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ACM Southeast Regional Conference (ACMSE)

Revisiting a limit on efficient quantum computation

Tarsem S. Purewal

Published Mar 10, 2006• 1 citation
Publisher

Abstract

In this paper, we offer an exposition of a theorem originally due to Adleman, Demarrais and Huang that shows that the quantum complexity class BQP (Bounded-error Quantum Polynomial time) is contained in the classical counting class PP (Probabilistic Polynomial time). Our proof follows the one given by Fortnow and Rogers that relates quantum computing to counting complexity classes by way of GapP functions. The contribution of this paper is an exposition of an important result that assumes a minimal background in computational complexity theory and no knowledge of quantum mechanics.

DOI
10.1145/1185448.1185502

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In this paper, we offer an exposition of a theorem originally due to Adleman, Demarrais and Huang that shows that the quantum complexity class BQP (Bounded-error Quantum Polynomial time) is contained in the classical counting class PP (Probabilistic Polynomial time). Our proof follows the one given by Fortnow and Rogers that relates quantum computing to counting complexity classes by way of GapP functions. The contribution of this paper is an exposition of an important result that assumes a minimal background in computational complexity theory and no knowledge of quantum mechanics.
I study the class of problems efficiently solvable by a quantum computer, given the ability to ‘postselect’ on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or probabilistic polynomial-time. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PP-complete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.
I study the class of problems efficiently solvable by a quantum computer, given the ability to "postselect" on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic Polynomial-Time. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PP-complete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.
Quantum computers are widely believed have an advantage over classical computers, and some have even published some empirical evidence that this is the case. However, these publications do not include a rigorous proof of this advantage, which would have to minimally state that the class of problems decidable by a quantum computer in polynomial time, BQP, contains problems that are not in the class of problems decidable by a classical computer with similar time bounds, P. Here, I provide the proof of a stronger result that implies this result: BQP contains problems that lie beyond the much larger classical computing class NP. This proves that quantum computation is able to efficiently solve problems which are far beyond the capabilities of classical computers.
Abstract. In this paper we study quantum computation from a complexity theoretic viewpoint. Our rst result is the existence of an ecient universal quantum Turing machine in Deutsch’s model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97{117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specied. We prove that O(log T) bits of precision suce to support a T step computation. This justies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the rst formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church{Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class BPP. The class BQP of languages that are eciently decidable (with small error-probability) on a quantum Turing machine satises BPP ⊆ BQP ⊆ P]P. Therefore, there is no possibility of giving a mathematical proof that quantum Turing machines are more powerful than classical probabilistic Turing machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory.
Alright, so now we've got this beautiful theory of quantum mechanics, and the possibly-even-more-beautiful theory of computational complexity. Clearly, with two theories this beautiful, you can't just let them stay single – you have to set them up, see if they hit it off, etc. And that brings us to the class BQP : Bounded-Error Quantum Polynomial-Time. We talked in Chapter 7 about BPP , or Bounded-Error Probabilistic Polynomial-Time. Informally, BPP is the class of computational problems that are efficiently solvable in the physical world if classical physics is true. Now we ask, what problems are efficiently solvable in the physical world if (as seems more likely) quantum physics is true? To me it’s sort of astounding that it took until the 1990s for anyone to really seriously ask this question, given that all the tools for asking it were in place by the 1960s or even earlier. It makes you wonder, what similarly obvious questions are there today that no one’s asking?