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Mathematical Foundations of Computer Science

A Common Algebraic Description for Probabilistic and Quantum Computations (Extended Abstract)

Martin Beaudry, José M. Fernandez, Markus Holzer

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Through the study of gate arrays we develop a unified frame- work to deal with probabilistic and quantum computations, where the former is shown to be a natural special case of the latter. On this ba- sis we show how to encode a probabilistic or quantum gate array into a sum-free tensor formula which satisfies the conditions of the partial trace problem, and vice-versa. In this way complete problems for the classes pr-BPP (promise BPP) and pr-BQP (promise BQP) are given when changing the semiring from (Q + , + , · ) to the field (Q, + , · ). Moreover, by variants of the problem under consideration, classes like ⊕P, NP, C=P, its complement co-C=P, the promise version of Valiant's class UP, its generalization promise SPP, and unique polytime US are captured as problem property and the semiring varies.

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arXiv:2201.01904•Jan 6, 2022•3 citations
Through the study of gate arrays we develop a unified frame- work to deal with probabilistic and quantum computations, where the former is shown to be a natural special case of the latter. On this ba- sis we show how to encode a probabilistic or quantum gate array into a sum-free tensor formula which satisfies the conditions of the partial trace problem, and vice-versa. In this way complete problems for the classes pr-BPP (promise BPP) and pr-BQP (promise BQP) are given when changing the semiring from (Q + , + , · ) to the field (Q, + , · ). Moreover, by variants of the problem under consideration, classes like ⊕P, NP, C=P, its complement co-C=P, the promise version of Valiant's class UP, its generalization promise SPP, and unique polytime US are captured as problem property and the semiring varies.
We prove that quantum computation is polynomially equivalent to classical probabilistic computation with an oracle for estimating the value of simple sums, quadratically signed weight enumerators. The problem of estimating these sums can be cast in terms of promise problems and has two interesting variants. An oracle for the unconstrained variant may be more powerful than quantum computation, while an oracle for a more constrained variant is efficiently solvable in the one-bit model of quantum computation. Thus, problems involving estimation of quadratically signed weight enumerators yield problems in BQP (bounded error quantum polynomial time) that are distinct from the ones studied so far, include a canonical BQP complete problem, and can be used to define and study complexity classes and their relationships to quantum computation.
We connect both discrete and algebraic complexity theory with the satisfiability problem in certain non-Boolean lattices. Specifically, quantum logic was introduced in 1936 by Garrett Birkhoff and John von Neumann as a framework for capturing the logical peculiarities of quantum observables: in the 1D case it coincides with Boolean propositional logic but, starting with dimension two, violates the distributive law. We introduce the weak and strong satisfiability problem for quantum logic propositional formulae. It turns out that in dimension two, both are also NP --complete. For higher-dimensional spaces ℝ d and ℂ d with d ≥ 3 fixed, on the other hand, we show both problems to be complete for the nondeterministic Blum-Shub-Smale (BSS) model of real computation. This provides a unified view on both Turing and real BSS complexity theory, and extends the (still relatively scarce) list of problems established NP ℝ --complete with one, perhaps, closest in spirit to the classical Cook-Levin Theorem. More precisely, strong satisfiability of ∧ ∨ ∧ --terms is complete, while that of ∧ ∨--terms (i.e., those in conjunctive form) can be decided in polynomial time in dimensions d ≥ 2. The decidability of the infinite-dimensional case being still open, we proceed to investigate the case of indefinite finite dimensions. Here, weak satisfiability still belongs to NP R and strong satisfiability is still hard; the latter, in fact, turns out as polynomial-time equivalent to the feasibility of noncommutative integer polynomial equations over matrix rings.
Quantum computing is a prolific research area, halfway between physics and computer science [27, 29, 52]. Most likely, its origins may be dated back to 70's, when some works on quantum information began to appear (see, e.g., [34, 37]). In early 80's, R.P. Feynman suggested that the computational power of quantum mechanical processes might be beyond that of traditional computation models [25]. Almost at the same time, P. Benioff already proved that such processes are at least as powerful as Turing machines [9]. In 1985, D. Deutsch [22] proposed the notion of a quantum Turing machine as a physically realizable model for a quantum computer. From the point of view of structural complexity, E. Bernstein and U. Vazirani introduced in [20] the class BQP of problems solvable in polynomial time on quantum Turing machines, focusing attention on relations with the corresponding deterministic and probabilistic classes P and BPP, respectively. Several works in the literature explored classical issues in complexity theory from the quantum paradigm perspective (see, e.g., [7, 60, 61]).
We define several models of computation based on permuting distinguishable particles (which we call balls) and characterize their computational complexity. In the quantum setting, we use the representation theory of the symmetric group to find variants of this model which are intermediate between BPP and DQC1 (the class of problems solvable with one clean qubit) and between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an exactly solvable scattering problem of particles moving on a line. Despite the simplicity of this model from the perspective of mathematical physics, we show that if we allow intermediate destructive measurements and specific input states, then the model cannot be efficiently simulated classically up to multiplicative error unless the polynomial hierarchy collapses. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP, and that a nondeterministic version of this model is NP-complete.
An important theoretical problem in the study of quantum computation, that is also practically relevant in the context of near-term quantum devices, is to understand the computational power of hybrid models, that combine poly-time classical computation with short-depth quantum computation. Here, we consider two such models: CQ_d which captures the scenario of a polynomial-time classical algorithm that queries a d-depth quantum computer many times; and QC_d which is more analogous to measurement-based quantum computation and captures the scenario of a d-depth quantum computer with the ability to change the sequence of gates being applied depending on measurement outcomes processed by a classical computation. Chia, Chung & Lai (STOC 2020) and Coudron & Menda (STOC 2020) showed that these models (with d=log^O(1) (n)) are strictly weaker than BQP (the class of problems solvable by poly-time quantum computation), relative to an oracle, disproving a conjecture of Jozsa in the relativised world. We show that, despite the similarities between CQ_d and QC_d, the two models are incomparable, i.e. CQ_d ⊈\nsubseteq⊈ QC_d and QC_d ⊈\nsubseteq⊈ CQ_d relative to an oracle. In other words, there exist problems that one model can solve but not the other and vice versa. We do this by considering new oracle problems that capture the distinctions between the two models and by introducing the notion of an intrinsically stochastic oracle, an oracle whose responses are inherently randomised, which is used for our second result. While we leave showing the second separation relative to a standard oracle as an open problem, we believe the notion of stochastic oracles could be of independent interest for studying complexity classes which have resisted separation in the standard oracle model. Our constructions also yield simpler oracle separations between the hybrid models and BQP, compared to earlier works.