Robin Kothari

Prior work of Beverland et al. has shown that any exact Clifford+T implementation of the n-qubit Toffoli gate must use at least n T gates. Here we show how to get away with exponentially fewer T gates, at the cost of incurring a tiny 1/poly(n) error that can be neglected in most practical situations. More precisely, the n-qubit Toffoli gate can be implemented to within error ϵ in the diamond distance by a randomly chosen Clifford+T circuit with at most O(log(1/ϵ)) T gates. We also give a matching Ω(log(1/ϵ)) lower bound that establishes optimality, and we show that any purely unitary implementation achieving even constant error must use Ω(n) T gates. We also extend our sampling technique to implement other Boolean functions. Finally, we describe upper and lower bounds on the T-count of Boolean functions in terms of non-adaptive parity decision tree complexity and its randomized analogue.

Abstract Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function f, deg(f) = O(~deg(f)^2): The degree of f is at most quadratic in the approximate degree of f. This is optimal as witnessed by the OR function. D(f) = O(Q(f)^4): The deterministic query complexity of f is at most quartic in the quantum query complexity of f. This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, Santha, and Smotrovs (2017). We apply these results to resolve the quantum analogue of the Aanderaa--Karp--Rosenberg conjecture. We show that if f is a nontrivial monotone graph property of an n-vertex graph specified by its adjacency matrix, then Q(f)=Omega(n), which is also optimal. We also show that the approximate degree of any read-once formula on n variables is Theta(sqrt{n}).

Abstract We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function f:R^n->R and its (sub)gradient. Our goal is to find an eps-approximate minimum of f starting from a point that is distance at most R from the true minimum. If f is G-Lipschitz, then the classic gradient descent algorithm solves this problem with O((GR/eps)^2) queries. Importantly, the number of queries is independent of the dimension n and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension n. In this paper we reprove the matching randomized lower bound using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved quadratically faster using quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need Omega((GR/eps)^2) queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family. In our second result, we consider the case of smooth functions. Here the optimal classical algorithm is not gradient descent, but accelerated gradient descent, which is also known to be optimal among all classical (randomized) algorithms. We show a matching quantum lower bound, showing that there is no quantum speedup over accelerated gradient descent either.

We study how efficiently a k-element set S subseteq [n] can be learned from a uniform superposition ket{S} of its elements. One can think of ket{S}=sum_{i in S} ket{i}/sqrt{|S|} as the quantum version of a uniformly random sample over S, as in the classical analysis of the “coupon collector problem.” We show that if k is close to n, then we can learn S using asymptotically fewer quantum samples than random samples. In particular, if there are n-k=O(1) missing elements then O(k) copies of ket{S} suffice, in contrast to the Theta(k log k) random samples needed by a classical coupon collector. On the other hand, if n-k=Omega(k), then Omega(k log k) quantum samples are necessary. More generally, we give tight bounds on the number of quantum samples needed for every k and n, and we give efficient quantum learning algorithms. We also give tight bounds in the model where we can additionally reflect through ket{S}. Finally, we relate coupon collection to a known example separating proper and improper PAC learning that turns out to show no separation in the quantum case.