Quantum majority vote

Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. While it can amplify the correctness of a quantum device with classical output, the analogous procedure for quantum output is not known. We introduce quantum majority vote as the following task: given a product state |ψ_1rangle otimes dots otimes |ψ_nrangle where each qubit is in one of two orthogonal states |ψrangle or |ψ^perprangle, output the majority state. We show that an optimal algorithm for this problem achieves worst-case fidelity of 1/2 + Θ\(1/sqrt{n}\). Under the promise that at least 2/3 of the input qubits are in the majority state, the fidelity increases to 1 - Θ(1/n) and approaches 1 as n increases. We also consider the more general problem of computing any symmetric and equivariant Boolean function f: \{0,1\}ⁿ → \{0,1\} in an unknown quantum basis, and show that a generalization of our quantum majority vote algorithm is optimal for this task. The optimal parameters for the generalized algorithm and its worst-case fidelity can be determined by a simple linear program of size O(n). The time complexity of the algorithm is O\(n⁴ log n\) where n is the number of input qubits.