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MethodLayer 0

Probabilistic error cancellation (PEC)

Write the inverse of the characterized noise channel as a quasi-probability distribution over implementable operations, sample circuits from it, and combine the results with signed weights. Unlike extrapolation it inverts the noise rather than fitting through it, so it is unbiased in principle.

Takes

A circuit, a target observable, a noisy device, a shot budget, and — for the model-based methods — a learned characterization of the device noise.

Returns

A bias-reduced expectation-value estimate, with a variance — equivalently a sampling overhead — that grows with circuit volume.

Same contract as the slot it fills.

This one, drawn

Probabilistic error cancellation (PEC)Zero-noise extrapolation (ZNE)Probabilistic error cancellation (PEC)Readout (measurement) error mitigationCheck the symmetries the answer must haveZero-noise extrapolation (ZNE) — click the name to read about itZero-noise extrapolation (ZNE)Probabilistic error cancellation (PEC) — click the name to read about itProbabilistic error cancellation (PEC)Readout (measurement) error mitigation — click the name to read about itReadout (measurement) error mitigationCheck the symmetries the answer must have — click the name to read about itSymmetry verificationNoisy expectation value — you start hereBias-reduced expectation value — you finish here

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From Noisy expectation value to Bias-reduced expectation value

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What it fills

  • Recover a noiseless expectation value by post-processing

    Estimate what an observable would have measured on a noiseless device by running modified or repeated circuits on the noisy one and combining the results classically. No qubits are spent on redundancy; the whole price is paid in shots.

When it applies

Requires an accurate characterization of the noise channel, and that is the binding constraint — PEC's experimental history is largely the history of noise-learning methods catching up. Van den Berg et al. make it work at scale by learning a sparse Pauli-Lindblad model that captures crosstalk, over twirled circuits, so that the noise really is stochastic Pauli noise; Endo, Benjamin and Li treat imperfect knowledge of the error model explicitly. Temme, Bravyi and Gambetta state that the size of the circuits to which these techniques can be applied is limited by the rate at which errors are introduced.

Requires

Every step this method names moves its route along, so there is nothing it needs alongside them.

Example

given  the ideal circuit, a characterization of the noise channel, and a
       target precision delta

# an accurate characterization of the noise channel is required, and that is
# the binding constraint -- PEC's experimental history is largely the history
# of noise-learning methods catching up
#   van den Berg et al. make it work at scale by learning a sparse
#   Pauli-Lindblad model that captures crosstalk, over twirled circuits, so
#   that the noise really is stochastic Pauli noise
#   Endo, Benjamin and Li treat imperfect knowledge of the error model
#   explicitly

write the inverse of the characterized noise channel as a quasi-probability
    distribution over implementable operations
sample circuits from that distribution
combine the results with signed weights

return the combined estimate

# unlike extrapolation this inverts the noise rather than fitting through it,
# so it is unbiased in principle

# what it costs -- two statements, one per generation, both from the full
# texts; neither abstract carries a formula:
#   Temme, Bravyi and Gambetta, their Eq. 9: a noisy basis simulates the
#   ideal circuit with overhead gamma_beta >= 1, and the approx delta^-2
#   runs that precision delta would cost without noise are multiplied by
#   gamma_beta^2
#   van den Berg et al., for the learned sparse Pauli-Lindblad model:
#   inverting one noise channel costs sampling overhead
#   gamma = exp(2 sum_k lambda_k), a circuit of l noisy layers costs
#   gamma(l) = prod_{i=1}^{l} gamma_i, and the estimator's variance scales
#   with the square of the sampling overhead

# Temme, Bravyi and Gambetta state that the size of the circuits to which
# these techniques can be applied is limited by the rate at which errors are
# introduced

Cost, as the source states it

Two statements, one per generation. Temme, Bravyi and Gambetta: a noisy basis simulates the ideal circuit with overhead γβ≥1\gamma_{\beta} \ge 1, and the ≈δ−2\approx \delta^{-2} runs that precision δ\delta would cost without noise are multiplied by γβ2\gamma_{\beta}^2 (their Eq. 9). Van den Berg et al., for the learned sparse Pauli–Lindblad model: inverting one noise channel costs sampling overhead γ=exp⁡(2∑kλk)\gamma = \exp(2\sum_k \lambda_k), a circuit of ll noisy layers costs γ(l)=∏i=1lγi\gamma(l) = \prod_{i=1}^{l} \gamma_i, and the estimator's variance scales with the square of the sampling overhead. Both stated in the full texts; neither abstract carries a formula.

Implementations

  • PEC at scale with a learned sparse Pauli-Lindblad model

    The demonstration that moved PEC from principle to practice. The abstract names the obstacle in the same terms this record's `conditions` does: "Learning correlated noise channels in large quantum circuits, however, has been a major challenge and has severely hampered experimental realizations."

    A protocol for learning and inverting a sparse noise model that captures correlated noise and scales to large quantum devices, demonstrated on a superconducting quantum processor with crosstalk errors.

    The paper presents the demonstration as "an important milestone in opening the way to quantum computing with noise-free observables at larger circuit volumes". The abstract reports no numerical result, so none is quoted here; the overhead figures this record does carry are in `cost`, from the papers that state them.

What it needs

Nobody has taken this apart yet. That is a gap in this graph, not a claim that the method has no parts.

Other ways to fill the same slot

Different approaches

  • Zero-noise extrapolation (ZNE)

    Deliberately amplify the device noise by a set of known factors, measure the observable at each, and extrapolate the resulting curve back to zero noise. The fit is Richardson's deferred approach to the limit, or another model.

  • Readout (measurement) error mitigation

    Correct the classical readout channel by deconvolving the assignment matrix that maps true bitstrings to observed ones. The naive form calibrates and inverts the full 2nx2n2^n x 2^n matrix; the scalable form never forms it.

  • Check the symmetries the answer must have

    The physics fixes quantities the true state cannot change — particle number, spin. Measure them alongside the run and discard, or post-process away, the outcomes that violate them: an error that moves the state out of the sector announces itself, and one that keeps it inside does not.

In the Atlas

No record in the Atlas covers this yet. The catalogue is circuits and primitives; this part of the literature is not in it.

Sources