Exponential complexity of the quantum adiabatic algorithm for certain satisfiability problems

Hen, Itay; Young, A. P.

doi:10.1103/physreve.84.061152

Cited by 84 publications

(96 citation statements)

References 45 publications

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“…This requires an analysis of the size dependence of the minimum gap for large sizes, presumably using Quantum Monte Carlo simulations, e.g. [46], to get to large enough sizes to see the trend.…”

Section: Conclusion and Future Researchmentioning

confidence: 99%

Solving the graph-isomorphism problem with a quantum annealer

Hen

Young

2012

Phys. Rev. A

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We propose a novel method using a quantum annealer -an analog quantum computer based on the principles of quantum adiabatic evolution -to solve the Graph Isomorphism problem, in which one has to determine whether two graphs are isomorphic (i.e., can be transformed into each other simply by a relabeling of the vertices). We demonstrate the capabilities of the method by analyzing several types of graph families, focusing on graphs with particularly high symmetry called strongly regular graphs (SRG's). We also show that our method is applicable, within certain limitations, to currently available quantum hardware such as "D-Wave One".

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Section: Conclusion and Future Researchmentioning

confidence: 99%

Solving the graph-isomorphism problem with a quantum annealer

Hen

Young

2012

Phys. Rev. A

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“…This implies that the system eventually reaches the ground state of the original Ising model representing the solution to the combinatorial optimization problem. There exists a large body of analytical, numerical, and experimental studies on quantum annealing, and active debates are going on to compare quantum annealing with the corresponding classical heuristic, simulated annealing, recent examples of which include Matsuda et al (2009), Young et al (2010), Hen and Young (2011), Farhi et al (2012), Boixo et al (2014), Katzgraber et al (2014Katzgraber et al ( , 2015, Rønnow et al (2014), Albash et al (2015), Heim et al (2015), Hen et al (2015), Isakov et al (2016), Martin-Mayor and Hen (2015), Steiger et al (2015), Venturelli et al (2015), Crosson and Harrow (2016), Denchev et al (2016), Kechedzhi and Smelyanskiy (2016), Mandrà et al (2016a,b), Marshall et al (2016), and Muthukrishnan et al (2016).…”

Section: Introductionmentioning

confidence: 99%

Exponential Enhancement of the Efficiency of Quantum Annealing by Non-Stoquastic Hamiltonians

2017

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Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like XX and XXXXX with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian nonstoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.

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“…Recent promising experimental research findings [11][12][13], as well as intensive theoretical work [see, e.g., [14][15][16][17][18][19][20][21][22][23] in the field of Adiabatic Quantum Computing (AQC) suggest the possibility that some of the experimental difficulties may be overcome by the use of "quantum annealing" devices, which implement the simple yet potentially-powerful quantum-adiabatic algorithmic approach proposed by Farhi et al [24] about a decade ago. AQC is an analog, continuous, quantum computing paradigm, and as such it has the potential of being easier to implement successfully, offering several advantages over the "traditional" gate model, in the form of inherent fault-tolerance and natural robustness against decoherence and dephasing [25,26].…”

Section: Introductionmentioning

confidence: 99%

Fourier-transforming with quantum annealers

Hen

2014

Front. Phys.

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We introduce a set of quantum adiabatic evolutions that we argue may be used as "building blocks," or subroutines, in the construction of an adiabatic algorithm that executes Quantum Fourier Transform (QFT) with the same complexity and resources as its gate-model counterpart. One implication of the above construction is the theoretical feasibility of implementing Shor's algorithm for integer factorization in an optimal manner, and any other algorithm that makes use of QFT, on quantum annealing devices. We discuss the possible advantages, as well as the limitations, of the proposed approach as well as its relation to traditional adiabatic quantum computation.Keywords: adiabatic quantum computing, quantum adiabatic algorithm, quantum fourier transform, integer factorization, Shor's algorithm

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Exponential complexity of the quantum adiabatic algorithm for certain satisfiability problems

Cited by 84 publications

References 45 publications

Solving the graph-isomorphism problem with a quantum annealer

Solving the graph-isomorphism problem with a quantum annealer

Exponential Enhancement of the Efficiency of Quantum Annealing by Non-Stoquastic Hamiltonians

Fourier-transforming with quantum annealers

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