Energy as Computing

We offer well-motivated and basis-independent definitions for the total amount of change occurring along any continuous trajectory of a time-dependent quantum state vector, as well as the amount of physical/computational "effort" required to carry out a given unitary transformation in an abstract setting, given a set of possible initial quantum states, and a set of available Hamiltonians. Our definitions are based on the action of the Hamiltonian, which we show is always exactly twice the area swept out in the complex plane by the state-vector coefficients in any basis. Using our definitions, we show that the rate of change of any state is exactly given by its quantum-average energy (relative to the given ground state), while the "rate of computing" in the abstract situation could be considered equal to the energy of the highest-energy state in the input set. The minimum effort required to carry out various types of quantum and classical logic operations is explored. Among other results, we show that the minimum effort to perform any 1-qubit unitary gate is $0 \leq |\theta|\hbar \leq h/2$, where we view the gate as being equivalent to a rotation of the Bloch sphere through an angle of $\theta$ (with $|\theta|\leq\pi$) about some axis in three-space