Classical and probabilistic computation

Ki-Hyuk Han, G.S. Park, Jeong Ung Ahn +7 more

Probabilistic computing, a class of physics-based computing, bridges the gap between quantum computing and the classical von Neumann architecture. This approach provides more efficient means of addressing NP problems, which are challenging for classical computers. In this work, we analyze the core concept of probabilistic computing which is based on the Ising model framework-including bit fluctuations and energy trends. In addition, we extend the traditional binary (two-level) system into a multi-level probabilistic framework, i.e. number partitioning problem to multiway number partitioning problem, as a case study.

A. Kitaev, Alexander Khanievich Shen', M. Vyalyi

CHRISTOF WETTERICH

A simple probabilistic cellular automaton is shown to be equivalent to a relativistic fermionic quantum field theory with interactions. Occupation numbers for fermions are classical bits or Ising spins. The automaton acts deterministically on bit configurations. The genuinely probabilistic character of quantum physics is realized by probabilistic initial conditions. In turn, the probabilistic automaton is equivalent to the classical statistical system of a generalized Ising model. For a description of the probabilistic information at any given time quantum concepts as wave functions and non-commuting operators for observables emerge naturally. Quantum mechanics can be understood as a particular case of classical statistics. This offers prospects to realize aspects of quantum computing in the form of probabilistic classical computing. This article is part of the theme issue 'Quantum technologies in particle physics'.

Xiaodong Jia, Andre Kornell, Bert Lindenhovius +2 more

We consider a programming language that can manipulate both classical and quantum information. Our language is type-safe and designed for variational quantum programming, which is a hybrid classical-quantum computational paradigm. The classical subsystem of the language is the Probabilistic FixPoint Calculus (PFPC), which is a lambda calculus with mixed-variance recursive types, term recursion and probabilistic choice. The quantum subsystem is a first-order linear type system that can manipulate quantum information. The two subsystems are related by mixed classical/quantum terms that specify how classical probabilistic effects are induced by quantum measurements, and conversely, how classical (probabilistic) programs can influence the quantum dynamics. We also describe a sound and computationally adequate denotational semantics for the language. Classical probabilistic effects are interpreted using a recently-described commutative probabilistic monad on DCPO. Quantum effects and resources are interpreted in a category of von Neumann algebras that we show is enriched over (continuous) domains. This strong sense of enrichment allows us to develop novel semantic methods that we use to interpret the relationship between the quantum and classical probabilistic effects. By doing so we provide a very detailed denotational analysis that relates domain-theoretic models of classical probabilistic programming to models of quantum programming.

Smita Krishnaswamy, Igor L. Markov, John P. Hayes