An accurate evaluation of the performance of quantum entanglement sources-based on various physical realizations and designed for application in numerous quantum protocols-requires a thorough characterization of the ensemble of the generated output quantum states. This presents a great challenge, as numerous measurements, such as full state tomography, need to be performed on the ensemble. This incentivizes us to reduce the overall duration of the characterization procedure, so that time is saved and the effect of setup drift reduced. For many physical realizations of the quantum states, including polarization and path-encoded qubits, a traveling salesman problem of finding the optimal order of tomographic projections can be found and solved [1]. The achieved tomography speedup, found and experimentally verified [2,3] for up to six-qubit systems, is seen to increase with the size of the quantum system in question, see Fig. 1(a). For three-qubit systems already, the time spent on setting the measurement projections can be halved. A step-by-step workflow [4] is introduced to suit various tomographic scenarios.
Quantum tomography requires repeated measurements of many copies of the physical system, all prepared by a source in the unknown state. In the limit of very many copies measured, the often-used maximum-likelihood (ML) method for converting the gathered data into an estimate of the state works very well. For smaller data sets, however, it often suffers from problems of rank deficiency in the estimated state. For many systems of relevance for quantum information processing, the preparation of a very large number of copies of the same quantum state is still a technological challenge, which motivates us to look for estimation strategies that perform well even when there is not much data. After reviewing the concept of minimax state estimation, we use minimax ideas to construct a simple estimator for quantum states. We demonstrate that, for the case of tomography of a single qubit, our estimator significantly outperforms the ML estimator for small number of copies of the state measured. Our estimator is always full-rank, and furthermore, has a natural dependence on the number of copies measured, which is missing in the ML estimator.
Abstract Two-qubit systems typically employ 36 projective measurements for high-fidelity tomographic estimation. The overcomplete nature of the 36 measurements suggests possible robustness of the estimation procedure to missing measurements. In this paper, we explore the resilience of machine-learning-based quantum state estimation techniques to missing measurements by creating a pipeline of stacked machine learning models for imputation, denoising, and state estimation. When applied to simulated noiseless and noisy projective measurement data for both pure and mixed states, we demonstrate quantum state estimation from partial measurement results that outperforms previously developed machine-learning-based methods in reconstruction fidelity and several conventional methods in terms of resource scaling. Notably, our developed model does not require training a separate model for each missing measurement, making it potentially applicable to quantum state estimation of large quantum systems where preprocessing is computationally infeasible due to the exponential scaling of quantum system dimension.
Estimating nonlinear properties of quantum states is a central task in quantum information science. Multivariate traces,
tr(ρ1⋯ρK), and nonlinear observables such as
tr(ρK), for integer
K, 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
n-qubit state to a reduced
q-qubit subspace, estimating
tr(ρK) requires approximately
O(2(n−q)(K−1)) copies of
ρ, with each qubit projected out increasing the sampling cost by a factor of
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.