Quantum Majority Vote

Authors Harry Buhrman, Noah Linden, Laura Mančinska , Ashley Montanaro , Maris Ozols

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Author Details

Harry Buhrman
  • QuSoft, CWI, Amsterdam, The Netherlands
  • University of Amsterdam, The Netherlands
Noah Linden
  • University of Bristol, UK
Laura Mančinska
  • University of Copenhagen, Denmark
Ashley Montanaro
  • Phasecraft Ltd., Bristol, UK
  • University of Bristol, UK
Maris Ozols
  • QuSoft, Amsterdam, The Nehterlands
  • University of Amsterdam, The Netherlands

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Harry Buhrman, Noah Linden, Laura Mančinska, Ashley Montanaro, and Maris Ozols. Quantum Majority Vote. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, p. 29:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


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_1⟩ ⊗ … ⊗ |ψ_n⟩ where each qubit is in one of two orthogonal states |ψ⟩ or |ψ^⟂⟩, output the majority state. We show that an optimal algorithm for this problem achieves worst-case fidelity of 1/2 + Θ(1/√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.

Subject Classification

ACM Subject Classification
  • Computer systems organization → Quantum computing
  • quantum algorithms
  • quantum majority vote
  • Schur-Weyl duality


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  1. Harry Buhrman, Noah Linden, Laura Mančinska, Ashley Montanaro, and Maris Ozols. Quantum majority vote. (Full version). URL: https://doi.org/10.48550/ARXIV.2211.11729.
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