector spin-1/2

“…This capability would then enable the experimental exploration of a number of potentially significant ideas that until now have remained out of reach of available physical qubit hardware. These include, for example: the use of strong, non-stoquastic driver Hamiltonians in quantum annealing [42][43][44][45][46][47][48][49] and quantum simulation [13][14][15][16][17]; Hamiltonians required for Hamiltonian and holonomic computing paradigms [18][19][20][21]9]; and engineered emulation of the full quantum Heisenberg model. If combined with one or more of the proposed schemes for realizing static [99][100][101][102] or pulsed [64] multiqubit interactions, an even wider range of possibilities would become accessible, including adiabatic topological quantum computation [12], adiabatic quantum chemistry [22][23][24], and quantum error suppression [8,9].…”

“…In particular, although these qubits are well-suited to emulation of the simpler, transverse-field Ising spin (which interacts with other spins only via its z-component), controllable vector spin interactions of the kind we discuss here are fundamentally beyond their capability. A direct consequence of this is an inability to realize the controllable 'non-stoquastic' qubit interactions [40] that have been the subject of intense interest in recent years [41][42][43][44][45][46][47][48][49][50][51][52], due to the potentially transformative role such interactions may have on the computational power of quantum annealing. Exploring this capability is of paramount importance, in light of the fact that while quantum annealing has many potential applications, its ability to enable beyond-classical capabilities has yet to be fully understood or even conclusively shown.…”