Interpreting Energy as the Rate of Quantum Computation

Over the last few decades, progress in fields such as the physical limits of computing and quantum computing has increasingly taught us that it can be helpful to think about physics itself in computational terms. For example, recent work 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 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 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, and we can also consider it to be a measure of “computational work,” which we will dub the effort. For any specific classical or quantum computational operation, we can (in principle at least) characterize the minimum effort required to perform it on a worst-case input state, and this in turn determines the minimum time required for quantum systems of a given energy to carry out that operation on worst-case inputs. We calculate the minimum effort required to carry out some basic 1-bit and n-bit quantum and classical operations in an simple unconstrained scenario.