The (negative-weighted) quantum adversary bound is a tight characterisation of the quantum query complexity for any partial function. We analyse the extent to which this bound can be generalised. Ambainis et al. [arXiv:1012.2112] and Lee et al. [arXiv:1011.3020] generalised this bound to the state generation and state conversion problems, respectively. Using the ideas by Lee et al., we get even further generalisations of the bound. We obtain a version of the bound for general input oracles, which are just arbitrary unitaries. We also generalise the bound to the problem of implementing arbitrary unitary transformations. Similarly to the bound by Lee et al., our bound is a lower bound for exact transformation and an upper bound for approximate transformation. This version of the bound possesses the tight composition property. Using this construction, we also obtain lower bounds on the quantum query complexity of functions and relations with general input oracles.
The (negative-weighted) quantum adversary bound is a tight characterisation of the quantum query complexity for any partial function. We analyse the extent to which this bound can be generalised. Ambainis et al. [arXiv:1012.2112] and Lee et al. [arXiv:1011.3020] generalised this bound to the state generation and state conversion problems, respectively. Using the ideas by Lee et al., we get even further generalisations of the bound. We obtain a version of the bound for general input oracles, which are just arbitrary unitaries. We also generalise the bound to the problem of implementing arbitrary unitary transformations. Similarly to the bound by Lee et al., our bound is a lower bound for exact transformation and an upper bound for approximate transformation. This version of the bound possesses the tight composition property. Using this construction, we also obtain lower bounds on the quantum query complexity of functions and relations with general input oracles.
State conversion generalizes query complexity to the problem of converting between two input-dependent quantum states by making queries to the input. We characterize the complexity of this problem by introducing a natural information-theoretic norm that extends the Schur product operator norm. The complexity of converting between two systems of states is given by the distance between them, as measured by this norm. In the special case of function evaluation, the norm is closely related to the general adversary bound, a semi-definite program that lower-bounds the number of input queries needed by a quantum algorithm to evaluate a function. We thus obtain that the general adversary bound characterizes the quantum query complexity of any function whatsoever. This generalizes and simplifies the proof of the same result in the case of boolean input and output. Also in the case of function evaluation, we show that our norm satisfies a remarkable composition property, implying that the quantum query complexity of the composition of two functions is at most the product of the query complexities of the functions, up to a constant. Finally, our result implies that discrete and continuous-time query models are equivalent in the bounded-error setting, even for the general state-conversion problem.
We propose a new definition of quantum Las Vegas query complexity. We show that it is exactly equal to the quantum adversary bound. This is achieved by a new and very simple way of transforming a feasible solution to the adversary optimisation problem into a quantum query algorithm. This allows us to generalise the bound to include unidirectional access, multiple input oracles, and input oracles that are not unitary. As an application, we demonstrate a separation between unidirectional and bidirectional access to an input oracle for a rather natural unitary permutation inversion problem.
One of the main reasons for query model's prominence in quantum complexity is the presence of concrete lower bounding techniques: polynomial method and adversary method. There have been considerable efforts to not just give lower bounds using these methods but even to compare and relate them. We explore the value of these bounds on quantum query complexity for the class of symmetric functions, arguably one of the most natural and basic set of Boolean functions. We show (using the recently introduced spectral sensitivity) that both these bounds (positive adversary and approximate degree) give the same value for every total symmetric Boolean function. We also look at the quantum query complexity of Gap Majority, a partial symmetric function. It has gained importance recently in regard to understanding the composition of randomized query complexity. We characterize the quantum query complexity of Gap Majority and show a lower bound on noisy randomized query complexity (Ben-David and Blais, FOCS 2020) in terms of quantum query complexity. In addition, we study how large certificate complexity and block sensitivity can be as compared to sensitivity (even up to constant factors) for symmetric functions. We show tight separations, i.e., give upper bound on possible separations and construct functions achieving the same.
In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows: * We develop a new technique for the construction of quantum algorithms: learning graphs. * We use learning graphs to improve quantum query complexity of the triangle detection and the k-distinctness problems. * We prove tight lower bounds for the k-sum and the triangle sum problems. * We construct quantum algorithms for some subgraph-finding problems that are optimal in terms of query, time and space complexities. * We develop a generalisation of quantum walks that connects electrical properties of a graph and its quantum hitting time. We use it to construct a time-efficient quantum algorithm for 3-distinctness.
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(nlogn) 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 xS=b for an arbitrary S∈Q and b∈{−1,1}S of its choice, at unit cost. We show near-tight bounds when Q is any of the following collections: bounded-size sets, contiguous blocks, prefixes, and only the full set.
All of these results are derived using a framework that we develop. Using symmetries of the task at hand we show that the quantum query complexity of learning x is characterized, up to a constant factor, by an optimization program, which is 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.
The polynomial and the adversary methods are the two main tools for proving lower bounds on query complexity of quantum algorithms. Both methods have found a large number of applications, some problems more suitable for one method, some for the other. It is known though that the adversary method, in its general negative-weighted version, is tight for bounded-error quantum algorithms, whereas the polynomial method is not. By the tightness of the former, for any polynomial lower bound, there ought to exist a corresponding adversary lower bound. However, direct reduction was not known. In this paper, we give a simple and direct reduction from the polynomial method (in the form of a dual polynomial) to the adversary method. This shows that any lower bound in the form of a dual polynomial is actually an adversary lower bound of a specific form.