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SIAM Journal on Computing

The Quantum Communication Complexity of Sampling

Andris Ambainis, Leonard J. Schulman, Amnon Ta‐Shma, Umesh V. Vazirani, Avi Wigderson

Published Jan 1, 2003• 44 citations
Publisher

Abstract

Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function $f: X \times Y \rightarrow \{0,1\}$ and a probability distribution ${\cal D}$ over $X \times Y$, we define the sampling complexity of $(f, {\cal D})$ as the minimum number of bits that Alice and Bob must communicate for Alice to pick $x \in X$ and Bob to pick $y \in Y$ as well as a value z such that the resulting distribution of $(x,y,z)$ is close to the distribution $({\cal D}, f({\cal D}))$. In this paper we initiate the study of sampling complexity, in both the classical and quantum models. We give several variants of a definition. We completely characterize some of these variants and give upper and lower bounds on others. In particular, this allows us to establish an exponential gap between quantum and classical sampling complexity for the set-disjointness function.

DOI
10.1137/s009753979935476

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Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f:X×Y→{0,1}f: X \times Y \rightarrow \{0,1\}f:X×Y→{0,1} and a probability distribution D{\cal D}D over X×YX \times YX×Y, we define the sampling complexity of (f,D)(f, {\cal D})(f,D) as the minimum number of bits that Alice and Bob must communicate for Alice to pick x∈Xx \in Xx∈X and Bob to pick y∈Yy \in Yy∈Y as well as a value z such that the resulting distribution of (x,y,z)(x,y,z)(x,y,z) is close to the distribution (D,f(D))({\cal D}, f({\cal D}))(D,f(D)). In this paper we initiate the study of sampling complexity, in both the classical and quantum models. We give several variants of a definition. We completely characterize some of these variants and give upper and lower bounds on others. In particular, this allows us to establish an exponential gap between quantum and classical sampling complexity for the set-disjointness function.
Sampling is an important primitive in probabilistic and quantum algorithms. In the spirit of communication complexity, given a function f: X/spl times/Y/spl rarr/{0,1} and a probability distribution D over X/spl times/Y, we define the sampling complexity of (f,D) as the minimum number of bits Alice and Bob must communicate for Alice to pick x/spl isin/X and Bob to pick y/spl isin/Y as well as a valve z s.t. the resulting distribution of (x,y,z) is close to the distribution (D,f(D)). In this paper we initiate the study of sampling complexity, in both the classical and quantum model. We give several variants of the definition. We completely characterize some of these tasks, and give upper and lower bounds on others. In particular this allows us to establish an exponential gap between quantum and classical sampling complexity, for the set disjointness function. This is the first exponential gap for any task where the classical probabilistic algorithm is allowed to err.
We study the communication complexity of a number of graph properties where the edges of the graph GGG are distributed between Alice and Bob (i.e., each receives some of the edges as input). Our main results are: * An Omega(n) lower bound on the quantum communication complexity of deciding whether an n-vertex graph G is connected, nearly matching the trivial classical upper bound of O(n log n) bits of communication. * A deterministic upper bound of O(n^{3/2}log n) bits for deciding if a bipartite graph contains a perfect matching, and a quantum lower bound of Omega(n) for this problem. * A Theta(n^2) bound for the randomized communication complexity of deciding if a graph has an Eulerian tour, and a Theta(n^{3/2}) bound for the quantum communication complexity of this problem. The first two quantum lower bounds are obtained by exhibiting a reduction from the n-bit Inner Product problem to these graph problems, which solves an open question of Babai, Frankl and Simon. The third quantum lower bound comes from recent results about the quantum communication complexity of composed functions. We also obtain essentially tight bounds for the quantum communication complexity of a few other problems, such as deciding if G is triangle-free, or if G is bipartite, as well as computing the determinant of a distributed matrix.
We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input xxx, Bob gets an input yyy, and their goal is to each produce an output a,ba,ba,b distributed according to some pre-specified joint distribution p(a,b∣x,y)p(a,b|x,y)p(a,b∣x,y). Our results apply to any non-signaling distribution, that is, those where Alice's marginal distribution does not depend on Bob's input, and vice versa.~~~By taking a geometric view of the non-signaling distributions, we introduce a simple new technique based on affine combinations of lower-complexity distributions, and we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. Specifically, we introduce two complexity measures, one which gives lower bounds on classical communication, and one for quantum communication. These measures can be expressed as convex optimization problems. We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, information theoretic methods are necessary to prove strong lower bounds. ~~~ Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence of this is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication.
Communication complexity is the amount of communication needed to compute a function when the function inputs are distributed over multiple parties. In its simplest form, one-way communication complexity, Alice and Bob compute a function f(x,y) , where x is given to Alice and y is given to Bob, and only one message from Alice to Bob is allowed. A fundamental question in quantum information is the relationship between one-way quantum and classical communication complexities, i.e., how much shorter the message can be if Alice is sending a quantum state instead of bit strings? We make some progress towards this question with the following results.Let f:X×Y→Z∪{⊥} be a partial function and μ be a distribution with support contained in f−1(Z) . Denote d=|Z| . Let Rϵ1,μ(f) be the classical one-way communication complexity of f ; Qϵ1,μ(f) be the quantum one-way communication complexity of f and Qϵ1,μ,∗(f) be the entanglement-assisted quantum one-way communication complexity of f , each with distributional error (average error over μ ) at most ϵ . We show:1) If μ is a product distribution, η>0 and 0≤ϵ≤1−1/d , then, R2ϵ−dϵ2/(d−1)+η1,μ(f)≤2Qϵ1,μ,∗(f)+O(log⁡log⁡(1/η)). 2)If μ is a non-product distribution and Z={0,1} , then ∀ϵ,η>0 such that ϵ/η+η<0.5 , R3η1,μ(f)=O(Qϵ1,μ(f)⋅CS(f)/η3), where CS(f)=maxyminz&am
We study an extension of the standard two-party communication model in which Alice and Bob hold probability distributions ppp and qqq over domains XXX and YYY, respectively. Their goal is to estimate Ex∼p, y∼q[f(x,y)] \mathbb{E}_{x \sim p,\, y \sim q}[f(x, y)] Ex∼p,y∼q​[f(x,y)] to within additive error ε\varepsilonε for a bounded function fff, known to both parties. We refer to this as the distributed estimation problem. Special cases of this problem arise in a variety of areas including sketching, databases and learning. Our goal is to understand how the required communication scales with the communication complexity of fff and the error parameter ε\varepsilonε. The random sampling approach -- estimating the mean by averaging fff over O(1/ε2)O(1/\varepsilon^2)O(1/ε2) random samples -- requires O(R(f)/ε2)O(R(f)/\varepsilon^2)O(R(f)/ε2) total communication, where R(f)R(f)R(f) is the randomized communication complexity of fff. We design a new debiasing protocol which improves the dependence on 1/ε1/\varepsilon1/ε to be linear instead of quadratic. Additionally we show better upper bounds for several special classes of functions, including the Equality and Greater-than functions. We introduce lower bound techniques based on spectral methods and discrepancy, and show the optimality of many of our protocols: the debiasing protocol is tight for general functions, and that our protocols for the equality and greater-than functions are also optimal. Furthermore, we show that among full-rank Boolean functions, Equality is essentially the easiest.
We study the communication complexity of computing functions F:{0,1}n×{0,1}n→{0,1}F:\{0,1\}^n\times \{0,1\}^n \rightarrow \{0,1\}F:{0,1}n×{0,1}n→{0,1} in the memoryless communication model. Here, Alice is given x∈{0,1}nx\in \{0,1\}^nx∈{0,1}n, Bob is given y∈{0,1}ny\in \{0,1\}^ny∈{0,1}n and their goal is to compute F(x,y) subject to the following constraint: at every round, Alice receives a message from Bob and her reply to Bob solely depends on the message received and her input x; the same applies to Bob. The cost of computing F in this model is the maximum number of bits exchanged in any round between Alice and Bob (on the worst case input x,y). In this paper, we also consider variants of our memoryless model wherein one party is allowed to have memory, the parties are allowed to communicate quantum bits, only one player is allowed to send messages. We show that our memoryless communication model capture the garden-hose model of computation by Buhrman et al. (ITCS'13), space bounded communication complexity by Brody et al. (ITCS'13) and the overlay communication complexity by Papakonstantinou et al. (CCC'14). Thus the memoryless communication complexity model provides a unified framework to study space-bounded communication models. We establish the following: (1) We show that the memoryless communication complexity of F equals the logarithm of the size of the smallest bipartite branching program computing F (up to a factor 2); (2) We show that memoryless communication complexity equals garden-hose complexity; (3) We exhibit various exponential separations between these memoryless communication models. We end with an intriguing open question: can we find an explicit function F and universal constant c>1 for which the memoryless communication complexity is at least clog⁡nc \log nclogn? Note that c≥2+εc\geq 2+\varepsilonc≥2+ε would imply a Ω(n2+ε)Ω(n^{2+\varepsilon})Ω(n2+ε) lower bound for general formula size, improving upon the best lower bound by Nečiporuk in 1966.