The complexity of stoquastic local Hamiltonian problems

Bravyi, Sergey; DiVincenzo, David P.; Oliveira, Roberto I.; Terhal, Barbara M.

doi:10.26421/qic8.5-1

Cited by 201 publications

(308 citation statements)

References 23 publications

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“…ij is the well known 2D Heisenberg model, for which the problem of computing the exact ground state has been extensively studied. While the bipartite graph case can be mapped to a stoquastic Hamiltonian [20], the QCM technique can be used in general for harder Hamiltonian models, in particular Heisenberg models which are QMA complete [21].…”

Section: Hamiltonian Problem Operator Reduction and Scalingmentioning

confidence: 99%

Quantum computed moments correction to variational estimates

Vallury

Jones

Hill

et al. 2020

Quantum

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The variational principle of quantum mechanics is the backbone of hybrid quantum computing for a range of applications. However, as the problem size grows, quantum logic errors and the effect of barren plateaus overwhelm the quality of the results. There is now a clear focus on strategies that require fewer quantum circuit steps and are robust to device errors. Here we present an approach in which problem complexity is transferred to dynamic quantities computed on the quantum processor – Hamiltonian moments, ⟨Hn⟩. From these quantum computed moments, an estimate of the ground-state energy can be obtained using the ``infimum'' theorem from Lanczos cumulant expansions which manifestly corrects the associated variational calculation. With higher order effects in Hilbert space generated via the moments, the burden on the trial-state quantum circuit depth is eased. The method is introduced and demonstrated on 2D quantum magnetism models on lattices up to 5×5 (25 qubits) implemented on IBM Quantum superconducting qubit devices. Moments were quantum computed to fourth order with respect to a parameterised antiferromagnetic trial-state. A comprehensive comparison with benchmark variational calculations was performed, including over an ensemble of random coupling instances. The results showed that the infimum estimate consistently outperformed the benchmark variational approach for the same trial-state. These initial investigations suggest that the quantum computed moments approach has a high degree of stability against trial-state variation, quantum gate errors and shot noise, all of which bodes well for further investigation and applications of the approach.

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Section: Hamiltonian Problem Operator Reduction and Scalingmentioning

confidence: 99%

Quantum computed moments correction to variational estimates

Vallury

Jones

Hill

et al. 2020

Quantum

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“…In addition, the quantum partition function of a stoquastic Hamiltonian can be expressed as a sum of nonnegative, easily computable weights, which implies that Markov chain Monte Carlo algorithms can be used to perform importance sampling of the quantum configuration space to calculate thermal averages of physical observables, using these weights as (unnormalized) probabilities. For this reason, it is said that stoquastic Hamiltonians do not suffer from the sign problem [11,10]. However, it is important to note that the absence of a sign problem does not necessarily imply polynomial-time convergence of standard Monte Carlo methods [16,23,12].…”

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confidence: 99%

“…From a computational complexity perspective, the problem of estimating ground state energies of stoquastic local Hamiltonians is considered easier than for general Hamiltonians [10,9]. Moreover, in the classification of the complexity of estimating ground state energies of local Hamiltonians, stoquastic Hamiltonians appear as the only intermediate class between classical Hamiltonians and general Hamiltonians [14].…”

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confidence: 99%

“…Remark. When dealing with k-local Hamiltonians with k > 2, it is important to note that two distinct notions of stoquasticity have been defined in [10], namely, there exist termwise-stoquastic Hamiltonians and globally stoquastic Hamiltonians. A globally stoquastic Hamiltonian is a symmetric Z-matrix, while a Hamiltonian which is k-local termwise-stoquastic is one which can be decomposed into k-local terms such that each term is a symmetric Z-matrix.…”

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confidence: 99%

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

Klassen¹,

Marvian²,

Piddock³

et al. 2020

SIAM J. Comput.

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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 \BbbR which is polynomial in the number of qubits, whether the sign problem of the Hamiltonian can be cured by single-qubit rotations.

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“…By trivial, we mean that the circuits in Boxworld only consist of making the local 'fiducial' measurements { j (x j , a j |} on a state and performing classical post-processing on the outcomes. This process can be simulated by the prover giving the verifier the classical string of measurement outcomes similar to the approach of lemma 2 in [44]. That is, while poly-size advice states in Boxworld can encode any Boolean function, the theory has no non-trivial dynamics to efficiently verify that this function is encoded in the state if the prover cannot be trusted.…”

Section: (B) Example: Boxworldmentioning

confidence: 99%

Bounds on the power of proofs and advice in general physical theories

Lee¹,

Hoban²

2016

Proc. R. Soc. A.

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Quantum theory presents us with the tools for computational and communication advantages over classical theory. One approach to uncovering the source of these advantages is to determine how computation and communication power vary as quantum theory is replaced by other operationally defined theories from a broad framework of such theories. Such investigations may reveal some of the key physical features required for powerful computation and communication. In this paper, we investigate how simple physical principles bound the power of two different computational paradigms which combine computation and communication in a non-trivial fashion: computation with advice and interactive proof systems. We show that the existence of non-trivial dynamics in a theory implies a bound on the power of computation with advice. Moreover, we provide an explicit example of a theory with no non-trivial dynamics in which the power of computation with advice is unbounded. Finally, we show that the power of simple interactive proof systems in theories where local measurements suffice for tomography is non-trivially bounded. This result provides a proof that QMA is contained in PP, which does not make use of any uniquely quantum structure—such as the fact that observables correspond to self-adjoint operators—and thus may be of independent interest.

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The complexity of stoquastic local Hamiltonian problems

Cited by 201 publications

References 23 publications

Quantum computed moments correction to variational estimates

Quantum computed moments correction to variational estimates

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

Bounds on the power of proofs and advice in general physical theories

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