[Paper Review] On the orthogonal rank of Cayley graphs and impossibility of quantum round elimination

Jop Briët, Jeroen Zuiddam|arXiv (Cornell University)|Aug 22, 2016

Quantum Computing Algorithms and Architecture11 references6 citations

TL;DR

This paper establishes an exponential lower bound on the orthogonal rank of Cayley graphs over the hypercube {0,1}^n with edges between vertices at Hamming distance at least n/2. Using Fourier analysis and the Lovász theta function, the authors prove that symmetric orthogonal embeddings cannot achieve sub-exponential dimension, resolving a conjecture on the impossibility of quantum round elimination in exact quantum communication complexity.

ABSTRACT

After Bob sends Alice a bit, she responds with a lengthy reply. At the cost of a factor of two in the total communication, Alice could just as well have given the two possible replies without listening and have Bob select which applies to him. Motivated by a conjecture stating that this form of "round elimination" is impossible in exact quantum communication complexity, we study the orthogonal rank and a symmetric variant thereof for a certain family of Cayley graphs. The orthogonal rank of a graph is the smallest number $d$ for which one can label each vertex with a nonzero $d$-dimensional complex vector such that adjacent vertices receive orthogonal vectors. We show an exp$(n)$ lower bound on the orthogonal rank of the graph on $\{0,1\}^n$ in which two strings are adjacent if they have Hamming distance at least $n/2$. In combination with previous work, this implies an affirmative answer to the above conjecture.

Motivation & Objective

• Address a conjecture in quantum communication complexity stating that round elimination is impossible in the exact quantum setting.
• Investigate the orthogonal rank and symmetric orthogonal rank of Cayley graphs derived from the hypercube {0,1}^n with edges at Hamming distance ≥ n/2.
• Establish a lower bound on the minimal dimension required for orthogonal embeddings of these graphs to resolve the conjecture.
• Use spectral and harmonic analysis techniques to analyze the structure of the graphs and their embedding constraints.

Proposed method

• Define the Cayley graph H₂ⁿ(d) on the group (ℤ₂)ⁿ with edges between vertices differing in at least d coordinates.
• Use the Lovász theta function as a relaxation to lower bound the orthogonal rank via duality with the complement graph.
• Apply Fourier analysis on the Boolean cube to re-express the theta function in terms of non-negative Fourier coefficients with zero constraints on high-weight sets.
• Leverage the Cauchy–Vandermonde identity and binomial coefficient bounds to control the decay of candidate functions.
• Use Bochner’s theorem to reduce the optimization over matrices to a function on the group depending only on differences.
• Derive an upper bound on the maximum value of the dual function g(0), leading to a lower bound on the orthogonal rank via ξ(G) ≥ 2ⁿ / ϑ(G).

Experimental results

Research questions

• RQ1 Can the orthogonal rank of the Cayley graph H₂ⁿ(n/2) be bounded below by an exponential function in n?
• RQ2 Is there a symmetric orthogonal embedding of this graph in sub-exponential dimension?
• RQ3 Does the exponential lower bound on orthogonal rank imply the impossibility of quantum round elimination in exact quantum communication?
• RQ4 What is the asymptotic behavior of the Lovász theta function for the complement of H₂ⁿ(n/2)?
• RQ5 Can Fourier-analytic techniques be used to derive tight bounds on the minimal embedding dimension of such Cayley graphs?

Key findings

• The orthogonal rank ξ(G) of the graph H₂ⁿ(n/2) is at least exp(εn − c) for some ε > 0 and constant c, establishing an exponential lower bound.
• The symmetric orthogonal rank ξ_sym(G) is also at least exp(εn − c), showing that symmetric embeddings do not yield sub-exponential dimension.
• The Lovász theta function ϑ(G) for the graph H₂ⁿ(n/2) is at most 2⁻(εn−c), which implies a lower bound on ϑ(Ḡ) ≥ 2^(εn−c) via duality.
• The maximum value of the dual function g(0) in the theta function relaxation is bounded above by 2⁻(εn−c), derived via binomial coefficient estimates and the Cauchy–Vandermonde identity.
• The analysis confirms that no symmetric orthogonal embedding exists in sub-exponential dimension, resolving the conjecture on quantum round elimination.
• The result implies that in the exact quantum communication model, removing a round of communication cannot be compensated by a constant-factor increase in communication, confirming the impossibility of quantum round elimination.