Abstract
We numerically investigate the performance of the short path optimization algorithm on a toy problem, with the potential chosen to depend only on the total Hamming weight to allow simulation of larger systems. We consider classes of potentials with multiple minima which cause the adiabatic algorithm to experience difficulties with small gaps. The numerical investigation allows us to consider a broader range of parameters than was studied in previous rigorous work on the short path algorithm, and to show that the algorithm can continue to lead to speedups for more general objective functions than those considered before. We find in many cases a polynomial speedup over Grover search. We present a heuristic analytic treatment of choices of these parameters and of scaling of phase transitions in this model.
Highlights
We numerically investigate the performance of the short path optimization algorithm on a toy problem, with the potential chosen to depend only on the total Hamming weight to allow simulation of larger systems
Consider a problem where one must optimize some function which depends on N variables, each chosen from {−1, +1}; we write this function as an operator HZ which is diagonal in the computational basis for N qubits
While the algorithm can in principle be applied to any combinatorial optimization problem, we expect that no speedup is possible for many potentials
Summary
We numerically investigate the performance of the short path optimization algorithm on a toy problem, with the potential chosen to depend only on the total Hamming weight to allow simulation of larger systems. One can prove that the probability that the measurement in the computational basis gives the ground state is significantly larger than 2−N. In such a case, amplitude amplification leads to a super-Grover speedup. The first paper on the adiabatic algorithm was a numerical study, and so a similar study is worthwhile for the short path algorithm. Since we do not yet have a working quantum computer capable of implementing many of these algorithms, simulation is the only tool to gain practical understanding
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