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arXiv:2310.17328

Shadow tomography with noisy readouts

H. Chau Nguyen

Published Oct 26, 2023• 1 citation
arXivHTMLPDF

Abstract

Shadow tomography is a scalable technique to characterise the quantum state of a quantum computer or quantum simulator. The protocol is based on the transformation of the outcomes of random measurements into the so-called classical shadows, which can later be transformed into samples of expectation values of the observables of interest. By construction, classical shadows are intrinsically sensitive to readout noise. In fact, the complicated structure of the readout noise due to crosstalk appears to be detrimental to its scalability. We show that classical shadows accept much more flexible constructions beyond the standard ones, which can eventually be made more conformable with readout noise. With this construction, we show that readout errors in classical shadows can be efficiently mitigated by randomly flipping the qubit before, and the classical outcome bit after the measurement, referred to as $X$-twirling. That a single $X$-gate is sufficient for mitigating readout noise for classical shadows is in contrast to Clifford-twirling, where the implementation of random Clifford gates is required.

DOI
10.48550/arxiv.2310.17328

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Shadow tomography is a scalable technique to characterise the quantum state of a quantum computer or quantum simulator. The protocol is based on the transformation of the outcomes of random measurements into the so-called classical shadows, which can later be transformed into samples of expectation values of the observables of interest. By construction, classical shadows are intrinsically sensitive to readout noise. In fact, the complicated structure of the readout noise due to crosstalk appears to be detrimental to its scalability. We show that classical shadows accept much more flexible constructions beyond the standard ones, which can eventually be made more conformable with readout noise. With this construction, we show that readout errors in classical shadows can be efficiently mitigated by randomly flipping the qubit before, and the classical outcome bit after the measurement, referred to as XXX-twirling. That a single XXX-gate is sufficient for mitigating readout noise for classical shadows is in contrast to Clifford-twirling, where the implementation of random Clifford gates is required.
Classical shadow tomography is a powerful randomized measurement protocol for predicting many properties of a quantum state with few measurements. Two classical shadow protocols have been extensively studied in the literature: the single-qubit (local) Pauli measurement, which is well suited for predicting local operators but inefficient for large operators; and the global Clifford measurement, which is efficient for low-rank operators but infeasible on near-term quantum devices due to the extensive gate overhead. In this work, we demonstrate a scalable classical shadow tomography approach for generic randomized measurements implemented with finite-depth local Clifford random unitary circuits, which interpolates between the limits of Pauli and Clifford measurements. The method combines the recently proposed locally-scrambled classical shadow tomography framework with tensor network techniques to achieve scalability for computing the classical shadow reconstruction map and evaluating various physical properties. The method enables classical shadow tomography to be performed on shallow quantum circuits with superior sample efficiency and minimal gate overhead and is friendly to noisy intermediate-scale quantum (NISQ) devices. We show that the shallow-circuit measurement protocol provides immediate, exponential advantages over the Pauli measurement protocol for predicting quasi-local operators. It also enables a more efficient fidelity estimation compared to the Pauli measurement.
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(2N)O\left(2^{N} \right)O(2N) samples to estimate a logical Pauli observable with [[N,k]][[N, k]][[N,k]] error correction code, our technique requires only O(4k)O\left(4^{k} \right)O(4k) 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 10510^5105 samples under 1% depolarizing noise.
Expectation values of observables are routinely estimated using so-called classical shadows-the outcomes of randomized bases measurements on a repeatedly prepared quantum state. In order to trust the accuracy of shadow estimation in practice, it is crucial to understand the behavior of the estimators under realistic noise. In this Letter, we prove that any shadow estimation protocol involving Clifford unitaries is stable under gate-dependent noise for observables with bounded stabilizer norm-originally introduced in the context of simulating Clifford circuits. In contrast, we demonstrate with concrete examples that estimation of "magic" observables can lead to highly misleading results in the presence of miscalibration errors and a worst case bias scaling exponentially in the system size. We further find that so-called robust shadows, aiming at mitigating noise, can introduce a large bias in the presence of gate-dependent noise compared to unmitigated classical shadows. Nevertheless, we guarantee the functioning of robust shadows for a more general noise setting than in previous works. On a technical level, we identify average noise channels that affect shadow estimators and allow for a more fine-grained control of noise-induced biases.
Given copies of a quantum state ρ, a shadow tomography protocol aims to learn all expectation values from a fixed set of observables, to within a given precision ε. We say that a shadow tomography protocol is triply efficient if it is sample- and time-efficient, and only employs measurements that entangle a constant number of copies of ρ at a time. The classical shadows protocol based on random single-copy measurements is triply efficient for the set of local Pauli observables. This and other protocols based on random singlecopy Clifford measurements can be understood as arising from fractional colorings of a graph G that encodes the commutation structure of the set of observables. Here we describe a framework for two-copy shadow tomography that uses an initial round of Bell measurements to reduce to a fractional coloring problem in an induced subgraph of G with bounded clique number. This coloring problem can be addressed using techniques from graph theory known as chi-boundedness. Using this framework we give the first triply efficient shadow tomography scheme for the set of local fermionic observables, which arise in a broad class of interacting fermionic systems in physics and chemistry. We also give a triply efficient scheme for the set of all n-qubit Pauli observables. Our protocols for these tasks use two-copy measurements, which is necessary: sample- efficient schemes are provably impossible using only single-copy measurements. Finally, we give a shadow tomography protocol that compresses an n-qubit quantum state into a poly(n )-sized classical representation, from which one can extract the expected value of any of the 4n Pauli observables in poly(n ) time, up to a small constant error.
Efficiently learning expectation values of unknown quantum states via classical shadows has become an important primitive in both theoretical and experimental aspects of quantum computation. Typically, classical shadow protocols involve randomised measurements induced by sampling uniformly randomly from a compact group, a situation which is now quite well understood. In this work we go beyond this standard assumption, studying the classical shadow protocols occasioned by sampling uniformly randomly from the so-called compact symmetric spaces. We uncover a unifying theory of such protocols, extending the extent to which the general theory of classical shadows is understood at a mathematical level. Interestingly, for the estimation of observables sampled from certain distributions we further find that some of these protocols allow for slight improvements in sample-complexity over existing shadow schemes. As part of our technical analysis we obtain ε\varepsilonε-approximate relative-error kkk-designs over the nnn-qubit symplectic group in depth O(nk+log⁡(1/ε))\mathcal{O}(nk+\log(1/\varepsilon))O(nk+log(1/ε)), which is optimal with respect to nnn, and may be of some independent interest.
The classical shadows protocol is an efficient strategy for estimating properties of an unknown state ρρρ using a small number of state copies and measurements. In its original form, it involves twirling the state with unitaries from some ensemble and measuring the twirled state in a fixed basis. It was recently shown that for computing local properties, optimal sample complexity (copies of the state required) is remarkably achieved for unitaries drawn from shallow depth circuits composed of local entangling gates, as opposed to purely local (zero depth) or global twirling (infinite depth) ensembles. Here we consider the sample complexity as a function of the depth of the circuit, in the presence of noise. We find that this noise has important implications for determining the optimal twirling ensemble. Under fairly general conditions, we i) show that any single-site noise can be accounted for using a depolarizing noise channel with an appropriate damping parameter fff; ii) compute thresholds fthf_{\text{th}}fth​ at which optimal twirling reduces to local twirling for arbitrary operators and iii) nthn^{\text{th}}nth order Renyi entropies (n≥2n \ge 2n≥2); and iv) provide a meaningful upper bound tmaxt_{\text{max}}tmax​ on the optimal circuit depth for any finite noise strength fff, which applies to all operators and entanglement entropy measurements. These thresholds strongly constrain the search for optimal strategies to implement shadow tomography and can be easily tailored to the experimental system at hand.