Ground State Connectivity of Local Hamiltonians

Gharibian, Sevag; Sikora, Jamie

doi:10.1007/978-3-662-47672-7_50

Cited by 22 publications

(47 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Adding clock-checking (16), we find that the positive semidefinite Hamiltonian H dw,L N + H dw clock-check has a unique, frustration-free (annihilated by all projector terms), zero-energy ground state |ψ dw = 1…”

Section: The Domain Wall (Unary) Clockmentioning

confidence: 99%

See 1 more Smart Citation

Clocks in Feynman's computer and Kitaev's local Hamiltonian: Bias, gaps, idling, and pulse tuning

2018

View full text Add to dashboard Cite

We present a collection of results about the clock in Feynman's computer construction and Kitaev's Local Hamiltonian problem. First, by analyzing the spectra of quantum walks on a line with varying endpoint terms, we find a better lower bound on the gap of the Feynman Hamiltonian, which translates into a less strict promise gap requirement for the QMA-complete Local Hamiltonian problem. We also translate this result into the language of adiabatic quantum computation. Second, introducing an idling clock construction with a large state space but fast Cesaro mixing, we provide a way for achieving an arbitrarily high success probability of computation with Feynman's computer with only a logarithmic increase in the number of clock qubits. Finally, we tune and thus improve the costs (locality, gap scaling) of implementing a (pulse) clock with a single excitation.

show abstract

Section: The Domain Wall (Unary) Clockmentioning

confidence: 99%

“…The Hilbert space thus splits into the invariant good subspace spanned by states with a single domain wall, and other invariant subspaces. In those, all states have energy at least a constant E ≥ 1, because each such state is "detected" by at least one of the clock-checking terms in (16).…”

Section: The Domain Wall (Unary) Clockmentioning

confidence: 99%

Clocks in Feynman's computer and Kitaev's local Hamiltonian: Bias, gaps, idling, and pulse tuning

2018

View full text Add to dashboard Cite

show abstract

“…Let us start with the definition of the QCMA-complete Ground State Connectivity (GSCON) problem [11] about the possibility of traversal between two low-energy states for a local Hamiltonian, using local unitary transformations, while remaining in a low-energy sector.…”

Section: The Ground Space Connectivity Problem (Gscon)mentioning

confidence: 99%

“…Decide, which of the two cases is true: 1 In general, this could be also l-local unitaries, we choose l = 2. This variant of the problem is still QCMA complete [11]. In this paper, we consider a specific version that we call frustration-free GSCON.…”

Section: A K-local Hamiltonianmentioning

confidence: 99%

Shorter unentangled proofs for ground state connectivity

Caha

Nagaj

Schwarz

2018

Quantum Inf Process

View full text Add to dashboard Cite

Can one considerably shorten a proof for a quantum problem by using a protocol with a constant number of unentangled provers? We consider a frustration-free variant of the QCMA-complete Ground State Connectivity (GSCON) problem for a system of size n with a proof of superlinear-size. We show that we can shorten this proof in QMA (2): there exists a two-copy, unentangled proof with length of order n, up to logarithmic factors, while the completeness-soundness gap of the new protocol becomes a small inverse polynomial in n.

show abstract

“…Few complete problems for QCMA are known; an interesting example is the Ground State Connectivity (GSCON) problem considered in [68].…”

Section: Chapter Notesmentioning

confidence: 99%

Quantum Proofs

Vidick

Watrous

2016

FNT in Theoretical Computer Science

View full text Add to dashboard Cite

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

show abstract

Ground State Connectivity of Local Hamiltonians

Cited by 22 publications

References 30 publications

Clocks in Feynman's computer and Kitaev's local Hamiltonian: Bias, gaps, idling, and pulse tuning

Clocks in Feynman's computer and Kitaev's local Hamiltonian: Bias, gaps, idling, and pulse tuning

Shorter unentangled proofs for ground state connectivity

Quantum Proofs

Contact Info

Product

Resources

About