Can Grover’s algorithm speed up search of a physical region—for example a 2-D grid of size √ n × √ n? The problem is that √ n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Benioff. In particular, we show how to search a d-dimensional hypercube in time O ( √ n) for d ≥ 3, or O ( √ nlog 5/2 n) for d = 2. More generally, we introduce a model of quantum query complexity on graphs, motivated by fundamental physical limits on information storage, particularly the holographic principle from black hole thermodynamics. Our results in this model include almost-tight upper and lower bounds for many search tasks; a generalized algorithm that works for any graph with good expansion properties, not just hypercubes; and relationships among several notions of ‘locality’ for unitary matrices acting on graphs. As an application of our results, we give an O (√ n)-qubit communication protocol for the disjointness problem, which improves an upper bound of Høyer and de Wolf and matches a lower bound of Razborov.

The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. We exhibit a function with polynomial degree M and quantum query complexity Ω(M 1.321...). This is the first superlinear separation between polynomial degree and quantum query complexity. The lower bound is shown by a new, more general version of quantum adversary method. 1

A strong direct product theorem says that if we want to compute k independent instances of a function, using less than k times the resources needed for one instance, then our overall success probability will be exponentially small in k. We establish such theorems for the classical as well as quantum query complexity of the OR function. This implies slightly weaker direct product results for all total functions. We prove a similar result for quantum communication protocols computing k instances of the Disjointness function. Our direct product theorems...

Abstract. We prove new lower bounds for bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [34] to the quantum case. Applying this method we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other lower bound methods based on the Fourier transform, notably showing that � ¯s(f) / log n, for the average sensitivity ¯s(f) of a function f, yields a lower bound on the bounded error quantum communication complexity of f(x ∧ y ⊕ z), where x is a Boolean word held by Alice and y, z are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are O(log n).

In a breakthrough result, Razborov (2003) gave optimal lower bounds on the communication complexity of every function f of the form f (x, y) = D(|x ∧ y|) for some D: {0, 1,..., n} → {0, 1}, in the bounded-error quantum model with and without prior entanglement. This was proved by the multidimensional discrepancy method. We give an entirely different proof of Razborov’s result, using the original, one-dimensional discrepancy method. This refutes the commonly held intuition (Razborov 2003) that the original discrepancy method fails for functions such as disjointness. More importantly, our communication lower bounds hold for a much broader class of functions for which no methods were available. Namely, fix an arbitrary function f: {0, 1} n/4 → {0, 1} and let A be the Boolean matrix whose columns are each an application of f to some subset of the variables x1, x2,..., xn. We prove that the communication complexity of A in the bounded-error quantum model with and without prior entanglement is Ω(d), where d is the approximate degree of f. From this result, Razborov’s lower bounds follow easily. Our result also establishes a large new class of total Boolean functions whose quantum communication complexity (regardless of prior entanglement) is at best polynomially smaller than their classical complexity. Our proof method is a novel combination of two ingredients. The first is a certain equivalence of approximation and orthogonality in Euclidean n-space, which follows by linear-programming duality. The second is a new construction of suitably structured matrices with low spectral norm, the pattern matrices, which we realize using matrix analysis and the Fourier transform over Z n 2. The method of this paper has recently inspired important progress in multiparty communication complexity. 1

We introduce a new method to derive lower bounds on randomized and quantum communication complexity. Our method is based on factorization norms, a notion from Banach Space theory. As we show, our bounds compare favorably with previously known bounds. Aside from the new results that we derive, our method yields new and more transparent proofs of some known results as well. Among our new results we extend some known lower bounds to the realm of quantum communication complexity with entanglement. 1

We present two results for computational models that allow error probabilities close to 1/2. First, most computational complexity classes have an analogous class in communication complexity. The class PP in fact has two, a version with weakly restricted bias called PP cc, and a version with unrestricted bias called UPP cc. Ever since their introduction by Babai, Frankl, and Simon in 1986, it has been open whether these classes are the same. We show that PP cc � UPP cc. Our proof combines a query complexity separation due to Beigel with a technique of Razborov that translates the acceptance probability of quantum protocols to polynomials. Second, we study how small the bias of minimal-degree polynomials that sign-represent Boolean functions needs to be. We show that the worst-case bias is at worst doubleexponentially small in the sign-degree (which was very recently shown to be optimal by Podolski), while the averagecase bias can be made single-exponentially small in the sign-degree (which we show to be close to optimal). 1

The polynomial method and Ambainis’s lower bound method are two main quantum lower bound techniques. Recently Ambainis showed that the polynomial method is not tight. The present paper aims at studying the limitation of Ambainis’s lower bounds. We first give a generalization of the three known Ambainis’s lower bound theorems. Then it is shown that all these four Ambainis’s lower bounds have an upper bound, which is in terms of certificate complexity. This implies that for some problems such as TRIANGLE, k-CLIQUE, and BIPARTITE/GRAPH MATCHING whose quantum query complexities are still open, the best known lower bounds cannot be further improved by using Ambainis’s techniques. Another consequence is that all the Ambainis’s lower bounds are not tight. Finally, we show that for total functions, this upper bound for Ambainis’s lower bounds can be further improved. This also implies limitation of Ambainis’s method on some specific problems such as AND-OR TREE, whose precise quantum complexity is still unknown. 1

A major open problem in communication complexity is whether or not quantum protocols can be exponentially more efficient than classical ones for computing a total Boolean function in the twoparty interactive model. The answer appears to be “No”. In 2002, Razborov proved this conjecture for so far the most general class of functions F (x, y) = fn(x1 · y1, x2 · y2,..., xn · yn), where fn is a symmetric Boolean function on n Boolean inputs, and xi, yi are the i’th bit of x and y, respectively. His elegant proof critically depends on the symmetry of fn. We develop a lower-bound method that does not require symmetry and prove the conjecture for a broader class of functions. Each of those functions F (x, y) is the “block-composition ” of a “building block ” gk: {0, 1} k × {0, 1} k → {0, 1}, and an fn: {0, 1} n → {0, 1}, such that F (x, y) = fn(gk(x1, y1), gk(x2, y2),..., gk(xn, yn)), where xi and yi are the i’th k-bit block of x, y ∈ {0, 1} nk, respectively. We show that as long as gk itself is “hard ” enough, its block-composition with an arbitrary fn has polynomially related quantum and classical communication complexities. Our approach gives an alternative proof for Razborov’s result (albeit with a slightly weaker parameter), and establishes new quantum lower bounds. For example, when gk is the Inner Product function with k = Ω(log n), the deterministic communication complexity of its block-composition with any fn is asymptotically at most the quantum complexity to the power of 7.

Abstract. We prove that the concept class of disjunctions cannot be pointwise approximated by linear combinations of any small set of arbitrary real-valued functions. That is, suppose there exist functions φ1,...,φr: {−1,1} n → R with the property that every disjunction f on n variables has � f − ∑ r i=1 αiφi� ∞ � 1/3 for some reals α1,...,αr. We prove that then r � 2 Ω( √ n). This lower bound is tight. We prove an incomparable lower bound for the concept class of linear-size DNF formulas. For the concept class of majority functions, we obtain a lower bound of Ω(2 n /n), which almost meets the trivial upper bound of 2 n for any concept class. These lower bounds substantially strengthen and generalize the polynomial approximation lower bounds of Paturi and show that the regression-based agnostic learning algorithm of Kalai et al. is optimal. Our techniques involve a careful application of results in communication complexity due to Razborov and Buhrman et al. 1