Abstract. We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs, several of which are tight for every space bound, also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices [Formula: see text], including the discrete Fourier transform matrix, we prove that quantum circuits with at most [Formula: see text] input queries and [Formula: see text] qubits of memory require [Formula: see text] to compute matrix-vector product [Formula: see text] for [Formula: see text]. We similarly prove that matrix multiplication for [Formula: see text] binary matrices requires [Formula: see text]. Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems with any space bound. We obtain matching lower bounds for the stronger notion of quantum cumulative memory complexity—the sum of the space per layer of a circuit. We also consider Boolean (i.e., AND-OR) matrix multiplication and matrix-vector products, improving the previous quantum time-space tradeoff lower bounds for [Formula: see text] Boolean matrix multiplication to [Formula: see text] from [Formula: see text]. Our improved lower bound for Boolean matrix multiplication is based on a new coloring argument that extracts more from the strong direct product theorem that was the basis for prior work. To obtain our tight lower bounds for linear algebra problems, we require much stronger bounds than strong direct product theorems. We obtain these bounds by adding a new bucketing method to the quantum recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.

We prove lower bounds on the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Using a novel way of applying recording query methods we show that for many linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices A, including the discrete Fourier transform (DFT) matrix, we prove that quantum circuits with at most T input queries and S qubits of memory require T=Ω(n2/S) to compute matrix-vector product Ax for x ∈ {0,1}n. We similarly prove that matrix multiplication for n× n binary matrices requires T=Ω(n3 / √S). Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems at any space bound. We also improve the previous quantum time-space tradeoff lower bounds for n× n Boolean (i.e. AND-OR) matrix multiplication from T=Ω(n2.5/S1/2) to T=Ω(n2.5/S1/4) which has optimal exponents for the powerful query algorithms to which it applies. Our method also yields improved lower bounds for classical algorithms.

Given a function $f$ on $\mathbb{F}_2^n$, we study the following problem. What is the largest affine subspace $\mathcal{U}$ such that when restricted to $\mathcal{U}$, all the non-trivial Fourier coefficients of $f$ are very small? For the natural class of bounded Fourier degree $d$ functions $f:\mathbb{F}_2^n \to [-1,1]$, we show that there exists an affine subspace of dimension at least $ \tildeΩ(n^{1/d!}k^{-2})$, wherein all of $f$'s nontrivial Fourier coefficients become smaller than $ 2^{-k}$. To complement this result, we show the existence of degree $d$ functions with coefficients larger than $2^{-d\log n}$ when restricted to any affine subspace of dimension larger than $Ω(dn^{1/(d-1)})$. In addition, we give explicit examples of functions with analogous but weaker properties. Along the way, we provide multiple characterizations of the Fourier coefficients of functions restricted to subspaces of $\mathbb{F}_2^n$ that may be useful in other contexts. Finally, we highlight applications and connections of our results to parity kill number and affine dispersers.

Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function f:{±1}ⁿ → {±1}: 1) We give a poly(k, 1/(ε)) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+ε)-far from k'-juntas, where k' = O(k/(ε²)). 2) In the non-relaxed setting, we extend our ideas to give a 2^{Õ(√{k/ε})} (adaptive) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ+ε)-far from k-juntas. To the best of our knowledge, this is the first subexponential-in-k query algorithm for approximating the distance of f to being a k-junta (previous results of Blais, Canonne, Eden, Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required exponentially many queries in k). Our techniques are Fourier analytical and make use of the notion of "normalized influences" that was introduced by Talagrand [Michel Talagrand, 1994].