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Experimental quantum state measurement with classical shadows

Ting Zhang, Jinzhao Sun, Xiao-Xu Fang, Xiao-Ming Zhang, Xiao Yuan, He Lu

arXiv:2106.10190v2quant-phphysics.data-anphysics.optics

TL;DR

The paper addresses the practical feasibility of advanced quantum-state measurements under realistic hardware noise and finite sampling. It experimentally compares classical-shadow strategies with conventional measurement methods on a four-qubit GHZ state, finding advantages for classical shadows and demonstrating their use in entanglement analysis.

  • Problem

    The feasibility and comparative performance of advanced measurement schemes on realistic noisy hardware remain under exploration.

    Source: p. 1

  • Method

    The authors use a four-qubit quantum processor to compare importance sampling, observable grouping, and uniform, biased, and derandomized classical shadows for observables, Hamiltonians, and nonlinear state properties.

    Source: pp. 1, 3

  • Results

    The experiment reports advantages for (derandomized) classical shadows over other advanced schemes and uses shadows to detect genuine multipartite and bipartite entanglement.

    Source: p. 4

  • Takeaways & Limitations

    The authors conclude that these measurement schemes may reduce measurement costs for larger, more complicated problems and can support energy estimation tasks on near-term devices.

    Source: p. 5

    The authors identify incorporating error mitigation into the measurement schemes as a direction for further research.

    Source: p. 5

Abstract

from arXiv · show

A crucial subroutine for various quantum computing and communication algorithms is to efficiently extract different classical properties of quantum states. In a notable recent theoretical work by Huang, Kueng, and Preskill [Nat. Phys. 16, 1050 (2020)], a thrifty scheme showed how to project the quantum state into classical shadows and simultaneously predict $M$ different functions of a state with only $\mathcal{O}(\log_2 M)$ measurements, independent of the system size and saturating the information-theoretical limit. Here, we experimentally explore the feasibility of the scheme in the realistic scenario with a finite number of measurements and noisy operations. We prepare a four-qubit GHZ state and show how to estimate expectation values of multiple observables and Hamiltonians. We compare the measurement strategies with uniform, biased, and derandomized classical shadows to conventional ones that sequentially measure each state function exploiting either importance sampling or observable grouping. We next demonstrate the estimation of nonlinear functions using classical shadows and analyze the entanglement of the prepared quantum state. Our experiment verifies the efficacy of exploiting (derandomized) classical shadows and sheds light on efficient quantum computing with noisy intermediate-scale quantum hardware.

I. METHODS

The section presents a unified framework for estimating Pauli-decomposed observables, then describes importance sampling, grouping, and classical-shadow strategies within it. It also explains how classical shadows estimate nonlinear state properties from single-copy measurements and can be used to analyze entanglement.

A. Framework for measuring quantum states

  • The framework estimates an observable by selecting measurement bases and probabilities, then processing outcomes with an estimator defined for the chosen scheme.Observables are decomposed into Pauli operators, and compatible operators can be measured in the same basis.Source: A. Framework for measuring quantum… · p. 7
  • Importance sampling selects an observable with probability proportional to its coefficient weight, while grouping assigns compatible observables to shared measurement bases.The grouping method described is largest degree first, with basis probabilities chosen uniformly or by group weight.Source: A. Framework for measuring quantum… · p. 7
  • Uniform classical shadows sample each local Pauli basis uniformly, while locally biased shadows optimize local measurement probabilities; uniform-shadow variance grows exponentially with observable support.The section also describes derandomized shadows, which choose bases deterministically using a greedy algorithm.Source: A. Framework for measuring quantum… · p. 8
  • Derandomized estimation can be biased when some observable terms are not hit by any selected measurement basis, and the procedure is not guaranteed to be globally optimal.When every observable receives at least one measurement basis, the stated estimator is unbiased.Source: A. Framework for measuring quantum… · pp. 8–9

B. Classical shadows

  • Classical shadows extend to nonlinear functions by estimating multi-copy observables, including subsystem purity and partially transposed density-matrix moments, from distinct shadows of single-copy measurements.The U-statistics estimator sums over distinct shadows; the section notes that multi-copy joint measurements may reduce sample complexity but can be challenging experimentally.Source: B. Classical shadows · pp. 10–11
  • The post-processing for partially transposed moments avoids forming an exponentially large density matrix, although estimator variance scales exponentially with subsystem size.The experiment uses random local Clifford operations and projective measurements to construct shadows.Source: B. Classical shadows · pp. 9, 11

C. Error analysis for higher order nonlinear function

Higher-order nonlinear estimates are built from multiple classical shadows, with uncertainty determined by both joint-state and lower-order variance contributions. Their sample-size behavior changes from 1/Ns in an intermediate regime to 1/√Ns asymptotically, while the required measurements scale exponentially with function order and target-system size.

C. Error analysis for higher order nonlinear function

  • Higher-order functions are represented as linear functions on tensor products of copies, estimated using symmetric products of distinct classical shadows.For experiments, the full density matrix can instead be stored directly, but that calculation is not scalable to large systems.Source: C. Error analysis for higher order… · pp. 11–12
  • The required number of measurements scales exponentially with both the nonlinear-function order and the target-system size.The sample bound is framed for achieving estimation error ε with failure probability δ.Source: C. Error analysis for higher order… · p. 11
  • In the intermediate-sample regime, the estimation error decays as 1/Ns, whereas it decays as 1/√Ns in the large-sample limit.The intermediate behavior applies when 4|AB| > 2p2^2|AB|; in the large-sample limit, the first variance term dominates.Source: C. Error analysis for higher order… · p. 12
  • The variance analysis combines joint-state variance with contributions from lower-order variance terms, which can be iteratively computed in canonical form.For nth-order PT moments, the lower-order terms are organized by configuration counts and permutations.Source: C. Error analysis for higher order… · p. 12

D. Derivation of the PT-moments

This section proves an equality used in the PT-moment derivation by expanding the left-hand side and reducing the resulting index expression. The final equality follows by a cyclic index replacement modulo n.

D. Derivation of the PT-moments

  • The derivation establishes the stated equality by expanding the left-hand side into indexed density-matrix elements and contractions.The displayed expansion supplies the terms that are contracted in the proof.Source: D. Derivation of the PT-moments · p. 12
  • The final equality follows by replacing k with k + 1 modulo n across the indices.The passage identifies this replacement as the last step completing the proof.Source: D. Derivation of the PT-moments · p. 12

II. EXPERIMENTAL IMPLEMENTATION

The implementation uses wave plates, a polarization beam splitter, and a beam displacer to control and route photons. A Sagnac source produces polarization-entangled photon pairs that are extended into a four-qubit GHZ state across polarization and path degrees of freedom.

II. EXPERIMENTAL IMPLEMENTATION

  • Half- and quarter-wave plates implement unitary transformations, with their angles measured relative to vertical polarization.The polarization beam splitter transmits horizontally polarized photons and reflects vertically polarized photons; the beam displacer routes polarization components into distinct paths.Source: §II · p. 13

A. Polarization-entangled photon source

  • The Sagnac-interferometer source generates maximally polarization-entangled photon pairs, while the right-hand setup targets a four-qubit GHZ state encoded in polarization and path.The source uses a periodically poled potassium titanyl phosphate crystal pumped by a 405 nm UV laser diode.Source: Figure 4 · p. 13, A. Polarization-entangled photon source · p. 13

B. Preparation of GHZ state

The experiment extends polarization-entangled photons into a four-qubit GHZ state using beam displacers and fiber-compensation optics. Tomography reconstructs the prepared state, whose measured fidelity is 0.9546 ± 0.0006.

B. Preparation of GHZ state

  • Two beam displacers extend polarization-entangled photons into the four-qubit GHZ state, with wave plates correcting fiber-induced unitary transformations.The state was reconstructed by quantum state tomography using 7.67424 × 10^5 coincidences; the figure displays its real and imaginary density-matrix parts.Source: B. Preparation of GHZ state · pp. 13–14, Figure 5 · p. 14
  • The prepared four-qubit GHZ state has a measured fidelity of 0.9546 ± 0.0006.The principal imperfections are attributed to high-order emission in spontaneous down-conversion and interference on the beam displacer.Source: B. Preparation of GHZ state · p. 14

C. Implementation and detection of classical shadow

The implementation realizes classical-shadow measurements with photonic polarization- and path-encoded qubits. Separate optical configurations apply single-qubit Clifford operations before Pauli-Z readout or measure directly in a chosen Pauli basis.

C. Implementation and detection of classical shadow

  • Single-qubit Clifford operations and Pauli-Z readout are implemented with wave plates, with up to three wave plates sufficient for any single-qubit Clifford unitary.The experimental setup detects the four polarization-path outcomes after the operations.Source: C. Implementation and detection of… · pp. 14–15

D. Measurement in Pauli basis

E. Data acquisition

Data acquisition identifies photon coincidences within a fixed time window and accumulates a specified number for each randomly selected unitary. The accompanying histogram reports the time required to accumulate different coincidence counts across 700 unitaries.

E. Data acquisition

  • A coincidence is a second photon arrival within a 3 ns window after fixing the first photon's time tag; otherwise, acquisition advances to the next time tag.The procedure repeats until one coincidence is obtained.Source: E. Data acquisition · p. 17
  • For each randomly selected unitary, the experiment accumulates Nr coincidences; the histogram shows statistical times for Nr = 2, 5, 10, and 20 across 700 unitaries.The caption defines the total coincidences as 700Nr.Source: Figure 7 · p. 17, E. Data acquisition · p. 17

A. Statistical error of the estimation

The reported statistical errors quantify variation across 20 independent repetitions of the full setup. For each measurement basis, the displayed results use Nr = 5 coincidences.

A. Statistical error of the estimation

B. Numerical simulation with noiseless state

In noiseless-state numerical simulations, derandomized classical shadows outperform the other measurement schemes, including for Hamiltonian estimation. The comparison defines estimation error against direct calculation on |GHZ4⟩.

B. Numerical simulation with noiseless state

  • Derandomized classical shadows outperform the other schemes in noiseless-state estimation, especially for ⟨H2⟩.The authors attribute the greater measurement cost of ⟨H2⟩ to its larger number of noncommuting terms.Source: B. Numerical simulation with noiseless… · p. 18, Figure 9 · p. 18
  • The noiseless-state estimation error is the difference from direct calculation with |GHZ4⟩, and the simulations also show derandomized classical shadows outperforming other schemes.The authors expect greater advantages for larger systems or more complex Hamiltonians.Source: B. Numerical simulation with noiseless… · p. 18

C. Estimation of cluster Hamiltonian with Ising interactions

For the cluster Hamiltonian, derandomized classical shadows performed slightly better experimentally than LDF grouping, with relatively small variance. The hydrogen and nonlinear-function demonstrations extend the comparisons to molecular energies, subsystem moments, and entanglement analysis.

C. Estimation of cluster Hamiltonian with Ising interactions

  • Derandomized classical shadows had relatively small variance versus LDF grouping and slightly better experimental performance for estimating the cluster Hamiltonian’s second-order moment.The authors suggest broader measurement-basis coverage may improve robustness to measurement noise in this case, while noting that the explanation is not definite.Source: C. Estimation of cluster Hamiltonian… · pp. 18–19, Figure 10 · p. 19

D. Estimation of hydrogen molecular Hamiltonian

  • For the four-qubit hydrogen molecule, the five measurement schemes showed similar energy-estimation performance across Jordan-Wigner, parity, and Bravyi-Kitaev encodings.The authors note that advanced schemes could outperform conventional measurements as problem size increases, as shown theoretically and numerically [47] [51].Source: D. Estimation of hydrogen molecular… · p. 19, Figure 11 · p. 19

E. Estimation of nonlinear function

  • The nonlinear-function demonstration estimates subsystem purity and p2 and p3 moments, but does not assure violation of the p3-PPT condition for the reduced subsystem.The authors present the partial-transpose moment estimates as proof of principle and say the experimental results can help infer entanglement structure.Source: E. Estimation of nonlinear function · p. 20, Figure 12 · p. 20, Figure 13 · p. 20

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