New Developments in Quantum Algorithms

In this survey, we describe two recent developments in quantum algorithms. The first new development is a quantum algorithm for evaluating a Boolean formula consisting of AND and OR gates of size N in time O(\sqrt{N}). This provides quantum speedups for any problem that can be expressed via Boolean formulas. This result can be also extended to span problems, a generalization of Boolean formulas. This provides an optimal quantum algorithm for any Boolean function in the black-box query model. The second new development is a quantum algorithm for solving systems of linear equations. In contrast with traditional algorithms that run in time O(N^{2.37...}) where N is the size of the system, the quantum algorithm runs in time O(\log^c N). It outputs a quantum state describing the solution of the system.Comment: 11 pages, 1 figure, to appear as an invited survey talk at MFCS'201

Span programs and quantum algorithms for st-connectivity and claw detection

We introduce a span program that decides st-connectivity, and generalize the span program to develop quantum algorithms for several graph problems. First, we give an algorithm for st-connectivity that uses O(n d^{1/2}) quantum queries to the n x n adjacency matrix to decide if vertices s and t are connected, under the promise that they either are connected by a path of length at most d, or are disconnected. We also show that if T is a path, a star with two subdivided legs, or a subdivision of a claw, its presence as a subgraph in the input graph G can be detected with O(n) quantum queries to the adjacency matrix. Under the promise that G either contains T as a subgraph or does not contain T as a minor, we give O(n)-query quantum algorithms for detecting T either a triangle or a subdivision of a star. All these algorithms can be implemented time efficiently and, except for the triangle-detection algorithm, in logarithmic space. One of the main techniques is to modify the st-connectivity span program to drop along the way "breadcrumbs," which must be retrieved before the path from s is allowed to enter t.Comment: 18 pages, 4 figure

Quantum algorithm for tree size estimation, with applications to backtracking and 2-player games

We study quantum algorithms on search trees of unknown structure, in a model where the tree can be discovered by local exploration. That is, we are given the root of the tree and access to a black box which, given a vertex $v$, outputs the children of $v$. We construct a quantum algorithm which, given such access to a search tree of depth at most $n$, estimates the size of the tree $T$ within a factor of $1\pm \delta$ in $\tilde{O}(\sqrt{nT})$ steps. More generally, the same algorithm can be used to estimate size of directed acyclic graphs (DAGs) in a similar model. We then show two applications of this result: a) We show how to transform a classical backtracking search algorithm which examines $T$ nodes of a search tree into an $\tilde{O}(\sqrt{T}n^{3/2})$ time quantum algorithm, improving over an earlier quantum backtracking algorithm of Montanaro (arXiv:1509.02374). b) We give a quantum algorithm for evaluating AND-OR formulas in a model where the formula can be discovered by local exploration (modeling position trees in 2-player games). We show that, in this setting, formulas of size $T$ and depth $T^{o(1)}$ can be evaluated in quantum time $O(T^{1/2+o(1)})$. Thus, the quantum speedup is essentially the same as in the case when the formula is known in advance.Comment: Fixed some typo

Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates

The class $FORMULA[s] \circ \mathcal{G}$ consists of Boolean functions computable by size-$s$ de Morgan formulas whose leaves are any Boolean functions from a class $\mathcal{G}$. We give lower bounds and (SAT, Learning, and PRG) algorithms for $FORMULA[n^{1.99}]\circ \mathcal{G}$, for classes $\mathcal{G}$ of functions with low communication complexity. Let $R^{(k)}(\mathcal{G})$ be the maximum $k$-party NOF randomized communication complexity of $\mathcal{G}$. We show: (1) The Generalized Inner Product function $GIP^k_n$ cannot be computed in $FORMULA[s]\circ \mathcal{G}$ on more than $1/2+\varepsilon$ fraction of inputs for $$s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right).$$ As a corollary, we get an average-case lower bound for $GIP^k_n$ against $FORMULA[n^{1.99}]\circ PTF^{k-1}$. (2) There is a PRG of seed length $n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right)$ that $\varepsilon$-fools $FORMULA[s] \circ \mathcal{G}$. For $FORMULA[s] \circ LTF$, we get the better seed length $O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right)$. This gives the first non-trivial PRG (with seed length $o(n)$) for intersections of $n$ half-spaces in the regime where $\varepsilon \leq 1/n$. (3) There is a randomized $2^{n-t}$-time $\#$SAT algorithm for $FORMULA[s] \circ \mathcal{G}$, where $$t=\Omega\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}.$$ In particular, this implies a nontrivial #SAT algorithm for $FORMULA[n^{1.99}]\circ LTF$. (4) The Minimum Circuit Size Problem is not in $FORMULA[n^{1.99}]\circ XOR$. On the algorithmic side, we show that $FORMULA[n^{1.99}] \circ XOR$ can be PAC-learned in time $2^{O(n/\log n)}$

Quantum Subroutine Composition

An important tool in algorithm design is the ability to build algorithms from other algorithms that run as subroutines. In the case of quantum algorithms, a subroutine may be called on a superposition of different inputs, which complicates things. For example, a classical algorithm that calls a subroutine $Q$ times, where the average probability of querying the subroutine on input $i$ is $p_i$, and the cost of the subroutine on input $i$ is $T_i$, incurs expected cost $Q\sum_i p_i E[T_i]$ from all subroutine queries. While this statement is obvious for classical algorithms, for quantum algorithms, it is much less so, since naively, if we run a quantum subroutine on a superposition of inputs, we need to wait for all branches of the superposition to terminate before we can apply the next operation. We nonetheless show an analogous quantum statement (*): If $q_i$ is the average query weight on $i$ over all queries, the cost from all quantum subroutine queries is $Q\sum_i q_i E[T_i]$. Here the query weight on $i$ for a particular query is the probability of measuring $i$ in the input register if we were to measure right before the query. We prove this result using the technique of multidimensional quantum walks, recently introduced in arXiv:2208.13492. We present a more general version of their quantum walk edge composition result, which yields variable-time quantum walks, generalizing variable-time quantum search, by, for example, replacing the update cost with $\sqrt{\sum_{u,v}\pi_u P_{u,v} E[T_{u,v}^2]}$, where $T_{u,v}$ is the cost to move from vertex $u$ to vertex $v$. The same technique that allows us to compose quantum subroutines in quantum walks can also be used to compose in any quantum algorithm, which is how we prove (*)