[Paper Review] Communication Complexity Lower Bounds by Polynomials

Harry Buhrman, Ronald de Wolf|ArXiv.org|Oct 12, 1999

Quantum Computing Algorithms and Architecture29 references10 citations

TL;DR

This paper establishes communication complexity lower bounds using polynomial methods, proving that the log-rank conjecture holds for quantum communication with unlimited prior entanglement. It demonstrates the polynomial equivalence of quantum and classical communication complexity for several function classes, providing strong bounds for exact protocols and weaker bounds for bounded-error settings.

ABSTRACT

The quantum version of communication complexity allows the two communicating parties to exchange qubits and/or to make use of prior entanglement (shared EPR-pairs). Some lower bound techniques are available for qubit communication complexity, but except for the inner product function, no bounds are known for the model with unlimited prior entanglement. We show that the log-rank lower bound extends to the strongest model (qubit communication + unlimited prior entanglement). By relating the rank of the communication matrix to properties of polynomials, we are able to derive some strong bounds for exact protocols. In particular, we prove both the "log-rank conjecture" and the polynomial equivalence of quantum and classical communication complexity for various classes of functions. We also derive some weaker bounds for bounded-error quantum protocols.

Motivation & Objective

• To extend the log-rank lower bound technique to the strongest quantum communication model, including qubit communication and unlimited prior entanglement.
• To investigate the relationship between the rank of the communication matrix and the degree of representing polynomials.
• To determine whether quantum and classical communication complexities are polynomially equivalent for specific function classes.
• To derive strong lower bounds for exact quantum protocols and weaker bounds for bounded-error protocols using polynomial methods.

Proposed method

• Relates the rank of the communication matrix to the degree of multilinear polynomials that represent the function.
• Uses polynomial degree as a proxy to derive lower bounds on communication complexity via rank-based arguments.
• Applies the log-rank conjecture to the quantum setting with unlimited entanglement, showing it holds under polynomial representation.
• Employs techniques from algebraic complexity and quantum information to analyze the structure of communication matrices.
• Analyzes the behavior of polynomials under quantum operations, including entanglement and qubit exchange.
• Establishes a hierarchy of complexity bounds by classifying functions based on their polynomial degree and matrix rank.

Experimental results

Research questions

• RQ1 Does the log-rank lower bound extend to quantum communication with unlimited prior entanglement?
• RQ2 Can polynomial degree be used to characterize the communication complexity of exact quantum protocols?
• RQ3 Are quantum and classical communication complexities polynomially equivalent for specific classes of functions?
• RQ4 What are the limitations of polynomial methods in bounding bounded-error quantum communication complexity?
• RQ5 How does the rank of the communication matrix relate to the degree of the representing polynomial in entangled settings?

Key findings

• The log-rank conjecture holds for quantum communication with unlimited prior entanglement, extending classical results to the strongest quantum model.
• For exact protocols, the paper proves strong lower bounds using polynomial degree and matrix rank, confirming the log-rank conjecture in this context.
• Quantum and classical communication complexities are polynomially equivalent for several function classes, including symmetric functions and functions with low-rank matrices.
• Weaker lower bounds are derived for bounded-error quantum protocols using polynomial methods, though the exact equivalence remains open in this regime.
• The rank of the communication matrix is tightly linked to the degree of the minimal-degree polynomial representing the function, enabling new complexity lower bounds.
• The results show that polynomial methods are powerful tools for analyzing quantum communication complexity, even in the presence of unlimited entanglement.