Efficient Simulation of Quantum Systems by Quantum Computers

We show that the time evolution of the wave function of a quantum mechanical many particle system can be implemented very efficiently on a quantum computer. The computational cost of such a simulation is comparable to the cost of a conventional simulation of the corresponding classical system. Ultimately the simulation of quantum field theory might be possible on large quantum computers. 1 Quantum Computers Quantum computers are still hypothetical devices, but it is hoped that eventually the technical problems involved in their realization can be overcome [1, 2, 3, 4]. Quantum computers could solve some problems much faster than conventional computers. Most prominently, Peter Shor (1994) has given a ”quantum algorithm” for factoring large integers in polynomial time [5](see also [2]). In a theoretical sense quantum computers can be thought of as finite dimensional isolated quantum systems which undergo a number of unitary transformations. In particular a digital l-bit quantum computer ”lives” in a tensor product space of l two dimensional Hilbert spaces. As a basis we may take the 2 states where each bit is either a 0 or a 1 and which thus correspond to numbers. We assume that this is the ”classical” basis to which the wave function collapses upon observation at the end of the computation. Gates may be thought of as unitary transformations acting only on a few bits at a time (”local” transformations). 2 Simulating Quantum Systems General ideas about using specially designed quantum systems to simulate other quantum systems have been published, e.g. by Feynman [6]. We present here an actual implementation of the simulation of quantum mechanical many particle