Shadow Distillation: Quantum Error Mitigation with Classical Shadows for Near-Term Quantum Processors

• Fault-tolerant error correction remains experimentally demanding because its hardware overhead pushes current capabilities, motivating error mitigation for near-term devices.Source: §I · p. 1
• Classical shadows and randomized measurements offer efficient extraction of many linear state properties, whereas tomography becomes exponentially costly with system size.Source: §I · p. 1
• The paper applies randomized measurements and classical shadows to multiple-copy error mitigation while trading quantum resources against single-qubit measurement resources.Source: §I · p. 1
• The study explores measurement-setting versus repetition trade-offs and applies the method to stabilizer measurements of a 5-qubit trapped-ion GHZ state.Source: §I · p. 2

• The noisy state is modeled as an ideal component mixed with an error density matrix orthogonal to the target state.Source: §II · p. 2
• ⟨O⟩(m) = tr(Oρ^m)/tr(ρ^m) suppresses state-preparation errors exponentially in m when the ideal state is the dominant eigenvector of ρ.The model assumes ρ = (1 − ε)|ψ⟩⟨ψ| + ερ_error with ρ_error orthogonal to |ψ⟩.Source: §II · p. 2
• Previous schemes prepare ρ^m using m copies and controlled permutation operators, requiring two-qubit gates between copies while needing coherent access to only two copies simultaneously.Source: §II · p. 2
• Dual-state methods remove inter-copy quantum operations but require a known state-preparation unitary and similar noise on the state and its dual.Their increased circuit depth can be problematic under non-Markovian errors.Source: §II · p. 2

• Shadow Distillation uses classical shadows to estimate tr(Oρ^m) and tr(ρ^m), trading circuit size for sample complexity in near-term devices.Source: §III · p. 2
• Random local measurements use NU settings and NS shots per setting, converting outcomes into classical snapshots that can be averaged to estimate state observables.Source: §III · pp. 2–3
• Classical shadows based on random Clifford or Pauli measurements can be stored and manipulated with time and memory polynomial in nq, NU, and NS.Source: §III · p. 3
• For m = 2, estimators of tr(Oρ^2) and tr(ρ^2) can be computed from snapshot pairs, with polynomial-time postprocessing for products of single-qubit Pauli operators.Source: §III · p. 3
• The sample complexity of nonlinear quantities can grow exponentially with qubit number, but second-order mitigation for Pauli observables performs favorably against full quantum state tomography.Source: §III · p. 3

Scope

• The reported ∆2 measures only shadow-distillation estimation error and excludes errors that the mitigation procedure does not correct.The simulations therefore assess statistical estimation performance rather than total experimental error.Source: §IV · p. 3

Resource scaling

• ∆2 scales as 1/NU and, at fixed NU, converges as 1/NS to a constant determined by NU.The simulations use random pure states with depolarization noise and average over 100 random states.Source: §IV · p. 3

Purity dependence

• ∆2 rapidly converges with purity to a value determined by NU and NS.The purity dependence is examined with NS = NU = 1024.Source: §IV · p. 4, Figure 2 · p. 4

Estimator limitations

• The ratio estimator ˆo2/ˆs2 is generally biased, lacks a closed-form variance, and can suffer large errors when NU is small.Errors in the purity estimate can dominate the ratio; incorporating a prior purity estimate can reduce these errors.Source: §IV · p. 4

System-size scaling

• NU scales as 2^(γnq) with γ ≈ 0.82, compared with 3^nq for full quantum state tomography.For ∆ = 0.01, the fitted exponent is γ = 0.82, favoring the near-term regime.Source: §IV · p. 4, Figure 2 · p. 4

Experimental setting

• The GHZ experiment uses stabilizer measurements whose expectation values can be used to estimate GHZ-state fidelity.The state is a 5-qubit GHZ state with generators including pairwise ZZ operators and the five-qubit X operator.Source: §V · p. 4

Resource optimization

• The resource optimization models measurement time as T = NU(1000 + NS), reflecting that changing bases takes roughly 1000 times longer than repeating a fixed-basis measurement.For fixed T, the simulations select NU and NS that minimize ∆2, but the simulated optimum may not exactly match the experiment.Source: §V · p. 4

Scope

• The resource-allocation heuristic may depend on the specific error channel and the purity of the experimental state.Consequently, the simulated optimum is useful for allocation but is not guaranteed to be experimentally optimal.Source: §V · p. 5

Experimental results

• Increasing NU from 1446 to 2666 reduces the bootstrap error bars of the mitigated stabilizer values under the optimized resource choices.These values correspond to optimal choices for total measurement times T = 1.5 × 10^7 and 2.8 × 10^7, respectively.Source: §V · p. 5
• The operator Q_i X_i, representing X1X2X3X4X5, benefits most from shadow distillation because it is most severely affected by experimental errors.For the ideal GHZ state, all listed stabilizer expectation values equal 1.Source: §V · pp. 4–5

Error analysis

• Analysis of the mitigated results identifies detection errors and dephasing as major noise sources in the trapped-ion experiment.This diagnosis is based on simulations and analysis of possible experimental error sources.Source: §V · p. 5

• Shadow Distillation provides numerical evidence of better sample complexity than full quantum state tomography for mitigating state-preparation errors.Source: §VI · p. 5, Figure 5 · p. 9, Appendix C · p. 9

Resource optimization

• Prior-knowledge estimators and experimentally constrained allocation of NU and NS provide practical heuristics for reducing measurement error.The work models experiment time as T = NU(1000 + NS) and optimizes NU and NS for fixed T.Source: §VI · p. 5

Limitations and outlook

• Classical postprocessing scales as O(M^2), while full-density-matrix mitigation is limited by exponential storage and manipulation costs.The authors place Shadow Distillation where storing shadows is feasible but storing and processing the full density matrix is not.Source: §VI · p. 5

Outlook

• Combining data from different experimental systems may help convert coherent errors into incoherent errors without requiring coherent operations between systems.Source: §VI · p. 6

• The numerical analysis averages squared estimation errors over random mixed states and studies their dependence on NU, NS, and state purity.The simulations use bootstrap resampling, 10,000 random bases with 10,000 shots each, and NR = 100 random mixed states.Source: Appendix B · p. 8

Numerical simulations

• For the 5-qubit GHZ case, the empirical fit uses c1 = 3384 and c2 = 22 and captures the observed behavior across a large parameter range.Source: Appendix C · p. 9

• The Bayesian construction assumes a normally distributed purity observation with known variance and uses σ2 proportional to 1/NU.Source: Appendix D · pp. 9–10
• The method is most useful when the experiment provides a good guess about the state purity.Source: Appendix D · p. 10
• With µ0 = 0.9 and α = 100, the true purity is 0.81, and the biased estimator improves errors for smaller NU despite more than 10% prior error.Source: Appendix D · p. 10, Figure 6 · p. 10

• The trapped-ion processor contains nine 171Yb+ ions confined in a Paul trap with blade electrodes.Typical single- and two-qubit gate fidelities are 99.5(2)% and 98–99%, respectively.Source: §1 · p. 10
• The GHZ state is prepared on five qubits using the circuit shown in Fig. 7, which employs the two-qubit gate RXX(θ).Source: §1 · p. 10

• The simulated circuit includes coherent over-rotation, single-qubit dephasing, and detection errors as distinct error channels.The figure compares direct estimation with second-order mitigation for δcoh = 0.15, pdeph = 0.1, and pdet = 0.01.Source: §2 · p. 11, Figure 8 · p. 12
• Detection errors are modeled by rotating into the measurement basis, extracting outcome probabilities, and applying an uncorrelated error matrix with p0 = p1 = pdet.Expectation values are then calculated from the modified probability distribution.Source: §2 · p. 11
• For second-order mitigation under detection errors, the simulation reconstructs the density matrix from measurements of all 4^nq Pauli strings.Source: §2 · p. 11
• Static coherent errors do not benefit from Shadow Distillation because they change the eigenstates of ρ while leaving its eigenvalues unchanged.Source: Figure 8 · p. 12

• Static coherent errors do not benefit from shadow distillation because they change ρ's eigenstates without changing its eigenvalues.Source: §3 · p. 12
• The Q_i X_i operator is most affected by the experimental errors, yet its substantial improvement suggests that the errors are mostly incoherent.Source: §3 · p. 12
• Dephasing and detection errors benefit from shadow distillation, unlike coherent errors, as supported by numerical simulations.Source: §3 · p. 12
• Detection errors are unlikely to be the sole error source; residual errors may involve higher-order incoherent errors, coherent mismatch, or two-qubit-gate under(over)-rotation.Source: §3 · p. 12
• Detection-calibrated expectation values remain below shadow-distilled values, indicating mitigation beyond the detection process.Source: §4 · p. 12, Figure 9 · p. 13
• The contrast between ZZ and Q_i X_i may arise from either dephasing or detection errors, while the absence of a feature in ⟨ZZ⟩s can be explained by dephasing.Source: §4 · p. 13