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Study on discrete quantum adiabatic computation in combinatorial search problems

Mohamed El-fiky

Published Jan 1, 2012• 0 citations
OpenAlex

Abstract

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.

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Comments on Adiabatic Quantum Algorithms

Mary Beth Ruskai

Recently a method for adiabatic quantum computation has been proposed and there has been considerable speculation about its efficiency for NP-complete problems. Heuristic arguments in its favor are based on the unproven assumption of an eigenvalue gap. We show that, even without the assumption of an eigenvalue gap, other standard arguments can be used to show that a large class of Hamiltonians proposed for adiabatic quantum computation have unique ground states. We also discuss some of the issues which arise in trying to analyze the behavior of the eigenvalue gap. In particular, we propose several mechanisms for modifying to final Hamiltonian to perform an adiabatic search with efficiency comparable to that for 3-SAT. We also propose the use of randomly defined final Hamiltonians as a mechanism for analyzing the generic spectral behavior of the interpolating Hamiltonians associated with problems which lack sufficent structure to be amenable to efficient classical algorithms.

arXiv:quant-ph/0203127•Mar 26, 2002•0 citations
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.
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 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.
We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are unable to estimate this gap in general. We give some special symmetric cases of the satisfiability problem where the symmetry allows us to estimate the gap and we show that, in these cases, our algorithm runs in polynomial time.
We give a quantum algorithm for solving instances of the satisfiability problem, based on adiabatic evolution. The evolution of the quantum state is governed by a time-dependent Hamiltonian that interpolates between an initial Hamiltonian, whose ground state is easy to construct, and a final Hamiltonian, whose ground state encodes the satisfying assignment. To ensure that the system evolves to the desired final ground state, the evolution time must be big enough. The time required depends on the minimum energy difference between the two lowest states of the interpolating Hamiltonian. We are unable to estimate this gap in general. We give some special symmetric cases of the satisfiability problem where the symmetry allows us to estimate the gap and we show that, in these cases, our algorithm runs in polynomial time.
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.
Recently a method for adiabatic quantum computation has been proposed and there has been considerable speculation about its efficiency for NP-complete problems. Heuristic arguments in its favor are based on the unproven assumption of an eigenvalue gap. We show that, even without the assumption of an eigenvalue gap, other standard arguments can be used to show that a large class of Hamiltonians proposed for adiabatic quantum computation have unique ground states. We also discuss some of the issues which arise in trying to analyze the behavior of the eigenvalue gap. In particular, we propose several mechanisms for modifying to final Hamiltonian to perform an adiabatic search with efficiency comparable to that for 3-SAT. We also propose the use of randomly defined final Hamiltonians as a mechanism for analyzing the generic spectral behavior of the interpolating Hamiltonians associated with problems which lack sufficent structure to be amenable to efficient classical algorithms.