Genetic programming for quantum computers

TL;DR: This paper exhibits the first evolved betterthan-classical quantum algorithm, for Deutsch’s “early promise” problem, and demonstrates a technique for evolving scalable quantum gate arrays and discusses other issues in the application of genetic programming to quantum computation and vice versa.

Abstract: Genetic programming can be used to automatically discover algorithms for quantum computers that are more efficient than any classical computer algorithms for the same problems. In this paper we exhibit the first evolved betterthan-classical quantum algorithm, for Deutsch’s “early promise” problem. We also demonstrate a technique for evolving scalable quantum gate arrays and discuss other issues in the application of genetic programming to quantum computation and vice versa. 1. Quantum Computing Quantum computers are computational devices that use atomic-scale objects, for example 2-state particles, to store and manipulate information (Steane, 1997; for an elementary on-line tutorial see Braunstein, 1995; for an introduction for the general reader see Milburn, 1997). The physics of these devices allows them to do things that common digital (henceforth “classical”) computers cannot. Although quantum computers and classical computers appear to be bound by the same limits of Turing computability, physicists argue that quantum computers can solve certain problems using less resources (time and space) than classical computers are thought to require (Jozsa, 1997). For example, Shor’s quantum algorithm finds the prime factors of an n-digit number in time O(n), while the best known classical factoring algorithms require at least time O(2 1 3 2 3 n n / / log( ) ) (Shor, 1994; Beckman et al. 1996). And Grover’s quantum database search algorithm can find an item in an unsorted list of n items in O( n ) steps, while classical algorithms clearly require O(n) (Grover, 1997). The full power of quantum computation is a subject of active investigation. The smallest unit of quantum information is the qubit, which is analogous to the classical bit. Whereas a classical system of n bits is at any time in one of 2 states, a quantum system of n qubits can be in any linear superposition of Genetic Programming for Quantum Computers Lee Spector Howard Barnum Herbert J. Bernstein lspector@hampshire.edu hbarnum@hampshire.edu hbernstein@hampshire.edu School of Cognitive Science School of Natural Science Hampshire College and Institute for Science and Interdisciplinary Studies (ISIS) Amherst, MA 01002 Hampshire College Amherst, MA 01002 these 2 states simultaneously. Although we cannot read the entire state (because measurement interferes with the system), it appears that this quantum parallelism can nonetheless be harnessed to perform real computational work. In physical terms, a qubit can be thought of as a twocomponent wave, where each component represents a classical value, 0 or 1. The height of a component wave gives the probability that the qubit will be found in a particular classical state, and the phase controls how the wave will interfere with other waves. As usual, a wave with height and phase can be represented by a single complex number. Unlike a classical bit, a qubit can be in both states at the same time, and these states may be in phase, out of phase, or somewhere in between, leading to constructive or destructive interference. To date, only very small quantum computers have been built, but the planning and construction of larger devices is in progress. Current experimental quantum computing hardware is based on the use of ion traps, cavity QED, and NMR techniques; many difficult problems must be solved before these techniques can be scaled up, but a discussion of these issues is beyond the scope of this paper (see Preskill, 1997). Quantum computers are different from classical computers in several ways, and it is not obvious how, in general, software can be developed to take advantage of their non-classical power. A variety of basic questions about this power are still open, for example whether or not there exist polynomial time quantum algorithms for classically NP complete problems. In order to assess the wisdom of expending resources on the physical realization of quantum computers, it is important that we develop a more solid understanding of their real computational powers. One way to further this understanding is to develop more quantum algorithms, either by hand or by use of automatic programming techniques (as below). And there are additional reasons, aside from guidance for research and development efforts, for wanting to determine the real powers of quantum computation. For example, Penrose has argued that quantum effects play an important role in human brains (Penrose, 1994), although his claims have been disputed. Because complexity-theoretic arguments play a major role in cognitive science, a rethinking of the computational complexity limits of brain processes could have a significant impact on the study of human cognition. Spector, L., H. Barnum, and H.J. Bernstein. 1998. Genetic Programming for Quantum Computers. In Genetic Programming 1998: Proceedings of the Third Annual Conference, edited by J.R. Koza, W. Banzhaf, K. Chellapilla, K. Deb, M. Dorigo, D.B. Fogel, M.H. Garzon, D.E. Goldberg, H. Iba, and R.L. Riolo. pp. 365-374. San Francisco, CA: Morgan Kaufmann.