[Paper Review] A note on the power of quantum fingerprinting

Alexander Golynski, Pranab Kumar Sen|ArXiv.org|Oct 12, 2005

Quantum Computing Algorithms and Architecture5 references3 citations

TL;DR

This paper improves upon Yao's quantum fingerprinting simulation of classical public coin simultaneous message protocols, achieving a near-quadratic reduction in communication cost. It introduces the row-column width (rcw) of a matrix as a tighter complexity measure, showing that quantum protocols can simulate classical ones with cost O(rcw(D)^4(R^‖,pub(f) + log n)), significantly improving prior exponential bounds.

ABSTRACT

In this short note, we improve and extend Yao's paper "On the power of quantum fingerprinting" about simulating a classical public coin simultaneous message protocol by a quantum simultaneous message protocol with no shared resource.

Motivation & Objective

• To improve the exponential overhead in simulating classical public coin simultaneous message protocols using quantum fingerprinting.
• To introduce and analyze a new complexity measure, row-column width (rcw), for referee matrices.
• To provide tighter upper bounds on quantum communication complexity by relating it to rcw and convex width (convw).
• To generalize and strengthen Yao's original simulation result with a near-quadratic improvement.
• To address the open question of whether exponential overhead in such simulations can be overcome.

Proposed method

• Define quantum fingerprints |u_x⟩ and |v_y⟩ as quantum states encoding classical messages using public coin randomness.
• Construct the inner product ⟨u_x|v_y⟩ to mirror the classical protocol's success probability, scaled by 1/√M_A.
• Use a decomposition of the referee matrix D into symmetric positive semidefinite matrices and permutation matrices to define convex width convw(D).
• Introduce row-column width rcw(D) as the minimum K such that D = EF with ‖E‖_F ≤ √K and ‖F‖_F ≤ √K, where E and F are real matrices.
• Prove that rcw(D) ≤ convw(D) and that rcw(D) ≤ √M for boolean M×M matrices.
• Apply Holder’s inequality and trace norm bounds to derive lower bounds on rcw(D), showing it is Ω(√M) for inner product matrices.

Experimental results

Research questions

• RQ1 Can the exponential overhead in simulating classical public coin simultaneous message protocols by quantum protocols be reduced?
• RQ2 What is the tightest possible upper bound on quantum communication complexity in terms of classical protocol complexity?
• RQ3 How does the new row-column width (rcw) measure compare to the previously used convex width (convw)?
• RQ4 Can rcw(D) provide a better characterization of quantum communication complexity than convw(D) for boolean matrices?
• RQ5 Is it possible to achieve a polynomial or even constant-factor improvement over classical protocols in the simultaneous message model?

Key findings

• The paper achieves a near-quadratic improvement in the simulation of classical public coin protocols by quantum protocols, showing Q^‖(f) ≤ O(2^{R^‖,pub(f)}(R^‖,pub(f) + log n + 1)).
• The row-column width rcw(D) is introduced as a new matrix complexity measure satisfying rcw(D) ≤ convw(D) and rcw(D) ≤ √M for M×M boolean matrices.
• For the inner product function on n-bit strings, the paper proves that rcw(D) ≥ (√M - 1)/2, showing that rcw(D) is Ω(√M), which is optimal up to constants.
• The quantum communication complexity is bounded by Q^‖(f) ≤ O(convw(D)^4(R^‖,pub(f) + log n + 1)), improving upon Yao's original bound.
• The result is shown to be tight up to logarithmic factors, as rcw(D) cannot be smaller than Ω(√M) for general boolean matrices.
• The paper establishes that rcw(D) provides a strictly tighter bound than convw(D) for some matrices, such as the all-ones-first-column matrix where rcw(D) ≤ 1 but convw(D) = M.