Estimates for the deviations of random walks and a stochastic method for solving the Schrodinger equation

Thomas L. Beck, Joel H. Dedrick

Wolfgang Wagner

Garnet N. Ord

Alán Aspuru‐Guzik

Torres, Frank

The Born rule asserts the probability distribution of eigenstates observed in unbiased quantum measurements, but the reason it holds remains elusive. This manuscript discusses how the Born rule might be explained by Schrodinger equation dynamics, if a measurement comprises a system responding to random fluctuations until it is within an arbitrarily small tolerance of a measurement eigenstate. We describe the random walk dynamics that produce this behavior in terms of a class of time-dependent, stochastic unitary matrices U(t). We also discuss the class of stochastic potential energies in the Schrodinger equation that is equivalent to this class of unitary matrices. This analysis raises some questions worth considering, including how to determine if any measurements actually follow the predicted random walk mechanism and whether a reliable measurement apparatus could be designed that deviates from Born rule probabilities. Interestingly, if any measurements do follow this random walk mechanism, then exposing a quantum system to stochastic 'noise' is an intrinsic part of such a measurement, not merely an unwanted side effect. This characteristic would have implications for reducing noise in quantum sensing and quantum computing. This is a draft of a work in progress. Questions and suggestions are welcome.

James B. Anderson

Two extensions of the fixed-node random walk method of solving the Schrödinger equation are described. A simple iterative procedure is found to reduce time-step error and the use of two trial wave functions in place of one for importance sampling is found to reduce computation effort. In combination the two modifications reduce computation effort for a fixed accuracy by a factor of about 10.