Quantum majority vote

• URL: http://arxiv.org/abs/2211.11729v1
• Date: Mon, 21 Nov 2022 18:47:46 GMT
• Title: Quantum majority vote
• Authors: Harry Buhrman and Noah Linden and Laura Man\v{c}inska and Ashley Montanaro and Maris Ozols
• Abstract summary: Majority vote is a basic method for amplifying correct outcomes that is widely used in computer science and beyond. We show that an optimal algorithm for this problem achieves worst-case fidelity of $1/2 + Theta (1/sqrtn)$. Under the promise that at least $2/3$ of the input qubits are in the majority state, the fidelity increases to $1 - Theta (1/n)$ and approaches $1$ as $n$ increases.
• Score: 1.43494686131174
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $|\psi_1\rangle \otimes \dots \otimes |\psi_n\rangle$ where each qubit is in one of two orthogonal states $|\psi\rangle$ or $|\psi^\perp\rangle$, output the majority state. We show that an optimal algorithm for this problem achieves worst-case fidelity of $1/2 + \Theta(1/\sqrt{n})$. Under the promise that at least $2/3$ of the input qubits are in the majority state, the fidelity increases to $1 - \Theta(1/n)$ and approaches $1$ as $n$ increases. We also consider the more general problem of computing any symmetric and equivariant Boolean function $f: \{0,1\}^n \to \{0,1\}$ 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 $O(n^4 \log n)$ where $n$ is the number of input qubits.

Related papers

• Towards Optimal Circuit Size for Sparse Quantum State Preparation [10.386753939552872]
  We consider the preparation for $n$-qubit sparse quantum states with $s$ non-zero amplitudes and propose two algorithms. The first algorithm uses $O(ns/log n + n)$ gates, improving upon previous methods by $O(log n)$. The second algorithm is tailored for binary strings that exhibit a short Hamiltonian path.
  arXiv  Detail & Related papers  (2024-04-08T02:13:40Z)
• Towards large-scale quantum optimization solvers with few qubits [59.63282173947468]
  We introduce a variational quantum solver for optimizations over $m=mathcalO(nk)$ binary variables using only $n$ qubits, with tunable $k>1$. We analytically prove that the specific qubit-efficient encoding brings in a super-polynomial mitigation of barren plateaus as a built-in feature.
  arXiv  Detail & Related papers  (2024-01-17T18:59:38Z)
• Mind the gap: Achieving a super-Grover quantum speedup by jumping to the end [114.3957763744719]
  We present a quantum algorithm that has rigorous runtime guarantees for several families of binary optimization problems. We show that the algorithm finds the optimal solution in time $O*(2(0.5-c)n)$ for an $n$-independent constant $c$. We also show that for a large fraction of random instances from the $k$-spin model and for any fully satisfiable or slightly frustrated $k$-CSP formula, statement (a) is the case.
  arXiv  Detail & Related papers  (2022-12-03T02:45:23Z)
• Quantum Goemans-Williamson Algorithm with the Hadamard Test and Approximate Amplitude Constraints [62.72309460291971]
  We introduce a variational quantum algorithm for Goemans-Williamson algorithm that uses only $n+1$ qubits. Efficient optimization is achieved by encoding the objective matrix as a properly parameterized unitary conditioned on an auxilary qubit. We demonstrate the effectiveness of our protocol by devising an efficient quantum implementation of the Goemans-Williamson algorithm for various NP-hard problems.
  arXiv  Detail & Related papers  (2022-06-30T03:15:23Z)
• Optimal (controlled) quantum state preparation and improved unitary synthesis by quantum circuits with any number of ancillary qubits [20.270300647783003]
  Controlled quantum state preparation (CQSP) aims to provide the transformation of $|irangle |0nrangle to |irangle |psi_irangle $ for all $iin 0,1k$ for the given $n$-qubit states. We construct a quantum circuit for implementing CQSP, with depth $Oleft(n+k+frac2n+kn+k+mright)$ and size $Oleft(2n+kright)$
  arXiv  Detail & Related papers  (2022-02-23T04:19:57Z)
• How to simulate quantum measurement without computing marginals [3.222802562733787]
  We describe and analyze algorithms for classically computation measurement of an $n$-qubit quantum state $psi$ in the standard basis. Our algorithms reduce the sampling task to computing poly(n)$ amplitudes of $n$-qubit states.
  arXiv  Detail & Related papers  (2021-12-15T21:44:05Z)
• Asymptotically Optimal Circuit Depth for Quantum State Preparation and General Unitary Synthesis [24.555887999356646]
  The problem is of fundamental importance in quantum algorithm design, Hamiltonian simulation and quantum machine learning, yet its circuit depth and size complexity remain open when ancillary qubits are available. In this paper, we study efficient constructions of quantum circuits with $m$ ancillary qubits that can prepare $psi_vrangle$ in depth. Our circuits are deterministic, prepare the state and carry out the unitary precisely, utilize the ancillary qubits tightly and the depths are optimal in a wide range of parameter regime.
  arXiv  Detail & Related papers  (2021-08-13T09:47:11Z)
• Low-depth Quantum State Preparation [3.540171881768714]
  A crucial subroutine in quantum computing is to load the classical data of $N$ complex numbers into the amplitude of a superposed $n=lceil logNrceil$-qubit state. Here we investigate this space-time tradeoff in quantum state preparation with classical data. We propose quantum algorithms with $mathcal O(n2)$ circuit depth to encode any $N$ complex numbers.
  arXiv  Detail & Related papers  (2021-02-15T13:11:43Z)
• An Optimal Separation of Randomized and Quantum Query Complexity [67.19751155411075]
  We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order $ellsqrtbinomdell (1+log n)ell-1,$ sum to at most $cellsqrtbinomdell (1+log n)ell-1,$ where $n$ is the number of variables, $d$ is the tree depth, and $c>0$ is an absolute constant.
  arXiv  Detail & Related papers  (2020-08-24T06:50:57Z)
• Maximizing Determinants under Matroid Constraints [69.25768526213689]
  We study the problem of finding a basis $S$ of $M$ such that $det(sum_i in Sv_i v_i v_itop)$ is maximized. This problem appears in a diverse set of areas such as experimental design, fair allocation of goods, network design, and machine learning.
  arXiv  Detail & Related papers  (2020-04-16T19:16:38Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.