Quantum Algorithms for Modeling and Simulation: A Grand Challenge for Modeling and Simulation

The end of the wild computational sleigh ride of Moore's Law is the basis for this grand challenge. The doubling of computing performance every 18 months predicted by Moore’s law nearly 50 years ago will soon end. The number of active atoms in a computational device reduce to “one” sometime between 2010 and 2020. Barring some unforeseen invention, this is the end of the road for the current approach. But the problem suggests the solution; atoms are inherently quantum by nature and use of quantum effects in computation provides another route to increased computational power. Quantum computers require very different algorithms and the grand challenge for modeling and simulation is to develop simulation algorithms adapted to the coming era of quantum computing. In a few years of research, quantum algorithms for searching and factoring large numbers have already been found. Applying these ideas to simulation and developing quantum algorithms specifically for simulation is the challenge. If quantum algorithms are not developed for modeling and simulation, modeling and simulation will never be able to make use of the exponential power provided by the quantum computer, and the end of Moore's Law will determine the maximum speed of modeling and simulation computation. Many modeling and simulation problems ranging from terrain correlation and intervisibility, to solving nonlinear differential equations, could certainly make use of effective quantum algorithms, if they can be developed. INTRODUCTION Simulation should pay close attention to developments in quantum computation. Where simulation has ridden the exponential growth in computer power described by Moore’s Law [Clarke, 1995], quantum computation researchers believe a truly quantum computer will provide an unprecedented leap forward in ability to compute [Bennet, 1995]. This “quantum leap” in computational ability may be necessary before one can hope to implement truly intelligent “computer generated forces”. But in any case, it will provide greater capability to simulate processes of all varieties [Lloyd, 1995]. There are also interesting connections between quantum computation and other logic and computational paradigms [Clarke, 1998]. There are many possible approaches to the actual implementation of quantum computers. However, the only approach to date that has actually resulted in a quantum mechanical computation, is quantum computation based on nuclear magnetic resonance technology (as in MRI scanners). The Stanford-Berkeley-MIT-IBM Quantum Computation Research Project and others are actively pursuing the possibility of using the precession of atomic nuclei in magnetic fields [Gershenfeld and Chuang, 1998]. The precession and rotation of nuclei can be described by quantum mechanics, so with the proper manipulation and measurement procedures, quantum computation could be achieved. In brief, quantum computation operates by utilizing qbits (quantum bit) instead of bits. A qbit is the state of a quantum entity like a nuclear spin that generally can be regarded as neither true or false, but as a combination, a superposition, of both. According to quantum mechanics, since 1925 it has been generally accepted that a quantum system is in a superposition of states, and only becomes fixed in any one state when it is measured. Properly utilizing this strange but true fact of nature would enable the quantum computer to simultaneously execute both branches of a conditional, and thereby potentially enormously expanding the power of a quantum computer. SURVEY OF QUANTUM SIMULATION In this section some recent articles pertaining to the use of quantum computation in simulation will be discussed along. Game Theory Training simulation uses computer generated force (CGF) software to provide challenging exercises to the trainee. It is essential that CGF behavior be realistic but not predictable. Game theory can provide a rational basis for CGF, but avoiding predictability can be a problem. Some of the most elegant applications of quantum computing have been to game theory, and a quantum game may provide just the unpredictably rational behavior needed for a CGF. Quantum games are small problems and have already been implemented on available quantum computing hardware so a quantum CGF may be a relatively near term possibility. The first paper on quantum games the author is aware of is "Quantum Strategies" by David Myer [1999]. In this article the redoubtable Captain Picard of the Enterprise is pitted against his nemesis Q in the traditional game of matching pennies. As might be expected Picard is limited to a classical strategy whereas Q can employ a quantum strategy. Perhaps not surprisingly, a quantum strategy can beat even an optimal classical strategy. Myer [2000] has since gone on to show how the concept of quantum game against a classical player can be viewed as a quantum algorithm for an oracle. He formalizes this correspondence and gives examples of games (and hence oracle problems) for which the quantum player can do better than would be possible classically. Since the introduction of the ideas of quantum games and quantum strategies the ideas have been taken in several directions. A recent comprehensive article is that by Eisert and Wilkens [2000] who study quantum games with classical analogs in order to highlight the peculiarities of quantum games, giving special emphasis to a detailed investigation of different sets of quantum strategies. Piotrowski and Sladkowski [2001] apply quantum-like descriptions to the analysis of markets and economics. Their paper focuses on quantum bargaining games, which are a special class of quantum market games without institutionalized clearinghouses. Dynamical Computation Feynman's early work on quantum computation was motivated by the difficulty of solving the dynamical equations of quantum mechanics. Simulating a quantum system on a classical computer is exponentially hard, but relatively easy (and in fact, natural) on a quantum computer. There is thus a natural match between quantum computation and quantum dynamical computation. Several recent papers have explored these applications and the authors believe that quantum computers will prove useful for solving propagation problems by making use of the correspondence between the parabolic wave equation and the Schrodinger equation. The utility of inverse scattering transforms for soliton problems, also suggests that quantum computers will be extremely useful for soliton and nonlinear wave problems. In "Efficient Quantum Computing of Complex Dynamics" Benenti et al [2001] propose a quantum algorithm which optimally uses qubits to efficiently simulate a physical model described by the quantum sawtooth map. This model has rich and complex dynamics, which is accurately reproduced up to a time scale which is polynomial in the number of qubits. However, the authors find that the errors generated by static imperfections in the quantum computer hardware to be more dangerous than the errors of random noise in gate operations.