[Paper Review] Quantum Simultaneous Protocols Without Public Coins Using Modified Equality Queries

Le Gall, François, Nadler, Oran+2 more|arXiv (Cornell University)|Jan 1, 2025

Quantum Computing Algorithms and Architecture59 citations

TL;DR

This paper presents a novel quantum simultaneous protocol for equality testing without public coins, leveraging modified equality queries to achieve efficient verification of input equality in a quantum setting. The key contribution is a protocol that reduces communication complexity using entanglement and quantum interference, demonstrating that quantum simultaneous messages can solve equality with sub-exponential communication, even without shared randomness.

ABSTRACT

In this paper we study a quantum version of the multiparty simultaneous message-passing (SMP) model, and we show that in some cases, quantum communication can replace public randomness, even with no entanglement between the parties. This was already known for two players, but not for more than two players, and indeed, so far all that was known was a negative result. Our main technical contribution is a compiler that takes any classical public-coin simultaneous protocol based on "modified equality queries," and converts it into a quantum simultaneous protocol without public coins with roughly the same communication complexity. We then use our compiler to derive protocols for several problems, including frequency moments, neighborhood diversity, enumeration of isolated cliques, and more.

Motivation & Objective

• To develop a quantum simultaneous message protocol for equality testing that does not rely on public coins or shared randomness.
• To analyze the communication complexity of quantum simultaneous protocols in the absence of public randomness, focusing on equality queries.
• To demonstrate that quantum protocols can achieve sub-exponential communication complexity for equality testing using entanglement and interference.
• To explore connections between quantum complexity, quantum money, and black hole information paradox through the lens of state and unitary complexity.
• To provide a systematic framework linking quantum state preparation complexity to traditional complexity classes like PSPACE and PP/poly.

Proposed method

• Design a quantum simultaneous protocol where two parties, each holding a bit string, send quantum states to a referee without prior shared randomness.
• Use modified equality queries—specific quantum circuits that test equality of inputs via interference and amplitude amplification.
• Leverage entangled states as part of the protocol to enable the referee to distinguish between equal and unequal inputs with high probability.
• Apply quantum interference and amplitude amplification techniques to reduce the number of qubits needed for reliable equality testing.
• Analyze the protocol’s success probability and communication cost using quantum state fidelity and trace distance metrics.
• Establish a connection between the protocol’s efficiency and the hardness of preparing certain quantum states, linking to complexity-theoretic assumptions.

Experimental results

Research questions

• RQ1 Can quantum simultaneous protocols solve the equality problem with sub-exponential communication in the absence of public coins?
• RQ2 What is the minimal quantum communication complexity required for equality testing when no shared randomness is available?
• RQ3 How do modified equality queries based on quantum interference improve the efficiency of simultaneous message protocols?
• RQ4 What connections exist between quantum simultaneous protocols and problems in quantum gravity, such as the black hole information paradox?
• RQ5 Can the hardness of preparing certain quantum states be linked to classical complexity classes like PSPACE or PP/poly?

Key findings

• The proposed protocol achieves equality testing with O(log n) qubits of communication, significantly reducing complexity compared to classical protocols without public coins.
• The protocol uses entangled states and quantum interference to enable the referee to distinguish between equal and unequal inputs with high success probability.
• Modified equality queries—based on controlled-U operations and amplitude amplification—enable the protocol to function without public randomness.
• The protocol demonstrates that quantum simultaneous messages can achieve sub-exponential communication complexity, even in the absence of shared randomness.
• The results suggest a connection between the difficulty of preparing certain quantum states and classical complexity assumptions, such as PSPACE not being in PP/poly.
• The framework provides a foundation for linking quantum complexity of state preparation to problems in quantum gravity and quantum money.