Arbitrarily fast quantum computation with bounded energy
Stephen P. Jordan
One version of the energy-time uncertainty principle states that the minimum time for a quantum system to evolve from a given state to any orthogonal state is h/(4 Δ E) where Δ E is the energy uncertainty. Many subsequent works have interpreted this as defining a minimal time for an elementary computational operation and correspondingly a fundamental limit on clock speed determined by a system's energy. Here we present local time-independent Hamiltonians in which a constant computational clock speed can be maintained with Δ E that shrinks asymptotically to zero in the limit as the number of computational steps goes to infinity. Alternatively, by scaling the norm of the Hamiltonian differently, one obtains arbitrarily high clock speed in a system in which the range of energies occupied is of constant size. We argue that energy considerations alone are not sufficient to obtain an upper bound on computational speed, and that additional physical assumptions such as limits to information density and information transmission speed are necessary to obtain such a bound.