Quantum Mechanics and Polynomials of a Discrete Variable

Stanley Gudder

Alex Maltesson

This thesis investigates two complementary computational approaches to obtaining the outcome statistics of bosonic quantum experiments without directly performing the experiments themselves. In particular, we consider how continuous-variable quantum computations can be approximated using discrete-variable quantum systems and how bosonic outcome probabilities can be estimated using classical computational methods.The first part of the thesis focuses on the relation between continuous-variable (CV) and discrete-variable (DV) quantum computation. A framework is presented for approximating realistic CV quantum computations, consisting of polynomially bounded-energy states and a polynomial number of elementary gates, using DV quantum hardware. The proposed approximation error scales polynomially with the number of modes and can be made arbitrarily small by increasing the dimension of the simulating qudits. However, some technical aspects of the derivation remain to be rigorously justified. If these can be resolved, the result would establish a computational equivalence between the two paradigms. The remaining potential issues and possible approaches to resolving them are discussed.In the second part, we develop a framework for classically estimating individual outcome probabilities of general bosonic experiments using Monte Carlo methods based on (s)-ordered phase-space quasi-probability distributions. The resulting sampling complexity depends on properties of the phase-space quasi-probability distributions, in particular, their negativity and range, showing that negativity alone does not prevent efficient classical estimation. The outcome probability can be decomposed in several ways, giving rise to distinct trade-offs between these quantities. We discuss how experimentally relevant error channels can be incorporated into the framework and how they affect the classical estimation complexity, using pure loss as an illustrative example.Together, these results contribute to understanding the resources required to reproduce the outcome statistics of bosonic quantum experiments using both DV quantum and classical computation.

Ryu Sasaki B

Many examples of exactly solvable birth and death processes, a typical stationary Markov chain, are presented together with the explicit expressions of the transition probabilities. They are derived by similarity transforming exactly solvable ‘matrix’ quantum mechanics, which is recently proposed by Odake and the author. The (q-)Askey-scheme of hypergeometric orthogonal polynomials of a discrete variable and their dual polynomials play a central role. The most generic solvable birth/death rates are rational functions of q x (x being the population) corresponding to the q-Racah polynomial. 1

Lay Nam Chang, Zachary Lewis, Djordje Minić +1 more

Kasra Asnaashari, D. Bondarenko, Roman V. Krems

Discrete variable representation (DVR) enables efficient quantum algorithms for molecular vibrations. DVR structure reduces VQE measurement complexity to polynomial scaling and yields shallow circuits that accurately reproduce vibrational spectra.

Richard M. Friedberg, T. D. Lee