Buhrman, Cleve and Wigderson (STOC'98) showed that for every Boolean function f : {-1,1}ⁿ → {-1,1} and G ∈ {AND₂, XOR₂}, the bounded-error quantum communication complexity of the composed function f∘G equals O(𝖰(f) log n), where 𝖰(f) denotes the bounded-error quantum query complexity of f. This is achieved by Alice running the optimal quantum query algorithm for f, using a round of O(log n) qubits of communication to implement each query. This is in contrast with the classical setting, where it is easy to show that 𝖱^{cc}(f∘G) ≤ 2𝖱(f), where 𝖱^{cc} and 𝖱 denote bounded-error communication and query complexity, respectively. Chakraborty et al. (CCC'20) exhibited a total function for which the log n overhead in the BCW simulation is required. This established the somewhat surprising fact that quantum reductions are in some cases inherently more expensive than classical reductions. We improve upon their result in several ways. - We show that the log n overhead is not required when f is symmetric (i.e., depends only on the Hamming weight of its input), generalizing a result of Aaronson and Ambainis for the Set-Disjointness function (Theory of Computing'05). Our upper bound assumes a shared entangled state, though for most symmetric functions the assumed number of entangled qubits is less than the communication and hence could be part of the communication. - In order to prove the above, we design an efficient distributed version of noisy amplitude amplification that allows us to prove the result when f is the OR function. This also provides a different, and arguably simpler, proof of Aaronson and Ambainis’s O(√n) communication upper bound for Set-Disjointness. - In view of our first result above, one may ask whether the log n overhead in the BCW simulation can be avoided even when f is transitive, which is a weaker notion of symmetry. We give a strong negative answer by showing that the log n overhead is still necessary for some transitive functions even when we allow the quantum communication protocol an error probability that can be arbitrarily close to 1/2 (this corresponds to the unbounded-error model of communication). - We also give, among other things, a general recipe to construct functions for which the log n overhead is required in the BCW simulation in the bounded-error communication model, even if the parties are allowed to share an arbitrary prior entangled state for free.

In 1974, Ralph Merkle proposed the first unclassified protocol for secure communications over insecure channels. When legitimate communicating parties are willing to spend an amount of computational effort proportional to some parameter N, an eavesdropper cannot break into their communication without spending a time proportional to $$N^2$$ , which is quadratically more than the legitimate effort. In a quantum world, however, Merkle’s protocol is immediately broken by Grover’s algorithm, but it is easily repaired if we are satisfied with a quantum protocol against which a quantum adversary needs to spend a time proportional to $$N^{3/2}$$ in order to break it. Can we do better? We give two new key establishment protocols in the spirit of Merkle’s. The first one, which requires the legitimate parties to have access to a quantum computer, resists any quantum adversary who is not willing to make an effort at least proportional to $$N^{5/3}$$ , except with vanishing probability. Our second protocol is purely classical, yet it requires any quantum adversary to work asymptotically harder than the legitimate parties, again except with vanishing probability. In either case, security is proved for a typical run of the protocols: the probabilities are taken over the random (or quantum) choices made by the legitimate participants in order to establish their key as well as over the random (or quantum) choices made by the adversary who is trying to be privy to it.

We propose a new framework for turning quantum search algorithms that decide into quantum algorithms for finding a solution. Consider we are given an abstract quantum search algorithm A that can determine whether a target g exists or not. We give a general construction of another operator U that both determines and finds the target, whenever one exists. Our amplification method at most doubles the cost over using A, has little overhead, and works by controlling the evolution of A. This is the first known general framework to the open question of turning abstract quantum search algorithms into quantum algorithms for finding a solution. We next apply the framework to random walks. We develop a new classical algorithm and a new quantum algorithm for finding a unique marked element. Our new random walk finds a unique marked element using H update operations and 1/eps checking operations. Here H is the hitting time, and eps is the probability that the stationary distribution of the walk is in the marked state. Our classical walk is derived via quantum arguments. Our new quantum algorithm finds a unique marked element using H^(1/2) update operations and 1/eps^(1/2) checking operations, up to logarithmic factors. This is the first known quantum algorithm being simultaneously quadratically faster in both parameters. We also show that the framework can simulate Grover's quantum search algorithm, amplitude amplification, Szegedy's quantum walks, and quantum interpolated walks.

We give a quantum algorithm for finding a marked element on the grid when there are multiple marked elements. Our algorithm uses quadratically fewer steps than a random walk on the grid, ignoring logarithmic factors. This is the first known quantum walk that finds a marked element in a number of steps less than the square-root of the extended hitting time. We also give a new tighter upper bound on the extended hitting time of a marked subset, expressed in terms of the hitting times of its members.

The quantum adversary method is one of the most successful techniques for proving lower bounds on quantum query complexity. It gives optimal lower bounds for many problems, has application to classical complexity in formula size lower bounds, and is versatile with equivalent formulations interms of weight schemes, eigen values, and Kolmogorov complexity. All these formulations rely on the principlethat if an algorithm successfully computes a function then, in particular, itis able to distinguish between inputs which map to different values.

Quantum algorithms for graph problems are considered, both in the adjacency matrix model and in an adjacency list-like array model. We give almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, Strong Connectivity, Minimum Spanning Tree, and Single Source Shortest Paths. For example, we show that the query complexity of Minimum Spanning Tree is in $\Theta(n^{3/2})$ in the matrix model and in $\Theta(\sqrt{nm})$ in the array model, while the complexity of Connectivity is also in $\Theta(n^{3/2})$ in the matrix model but in $\Theta(n)$ in the array model. The upper bounds utilize search procedures for finding minima of functions under various conditions.

We present several applications of quantum amplitude amplification for deciding whether all elements in the image of a given function are distinct, for finding an intersection of two sorted tables, and for finding a triangle in a graph. Our techniques generalize and improve those of Brassard, Hoyer, and Tapp [ACM SIGACT News, 28 (1997), pp. 14--19]. This shows that in the quantum world element distinctness is significantly easier than sorting, in contrast to the classical world.

This paper investigates various aspects of the nonlocal effects that can arise when entangled quantum information is shared between two parties. A natural framework for studying nonlocality is that of cooperative games with incomplete information, where two cooperating players may share entanglement. Here, nonlocality can be quantified in terms of the values of such games. We review some examples of non-locality and show that it can profoundly affect the soundness of two-prover interactive proof systems. We then establish limits on nonlocal behavior by upper-bounding the values of several of these games. These upper bounds can be regarded as generalizations of the so-called Tsirelson inequality. We also investigate the amount of entanglement required by optimal and nearly optimal quantum strategies.

We present several applications of quantum amplitude amplification to finding claws and collisions in ordered or unordered functions. Our algorithms generalize those of Brassard, Hoyer, and Tapp (1998), and imply an O(N/sup 3/4/ log N) quantum upper bound for the element distinctness problem in the comparison complexity model. This contrasts with /spl Theta/(N log N) classical complexity. We also prove a lower bound of /spl Omega/(/spl radic/N) comparisons for this problem and derive bounds for a number of related problems.