Classical shadows with Pauli-invariant unitary ensembles

Bu, Kaifeng; Koh, Dax Enshan; Garcia, Roy J.; Jaffe, Arthur

doi:10.48550/arxiv.2202.03272

Cited by 6 publications

(9 citation statements)

References 38 publications

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“…( 3) is the inverse of the channel C E . It has been observed that for certain classes of measurements, the inverse channel C −1 E is particularly simple [7,27]. We are to show that behind this simplicity is the symmetry of the generalised measurement, which can be generalised to a much broader class of measurements.…”

Section: Symmetric Generalised Measurements and The Computation Of Th...mentioning

confidence: 88%

Optimising shadow tomography with generalised measurements

Nguyen¹,

Lennart²,

Steinberg³

et al. 2022

Preprint

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Advances in quantum technology require scalable techniques to efficiently extract information from a quantum system, such as expectation values of observables or its entropy. Traditional tomography is limited to a handful of qubits and shadow tomography has been suggested as a scalable replacement for larger systems. Shadow tomography is conventionally analysed based on outcomes of ideal projective measurements on the system upon application of randomised unitaries. Here, we suggest that shadow tomography can be much more straightforwardly formulated for generalised measurements, or positive operator valued measures. Based on the idea of the least-square estimator, shadow tomography with generalised measurements is both more general and simpler than the traditional formulation with randomisation of unitaries. In particular, this formulation allows us to analyse theoretical aspects of shadow tomography in detail. For example, we provide a detailed study of the implication of symmetries in shadow tomography. Shadow tomography with generalised measurements is also indispensable in realistic implementation of quantum mechanical measurements, when noise is unavoidable. Moreover, we also demonstrate how the optimisation of measurements for shadow tomography tailored toward a particular set of observables can be carried out.

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Section: Symmetric Generalised Measurements and The Computation Of Th...mentioning

confidence: 88%

Optimising shadow tomography with generalised measurements

Nguyen¹,

Lennart²,

Steinberg³

et al. 2022

Preprint

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show abstract

“…In our analysis, we assumed the Clifford circuit in the classical shadow part is noise-free. If there are noise in the Clifford circuit part, it can be mitigated if the noise is independent of the Clifford gates, as in [27][28][29], where similar idea was used for randomized benchmarking [30,31].…”

Section: Discussionmentioning

confidence: 99%

Logical shadow tomography: Efficient estimation of error-mitigated observables

Hu¹,

LaRose²,

You³

et al. 2022

Preprint

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We introduce a technique to estimate error-mitigated expectation values on noisy quantum computers. Our technique performs shadow tomography on a logical state to produce a memory-efficient classical reconstruction of the noisy density matrix. Using efficient classical post-processing, one can mitigate errors by projecting a general nonlinear function of the noisy density matrix into the codespace. The subspace expansion and virtual distillation can be viewed as special cases of the new framekwork. We show our method is favorable in the quantum and classical resources overhead. Relative to subspace expansion which requires O 2 N samples to estimate a logical Pauli observable with [[N, k]] error correction code, our technique requires only O 4 k samples. Relative to virtual distillation, our technique can compute powers of the density matrix without additional copies of quantum states or quantum memory. We present numerical evidence using logical states encoded with up to sixty physical qubits and show fast convergence to error-free expectation values with only 10 5 samples under 1% depolarizing noise.

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“…Subsequent to version 1 [53] of our manuscript, several new extensions and applications of classical shadows have been developed [48,49,. Amongst these are extensions of the classical shadows framework to quantum channels [71,72] and to more general ensembles, like locally scrambled unitary ensembles [69] and Pauli-invariant unitary ensembles [77]. Additional applications of classical shadows include avoiding barren plateaus in variational quantum algorithms [76], quantifying information scrambling [78], and estimating gate set properties [73].…”

Section: Classical Shadowsmentioning

confidence: 99%

Classical Shadows With Noise

Koh

Grewal

2022

Quantum

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The classical shadows protocol, recently introduced by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], is a quantum-classical protocol to estimate properties of an unknown quantum state. Unlike full quantum state tomography, the protocol can be implemented on near-term quantum hardware and requires few quantum measurements to make many predictions with a high success probability. In this paper, we study the effects of noise on the classical shadows protocol. In particular, we consider the scenario in which the quantum circuits involved in the protocol are subject to various known noise channels and derive an analytical upper bound for the sample complexity in terms of a shadow seminorm for both local and global noise. Additionally, by modifying the classical post-processing step of the noiseless protocol, we define a new estimator that remains unbiased in the presence of noise. As applications, we show that our results can be used to prove rigorous sample complexity upper bounds in the cases of depolarizing noise and amplitude damping.

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Classical shadows with Pauli-invariant unitary ensembles

Cited by 6 publications

References 38 publications

Optimising shadow tomography with generalised measurements

Optimising shadow tomography with generalised measurements

Logical shadow tomography: Efficient estimation of error-mitigated observables

Classical Shadows With Noise

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