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Journal of Modern Optics

Oracle Quantum Computing

André Berthiaume, Gilles Brassard A

Published Dec 1, 1994• 99 citations
Publisher

Abstract

Building on the work of Deutsch and Jozsa, we construct oracles relative to which (1) there is a decision problem that can be solved with certainty in worst-case polynomial time on the quantum computer, yet it cannot be solved classically in probabilistic expected polynomial time if errors are not tolerated, nor even in nondeterministic polynomial time, and (2) there is a decision problem that can be solved in exponential time on the quantum computer, which requires double exponential time on all but finitely many instances on any classical deterministic computer.

DOI
10.1080/09500349414552351

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Building on the work of Deutsch and Jozsa, we construct oracles relative to which (1) there is a decision problem that can be solved with certainty in worst-case polynomial time on the quantum computer, yet it cannot be solved classically in probabilistic expected polynomial time if errors are not tolerated, nor even in nondeterministic polynomial time, and (2) there is a decision problem that can be solved in exponential time on the quantum computer, which requires double exponential time on all but finitely many instances on any classical deterministic computer.
Building on work of Deutsch and Jozsa, we construct oracles relative to which (1) there is a decision problem that can be solved with certainty in worst-case polynomial time on the quantum computer, yet it cannot be solved classically in probabilistic expected polynomial time if errors are not tolerated nor even in nondeterministic polynomial time, and (2) there is a decision problem that can be solved in exponential time on the quantum computer, which requires double exponential time on all but finitely many instances on any classical deterministic computer. The results given in this paper have been presented at the 7th Annual IEEE Symposium on Structure in Complexity Theory, Boston, June 1992, and at the Workshop on Physics and Computation, PhysComp '92, Dallas, October 1992; they have appeared in preliminary form in those Proceedings [7, 8]. y Supported in part by an Nserc postgraduate fellowship. z Supported in part by Canada's Nserc E. W. R. Steacie Memorial Fellowship an...
This paper continues the study of the power of oracles to separate quantum com.plexity classes from classical (including probabilistic and nondeterministic) complexity classes, which we initiated in an earlier paper [5]. The new results are that, under appropriate oracles, (1) there is a decision problem that can be solved in polynomial time on the quantum computer, which would classically require exponential time in the worst case even on a nondeterministic computer (in particular QP g NP relative to this oracle), and (2) there is a decision problem that can be solved in exponential time on the quantum computer for which any deterministic classical computer would require double exponential time on all but finitely many instances.
Turning to quantum mechanics … secret, secret, close the doors! we always have had a great deal of difficulty in understanding the world view that quantum mechanics represents. … It has not yet become obvious to me that there's no real problem. I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem. So that's why I like to investigate things. – Richard Feynman, 1964 The only difference between a probabilistic classical world and the equations of the quantum world is that somehow or other it appears as if the probabilities would have to go negative. – Richard Feynman, in “Simulating Physics with Computers,” 1982 Our first result is the existence of an efficient universal quantum Turning machine in Deutsch's model. … We give the first formal evidence that quantum Turning machines violate the modern complexity theoretic formulation of the Church Turning thesis. We show the existence of a problem relative to an oracle that can be solved in polynomial time on a quantum Turning machine but require super polynomial time on a bounded error probabilistic Turning machine. – E. Bernstein and U. Vazirani, “Quantum Complexity Theory,” 1997 Quantum computing is a new computational model that may be physically realizable and may provide an exponential advantage over “classical” computational models such as probabilistic and deterministic Turing machines.
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.