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ACM Symposium on Theory of Computing (STOC)

Formula lower bounds via the quantum method

Avishay Tal

Published Jun 15, 2017• 17 citations
Publisher

Abstract

A de Morgan formula over Boolean variables x1,…,xn is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves are marked with variables or their negation. We define the size of the formula as the number of leaves in it. Proving that some explicit function (in P or NP) requires a large formula is a central open question in computational complexity. While we believe that some explicit functions require exponential formula size, currently the best lower bound for an explicit function is the Ω(n3) lower bound for Andreev's function.

DOI
10.1145/3055399.3055472

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A de Morgan formula over Boolean variables x1,…,xn is a binary tree whose internal nodes are marked with AND or OR gates and whose leaves are marked with variables or their negation. We define the size of the formula as the number of leaves in it. Proving that some explicit function (in P or NP) requires a large formula is a central open question in computational complexity. While we believe that some explicit functions require exponential formula size, currently the best lower bound for an explicit function is the Ω(n3) lower bound for Andreev's function.
We show that Nechiporuk's method [I. Wegener, The Complexity of Boolean Functions, Teubner-Wiley, New York, 1987] for proving lower bounds for Boolean formulas can be extended to the quantum case. This leads to an Ω(n2/log⁡2n)\Omega(n^2/\log^2 n)Ω(n2/log2n) lower bound for quantum formulas computing an explicit function. The only known previous explicit lower bound for quantum formulas [A. Yao, Proceedings of 34th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1993, pp. 352--361] states that the majority function does not have a linear-size quantum formula. We also show that quantum formulas can be simulated by Boolean circuits of almost the same size.
The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. This is why Boolean functions are needed with a high number of essential variables and a low polynomial degree. Unfortunately, it is a well-known problem to construct such functions. The best separation between these two complexity measures of a Boolean function was exhibited by Ambai- nis [5]. He constructed functions with polynomial degree M and number of variables Omega(M2). We improve such a separation to become exponential. On the other hand, we use a computerized exhaustive search to prove tightness of this bound.
We have seen in the previous chapters that quantum computers seem to be more powerful than classical computers for certain problems. There are limits on the power of quantum computers, however. Since a classical computer can simulate a quantum computer, a quantum computer can only compute the same set of functions that a classical computer can. The advantage of using a quantum computer is that the amount of resources needed by a quantum algorithm might be much less than what is needed by the best classical algorithm. In Section 9.1 we briefly define some classical and quantum complexity classes and give some relationships between them. Most of the interesting questions relating classical and quantum complexity classes remain open. For example, we do not yet know if a quantum computer is capable of efficiently solving an NP-complete problem (defined later). One can prove upper bounds on the difficulty of a problem by providing an algorithm that solves that problem, and proving that it will work within in a given running time. But how does one prove a lower bound on the computational complexity of a problem? For example, if we wish to find the product of two n-bit numbers, computing the answer requires outputting roughly 2n bits and that requires Ω(n) steps (in any computing model with finite-sized gates). The best-known upper bound for integer multiplication is O(n log n log log n) steps. It has proved extremely difficult to derive non-trivial lower bounds on the computational complexity of a problem. Most of the known non-trivial lower bounds are in the ‘black-box’ model (for both classical and quantum computing), where we only query the input via a ‘black-box’ of a specific form. We discuss the black-box model in more detail in Section 9.2. We then sketch several approaches for proving black-box lower bounds. The first technique has been called the ‘hybrid method’ and was used to prove that quantum searching requires Ω(√n) queries to succeed with constant probability. The second technique is called the ‘polynomial method’. We then describe a technique based on ‘block sensitivity’, and conclude with a technique known as the ‘adversary method’.
Exact and point-wise approximating representations of Boolean functions by real polynomials have been of great interest in the theory of computing. We focus on the study of sparsity of such representations. Our results include the following: First, we show that for every total Boolean function, its exact and approximate sparsity in the De Morgan basis are polynomially related to each other in the log scale, ignoring poly-log(n) factors. This answers an open question posed by Knop, Lovett, McGuire and Yuan (STOC 2021). It builds on and is analogous to the seminal result of Nisan and Szegedy (Computational Complexity 1994) who proved the same for degree and approximate degree. Second, we consider more powerful representations using generalized monomials, where each monomial is an indicator of a sub-cube. There are 3n such monomials, where n is the number of variables. We prove that even for these representations, the sparsity and approximate sparsity of total Boolean functions remain polynomially related to each other in the log scale, ignoring poly-log(n) factors. Third, we show that for every total Boolean function f, the log of its De Morgan sparsity characterizes up to polynomial loss and ignoring poly-log(n) factors, the quantum and classical 2-party bounded-error communication complexity of f ∘ EQ4, where EQ4 is Equality of two 2-bit strings, one held by Alice and the other by Bob. As a consequence, we show that bounded-error quantum protocols cannot exhibit super-polynomial cost advantage over their classical counterparts, for computing such functions.
We examine the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0,1} N in the black-box model. We show that the exponential quantum speed-up obtained for partial functions (i.e., problems involving a promise on the input) by Deutsch and Jozsa, Simon, and Shor cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with small error probability using T black-box queries, then there is a classical deterministic algorithm that computes f exactly with O ( Ts 6 ) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.