Quantum 3-SAT Is QMA1-Complete

Gosset, David; Nagaj, Daniel

doi:10.1109/focs.2013.86

Cited by 34 publications

(36 citation statements)

References 19 publications

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“…Exact reductions have also been achieved for certain Hamiltonians. For instance, "frustration-free" gadgets have been used in proofs of the QMA-Completeness of quantum satisfiability, and in restricting the necessary terms for embedding quantum circuits in Local Hamiltonian [58][59][60].…”

Section: Estimates Of Spectral Gap Scalingmentioning

confidence: 99%

Exploiting Locality in Quantum Computation for Quantum Chemistry

McClean

Babbush

Love

et al. 2014

J. Phys. Chem. Lett.

128

190

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Accurate prediction of chemical and material properties from first principles quantum chemistry is a challenging task on traditional computers. Recent developments in quantum computation offer a route towards highly accurate solutions with polynomial cost, however this solution still carries a large overhead. In this perspective, we aim to bring together known results about the locality of physical interactions from quantum chemistry with ideas from quantum computation. We show that the utilization of spatial locality combined with the Bravyi-Kitaev transformation offers an improvement in the scaling of known quantum algorithms for quantum chemistry and provide numerical examples to help illustrate this point. We combine these developments to improve the outlook for the future of quantum chemistry on quantum computers.

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Section: Estimates Of Spectral Gap Scalingmentioning

confidence: 99%

Exploiting Locality in Quantum Computation for Quantum Chemistry

McClean

Babbush

Love

et al. 2014

J. Phys. Chem. Lett.

128

190

View full text Add to dashboard Cite

show abstract

“…Quantum k-SAT was introduced in [39], where it was shown that the problem is in P for k = 2 and QMA 1 -complete for k ≥ 4; QMA 1 -completeness for k = 3 is due to [74]. The Consistency of Density Operators problem was shown to be QMA-complete with respect to Cook reductions in [123].…”

Section: Chapter Notesmentioning

confidence: 99%

Quantum Proofs

Vidick

Watrous

2016

FNT in Theoretical Computer Science

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Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class \class{NP}, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class \class{QSZK}, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.Comment: Survey published by NOW publisher

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“…This problem is proven to be QMA 1 complete for K ≥ 3 [7], with QMA 1 the quantum analog of NP [8]. A random ensemble of K-QSAT was introduced and studied in [9][10][11], where it has been shown to have a SAT-UNSAT phase transition.…”

Section: Introductionmentioning

confidence: 99%

Satisfiability-unsatisfiability transition in the adversarial satisfiability problem

2014

Self Cite

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Adversarial SAT (AdSAT) is a generalization of the satisfiability (SAT) problem in which two players try to make a boolean formula true (resp. false) by controlling their respective sets of variables. AdSAT belongs to a higher complexity class in the polynomial hierarchy than SAT and therefore the nature of the critical region and the transition are not easily paralleled to those of SAT and worth of independent study. AdSAT also provides an upper bound for the transition threshold of the quantum satisfiability problem (QSAT). We present a complete algorithm for AdSAT, show that 2-AdSAT is in P, and then study two stochastic algorithms (simulated annealing and its improved variant) and compare their performances in detail for 3-AdSAT. Varying the density of clauses α we find a sharp SAT-UNSAT transition at a critical value whose upper bound is αc < ∼ 1.5, thus providing a much stricter upper bound for the QSAT transition than those previously found.

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Quantum 3-SAT Is QMA1-Complete

Cited by 34 publications

References 19 publications

Exploiting Locality in Quantum Computation for Quantum Chemistry

Exploiting Locality in Quantum Computation for Quantum Chemistry

Quantum Proofs

Satisfiability-unsatisfiability transition in the adversarial satisfiability problem

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