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World Congress on Nature and Biologically Inspired Computing (NaBIC)

Discrete adiabatic quantum computation with quadric variation

Mohamed El-fiky, Satoshi Ono, Shigeru Nakayama

Published Dec 1, 2010• 0 citations
Publisher

Abstract

Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state with a slowly varying Hamiltonian to reach the required output state. This paper proposes a new variation method for the phase functions, quadric variation method. The experiments are carried out solving random instances of 3-SAT problems with three variation methods: linear, cubic, and quadric. The experiment's result has revealed that the overall search costs with the quadric variation are less than those for other methods with higher probability of finding the solutions on average.

DOI
10.1109/nabic.2010.5716272

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Quantum adiabatic algorithm is a method of solving computational problems by evolving the ground state with a slowly varying Hamiltonian to reach the required output state. This paper proposes a new variation method for the phase functions, quadric variation method. The experiments are carried out solving random instances of 3-SAT problems with three variation methods: linear, cubic, and quadric. The experiment's result has revealed that the overall search costs with the quadric variation are less than those for other methods with higher probability of finding the solutions on average.
The model of adiabatic quantum computation is a new paradigm for designing quantum algorithms. This model is based on the quantum adiabatic theorem, where a quantum computer evolves the quantum system slowly to switch gradually from an initial Hamiltonian with a ground state which is easy to construct, to a final Hamiltonian whose ground state encodes the solution of the problem to be solved. This dissertation presents an experimental study on the discrete quantum adiabatic computation in combinatorial search problems, we take k-Satisfiability problem as target problem to be examined. For k ? 3, k-SAT is NP-complete, i.e., among the most difficult NP problems in the worst case. Here we proposes a new monotonic variation method for the phase shift and mixing functions in the adiabatic quantum algorithm called the Quadric variation method, in order to speed up the algorithm search and decrease the overall resulting search cost. In addition, we present a better parameter configuration for the algorithm to be used with quadric variation as well as the previously proposed methods. The experiment are carried out in solving random instances of 3-SAT problems. The results indicate that the proposed method of using monotonic quadric variation in the phase functions of the adiabatic algorithm greatly improve the search behavior of the adiabatic algorithm and reduces the resulting search cost, also the results indicate that the better configuration of the algorithm parameters could greatly enhance the search behavior for the proposed method as well as the previously proposed methods.
The discrete formulation of adiabatic quantum computing is compared with other search methods, classical and quantum, for random satisfiability (SAT) problems. With the number of steps growing only as the cube of the number of variables, the adiabatic method gives solution probabilities close to 1 for problem sizes feasible to evaluate via simulation on current computers. However, for these sizes the minimum energy gaps of most instances are fairly large, so the good performance scaling seen for small problems may not reflect asymptotic behavior where costs are dominated by tiny gaps. Moreover, the resulting search costs are much higher than for other methods. Variants of the quantum algorithm that do not match the adiabatic limit give lower costs, on average, and slower growth than the conventional GSAT heuristic method.
In quantum adiabatic evolution algorithms, the quantum computer follows the ground state of a slowly varying Hamiltonian. The ground state of the initial Hamiltonian is easy to construct; the ground state of the final Hamiltonian encodes the solution of the computational problem. These algorithms have generally been studied in the case where the "straight line" path from initial to final Hamiltonian is taken. But there is no reason not to try paths involving terms that are not linear combinations of the initial and final Hamiltonians. We give several proposals for randomly generating new paths. Using one of these proposals, we convert an algorithmic failure into a success.
A quantum system will stay near its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. This quantum adiabatic behavior is the basis of a new class of algorithms for quantum computing. We tested one such algorithm by applying it to randomly generated hard instances of an NP-complete problem. For the small examples that we could simulate, the quantum adiabatic algorithm worked well, providing evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.
Quantum computation by adiabatic evolution, as described in quant-ph/0001106, will solve satisfiability problems if the running time is long enough. In certain special cases (that are classically easy) we know that the quantum algorithm requires a running time that grows as a polynomial in the number of bits. In this paper we present numerical results on randomly generated instances of an NP-complete problem and of a problem that can be solved classically in polynomial time. We simulate a quantum computer (of up to 16 qubits) by integrating the Schrodinger equation on a conventional computer. For both problems considered, for the set of instances studied, the required running time appears to grow slowly as a function of the number of bits.
Adiabatic quantum computation (AQC) is a universal model for quantum computation which seeks to transform the initial ground state of a quantum system into a final ground state encoding the answer to a computational problem. AQC initial Hamiltonians conventionally have a uniform superposition as ground state. We diverge from this practice by introducing a simple form of heuristics: the ability to start the quantum evolution with a state which is a guess to the solution of the problem. With this goal in mind, we explain the viability of this approach and the needed modifications to the conventional AQC (CAQC) algorithm. By performing a numerical study on hard-to-satisfy 6 and 7 bit random instances of the satisfiability problem (3-SAT), we show how this heuristic approach is possible and we identify that the performance of the particular algorithm proposed is largely determined by the Hamming distance of the chosen initial guess state with respect to the solution. Besides the possibility of introducing educated guesses as initial states, the new strategy allows for the possibility of restarting a failed adiabatic process from the measured excited state as opposed to restarting from the full superposition of states as in CAQC. The outcome of the measurement can be used as a more refined guess state to restart the adiabatic evolution. This concatenated restart process is another heuristic that the CAQC strategy cannot capture.