2 research outputs found
On the Interpretation of Energy as the Rate of Quantum Computation
Over the last few decades, developments in the physical limits of computing
and quantum computing have increasingly taught us that it can be helpful to
think about physics itself in computational terms. For example, work over the
last decade has shown that the energy of a quantum system limits the rate at
which it can perform significant computational operations, and suggests that we
might validly interpret energy as in fact being the speed at which a physical
system is "computing," in some appropriate sense of the word. In this paper, we
explore the precise nature of this connection. Elementary results in quantum
theory show that the Hamiltonian energy of any quantum system corresponds
exactly to the angular velocity of state-vector rotation (defined in a certain
natural way) in Hilbert space, and also to the rate at which the state-vector's
components (in any basis) sweep out area in the complex plane. The total angle
traversed (or area swept out) corresponds to the action of the Hamiltonian
operator along the trajectory, and we can also consider it to be a measure of
the "amount of computational effort exerted" by the system, or effort for
short. For any specific quantum or classical computational operation, we can
(at least in principle) calculate its difficulty, defined as the minimum effort
required to perform that operation on a worst-case input state, and this in
turn determines the minimum time required for quantum systems to carry out that
operation on worst-case input states of a given energy. As examples, we
calculate the difficulty of some basic 1-bit and n-bit quantum and classical
operations in an simple unconstrained scenario.Comment: Revised to address reviewer comments. Corrects an error relating to
time-ordering, adds some additional references and discussion, shortened in a
few places. Figures now incorporated into tex