QIP2021 | Quantum majority and other Boolean functions with quantum inputs (Maris Ozols)

Authors: Harry Buhrman, Noah Linden, Laura Mančinska, Ashley Montanaro and Maris Ozols
Affiliations: Centrum Wiskunde & Informatica | University of Bristol | University of Copenhagen | PhaseCraft Ltd and University of Bristol | University of Amsterdam

Abstract
Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. It can, for example, be used to amplify the correctness of a quantum device whose output is classical. However, when the output of a device is a quantum state, it is not apriori clear how to implement an analogous \emph{quantum} majority vote. To this end, we consider an extension of majority vote to quantum inputs and outputs: given a product state of the form $\ket{\phi_1, \phi_2, \dotsc ,\phi_n}$ where each qubit $\ket{\phi_i}$ is in one of two orthogonal states $\ket{\psi_0}$ or $\ket{\psi_1}$, output the majority state $\ket{\psi_0}$ or $\ket{\psi_1}$. We provide an optimal algorithm for this problem that achieves worst-case fidelity of $1/2 + \Theta(1/\sqrt{n})$. Under the promise that at least $2/3$ of the qubits are in the majority state, the fidelity increases to $1 - \Theta(1/n)$ and approaches one in the limit. More generally, we initiate the study of covariant and symmetric Boolean functions $f: \set{0,1^n} \to \set{0,1}$ with quantum inputs and outputs. We provide a simple linear program of size roughly $n/2$ for computing the optimal worst-case fidelity and show that a generalization of our algorithm is optimal for computing $f$. Our algorithm has complexity $O(n^4 \log n)$ where $n$ is the number of qubits.

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