Abstract
High-quality random samples of quantum states are needed for a variety of tasks in quantum information and quantum computation. Searching the high-dimensional quantum state space for a global maximum of an objective function with many local maxima or evaluating an integral over a region in the quantum state space are but two exemplary applications of many. These tasks can only be performed reliably and efficiently with Monte Carlo methods, which involve good samplings of the parameter space in accordance with the relevant target distribution. We show how the Markov-chain Monte Carlo method known as Hamiltonian Monte Carlo, or hybrid Monte Carlo, can be adapted to this context. It is applicable when an efficient parameterization of the state space is available. The resulting random walk is entirely inside the physical parameter space, and the Hamiltonian dynamics enable us to take big steps, thereby avoiding strong correlations between successive sample points while enjoying a high acceptance rate. We use examples of single and double qubit measurements for illustration.
Highlights
Our companion paper [1] states the motivation for this work and introduces the terminology and notational conventions we are using; when referring to an equation or figure in that paper, the respective number is preceded by ‘I-’
Like the Markov-chain Monte Carlo (MCMC) method discussed in I, Hamiltonian Monte Carlo (HMC) involves a ‘walk’ around the parameter space
While this conveys the idea of HMC, its actual implementation is, not in terms of a trial sample that is iteratively updated by the mapping (6), but by a Markovian random walk
Summary
Searching the high-dimensional quantum state space for a attribution to the author(s) and the title of global maximum of an objective function with many local maxima or evaluating an integral over a the work, journal citation and DOI. Region in the quantum state space are but two exemplary applications of many. These tasks can only be performed reliably and efficiently with Monte Carlo methods, which involve good samplings of the parameter space in accordance with the relevant target distribution. Monte Carlo method known as Hamiltonian Monte Carlo, or hybrid Monte Carlo, can be adapted to this context. It is applicable when an efficient parameterization of the state space is available. We use examples of single and double qubit measurements for illustration
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