Quantum Search with Generalized Wildcards

Abstract

In the "search with wildcards" problem [Ambainis, Montanaro, Quantum Inf. Comput.'14], one’s goal is to learn an unknown bit-string x ∈ {-1,1}ⁿ. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost O(√n log n) and a near-matching lower bound of Ω(√n). Belovs [Comput. Comp.'15] subsequently showed a tight O(√n) upper bound.
We consider a natural generalization of this problem, parametrized by a subset Q ⊆ 2^{[n]}, where an algorithm may test whether x_S = b for an arbitrary S ∈ Q and b ∈ {-1,1}^S of its choice, at unit cost. We show the following:  
- For all k ∈ [n], when Q is the collection of all sets of size at most k, the quantum query complexity is Θ(n/√k). In particular when k = n, this corresponds to the standard search with wildcards setting. This recovers and generalizes the tight characterization of Belovs, and Ambainis and Montanaro, using completely different techniques. 
- When Q is the collection of contiguous blocks, the quantum query complexity is Θ̃(n). 
- When Q is the collection of prefixes, the quantum query complexity is Θ(n).  All of these results are derived using a framework that we develop. We apply a symmetry reduction to the primal version of the negative-weight adversary bound, and show that the quantum query complexity of learning x is characterized, up to a constant factor, by a particular optimization program, which can be succinctly described as follows: `maximize over all odd functions f : {-1,1}ⁿ → ℝ the ratio of the maximum value of f to the maximum (over T ∈ Q) standard deviation of f on a subcube whose free variables are exactly T.'
To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.