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arXiv:2004.06439

The quantum query complexity of composition with a relation

Aleksandrs Belovs, Troy Lee

Published Apr 14, 2020• 3 citations
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Abstract

The negative weight adversary method, $\mathrm{ADV}^\pm(g)$, is known to characterize the bounded-error quantum query complexity of any Boolean function $g$, and also obeys a perfect composition theorem $\mathrm{ADV}^\pm(f \circ g^n) = \mathrm{ADV}^\pm(f) \mathrm{ADV}^\pm(g)$. Belovs gave a modified version of the negative weight adversary method, $\mathrm{ADV}_{rel}^\pm(f)$, that characterizes the bounded-error quantum query complexity of a relation $f \subseteq \{0,1\}^n \times [K]$, provided the relation is efficiently verifiable. A relation is efficiently verifiable if $\mathrm{ADV}^\pm(f_a) = o(\mathrm{ADV}_{rel}^\pm(f))$ for every $a \in [K]$, where $f_a$ is the Boolean function defined as $f_a(x) = 1$ if and only if $(x,a) \in f$. In this note we show a perfect composition theorem for the composition of a relation $f$ with a Boolean function $g$ \[ \mathrm{ADV}_{rel}^\pm(f \circ g^n) = \mathrm{ADV}_{rel}^\pm(f) \mathrm{ADV}^\pm(g) \enspace . \] For an efficiently verifiable relation $f$ this means $Q(f \circ g^n) = Θ( \mathrm{ADV}_{rel}^\pm(f) \mathrm{ADV}^\pm(g) )$.

DOI
10.48550/arxiv.2004.06439

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The negative weight adversary method, ADV±(g)\mathrm{ADV}^\pm(g)ADV±(g), is known to characterize the bounded-error quantum query complexity of any Boolean function ggg, and also obeys a perfect composition theorem ADV±(f∘gn)=ADV±(f)ADV±(g)\mathrm{ADV}^\pm(f \circ g^n) = \mathrm{ADV}^\pm(f) \mathrm{ADV}^\pm(g)ADV±(f∘gn)=ADV±(f)ADV±(g). Belovs gave a modified version of the negative weight adversary method, ADVrel±(f)\mathrm{ADV}_{rel}^\pm(f)ADVrel±​(f), that characterizes the bounded-error quantum query complexity of a relation f⊆{0,1}n×[K]f \subseteq \{0,1\}^n \times [K]f⊆{0,1}n×[K], provided the relation is efficiently verifiable. A relation is efficiently verifiable if ADV±(fa)=o(ADVrel±(f))\mathrm{ADV}^\pm(f_a) = o(\mathrm{ADV}_{rel}^\pm(f))ADV±(fa​)=o(ADVrel±​(f)) for every a∈[K]a \in [K]a∈[K], where faf_afa​ is the Boolean function defined as fa(x)=1f_a(x) = 1fa​(x)=1 if and only if (x,a)∈f(x,a) \in f(x,a)∈f. In this note we show a perfect composition theorem for the composition of a relation fff with a Boolean function ggg [ \mathrm{ADV}{rel}^\pm(f \circ g^n) = \mathrm{ADV}{rel}^\pm(f) \mathrm{ADV}^\pm(g) \enspace . ] For an efficiently verifiable relation fff this means Q(f∘gn)=Θ(ADVrel±(f)ADV±(g))Q(f \circ g^n) = Θ( \mathrm{ADV}_{rel}^\pm(f) \mathrm{ADV}^\pm(g) )Q(f∘gn)=Θ(ADVrel±​(f)ADV±(g)).
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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.
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.
Quantum strong direct-product theorems for specific functions have been known for nearly two decades. These have been extended to general results for function computation and state generation. The proofs of these results use a version of the multiplicative adversary method that does not naturally extend to relations. Standard strong direct-product theorems apply when algorithms must correctly answer every given question. Prior work extended them to equivalent threshold direct-product theorems, which require answers to all questions but only require that most answers are correct. We focus on two further generalizations. Strong selective direct-products apply to algorithms that adaptively choose, based on what they learn from queries, which questions from a large list to answer. This generalization is relational and useful for proving time-space tradeoffs. We prove a quantum strong selective direct-product theorem for all functions using a relational formulation of the multiplicative adversary method by Jeffery and Zur which we prove, via an equivalent formulation, satisfies a strong selective direct product property and captures any query lower bound for functions proven by negative-weights adversaries. The second generalization is list-decoding direct product problems introduced by Ben-David and Blais for classical query complexity. These allow an algorithm to produce a large list of possible output vectors such that one of them is fully correct. They proved that such theorems hold for randomized complexity of all Boolean functions. We prove a quantum analogue of this theorem for all partial Boolean-valued functions. We show that strong list-decoding direct-product theorems are implied by a special case of multiplicative adversaries which we show, via a new reduction, can be obtained from negative-weights adversaries for any Boolean-valued function.
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.