At the end of this introductory chapter, we mention some other results in quantum communication complexity or related models:
• Zero-error protocols. We have seen quantum-classical separations in the exact and the bounded-error settings. What about the zero-error setting?
It was observed in [43] that we can combine Lemma 6.4.1 with our zero-error quantum algorithms for AND-OR trees (Section 2.7.2) to get quantum zero-error protocols for the total function which is the dth-level AND-OR tree of x∧ y. These protocols use O(n1/2+1/dlog n) qubits of communication. We conjecture that classical zero-error protocols need Ω(n) communication for these functions (for fixed d), but were unfortunately unable to prove this.
Klauck [101] later constructed a similar function f for which he could prove a good lower bound on Rcc0(f ), thus establishing the first quantum-classical separation between Qcc0(f ) and Rcc0(f ) for a total function.
• One-way communication. Suppose the communication is one-round: Al-ice just sends qubits to Bob. Klauck [101] showed for all total functions that quantum communication is not significantly better than classical communi-cation for one-way communicommuni-cation in the exact or zero-error settings.
• Rounds. It is well known in classical communication complexity that al-lowing Alice and Bob k + 1 rounds of communication instead of k reduces
6.7. Summary 113 the required communication exponentially for some functions. An analo-gous result has recently been shown for quantum communication [104].
• Quantum sampling. For the sampling problem, Alice and Bob do not want to compute some f (x, y), but instead want to sample an (x, y)-pair according to some known joint probability distribution, using as little com-munication as possible. Ambainis et al. [12] give a tight algebraic charac-terization of quantum sampling complexity, and exhibit an exponential gap between the quantum and classical communication required for a sampling problem related to disjointness.
• Spooky communication. Brassard, Cleve, and Tapp [33] exhibit tasks that can be achieved in the quantum world with entanglement and no communication, but which would require communication in the classical world. They call such quantum protocols “spooky” in reference to Ein-stein’s description of certain quantum effects as “spooky actions at a dis-tance” (“spukhafte Fernwirkungen”). Brassard, Cleve, and Tapp also give upper and lower bounds on the amount of classical communication needed to “simulate” EPR-pairs. Their results may be viewed as quantitative ex-tensions of the famous Bell inequalities [20].
6.7 Summary
The basic problem of communication complexity is the following: Alice receives an input x and Bob receives an input y (usually of n bits each), and together they want to compute some function f (x, y) using as little communication be-tween them as possible. This model of distributed computation has found many applications in classical computing. Quantum communication complexity asks whether the amount of communication of such a problem can be reduced sig-nificantly if Alice and Bob can communicate qubits and/or make use of shared entanglement. The answer is ‘yes’ (sometimes). In this chapter we described the main examples known where quantum communication complexity is significantly less than classical communication complexity, as well as some applications.
Chapter 7
Lower Bounds for Quantum Communication Complexity
This chapter is based on the papers
• H. Buhrman and R. de Wolf. Communication Complexity Lower Bounds by Polynomials. In Proceedings of 16th IEEE Annual Conference on Com-putational Complexity (CCC 2001), pages 120–130, 2001.
• R. de Wolf. Characterization of Non-Deterministic Quantum Query and Quantum Communication Complexity. In Proceedings of 15th IEEE An-nual Conference on Computational Complexity (CCC 2000), pages 271–278, 2000.
7.1 Introduction
To repeat the previous chapter, the field of communication complexity deals with the following kind of problem: Alice receives some input x ∈ X, Bob receives some y ∈ Y , and together they want to compute some (usually Boolean) func-tion f (x, y) which depends on both x and y. At the end of the protocol they should both have the same output. We are interested in the minimum amount of communication that Alice and Bob need. The communication may be classical or quantum, and the protocols may be exact, zero-error, or bounded-error.
In Section 6.4, we saw some examples where quantum communication com-plexity was exponentially smaller than classical communication comcom-plexity. The question arises how big the gaps between quantum and classical can be for vari-ous (classes of) functions. In order to answer this, we need to exhibit limits on the power of quantum communication complexity, i.e., establish lower bounds on quantum communication complexity. Few such lower bound techniques are cur-rently known. Some lower bound methods are available for QccE(f ) [165, 108, 54,
115
12], but the only lower bound known for the entanglement-enhanced complexity Qcc∗E(f ) is for the inner product function [54].1 For the case of lower bounds on bounded-error protocols, our current techniques are even more limited. The main purpose of this chapter is to develop new tools for proving lower bounds on quantum communication protocols.
The tools we will develop are quite successful for proving lower bounds on exact quantum protocols. A strong and well known lower bound for the classical deterministic complexity Dcc(f ) is given by the logarithm of the rank (over the field of real numbers) of the communication matrix for f [121]. As first noted in [42], techniques of Yao [165] and Kremer [108] imply that an Ω(log rank(f ))-bound also holds for QccE(f ). Our first result in this chapter is to extend this bound to the entanglement-enhanced complexity Qcc∗E(f ) and to derive the op-timal constant:2
Qcc∗E(f ) ≥ log(rank(f )− 1)
2 .
This implies n/2 lower bounds for the Qcc∗E-complexity of the equality and dis-jointness problems, for which no good bounds were known prior to this work.
This n/2 is tight up to 1 bit, since Alice can send her n-bit input to Bob with n/2 qubits and n/2 EPR-pairs using superdense coding [24]; Bob can then com-pute f (x, y) and send back the 1-bit answer. Our n/2 lower bound also provides a new proof of optimality of superdense coding: if we were able to send more than 2 classical bits via 1 qubit of communication, then we would violate our communication complexity lower bounds. The same n/2 bound can be shown to hold for almost all functions (which, however, does not preclude the existence of interesting problems with large quantum-classical gaps).
In another direction, proof of the well known “log-rank conjecture” (Dcc(f )≤ (log rank(f ))k for some k) would now imply polynomial equivalence between Dcc(f ) and Qcc∗E(f ) (as already noted for Dcc(f ) and QccE(f ) in [12]). How-ever, this conjecture is a long standing open question which is probably hard to solve in full generality. In order to get a better handle on rank(f ), we re-late it to a property of polynomials. If our communication problem is of the form f (x, y) = g(x∧ y) for some Boolean function g (where x ∧ y is the n-bit string obtained by bitwise ANDing x and y), then we prove that rank(f ) equals the number of monomials mon(g) in the unique representing polynomial for g.
Since mon(g) is often easy to count, this relation allows us to prove polynomial equivalence of Dcc(f ) and Qcc∗E(f ) for the special cases where g is monotone or
1Recall from the previous chapter that for the Q∗ and C∗ complexities we only count the number of communicated qubits, not the number of prior EPR-pairs consumed by the protocol.
2During discussions we had with Michael Nielsen in Cambridge (UK) in the summer of 1999, it appeared that an equivalent result can be derived from results about Schmidt numbers in [129, Section 6.4.2].
Actually, in the conference version of this work [48], the lower bound was stated without the
‘−1’, but that proof contained a bug.
7.2. Lower Bounds for Exact Protocols 117