Does Adiabatic Quantum Optimization Fail for NP-Complete Problems?

Abstract

It has been recently argued that adiabatic quantum optimization would fail in solving NP-complete problems because of the occurrence of exponentially small gaps due to crossing of local minima of the final Hamiltonian with its global minimum near the end of the adiabatic evolution. Using perturbation expansion, we analytically show that for the NP-hard problem of maximum independent set there always exist adiabatic paths along which no such crossings occur. Therefore, in order to prove that adiabatic quantum optimization fails for any NP-complete problem, one must prove that it is impossible to find any such path in polynomial time.Adiabatic quantum optimization (AQO) was originally proposed [1] as a possible means for solving NP-complete problems faster than classical computation. In AQO, the Hamiltonian of the system is evolved from an initial form, H B , whose ground state defines the initial state of the system, to a final Hamiltonian H P , whose ground state is the optimal solution to an optimization problem. To ensure a large amplitude of the ground state at the end of the evolution, the computation time t f should increase as t f ∼ g −2 min , where g min is the minimum energy gap between the ground and first excited states during the evolution. The complexity of AQO is therefore determined by the scaling of g min with the problem size.Since its proposal, the complexity of AQO for solving NP-complete problems has been a subject of controversy. Early works suggested possibility of polynomial scaling with the size of the problem [1], but soon counterexamples were found [2,3]. Later it was shown that the size of the gap in those counterexamples can be increased significantly by changing the adiabatic path [4] or the initial Hamiltonian [5]. However, until recently, little insight existed on the physical process that can lead to exponentially small gaps.Using perturbation expansion, it was shown that a local minimum of H P crossing with the global minimum near the end of the evolution, sometimes called a first order quantum phase transition, can result in an extremely small g min , exponential in the Hamming distance between the two minima [6]. Using the same perturbation argument, Altshuler et al. [7] showed that for random exact cover instances, the probability of having such crossings increases with the system size and the crossing point moves toward the end of the evolution leading to an exponentially small gap. They, therefore, concluded that AQO would fail in solving random exact cover problems and possibly all NP-complete problems. Others also came to similar conclusions in different ways [8,9]. Later, Knysh et al. [10] questioned the result of [7] based on neglecting degeneracies of the minima and correlations between them. Moreover, the possibility of avoiding small gaps by changing adiabatic path was again pointed out by Farhi et al. [11], and the fact that one problem can be mapped into many different Hamiltonians with different gap behavior was mentioned by Choi [12]. Those arguments, however, wer...

Thermally assisted quantum annealing of a 16-qubit problem

“…2a and further described in the Methods section and Supplementary Note 1. The same type of anticrossing has been argued 28,29 to render QA ineffective because of the extremely small g min , though methods have been proposed to eliminate such anticrossings [30][31][32] .…”

An integrated programming and development environment for adiabatic quantum optimization

“…More generally, it has proven difficult to predict the run times of particular problem instances due to the complexity of the underlying quantum dynamics. An essential step in understanding these behaviors is to capture the influence that different programming choices have on observed run times [7,30,29,31,32,33,34,35,36]. A significant source of the complexity in analyzing implementations of the AQO algorithm arises from the multiple steps undertaken to synthesize the adiabatic quantum program.…”

Energy spectrum and exact cover in an extended quantum Ising model

“…The relation between adiabatic quantum evolution and quantum phase transitions is an ongoing topic of research [11,12]. Also, recently the statistics and scaling of energy gaps between the ground state and excited states-which form the limiting factor for the efficiency of adiabatic quantum computation-as well as the role played by the choice of H final have been investigated [13][14][15].…”