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arXiv:1806.05405 · Quantum

Two-local qubit Hamiltonians: when are they stoquastic?

Joel Klassen, Barbara M. Terhal

Published May 6, 2019• 29 citations
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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 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.

DOI
10.22331/q-2019-05-06-139

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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 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 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.
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 nnn labelled vertices in O~(n6.38)\widetilde O(n^{6.38})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)\widetilde O(n^5)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.
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.
Stoquastic Hamiltonians play a role in the computational complexity of the local Hamiltonian problem as well as the study of classical simulability. In particular, stoquastic Hamiltonians can be straightforwardly simulated using Monte Carlo techniques. We address the question of whether two or more Hamiltonians may be made simultaneously stoquastic via a unitary transformation. This question has important implications for the complexity of simulating quantum annealing where quantum advantage is related to the stoquasticity of the Hamiltonians involved in the anneal. We find that for almost all problems no such unitary exists and show that the problem of determining the existence of such a unitary is equivalent to identifying if there is a solution to a system of polynomial (in)equalities in the matrix elements of the initial and transformed Hamiltonians. Solving such a system of equations is NP-hard. We highlight a geometric understanding of this problem in terms of a collection of generalized Bloch vectors.
In the range of applications opened by quantum technology, often a highly entangled source state is needed as an input for a protocol (target state). The easiest example of such an application is QKD, other examples are quantum secret sharing and measurement based quantum computing. Generating a non-local entangled state is however not trivial. Let us assume that there is an entangled state (source state) present in the network before the start of the protocol. One can ask than whether this source state can be transformed to the target state with only local operations. Local operations usually have a lower error rate compared to non-local operations. As the set of all quantum states increases exponentially, we will restrict ourselves to graph states (quantum states described by a simple graph). This allows one to study the graphs and local operations in a graph theory perspective, instead of in a quantum information perspective. In this thesis we investigate the complexity of deciding equivalence of two graph states under local multi-qubit operations (T-LMQC-equivalent), where T refers to the partition denoting which qubits are on the same device which allows multi-qubit operations. When T consist only of single qubit nodes, this problem reduces to the single qubit Clifford equivalence problem (SQC-EQUIV). It is known that SQC-EQUIV can be solved in O(N^4). On the other side, if all qubits are in the same node, the problem can be solved in O(1) as every source state can be transformed to any graph state on the same vertex set. We will present three algorithms to solve T-LMQC equivalence. The first algorithm solves the problem in general but the running time scales exponentially in the number of multi-qubit nodes and the size of the graph. The other two scale polynomially in the size of the graph, but do not allow for nodes with more than 2 qubits. The remaining question is indeed whether deciding T-LMQC equivalence is an easy (in P) problem or a hard problem (NP-complete).