Robust and Efficient High-Dimensional Quantum State Tomography

Rambach, Markus; Qaryan, Mahdi; Kewming, Michael; Ferrie, Christopher; White, A. G.; Romero, Jacquiline

doi:10.1103/physrevlett.126.100402

Cited by 47 publications

(33 citation statements)

References 36 publications

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“…Robust low rank matrix recovery problems are ubiquitous in computer vision [1], signal processing [2], quantum state tomography [3], etc. In the vanilla setting, the task is to recover a rank r positive semidefinite (PSD) matrix X ∈ R d×d (r d) from a few corrupted linear measurements {(y i , A i )} n i=1 of the form…”

mentioning

confidence: 99%

Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery

Ding¹,

Jiang²,

Chen³

et al. 2021

Preprint

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We study the robust recovery of a low-rank matrix from sparsely and grossly corrupted Gaussian measurements, with no prior knowledge on the intrinsic rank. We consider the robust matrix factorization approach. We employ a robust 1 loss function and deal with the challenge of the unknown rank by using an overspecified factored representation of the matrix variable. We then solve the associated nonconvex nonsmooth problem using a subgradient method with diminishing stepsizes. We show that under a regularity condition on the sensing matrices and corruption, which we call restricted direction preserving property (RDPP), even with rank overspecified, the subgradient method converges to the exact low-rank solution at a sublinear rate. Moreover, our result is more general in the sense that it automatically speeds up to a linear rate once the factor rank matches the unknown rank. On the other hand, we show that the RDPP condition holds under generic settings, such as Gaussian measurements under independent or adversarial sparse corruptions, where the result could be of independent interest. Both the exact recovery and the convergence rate of the proposed subgradient method are numerically verified in the overspecified regime. Moreover, our experiment further shows that our particular design of diminishing stepsize effectively prevents overfitting for robust recovery under overparameterized models, such as robust matrix sensing and learning robust deep image prior. This regularization effect is worth further investigation.

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mentioning

confidence: 99%

Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery

Ding¹,

Jiang²,

Chen³

et al. 2021

Preprint

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show abstract

“…In addition to well-established quantum tomography tools, there are several new methods for characterising, measuring and extracting information about different quantum states 10 – 12 . Each method has its own set of pros and cons, regarding measurement time, the total number of required measurement settings and the ease of implementation.…”

Section: Discussionmentioning

confidence: 99%

“…Consequently, many techniques have been developed to witness, bound and attempt to quantify high-dimensional quantum states. These include approximating the density matrix via quantum state tomography (QST) with multiple qubit state projections 8 , using mutually unbiased bases 9 , 10 to probe the states or incorporating self-guided approaches 11 , 12 , and testing non-local bi-photon correlations by generalised Bell tests in higher dimensions 13 – 15 . However, the spectrum measurements do not confirm entanglement, the QST approach scales unfavourably with dimension, only bounds or witnesses are possible with the mutually unbiased bases method and the dimension to be probed must be known a priori (e.g.…”

Section: Introductionmentioning

confidence: 99%

Measuring dimensionality and purity of high-dimensional entangled states

et al. 2021

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High-dimensional entangled states are promising candidates for increasing the security and encoding capacity of quantum systems. While it is possible to witness and set bounds for the entanglement, precisely quantifying the dimensionality and purity in a fast and accurate manner remains an open challenge. Here, we report an approach that simultaneously returns the dimensionality and purity of high-dimensional entangled states by simple projective measurements. We show that the outcome of a conditional measurement returns a visibility that scales monotonically with state dimensionality and purity, allowing for quantitative measurements for general photonic quantum systems. We illustrate our method using two separate bases, the orbital angular momentum and pixels bases, and quantify the state dimensionality by a variety of definitions over a wide range of noise levels, highlighting its usefulness in practical situations. Importantly, the number of measurements needed in our approach scale linearly with dimensions, reducing data acquisition time significantly. Our technique provides a simple, fast and direct measurement approach.

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“…The parameterized quantum circuits for constructing quantum generators and discriminators do not require accurate implementations of specific quantum logics and can be achieved on the near-term quantum devices across different physical platforms. QGAN has far-reaching effects in solving the quantum many-body problem, which can directly extend to the optimal control and self-guided quantum tomography, especially when the system size goes large 27,28 . Our implementation paves the way to the much-anticipated computing paradigm with combined quantum-classical processors, and holds the intriguing potential to realize practical quantum supremacy 9 with noisy intermediate-scale quantum devices 29 .…”

Section: Discussionmentioning

confidence: 99%

Quantum generative adversarial networks with multiple superconducting qubits

et al. 2021

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Generative adversarial networks are an emerging technique with wide applications in machine learning, which have achieved dramatic success in a number of challenging tasks including image and video generation. When equipped with quantum processors, their quantum counterparts—called quantum generative adversarial networks (QGANs)—may even exhibit exponential advantages in certain machine learning applications. Here, we report an experimental implementation of a QGAN using a programmable superconducting processor, in which both the generator and the discriminator are parameterized via layers of single- and two-qubit quantum gates. The programmed QGAN runs automatically several rounds of adversarial learning with quantum gradients to achieve a Nash equilibrium point, where the generator can replicate data samples that mimic the ones from the training set. Our implementation is promising to scale up to noisy intermediate-scale quantum devices, thus paving the way for experimental explorations of quantum advantages in practical applications with near-term quantum technologies.

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Robust and Efficient High-Dimensional Quantum State Tomography

Cited by 47 publications

References 36 publications

Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery

Rank Overspecified Robust Matrix Recovery: Subgradient Method and Exact Recovery

Measuring dimensionality and purity of high-dimensional entangled states

Quantum generative adversarial networks with multiple superconducting qubits

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