Quantum Counterfeit Coin Problems

Longest common substring (LCS) is an important text processing problem, which has recently been investigated in the quantum query model. The decision version of this problem, LCS with threshold \(d\) , asks whether two length- \(n\) input strings have a common substring of length \(d\) . The two extreme cases, \(d=1\) and \(d=n\) , correspond, respectively to Element Distinctness and Unstructured Search, two fundamental problems in quantum query complexity. However, the intermediate case \(1\ll d\ll n\) was not fully understood. We show that the complexity of LCS with threshold \(d\) smoothly interpolates between the two extreme cases up to \(n^{o(1)}\) factors: — LCS with threshold \(d\) has a quantum algorithm in \(n^{2/3+o(1)}/d^{1/6}\) query complexity and time complexity, and requires at least \(\Omega(n^{2/3}/d^{1/6})\) quantum query complexity. Our result improves upon previous upper bounds \(\widetilde{O}(\min\{n/d^{1/2},n^{2/3}\})\) (Le Gall and Seddighin ITCS 2022, Akmal and Jin SODA 2022), and answers an open question of Akmal and Jin. Our main technical contribution is a quantum speed-up of the powerful String Synchronizing Set technique introduced by Kempa and Kociumaka (STOC 2019). It consistently samples \(n/\tau^{1-o(1)}\) synchronizing positions in the string depending on their length- \(\Theta(\tau)\) contexts, and each synchronizing position can be reported by a quantum algorithm in \(\widetilde{O}(\tau^{1/2+o(1)})\) time. Our quantum string synchronizing set also yields a near-optimal LCE data structure in the quantum setting. As another application of our quantum string synchronizing set, we study the \(k\) -mismatch Matching problem, which asks if the pattern has an occurrence in the text with at most \(k\) Hamming mismatches. Using a structural result of Charalampopoulos et al. (FOCS 2020), we obtain: — \(k\) -mismatch matching has a quantum algorithm with \(k^{3/4}n^{1/2+o(1)}\) query complexity and \(\widetilde{O}(kn^{1/2})\) time complexity. We also observe a non-matching quantum query lower bound of \(\Omega(\sqrt{kn})\) .

Inspired by the Elitzur-Vaidman bomb testing problem [19], we introduce a new query complexity model, which we call bomb query complexity B(f). We investigate its relationship with the usual quantum query complexity Q(f), and show that B(f) = Θ(Q(f)2).This result gives a new method to upper bound the quantum query complexity: we give a method of finding bomb query algorithms from classical algorithms, which then provide nonconstructive upper bounds on [EQUATION]. We subsequently were able to give explicit quantum algorithms matching our upper bound method. We apply this method on the single-source shortest paths problem on unweighted graphs, obtaining an algorithm with O(n1.5) quantum query complexity, improving the best known algorithm of [EQUATION] [21]. Applying this method to the maximum bipartite matching problem gives an O(n1.75) algorithm, improving the best known trivial O(n2) upper bound.

It is known that the classical and quantum query complexities of a total Boolean function f are polynomially related to the degree of its representing polynomial, but the optimal exponents in these relations are unknown. We show that the quantum query complexity of f is linearly related to the degree of a non-deterministic polynomial for f. We also prove a quantum-classical gap of 1 vs. N for query complexity for a total f. In the case of quantum communication complexity there is a (partly undetermined) relation between the complexity of f and the logarithm of the rank of its communication matrix. We show that the quantum communication complexity of f is linearly related to the logarithm of the rank of a version of the communication matrix and that it can be exponentially smaller than its classical counterpart.

It has recently been shown that quantum computers can efficiently solve the Heisenberg hidden subgroup problem, a problem whose classical query complexity is exponential. This quantum algorithm was...

We present several results on the complexity of various forms of Sperner's Lemma in the black-box model of computing. We give a deterministic algorithm for Sperner problems over pseudo-manifolds of arbitrary dimension. The query complexity of our algorithm is linear in the separation number of the skeleton graph of the manifold and the size of its boundary. As a corollary we get an $O(\sqrt{n})$ deterministic query algorithm for the black-box version of the problem 2D-SPERNER, a well studied member of Papadimitriou's complexity class PPAD. This upper bound matches the $\Omega(\sqrt{n})$ deterministic lower bound of Crescenzi and Silvestri. The tightness of this bound was not known before. In another result we prove for the same problem an $\Omega(^{4}\sqrt{n})$ lower bound for its probabilistic, and an $\Omega(^{8}\sqrt{n})$ lower bound for its quantum query complexity, showing that all these measures are polynomially related.

The oracle identification problem (OIP) is, given a set S of M Boolean oracles out of 2N ones, to determine which oracle in S is the current black-box oracle. We can exploit the information that candidates of the current oracle is restricted to S. The OIP contains several concrete problems such as the original Grover search and the Bernstein-Vazirani problem. Our interest is in the quantum query complexity, for which we present several upper bounds. They are quite general and mostly optimal: (i) The query complexity of OIP is \(O(\sqrt{N {\rm log} M {\rm log} N}{\rm log log} M)\) for anyS such that M = |S| > N, which is better than the obvious bound N if M \(< 2^{N/log^3 N}\). (ii) It is \(O(\sqrt{N})\) for anyS if |S| = N, which includes the upper bound for the Grover search as a special case. (iii) For a wide range of oracles (|S| = N) such as random oracles and balanced oracles, the query complexity is \(O(\sqrt{N/K})\), where K is a simple parameter determined by S.KeywordsBoolean FunctionQuantum ComputationQuantum AlgorithmRandom OracleQuery ComplexityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Let H be a (non-empty) graph on n vertices, possibly containing isolated vertices. Let f_H(G) = 1 iff the input graph G on n vertices contains H as a (not necessarily induced) subgraph. Let alpha_H denote the cardinality of a maximum independent set of H. In this paper we show: Q(f_H) = Omega( sqrt{alpha_H * n}), where Q(f_H) denotes the quantum query complexity of f_H. As a consequence we obtain lower bounds for Q(f_H) in terms of several other parameters of H such as the average degree, minimum vertex cover, chromatic number, and the critical probability. We also use the above bound to show that Q(f_H) = Omega(n^{3/4}) for any H, improving on the previously best known bound of Omega(n^{2/3}) [M. Santha/A. Chi-Chih Yao, unpublished manuscript]. Until very recently, it was believed that the quantum query complexity is at least square root of the randomized one. Our Omega(n^{3/4}) bound for Q(f_H) matches the square root of the current best known bound for the randomized query complexity of f_H, which is Omega(n^{3/2}) due to Groger. Interestingly, the randomized bound of Omega(alpha_H * n) for f_H still remains open. We also study the Subgraph Homomorphism Problem, denoted by f_{[H]}, and show that Q(f_{[H]}) = Omega(n). Finally we extend our results to the 3-uniform hypergraphs. In particular, we show an Omega(n^{4/5}) bound for quantum query complexity of the Subgraph Isomorphism, improving on the previously known Omega(n^{3/4}) bound. For the Subgraph Homomorphism, we obtain an Omega(n^{3/2}) bound for the same.

We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be a Boolean function and consider a function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q(f), we give an algorithm for evaluating F using [MATH HERE]) quantum queries. This improves on the bound of [MATH HERE]) that follows by treating each conjunction independently, and is tight for worst-case choices of f. Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f.By recursively applying our composition theorems, we obtain a nearly optimal O(n1−2−d) upper bound on the quantum query complexity and approximate degree of linear-size depth-d AC0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC0 circuits. We also show that any substantially faster learning algorithm will require fundamentally new techniques.

Experimental study of Forrelation in nuclear spins

This article initiates the study of testing properties of directed graphs. In particular, the article considers the most basic property of directed graphs--acyclicity. Because the choice of representation affects the choice of algorithm, the two main representations of graphs are studied. For the adjacency-matrix representation, most appropriate for dense graphs, a testing algorithm is developed that requires query and time complexity of O(1/e2), where e is a distance parameter independent of the size of the graph. The algorithm, which can probe the adjacency matrix of the graph, accepts every graph that is acyclic, and rejects, with probability at least 2/3, every graph whose adjacency matrix should be modified in at least e fraction of its entries so that it becomes acyclic. For the incidence list representation, most appropriate for sparse graphs, an Ω(|V|1/3) lower bound is proved on the number of queries and the time required for testing, where V is the set of vertices in the graph. Along with acyclicity, this article considers the property of strong connectivity. Contrasting upper and lower bounds are proved for the incidence list representation. In particular, if the testing algorithm can query on both incoming and outgoing edges at each vertex, then it is possible to test strong connectivity in O(1/e) time and query complexity. On the other hand, if the testing algorithm only has access to outgoing edges, then Ω(√N) queries are required to test for strong connectivity.

The approximate degree of a Boolean function f(x1,x2,…,xn) is the minimum degree of a real polynomial that approximates f pointwise within 1/3. Upper bounds on approximate degree have a variety of applications in learning theory, differential privacy, and algorithm design in general. Nearly all known upper bounds on approximate degree arise in an existential manner from bounds on quantum query complexity.

Query complexity of generalized Simon's problem

The celebrated minimax principle of Yao (1977) says that for any Boolean-valued function $f$ with finite domain, there is a distribution $\mu$ over the domain of $f$ such that computing $f$ to error $\epsilon$ against inputs from $\mu$ is just as hard as computing $f$ to error $\epsilon$ on worst-case inputs. Notably, however, the distribution $\mu$ depends on the target error level $\epsilon$ : the hard distribution which is tight for bounded error might be trivial to solve to small bias, and the hard distribution which is tight for a small bias level might be far from tight for bounded error levels. In this work, we introduce a new type of minimax theorem which can provide a hard distribution $\mu$ that works for all bias levels at once. We show that this works for randomized query complexity, randomized communication complexity, some randomized circuit models, quantum query and communication complexities, approximate polynomial degree, and approximate logrank. We also prove an improved version of Impagliazzo's hardcore lemma. Our proofs rely on two innovations over the classical approach of using Von Neumann's minimax theorem or linear programming duality. First, we use Sion's minimax theorem to prove a minimax theorem for ratios of bilinear functions representing the cost and score of algorithms. Second, we introduce a new way to analyze low-bias randomized algorithms by viewing them as “forecasting algorithms” evaluated by a certain proper scoring rule. The expected score of the forecasting version of a randomized algorithm appears to be a more fine-grained way of analyzing the bias of the algorithm. We show that such expected scores have many elegant mathematical properties: for example, they can be amplified linearly instead of quadratically. We anticipate forecasting algorithms will find use in future work in which a fine-grained analysis of small-bias algorithms is required.

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function f on n=2k bits defined by a complete binary tree of NAND gates of depth k, which achieves R0(f) = O(D(f)0.7537…). We show this is false by giving an example of a total boolean function f on n bits whose deterministic query complexity is Ω(n/log(n)) while its zero-error randomized query complexity is O(√n). We further show that the quantum query complexity of the same function is O(n1/4), giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total boolean function g on n variables that has zero-error randomized query complexity Ω(n/log(n)) and bounded-error randomized query complexity R(g) = O(√n). This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is QE(g) = O(√n). These functions show that the relations D(f) = O(R1(f)2) and R0(f) = O(R(f)2) are optimal, up to poly-logarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between Q and R0, a 3/2-power separation between QE and R, and a 4th power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by Goos, Pitassi, and Watson which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.

Focusing on property testing tasks that have query complexity that is independent of the size of the tested object (i.e., depends on the proximity parameter only), we prove the existence of a rich hierarchy of the corresponding complexity classes. That is, for essentially any function $$q : (0, 1] \rightarrow \mathbb{N}$$ , we prove the existence of properties for which $$\epsilon$$ -testing has query complexity $$\Theta(q(\Theta(\epsilon)))$$ . Such results are proved in three standard domains that are often considered in property testing: generic functions, adjacency predicates describing (dense) graphs, and incidence functions describing bounded-degree graphs. These results complement hierarchy theorems of Goldreich, Krivelevich, Newman, and Rozenberg (Computational Complexity, 2012), which refer to the dependence of the query complexity on the size of the tested object, and focus on the case that the proximity parameter is set to some small positive constant. We actually combine both flavors and get tight results on the query complexity of testing when allowing the query complexity to depend on both the size of the object and the proximity parameter.