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arXiv:0911.3438 · Physical Review Letters

First-Order Transitions and the Performance of Quantum Algorithms in Random Optimization Problems

Thomas Jörg, Florent Krząkała, Guilhem Semerjian, Francesco Zamponi

Published May 20, 2010• 100 citations
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Abstract

We present a study of the phase diagram of a random optimization problem in the presence of quantum fluctuations. Our main result is the characterization of the nature of the phase transition, which we find to be a first-order quantum phase transition. We provide evidence that the gap vanishes exponentially with the system size at the transition. This indicates that the quantum adiabatic algorithm requires a time growing exponentially with system size to find the ground state of this problem.

DOI
10.1103/physrevlett.104.207206

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We present a study of the phase diagram of a random optimization problem in the presence of quantum fluctuations. Our main result is the characterization of the nature of the phase transition, which we find to be a first-order quantum phase transition. We provide evidence that the gap vanishes exponentially with the system size at the transition. This indicates that the quantum adiabatic algorithm requires a time growing exponentially with system size to find the ground state of this problem.
Using the quantum cavity method, we present an analytical study of the phase diagram of a random optimization problem in a transverse field, in the thermodynamic limit and for a uniformly random distribution of instances. We include the effect of replica symmetry breaking, and unveil the occurrence of a first-order quantum phase transition. We back up our results with Quantum Monte Carlo and Exact Diagonalization methods, and provide evidence that the gap vanishes exponentially with the system size at the transition. This suggests that the Quantum Adiabatic Algorithm requires an exponential time to find the ground state of such problems.
Motivated by the quantum adiabatic algorithm (QAA), we consider the scaling of the Hamiltonian gap at quantum first-order transitions, generally expected to be exponentially small in the size of the system. However, we show that a quantum antiferromagnetic Ising chain in a staggered field can exhibit a first-order transition with only an algebraically small gap. In addition, we construct a simple classical translationally invariant one-dimensional Hamiltonian containing nearest-neighbor interactions only, which exhibits an exponential gap at a thermodynamic quantum first-order transition of essentially topological origin. This establishes that (i) the QAA can be successful even across first-order transitions but also that (ii) it can fail on exceedingly simple problems readily solved by inspection, or by classical annealing.
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In this paper we show that the performance of the quantum adiabatic algorithm is determined by phase transitions in underlying problem in the presence of transverse magnetic field ΓΓΓ. We show that the quantum version of random Satisfiability problem with 3 bits in a clause (3-SAT) has a first-order quantum phase transition. We analyze the phase diagram γ=γ(Γ)γ=γ(Γ)γ=γ(Γ) where γγγ is an average number of clauses per binary variable in 3-SAT. The results are obtained in a closed form assuming replica symmetry and neglecting time correlations at small values of the transverse field ΓΓΓ. In the limit of Γ=0Γ=0Γ=0 the value of γ(0)≈γ(0)\approxγ(0)≈ 5.18 corresponds to that given by the replica symmetric treatment of a classical random 3-SAT problem. We demonstrate the qualitative similarity between classical and quantum versions of this problem.
In Amin and Choi \cite{AC09}, we show that an adiabatic quantum algorithm for the NP-hard maximum independent set (MIS) problem on a set of special family of graphs in which there are exponentially many local maxima would have the exponentially small minimum spectral gap and thus would require the exponential time, due to the first order quantum phase transition (FQPT). The problem Hamiltonian of the adiabatic quantum algorithm for MIS is based on the reduction to the Ising problem and has flexible parameters. In this paper, we show numerically on the 15-vertex graph that by choosing the parameters appropriately in the problem Hamiltonian (without changing the problem to be solved) for MIS, we can prevent the FQPT and drastically increase the minimum spectral gap. The result is further supported by visualization from the Decomposed State Evolution Visualization (\desev) --- a visualization tool we introduced. Furthermore, our result also serves to concretely clarify that it is not sufficient to consider one specific problem Hamiltonian for proving the failure of adiabatic quantum optimization for a problem. We also raise the basic question about what the appropriate formulation of adiabatic running time should be.
We study both analytically and numerically the complexity of the adiabatic quantum evolution algorithm applied to random instances of combinatorial optimization problems. We use as an example the NP-complete set partition problem and obtain an asymptotic expression for the minimal gap separating the ground and exited states of a system during the execution of the algorithm. We show that for computationally hard problem instances the size of the minimal gap scales exponentially with the problem size. This result is in qualitative agreement with the direct numerical simulation of the algorithm for small instances of the set partition problem. We describe the statistical properties of the optimization problem that are responsible for the exponential behavior of the algorithm.