A quantum algorithm to efficiently sample the work distribution and to estimate the free energy of quantum systems

Augusto J. Roncaglia, Federico Cerisola, Juan Pablo Paz

Published 2014-09-12Version 1

We present a new method to measure work and to efficiently sample its probability distribution with fixed precision. The method can be used to estimate free energies on a quantum computer. It is based on three facts: (i) The probability to detect work $w$ in the state $\rho$ is $P(w)={\rm tr}[\rho \,W(w)]$, where $W(w)$ are positive operators satisfying $\int dw \,W(w)=I$. As $W(w)$ define a POVM (positive operator valued measure), work measurement always reduces to a projective measurement performed at a single time on an enlarged system. (ii) Work can be estimated using a variant of the "phase estimation algorithm" which is such that work $w$ is detected as the outcome of the single time measurement with probability $P(w)$. (iii) The efficient sampling of $P(w)$ can be combined with fluctuation theorems to estimate differences between the free energy of quantum states.