Fig. 1: The concept of quantum equilibrium propagation.
The goal is to efficiently and in a physical way obtain the gradient of some loss function (depending on expectation values measured at the 'output' degrees of freedom of a quantum system) with respect to tuneable parameters. a Rather than shifting N parameters separately and measuring the output expectation value for each shift (parameter-shift method), Onsager reciprocity dictates that the same information can be extracted by (b) shifting, i.e. nudging, only the parameters coupling to the output observables and (in a single go) measuring the response of all N operators coupled to the training parameters. This procedure, termed quantum equilibrium propagation, is more efficient as it requires only a single response experiment (or at most a small number of order 1, when some non-commuting observables have to be measured) independent of the system size, whereas the parameter shift method requires a number of experiments scaling linearly with the number of parameters. c Overview inspired by ref. 60 of digital and analogue neuromorphic computing schemes and platforms in the classical and quantum regime. Quantum equilibrium propagation can be applied to analogue quantum platforms such as quantum simulators with atoms and superconductors.
