Uselessness for an Oracle Model with Internal Randomness

Uselessness for an Oracle Model with Internal Randomness Aram W. Harrow Center for Theoretical Physics Massachusetts Institute of Technology David J. Rosenbaum∗ Department of Computer Science & Engineering University of Washington November 2, 2018 Abstract We consider a generalization of the standard oracle model in which the oracle acts on the target with a permutation selected according to internal random coins. We describe several problems that are impossible to solve classically but can be solved by a quantum algorithm using a single query; we show that such infinity-vs-one separations between classical and quantum query complexities can be constructed from much weaker separations. We also give conditions to determine when oracle problems—either in the standard model, or in any of the generalizations we consider—cannot be solved with success probability better than random guessing would achieve. In the oracle model with internal randomness where the goal is to gain any nonzero advantage over guessing, we prove (roughly speaking) that k quantum queries are equivalent in power to 2k classical queries, thus extending results of Meyer and Pommersheim. ∗Corresponding author: djr@cs.washington.edu arXiv:1111.1462v2 [quant-ph] 23 Sep 2013 1 Introduction Oracles are an important conceptual framework for understanding quantum speedups. They may represent subroutines whose code we cannot usefully examine, or an unknown physical system whose properties we would like to estimate. When used by a quantum computer, the most general form of an oracle is a possibly noisy quantum operation that can be applied to an n-qubit input. However, oracles this general have no obvious classical analogue, which makes it difficult to compare the ability of classical and quantum computers to efficiently interrogate oracles. This was the original motivation of the standard oracle model, in which f is a function from [N] = {1, . . . , N} to {0, 1}, and the oracle Of acts for a classical computer by mapping x, y to x, y ⊕f(x), and for a quantum computer as a unitary that maps |x, y⟩to |x, y ⊕f(x)⟩. One way to justify the standard oracle model is that if there is a (not necessarily reversible) classical circuit computing f, then Of can be simulated by computing f, XORing the answer onto the target, and uncomputing f. In this paper, we consider other forms of oracles that are more general than the standard oracle model, but nevertheless permit comparison between classical and quantum query complexities. Meyer and Pommersheim [6] generalized the standard model by letting A be a deterministic classical algorithm that takes the control x of the oracle and computes a value A(x). The oracle then acts by applying a permutation πA(x) to the target. We will further generalize the model by replacing A with a randomized classical algorithm. The random coins used by A are internal to the oracle and cannot be accessed externally. We call this concept an oracle with internal randomness. Note that even if A takes no input, the oracle can still be interesting since it may apply different permutations depending on its internal coin flips. Oracles with internal randomness correspond naturally to the situation in which a (quantum or classical) computer seeks to determine properties of a device that acts in a noisy or otherwise non-deterministic manner. One simple example is an oracle that “misfires”, i.e. when queried, the oracle does nothing with probability p and responds according to the standard oracle model with probability 1 −p. This model was considered in [9], which found, somewhat surprisingly, that the square-root advantage of Grover search disappears (i.e. there is an Ω(N) quantum query lower bound for computing the OR function) for any constant p > 0. The rest of our paper is divided into two parts. First, we explore various examples of oracles with internal randomness that demonstrate the power of the model. We will see that in some cases (e.g. Theorems 1 and 2), this can even result in problems solvable with one quantum query that are completely unsolvable using classical queries. In the second part, we consider the question of when oracle problems can be solved with any nontrivial advantage; i.e. a probability of success better than could be obtained by simply guessing the answer according to the prior distribution. For an example of when such advantage is not possible, consider the parity function on N bits. If these bits are drawn from the uniform distri- bution, then any classical algorithm making ≤N −1 queries—or any quantum algorithm making ≤N 2 −1 queries—will not be able to guess the parity with any nontrivial advantage. In Section 3, we consider the problem of when some number of queries are useless for solving an oracle problem. Informally, our main result is roughly that k quantum queries are useless if and only if 2k classical queries are useless (this is formalized in Theorem 9). However, a subtlety arises in our theorem when oracles h

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