Quantum Simultaneous Protocols without Public Coins using Modified Equality Queries

TL;DR

The paper develops a quantum multiparty SMP framework that replaces public randomness with quantum communication for certain problems, even without entanglement. It introduces modified equality queries (MEQ) and a compiler that converts MEQ decision trees of depth D into quantum SMP protocols with cost roughly O(k (log D + log(1/δ)) log n) qubits and error δ. The approach leverages linear quantum fingerprints, SWAP tests, gentle measurements, and the quantum union bound to reuse quantum information across sequential MEQ queries. The resulting protocols solve a range of problems, including frequency moments and various graph-related tasks, achieving polylogarithmic quantum communication in the NIH model and matching or surpassing public-coin classical benchmarks in several cases, with clear practical implications for distributed quantum computation.

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.

Figures (6)

• Figure 1: Description of the referee's procedure for Lemma \ref{['lemma:MEQ']}.
• Figure 2: An $\mathrm{MEQ}_{k,n}$ decision tree for $\mathrm{GroupByEQ}_{k,n}$ with $k = 3$ players. Each inner node is labeled with a query of the form "$x_i = x_j$?", which is short-hand notation for $\mathrm{MEQ}_{k,n}(i, j, 0^n, 0^n)$. The leaves are labeled with output partitions.
• Figure 3: A 2-outcome measurement of a quantum register $\mathsf{R}$.
• Figure 4: Conversion from a quantum circuit implementing a (non-projective) 2-outcome measurement associated with the unitary $U$ (left) to a quantum circuit implementing a 2-outcome projective measurement (right). Register $\sf{R}$ stores the initial state and Register $\sf{R}'$ stores the postmeasurement state. In the right picture, the 2-qubit gate between $U$ and $U^{-1}$ represents the CNOT gate, where the $X$ gate (also called NOT gate) is applied on the $\oplus$-part conditioned on the content of the black-circle part being $1$.
• Figure 5: Description of the SWAP test.

Theorems & Definitions (27)

• lemma 1
• theorem 1
• proposition 1
• theorem 2: Quantum union bound Gao15
• definition 1
• lemma 2
• proof : Proof of Lemma \ref{['lemma:MEQ']}
• theorem -1: repeated
• proof
• theorem 4