Finding the optimal solution to a complex optimization problem is of great
importance in practically all fields of science, technology, technical design
and econometrics. We demonstrate that a modified Grover's quantum algorithm can
be applied to real problems of finding a global minimum using modest numbers of
quantum bits. Calculations of the global minimum of simple test functions and
Lennard-Jones clusters have been carried out on a quantum computer simulator
using a modified Grover's algorithm. The number of function evaluations $N$
reduced from O(N) in classical simulation to $O(\sqrt{N})$ in quantum
simulation. We also show how the Grover's quantum algorithm can be combined
with the classical Pivot method for global optimization to treat larger
systems.Comment: 6 figures. Molecular Physics, in pres

Das R

Garey M

Goldberg DE

Hirvensalo M

Jing Zhu

Nelson RJ

Nielsen MA

Sabre Kais

Shor PW

Zhen Huang

English

arXiv

Can quantum computers solve optimization problems much more quickly than classical computers? One major piece of evidence for this proposition has been the fact that Quantum Annealing (QA) finds the minimum of some cost functions exponentially more quickly than classical Simulated Annealing (SA). One such cost function is the simple “Hamming weight with a spike” function in which the input is an n-bit string and the objective function is simply the Hamming weight, plus a tall thin barrier centered around Hamming weight n/4. While the global minimum of this cost function can be found by inspection, it is also a plausible toy model of the sort of local minima that arise in realworld optimization problems. It was shown by Farhi, Goldstone and Gutmann [1] that for this example SA takes exponential time and QA takes polynomial time, and the same result was generalized by Reichardt [2] to include barriers with width n^ζ and height n^α for ζ + α ≤ 1/2. This advantage could be explained in terms of quantummechanical “tunneling.” Our work considers a classical algorithm known as Simulated Quantum Annealing (SQA) which relates certain quantum systems to classical Markov chains. By proving that these chains mix rapidly, we show that SQA runs in polynomial time on the Hamming weight with spike problem in much of the parameter regime where QA achieves an exponential advantage over SA. While our analysis only covers this toy model, it can be seen as evidence against the prospect of exponential quantum speedup using tunneling. Our technical contributions include extending the canonical path method for analyzing Markov chains to cover the case when not all vertices can be connected by low-congestion paths. We also develop methods for taking advantage of warm starts and for relating the quantum state in QA to the probability distribution in SQA. These techniques may be of use in future studies of SQA or of rapidly mixing Markov chains in general

Crosson, Elizabeth

Harrow, Aram W.

Caltech Authors - Main

Simulated Quantum Annealing Can Be Exponentially Faster than Classical Simulated Annealing

Finding the optimal solution to a complex optimisation problem is of great importance in practically all fields of science, technology, technical design and econometrics. We demonstrate that a modified Grover\u27s quantum algorithm can be applied to real problems of finding a global minimum using modest numbers of quantum bits. Calculations of the global minimum of simple test functions and Lennard-Jones clusters have been carried out on a quantum computer simulator using a modified Grover\u27s algorithm. The number of function evaluations N reduced from O(N) in classical simulation to O(N1/2) in quantum simulation. We also show how the Grover\u27s quantum algorithm can be combined with the classical Pivot method for global optimisation to treat larger systems

Zhu, Jing

Huang, Zhen

Kais, Sabre

Purdue E-Pubs