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A complete Randomized Benchmarking Protocol accounting for Leakage Errors

T. Chasseur, F. K. Wilhelm

arXiv:1505.00580v2quant-ph

TL;DR

Standard Randomized Benchmarking does not account for leakage into additional non-computational levels, and its treatment of gate dependence is limited to small deviations from predominantly gate-independent errors. The analysis expands the resulting Λ in powers of small ∆; the zeroth-order term is the identity, the first order vanishes, and leading matrix elements are second order. The protocol yields an average single-gate error estimate, supports strongly gate-dependent channels, and remains compatible with interleaved benchmarking and gates outside a unitary 2-design.

  • Problem

    Standard Randomized Benchmarking does not account for leakage into additional non-computational levels, and its treatment of gate dependence is limited to small deviations from predominantly gate-independent errors.

    Source: §I · p. 1, §III · p. 2

  • Method

    The analysis expands the resulting Λ in powers of small ∆; the zeroth-order term is the identity, the first order vanishes, and leading matrix elements are second order.

    Source: §III · p. 3

  • Results

    The protocol yields an average single-gate error estimate, supports strongly gate-dependent channels, and remains compatible with interleaved benchmarking and gates outside a unitary 2-design.

    Source: §I · p. 1, §VI · p. 6

  • Takeaways & Limitations

    Leakage-aware RB provides a scalable benchmarking framework for physical qubit systems with additional levels, including settings where the randomized gate set is not a twirl design.

    Source: §I · p. 1, A. RB of Clifford sets that are not… · p. 4, §VII · p. 7

Abstract

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Randomized Benchmarking allows to efficiently and scalably characterize the average error of an unitary 2-design such as the Clifford group $\mathcal{C}$ on a physical candidate for quantum computation, as long as there are no non-computational leakage levels in the system. We investigate the effect of leakage errors on Randomized Benchmarking induced from an additional level per physical qubit and provide a modified protocol that allows to derive reliable estimates for the error per gate in their presence. We assess the variance of the sequence fidelity corresponding to the number of random sequences needed for valid fidelity estimation. Our protocol allows for gate dependent error channels without being restricted to perturbations. We show that our protocol is compatible with Interleaved Randomized Benchmarking and expand to benchmarking of arbitrary gates. This setting is relevant for superconducting transmon qubits, among other systems.

I. INDRODUCTION

The paper motivates leakage-aware Randomized Benchmarking because standard methods are insufficient for non-two-level physical qubits and leakage can evade standard RB assumptions. It therefore develops a generalized protocol intended to handle leakage, gate dependence, sequence-sampling variance, and interleaved benchmarking.

Motivation

  • Quantum process tomography is unreliable because it cannot separate gate errors from state-preparation and measurement errors, and it requires exponentially many measurements.Source: §I · p. 1
  • Randomized Benchmarking provides a scalable, SPAM-independent way to characterize average fidelity for unitary 2-designs such as Clifford gates.Interleaved Randomized Benchmarking estimates the average fidelity of a single gate up to an uncertainty set by the group fidelity.Source: §I · p. 1
  • Leakage into additional levels is viable in superconducting qubits, NV-centers, trapped ions, and neutral atoms, while standard RB does not account for it because leakage can be non-Markovian.The paper also notes that standard RB treats gate dependencies only for small deviations from predominantly gate-independent errors.Source: §I · p. 1

Scope

  • The work introduces a leakage-aware RB generalization that also permits strongly gate-dependent error channels and supports IRB for gates outside a unitary 2-design.Source: §I · p. 1

Paper structure

  • The protocol is organized around a modified sequence model, a fidelity derivation, SPAM effects, gate-dependent errors, and interleaved benchmarking.Source: §I · p. 1

II. MODIFIED PROTOCOL

The modified protocol randomizes leakage-level phases independently across qubits, extending the Clifford set so sequence errors can be modeled despite non-computational levels. Its fitted fidelity model combines multiple decay modes whose coefficients include SPAM contributions and whose eigenvalues characterize the averaged error.

Leakage-aware protocol

  • Randomly inserting ±1 phase factors on every leakage level independently destroys phase relations and extends the Clifford group C into the altered Clifford set C∗.Source: §II · p. 1
  • The altered set C∗ has 2^N times as many elements as the N-qubit Clifford group C_N, but it need not be closed or contain inverses or a neutral element.Source: §II · pp. 1–2

Fidelity model

  • The protocol fits a multi-exponential fidelity model whose coefficients encode the measurement functional and SPAM errors, while at least one eigenvalue is unity and fewer than 3^N distinct eigenvalues occur.Source: §II · p. 2
  • The resulting sum of coefficients, P_i a_i, is interpreted as the average error of applying a single gate.Source: §II · p. 2

III. DERIVING THE FIDELITY MODEL

The modified model handles leakage by extending the Clifford protocol to an enlarged state space and fitting the resulting fidelity decay as a multi-exponential form. It remains usable without exact knowledge of the twirled channel, while assumptions about leakage-level phases and complex eigenvalues define important scope limits.

III. DERIVING THE FIDELITY MODEL

  • Leakage invalidates the standard Clifford-group twirl because the extended gate set need not be closed or contain inverses or a neutral element.The protocol restores the needed structure by randomizing leakage-level phases, under additional assumptions about the gates.Source: §III · p. 2
  • For multiple qubits, treating the extended Clifford set as a twirl design requires entangling gates to act diagonally on the non-computational subspace, with leakage phases randomized away.The same argument does not hold automatically for more than one qubit.Source: §III · p. 2
  • The twirled leakage channel is represented by a nonnegative transition matrix whose leading eigenvalue is one and whose eigenvector gives the long-sequence limiting state.Trace preservation makes every column sum to one in the eigenbasis used for the model.Source: §III · pp. 2–3
  • Small unitary errors make the channel Hermitian through second order, ensuring real eigenvalues unless higher-order effects encounter near-degenerate second-order eigenvalues.The expansion uses U = exp(-i∆H), with the zeroth-order identity and vanishing first-order contribution leaving second-order terms as the leading terms.Source: §III · p. 3
  • The measured fidelity is a linear functional of the twirled channel and therefore fits a multi-exponential decay with coefficients that include SPAM effects.The same model can characterize computational-subspace population and total leakage without requiring control over the leakage space.Source: §III · p. 3
  • Complex decay parameters can make fidelity oscillations difficult to fit, but population returning to the initial state means the fitted fidelity captures at least 1 − 2^-N of the actual error.This provides a conservative bound when oscillatory contributions are small or overlap with natural fluctuations.Source: §III · p. 3

A. RB of Clifford sets that are not twirl designs

When the Clifford set is not a twirl design, the protocol loses a uniform long-sequence limit, but computational-subspace invariance still permits fidelity estimation. Simulations show only a small difference in estimated average fidelity for the tested unitary error.

A. RB of Clifford sets that are not twirl designs

  • The modified analysis extends sequence-length evolution through a linear positive map even when the full Clifford operations lack ordinary group closure.The resulting fidelity is represented as a linear functional of a matrix exponential.Source: A. RB of Clifford sets that are not… · p. 3
  • The protocol may lack a uniform large-y limit because the Clifford set does not connect computational and leakage states, although computational-subspace invariance remains.Only real eigenvalues contribute a factor 1 − 2^-N that vanishes for long sequences.Source: A. RB of Clifford sets that are not… · pp. 3–4
  • For the tested unitary error, twirl-design and non-twirl Clifford sets produce only a small difference in average fidelity because the computational-subspace twirl remains intact.The comparison uses 40 random sequences per sequence length.Source: A. RB of Clifford sets that are not… · p. 4, Figure 1 · p. 4

B. State Preparation and Measurement Errors – SPAM

SPAM errors can be incorporated into the leakage-aware fidelity model without changing its sequence-length dependence. They mainly contribute an offset and a small, sequence-length-independent deviation from the idealized model.

B. State Preparation and Measurement Errors – SPAM

  • SPAM errors have very little influence on the protocol and can be isolated as a sequence-length-independent offset in the fidelity curve.The model remains applicable when preparation and measurement occur in the computational subspace.Source: B. State Preparation and Measurement… · p. 4
  • The effective prepared and measured states preserve the matrix-exponential fidelity model, with SPAM determining the average fidelity of the Clifford set.The actual average fidelity of a single random Clifford acts as a factor reducing the highest possible effective fidelity.Source: B. State Preparation and Measurement… · p. 4
  • For non-twirl Clifford sets, replacing the final operation by the actual inverse is valid in the computational subspace, while the remaining deviation is negligible and does not scale with sequence length.The deviation includes state-preparation, measurement, and gate-error contributions.Source: B. State Preparation and Measurement… · p. 4

IV. VARIANCE OF THE FIDELITY

The fidelity variance from sampling random Clifford sequences is analyzed perturbatively around the twirled channel. The resulting variance is bounded by an exponential decay times sequence length, and simulations show standard-RB scaling.

IV. VARIANCE OF THE FIDELITY

  • The variance analysis expands the finite-sequence channel around the twirled channel and retains terms through second order in the sequence-sampling perturbation.Single perturbations twirl to zero, so odd orders and cross terms between different errors vanish.Source: §IV · p. 5
  • The variance is bounded by an exponential decay multiplied by sequence length, with decay and error terms comparable to single-gate errors.The analysis proceeds under the simplifying assumptions s1 = s2 and s3 = s4.Source: §IV · p. 5
  • Figure 2 examines variance across possible Clifford-gate combinations for a small error of 7.42 × 10^-4 using 30 sequences.The figure probes how sequence sampling affects fidelity estimates.Source: Figure 2 · p. 5
  • For states with eigenvalue λ = 1, the perturbative E terms vanish, ensuring an exponential decay whose estimate has the same order as standard RB.These states are preserved by every quantum channel.Source: §IV · p. 5

V. GATE DEPENDENT ERRORS

The protocol accommodates gate-dependent leakage errors without altering the randomized-benchmarking sequence, while controlling their approximation error. It also identifies residual effects on phase randomization, eigenvalues, and initial-state dependence.

V. GATE DEPENDENT ERRORS

  • Gate-dependent leakage errors do not require altering the randomized-benchmarking protocol, although the resulting approximation incurs an overall error of order ∥Λ − Λ_j∥^y ≪ 1.The correction is not restricted to perturbative gate dependence in the same way as earlier leakage-free treatments.Source: §V · p. 6
  • Phase randomization becomes imperfect under gate-dependent errors, potentially enlarging the transfer-matrix dimension and producing more complex eigenvalues.The text characterizes these effects as small relative to the average error when they arise from second-order single-qubit errors.Source: §V · p. 6
  • The protocol can be approximated as independent of the initial and measured states up to a very small error, allowing real-eigenvalue fitting to account for at least 1 − 2^-N of the actual error.Gate-dependent channels nevertheless produce a fundamentally different error-associated matrix, so real eigenvalues are not generally guaranteed.Source: §V · p. 6

VI. INTERLEAVED RANDOMIZED BENCHMARKING

Interleaved Randomized Benchmarking extends the leakage-aware framework from average Clifford performance to individual arbitrary unitary gates. It estimates the target gate by comparing the combined interleaved error with the Clifford-set error, provided the final inverse is sufficiently accurate.

VI. INTERLEAVED RANDOMIZED BENCHMARKING

  • Alternating the gate of interest with random Clifford gates preserves the framework’s fit model because the appended interleaved operation remains a linear operation on the sequence.The construction uses the final inverse for the sequence without the first interleaved gate.Source: §VI · p. 6
  • The method applies beyond gates in the randomized Clifford set when the final inverting gate’s error is small relative to sequence errors, though it may be an order of magnitude larger than the gate error.Under this assumption, the protocol is stated to hold for every unitary operation.Source: §VI · p. 6
  • The interleaved protocol estimates an arbitrary gate’s error by subtracting the Clifford-set error from the combined error, yielding 5.50 × 10^-4 versus an actual 5.44 × 10^-4 in the example.The combined error in the example is 1.304 × 10^-3, while the gate-independent Clifford error is 7.57 × 10^-4.Source: §VI · p. 6, Figure 3 · p. 7

VII. CONCLUSION

The conclusion reports that leakage errors affect Randomized Benchmarking and motivates an altered protocol to account for them. The supplied conclusion passage does not provide further details on the protocol or its results.

VII. CONCLUSION

  • The work investigates how leakage errors affect Randomized Benchmarking.Source: §VII · p. 7
  • The authors provide an altered protocol intended to account for leakage errors.Source: §VII · p. 7
  • The conclusion passage frames leakage accounting as the paper’s central protocol contribution.Source: §VII · p. 7
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