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DOI: 10.4230/LIPIcs.ITCS.2023.29

Quantum Majority Vote

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

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract

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.

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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)

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@InProceedings{buhrman_et_al:LIPIcs.ITCS.2023.29,
  author =	{Buhrman, Harry and Linden, Noah and Man\v{c}inska, Laura and Montanaro, Ashley and Ozols, Maris},
  title =	{{Quantum Majority Vote}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{29:1--29:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.29},
  URN =		{urn:nbn:de:0030-drops-175321},
  doi =		{10.4230/LIPIcs.ITCS.2023.29},
  annote =	{Keywords: quantum algorithms, quantum majority vote, Schur-Weyl duality}
}

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DOI: 10.4230/LIPIcs.TQC.2013.270

The Quantum Entropy Cone of Stabiliser States

Authors: Noah Linden, Frantisek Matus, Mary Beth Ruskai, and Andreas Winter

Published in: LIPIcs, Volume 22, 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)

Abstract

We investigate the universal linear inequalities that hold for the von Neumann entropies in a multi-party system, prepared in a stabiliser state. We demonstrate here that entropy vectors for stabiliser states satisfy, in addition to the classic inequalities, a type of linear rank inequalities associated with the combinatorial structure of normal subgroups of certain matrix groups. In the 4-party case, there is only one such inequality, the so-called Ingleton inequality. For these systems we show that strong subadditivity, weak monotonicity and Ingleton inequality exactly characterize the entropy cone for stabiliser states.

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Noah Linden, Frantisek Matus, Mary Beth Ruskai, and Andreas Winter. The Quantum Entropy Cone of Stabiliser States. In 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 22, pp. 270-284, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{linden_et_al:LIPIcs.TQC.2013.270,
  author =	{Linden, Noah and Matus, Frantisek and Ruskai, Mary Beth and Winter, Andreas},
  title =	{{The Quantum Entropy Cone of Stabiliser States}},
  booktitle =	{8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013)},
  pages =	{270--284},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-55-2},
  ISSN =	{1868-8969},
  year =	{2013},
  volume =	{22},
  editor =	{Severini, Simone and Brandao, Fernando},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2013.270},
  URN =		{urn:nbn:de:0030-drops-43278},
  doi =		{10.4230/LIPIcs.TQC.2013.270},
  annote =	{Keywords: Entropy inequalities, Stabiliser states, Ingleton inequality}
}

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Quantum Majority Vote

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The Quantum Entropy Cone of Stabiliser States

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