[Paper Review] The One-Way Communication Complexity of Group Membership

Scott Aaronson, François Le Gall+2 more|arXiv (Cornell University)|Feb 18, 2009

Complexity and Algorithms in Graphs10 references3 citations

TL;DR

This paper establishes upper bounds on the one-way classical communication complexity of the subgroup membership problem in finite groups, showing that it can be solved with O(log |G|) communication when the subgroup is normal, and with O(d_max · log |G|) communication in general, where d_max is the maximum dimension of irreducible complex representations. The key contribution is a representation-theoretic protocol leveraging spectral gaps in Cayley graphs to achieve efficient verification with bounded error.

ABSTRACT

This paper studies the one-way communication complexity of the subgroup membership problem, a classical problem closely related to basic questions in quantum computing. Here Alice receives, as input, a subgroup $H$ of a finite group $G$; Bob receives an element $x \in G$. Alice is permitted to send a single message to Bob, after which he must decide if his input $x$ is an element of $H$. We prove the following upper bounds on the classical communication complexity of this problem in the bounded-error setting: (1) The problem can be solved with $O(\log |G|)$ communication, provided the subgroup $H$ is normal; (2) The problem can be solved with $O(d_{\max} \cdot \log |G|)$ communication, where $d_{\max}$ is the maximum of the dimensions of the irreducible complex representations of $G$; (3) For any prime $p$ not dividing $|G|$, the problem can be solved with $O(d_{\max} \cdot \log p)$ communication, where $d_{\max}$ is the maximum of the dimensions of the irreducible $\F_p$-representations of $G$.

Motivation & Objective

• To determine the classical one-way communication complexity of the subgroup membership problem in finite groups.
• To investigate whether superlinear gaps between classical and quantum one-way communication complexity can exist for total functions.
• To develop efficient protocols for subgroup membership using group representation theory and spectral graph properties.
• To characterize groups for which the communication complexity is logarithmic in |G|, particularly those with abelian subgroups of constant index.
• To provide tight bounds on communication cost based on group structure and representation dimensions.

Proposed method

• Leverages representation theory of finite groups, focusing on irreducible complex and modular representations over F_p.
• Constructs a protocol where Alice sends a quantum-like state (classically encoded) representing a group algebra element, using a symmetric generating set A for the subgroup H.
• Employs the spectral gap of Cayley graphs generated by yA ∪ Ay^{-1} to bound the inner product between vectors in the representation space.
• Uses Babai's theorem on the second eigenvalue of Cayley graphs to establish a lower bound on the operator norm of the averaging operator S_A.
• Applies the Erdős–Rényi result to select generating sets A of size O(log |H|) with diameter O(log |H|), ensuring fast mixing and spectral expansion.
• Analyzes the inner product ⟨v, ρ(y)v⟩ to bound the probability of error, ensuring soundness via concentration of measure and norm control.

Experimental results

Research questions

• RQ1 Can the one-way communication complexity of subgroup membership be bounded in terms of group representation theory?
• RQ2 What is the communication cost for solving the subgroup membership problem when the subgroup H is normal?
• RQ3 How does the maximum dimension d_max of irreducible representations affect the communication complexity?
• RQ4 Can the protocol achieve O(log |G|) communication for groups with abelian subgroups of constant index?
• RQ5 What is the role of spectral gaps in Cayley graphs in ensuring soundness of the protocol?

Key findings

• For normal subgroups H of a finite group G, the one-way communication complexity of subgroup membership is O(log |G|).
• For any finite group G, the complexity is bounded by O(d_max · log |G|), where d_max is the maximum dimension of irreducible complex representations of G.
• For primes p not dividing |G|, the complexity is O(d_max · log p), where d_max is the maximum dimension of irreducible F_p-representations.
• The protocol achieves soundness error bounded by 1 - Ω(1/([K:H]^2 log^3 |H|)) when using a generating set of size O(log |H|).
• When d_max is constant, the communication complexity is O(log |G|), which holds precisely for groups with an abelian subgroup of constant index.
• The result implies that for such groups, the classical one-way communication complexity is logarithmic in |G|, matching the quantum lower bound up to constants.