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arXiv:2107.08975 · Physical Review A

Macroscopic features of quantum fluctuations in large- N qubit systems

A. B. Klimov, Carlos A Muñoz

Published May 23, 2014• 10 citations
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Abstract

We introduce a discrete $Q$ function of an $N$-qubit system projected into the space of symmetric measurements as a tool for analyzing general properties of quantum systems in the macroscopic limit. For known states the projected $Q$ function helps to visualize the results of collective measurements, and for unknown states it can be approximately reconstructed by measuring the lowest moments of the collective variables.

DOI
10.1103/physreva.89.052130

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Estimating nonlinear properties of quantum states is a central task in quantum information science. Multivariate traces, tr(ρ1⋯ρK)\mathrm{tr}(ρ_1 \cdots ρ_K)tr(ρ1​⋯ρK​), and nonlinear observables such as tr(ρK)\mathrm{tr}(ρ^K)tr(ρK), for integer KKK, can be accessed through collective measurements on multiple state copies, but standard protocols based on swap tests require coherent operations on the full Hilbert space and become experimentally unfeasible for large systems. In this work, we introduce a framework for multi-copy measurements based on random projections onto lower-dimensional subspaces prior to the collective measurement, which is then performed only on the reduced Hilbert space. This procedure yields a tunable tradeoff between coherent quantum resources and statistical sampling overhead, allowing the amount of coherent processing to be matched to the capabilities of the underlying hardware. We derive explicit formulas relating the Haar-averaged projected moments to multivariate traces of the original states and analyze the sampling overhead induced by the projection procedure. Specifically, after compressing an nnn-qubit state to a reduced qqq-qubit subspace, estimating tr(ρK)\mathrm{tr}(ρ^K)tr(ρK) requires approximately O(2(n−q)(K−1))O(2^{(n-q)(K-1)})O(2(n−q)(K−1)) copies of ρρρ, with each qubit projected out increasing the sampling cost by a factor of 2K−12^{K-1}2K−1. Our results establish how coherent multi-copy operations can be traded for additional state copies, enabling multi-copy quantum protocols to be optimized for the available hardware resources.