The problem Hamiltonian of the adiabatic quantum algorithm for the maximum-weight independent set problem (MIS) that is based on the reduction to the Ising problem (as described in [Choi08]) has flexible parameters. We show that by choosing the parameters appropriately in the problem Hamiltonian (without changing the problem to be solved) for MIS on CK graphs, we can prevent the first order quantum phase transition and significantly change the minimum spectral gap. We raise the basic question about what the appropriate formulation of adiabatic running time should be. We also describe adiabatic quantum algorithms for Exact Cover and 3SAT in which the problem Hamiltonians are based on the reduction to MIS. We point out that the argument in Altshuler et al.(arXiv:0908.2782 [quant-ph]) that their adiabatic quantum algorithm failed with high probability for randomly generated instances of Exact Cover does not carry over to this new algorithm.
Adiabatic quantum computation (AQC) was proposed by Farhi et al. [12] in 2000 as an alternative quantum paradigm to solve NP-hard optimization problems, which are believed to be classically intractable. Later, it was shown by Aharonov et al. [1] that AQC is not just limited to optimization problems, and is polynomially equivalent to conventional quantum computation (quantum circuit model). A quantum computer promises extraordinary power over a classical computer, as demonstrated by Shor [22] 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, we are interested in whether an adiabatic quantum computer can solve NP-complete problems more efficiently than a classical computer.
Unlike classical computation or conventional quantum model in which an algorithm is specified by a finite sequence of discrete operations via classical/quantum gates, the adiabatic quantum algorithm is continuous. It has been assumed (see Section 2 for more discussion) that, according to the adiabatic theorem, the dominant factor of the adiabatic running time (ART) of the algorithm scales polynomially with the inverse of the minimum spectral gap g min of the system Hamiltonian (that describes the algorithm). Therefore, in order to analyze the running time of an adiabatic algorithm, it is necessary to be able to bound g min analytically. However, g min is in general difficult to compute (it is as hard as solving the original problem if computed directly). Rigorous analytical analysis of adiabatic algorithms remains challenging. Most of studies have to resort to numerical calculations. These include 1 arXiv:1004.2226v1 [quant-ph] 13 Apr 2010 numerical integration of Schrödinger equation [12,5], eigenvalue computation (or exact diagonization) [28,21], and quantum Monte Carlo (QMC) technique [25,26]. However, not only are these methods limited to small sizes (as the simulations of quantum systems grow exponentially with the system size), but also little insight can be gained from these numbers to design and analyze the time complexity of the algorithm.
Perhaps, from the algorithmic design point of view, it is more important to unveil the quantum evolution black-box and thus enable us to obtain insight for designing efficient adiabatic quantum algorithms. For this purpose, we devise a visualization tool, called Decomposed State Evolution Visualization (DESEV). Through the aid of this tool, we constructed a family of instances of MIS, called CK graphs [7]. The numerical results of an adiabatic algorithm for MIS on these graphs suggested that g min is exponentially small and thus the algorithm requires exponential time. These results were then explained by the first order quantum phase transition (FQPT) in [4]. Since then, there have been some other papers (Altshuler et al., [2] ; Farhi et al., [14]; Young et al., [26]; Jorg et al., [18,19]) investigating the same phenomenon, i.e., first order quantum phase transition. In particular, Farhi et al. in [14] suggested that the exponential small gap caused by the FQPT could be overcome (for the set of instances they consider) by randomizing the choice of initial Hamiltonian. In this paper, we show that by changing the parameters in the problem Hamiltonian (without changing the problem to be solved) of the adiabatic algorithm for MIS on CK graphs, we prevent the FQPT from occurring and significantly increase g min . We do so by scaling the vertex-weight of the graph, namely, multiplying the weights of vertices by a scaling factor. In order to determine the best scaling factor, we raise the basic question about what the appropriate formulation of adiabatic running time should be.
We also describe adiabatic quantum algorithms for Exact Cover and 3SAT in which the problem Hamiltonians are based on the reduction to MIS. In [2], Altshuler et al. claimed that a particular adiabatic quantum algorithm failed with high probability for randomly generated instances of Exact Cover. They claimed that the correctness of their argument did not rely on the specific form of the problem Hamiltonian for Exact Cover. We demonstrate an adiabatic algorithm for Exact Cover in which the problem Hamiltonian is based on the reduction to MIS that questions the generality of this claim.
This paper is organized as follows. In Section 2, we review the adiabatic quantum algorithm, and adiabatic running time. In Section 3, we recall the adiabatic quantum algorithm for MIS based on the reduction to the Ising problem. In Section 4, we describe the visualization tool DESEV and the CK graphs. We show examples of DESEV on the MIS adiabatic algorithm for CK graphs. In Section 5, we describe how changing the parameters affects g min , and raise the question about ART. In Section 6, we describe adiabatic algorithms for Exact Cover and 3SAT that are based on MIS reduction. We concl
This content is AI-processed based on open access ArXiv data.
