One of the most important questions in studying quantum computation is: whether a quantum computer can solve NP-complete problems more efficiently than a classical computer? In 2000, Farhi, et al. (Science, 292(5516):472--476, 2001) proposed the adiabatic quantum optimization (AQO), a paradigm that directly attacks NP-hard optimization problems. How powerful is AQO? Early on, van Dam and Vazirani claimed that AQO failed (i.e. would take exponential time) for a family of 3SAT instances they constructed. More recently, Altshuler, et al. (Proc Natl Acad Sci USA, 107(28): 12446--12450, 2010) claimed that AQO failed also for random instances of the NP-complete Exact Cover problem. In this paper, we make clear that all these negative results are only for a specific AQO algorithm. We do so by demonstrating different AQO algorithms for the same problem for which their arguments no longer hold. Whether AQO fails or succeeds for solving the NP-complete problems (either the worst case or the average case) requires further investigation. Our AQO algorithms for Exact Cover and 3SAT are based on the polynomial reductions to the NP-complete Maximum-weight Independent Set (MIS) problem.
A quantum computer promises extraordinary power over a classical computer, as demonstrated by Shor [1] in 1994 with the polynomial quantum algorithm for solving the factoring problem, for which the best known classical algorithms are exponential. Just how much more powerful are quantum computers? In particular, can a quantum computer solve NP-complete problems [2] more efficiently than a classical computer? NP-complete problems are the "hardest" problems in NP -in the sense that if one can solve one NP-complete problem efficiently (i.e., in polynomial time) then one can solve all the problems in NP in polynomial time. In 2000, Farhi et al. [3,4] proposed an adiabatic quantum algorithm as an alternative quantum paradigm to directly solve NP-hard optimization problems, which are polynomially equivalent to their corresponding NP-complete decision problems. Apparently, the same idea to the adiabatic quantum optimization, under a different name of quantum annealing, was first put forward by Apolloni et al. in 1988, see [5,6] and references therein for a history of the field.
An adiabatic quantum algorithm is described by a time-dependent system Hamiltonian H(t) = (1 -s(t))H init + s(t)H problem (1) for t ∈ [0, T ], s(0) = 0, s(T ) = 1. There are three components of H(.): (1) initial Hamiltonian: H(0) = H init ;
(2) problem Hamiltonian: H(T ) = H problem ; and (3) evolution path: s : [0, T ] -→ [0, 1], e.g., s(t) = t T . H(t) is an adiabatic algorithm for a problem if we encode the problem into the problem Hamiltonian H problem such that the ground state of H problem corresponds to the answer to the problem. The initial Hamiltonian H init is chosen to be non-commutative with H problem and its ground state must be known and experimentally constructable, e.g.,
Here T is the running time of the algorithm. According to the adiabatic theorem, if H(t) evolves “slowly” enough, or equivalently, if T is “large” enough, which scales polynomially with the inverse of the minimum spectral gap g min (the difference between the two lowest energy levels) of the system Hamiltonian, the system remains in the ground state of H(t), and consequently, ground state of H(T ) = H problem gives the solution to the problem. This computational model is referred as the Adiabatic Quantum Computation (AQC). It has been shown [7,8] that AQC is polynomially equivalent to conventional quantum computation (quantum circuit model). For the optimization problem, the problem Hamiltonian can be expressed as a diagonal matrix in the computational basis. That is, let f problem : {0, 1} n -→ R be a cost function of the optimization problem such that the minimum of the f problem corresponds to the solution of the optimization problem, then the corresponding problem Hamiltonian H problem is the Hamiltonian with f problem as the energy function: namely, H problem = x∈{0,1} n f problem (x)|x x| (which needs to be expressible in polynomial resources, such as Eq. ( 4) ). Hereafter we use the cost function and the energy function of the problem Hamiltonian interchangeably. It is worthwhile to emphasize here that given a problem, there can be many possible cost functions and thus many possible problem Hamiltonians for the same problem. This restricted model is referred as the Adiabatic Quantum Optimization (AQO) (which is no longer polynomially equivalent to the quantum circuit model). We remark that this distinction between AQC and AQO was not made by Altshuler et al. [9] (They referred both as AQO). In this paper, the focus is on this restricted model.
The NP-complete problems that were initially proposed for AQO by Farhi et al. [3,4] were 3SAT and a special case of 3SAT -Exact Cover 3 (EC3). As an example, they proposed a clause-violation cost function as the energy function of the problem Hamiltonian. Namely, f problem (x) = number of clauses violated by the assignment x. This cost function (with perhaps constant difference) has been adopted by almost all the other adiabatic quantum computation works. In particular, van Dam and Vazirani [10] claimed that AQO failed to solve a family of 3SAT instances by showing that the AQO algorithm with this specific clause-violation cost function had the exponentially small minimum spectral gap. See Discussion for more discussion on some other similar claims( [11,12]). Recently, Altshuler et al. [9] claimed that AQO failed for the average case of NP-complete problems by arguing this specific clause-violation cost function based AQO algorithm failed for random instances of EC3, because of the Anderson localization phenomenon. While the cost function proposed is a “natural” one, nevertheless, it is not the only possible one. Recall that, according to the formulation of an AQO algorithm, the requirement of the problem Hamiltonian is that the ground state corresponds to the solution. There can be other cost functions with the same minimum (solution), e.g., see the reduction below. That is, there are other problem Hamiltoni
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