Optimal Strategies for Winning Certain Coset-Guessing Quantum Games

TL;DR

This paper analyzes a quantum coset guessing game, deriving optimal strategies and bounds for winning probabilities, and introduces an encoding circuit relevant for quantum coding systems.

Contribution

It provides the first tight upper bound on the joint guessing probability and designs an optimal encoding circuit using CNOT and Hadamard gates.

Findings

Joint guessing probability decreases exponentially with m.

Optimal strategy achieves the derived upper bound.

Encoding circuit uses only CNOT and Hadamard gates.

Abstract

In a recently introduced coset guessing game, Alice plays against Bob and Charlie, aiming to meet a joint winning condition. Bob and Charlie can only communicate before the game starts to devise a joint strategy. The game we consider begins with Alice preparing a 2m-qubit quantum state based on a random selection of three parameters. She sends the first m qubits to Bob and the rest to Charlie and then reveals to them her choice for one of the parameters. Bob is supposed to guess one of the hidden parameters, Charlie the other, and they win if both guesses are correct. From previous work, we know that the probability of Bob's and Charlie's guesses being simultaneously correct goes to zero exponentially as m increases. We derive a tight upper bound on this probability and show how Bob and Charlie can achieve it. While developing the optimal strategy, we devised an encoding circuit using only CNOT and Hadamard gates, which could be relevant for building efficient CSS-coded systems. We found that the role of quantum information that Alice communicates to Bob and Charlie is to make their responses correlated rather than improve their individual (marginal) correct guessing rates.