a Computational Role for the One-Dimensional Lieb Shultz-Mattis Spin Model

Regarding individual quantum-mechanical degrees of freedom as elements of a parallel computation, a simple but fully interacting quantum many-body system is analyzed in order to derive a fundamental quantum-mechanical limit to massively parallel computation. Exact solutions for all eigenvalues and eigenvectors of an operator Gamma that the characterizes parallel computational velocity are found. A symmetry between the Hamiltonian and Gamma is described, and its consequences investigated. The largest eigen-value of the computational velocity operator is found to scale with system size, N, as gamma _{rm max} ~ 2N/pi < N, a result which implies the impossibility of using additional quantum-mechanical parallel processors to obtain the same ideal speedup as in the classical case. A variational argument shows that this is a special case of a more fundamental limit to parallel quantum computation. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253 -1690.).