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arXiv:2007.11964

Termwise versus globally stoquastic local Hamiltonians: questions of complexity and sign-curing

Marios Ioannou, Stephen Piddock, Milad Marvian, Joel Klassen, Barbara M. Terhal

Published Jul 23, 2020• 0 citations
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Abstract

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 $\textbf{StoqMA}$-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 $\textbf{coNP}$-hardness of deciding global stoquasticity in a fixed basis and $Σ_2^p$-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.

DOI
10.48550/arxiv.2007.11964

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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\textbf{StoqMA}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\textbf{coNP}coNP-hardness of deciding global stoquasticity in a fixed basis and Σ2pΣ_2^pΣ2p​-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 XYZXYZXYZ Hamiltonians.
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\Sigma_2^pΣ2p​-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.
The QMA-completeness of the local Hamiltonian problem is a landmark result of the field of Hamiltonian complexity that studies the computational complexity of problems in quantum many-body physics. Since its proposal, substantial effort has been invested in better understanding the problem for physically motivated important families of Hamiltonians. In particular, the QMA-completeness of approximating the ground state energy of local Hamiltonians has been extended to the case where the Hamiltonians are geometrically local in one and two spatial dimensions. Among those physically motivated Hamiltonians, stoquastic Hamiltonians play a particularly crucial role, as they constitute the manifestly sign-free Hamiltonians in Monte Carlo approaches. Interestingly, for such Hamiltonians, the problem at hand becomes more ''classical'', being hard for the class MA (the randomized version of NP) and its complexity has tight connections with derandomization. In this work, we prove that both the two- and one-dimensional geometrically local analogues remain MA-hard with high enough qudit dimension. Moreover, we show that related problems are StoqMA-complete.
Despite having an unnatural definition, StoqMA\mathsf{StoqMA}StoqMA plays a central role in Hamiltonian complexity, e.g., in the classification theorem of the complexity of Hamiltonians by Cubitt and Montanaro (SICOMP 2016). Moreover, it lies between the two randomized extensions of NP\mathsf{NP}NP, MA\mathsf{MA}MA and AM\mathsf{AM}AM. Therefore, understanding the exact power of StoqMA\mathsf{StoqMA}StoqMA (and hopefully collapsing it with more natural complexity classes) is of great interest for different reasons. In this work, we take a step further in understanding this complexity class by showing that the Stoquastic Sparse Hamiltonians problem (StoqSH\mathsf{StoqSH}StoqSH) is in StoqMA\mathsf{StoqMA}StoqMA. Since Stoquastic Local Hamiltonians are StoqMA\mathsf{StoqMA}StoqMA-hard, this implies that StoqSH\mathsf{StoqSH}StoqSH is StoqMA\mathsf{StoqMA}StoqMA-complete. We complement this result by showing that the separable version of StoqSH\mathsf{StoqSH}StoqSH is StoqMA(2)\mathsf{StoqMA}(2)StoqMA(2)-complete, where StoqMA(2)\mathsf{StoqMA}(2)StoqMA(2) is the version of StoqMA\mathsf{StoqMA}StoqMA that receives two unentangled proofs.
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.
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\mathbb{R}R which is polynomial in the number of qubits, whether the sign problem of the Hamiltonian can be cured by single-qubit rotations.
We study the complexity of Local Hamiltonian problems under the restriction that every local term is an unweighted occurrence of the same fixed interaction. In the resulting Single-Interaction Hamiltonian (SIH) model, once the interaction is fixed, the Hamiltonian is specified entirely by its ordered interaction hypergraph. Our main technical tool is an exact compiler that converts any fixed finite alphabet of local interactions into a single interaction. On a designated auxiliary subspace, the compiled Hamiltonian reproduces the source Hamiltonian exactly, while the complete spectrum below a chosen separation scale is preserved with multiplicity. Combined with a finite-alphabet normalization, this gives a universal polynomial-time reduction from kkk-Local Hamiltonian to SIH3k+3_{3k+3}3k+3​ up to a known global energy scale and inverse-polynomial approximation. Sharper constructions give QMA-completeness of SIH3_33​ and of geometrically local SIH8_88​ with bounded degree and interaction range. In the stoquastic setting, we obtain StoqMA-completeness of Stoq-SIH3_33​ and QMA-completeness of 1-Pinned Stoq-SIH4_44​. We also establish hardness results for uniformly system-size-dependent interactions and derive single-interaction formulations of the Hamiltonian quantum PCP conjecture, frustration-free, exponentially precise, and guided Local Hamiltonian problems. These results show that independently chosen local matrices and independently tunable coupling strengths are not necessary for a broad range of Hamiltonian-complexity phenomena.