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Randomized benchmarking with gate-dependent noise
Joel J. Wallman
TL;DR
Gate-dependent noise can affect randomized-benchmarking decay, while earlier rigorous analyses relied on gate-independent noise assumptions. The paper proves that, for Markovian gate-dependent noise, its effect is captured by a decaying perturbation and the decay parameter retains an average-noise fidelity interpretation. For noise close to the identity, the perturbation is generally negligible at sequence lengths m≥3.
Problem
Earlier rigorous RB analyses assume gate-independent noise, while gate-dependent fluctuations can change the decay parameter relative to standard average-noise fidelity.
Source: §1 · p. 2Method
Results
For noise close to the identity, the perturbation is generally negligible for m≥3, and the decay parameter is the fidelity of the average noise between ideal gates.
Source: §5 · p. 8, §8 · p. 14Takeaways & Limitations
By reducing gate-dependent errors' potential impact, the analysis supports attributing statistically significant increases in survival probability with sequence length directly to non-Markovian noise.
Source: §8 · p. 14The zeroth-order model's higher-order terms are negligible only when mγ ≪1, creating tension with the sequence lengths needed for a reasonable fit.
Source: §7.1 · p. 12
Abstract
from arXiv · showhide
We analyze randomized benchmarking for arbitrary gate-dependent noise and prove that the exact impact of gate-dependent noise can be described by a single perturbation term that decays exponentially with the sequence length. That is, the exact behavior of randomized benchmarking under general gate-dependent noise converges exponentially to a true exponential decay of exactly the same form as that predicted by previous analysis for gate-independent noise. Moreover, we show that the operational meaning of the decay parameter for gate-dependent noise is essentially unchanged, that is, we show that it quantifies the average fidelity of the noise between ideal gates. We numerically demonstrate that our analysis is valid for strongly gate-dependent noise models. We also show why alternative analyses do not provide a rigorous justification for the empirical success of randomized benchmarking with gate-dependent noise.
1 Introduction
Randomized benchmarking (RB) efficiently estimates average fidelity, but its standard rigorous analysis assumes noise is Markovian and gate- and time-independent. This paper treats gate-dependent noise under Markovian noise and characterizes its effect on RB decay.
1 Introduction
- RB scales efficiently with qubit number and is robust to state-preparation and measurement errors, making it an important baseline for quantum-operation validation.It has also been used to optimize experimental control procedures.Source: §1 · pp. 1–2
- Earlier analyses assume Markovian, time- and gate-independent noise, while simulations found approximate single-exponential decays under some gate-dependent and non-Markovian models, sometimes with true and estimated fidelity differing by about a factor of two.Gate-dependent fluctuations were also reported to dominate the gate-independent decay curve and alter the inferred decay parameter relative to standard average-noise fidelity.Source: §1 · p. 2
- For gate-dependent, trace-preserving Markovian noise, the full fluctuation effect is captured by a single perturbation term that decays exponentially with sequence length.The paper proves its theorems without restricting noise to be trace-preserving, anticipating loss from encoded qubits due to imperfect error correction [27].Source: §1 · p. 2
- The RB decay parameter is the fidelity of a suitably defined average noise process, and the paper argues alternative analyses do not rigorously justify single-exponential fitting.The paper also addresses ambiguity in decomposing noise and uses it to cancel most gate-dependent fluctuations.Source: §1 · p. 2
2 Notation
The paper fixes notation for channels, their representations, and the norms and averages used in its analysis.
2 Notation
- Ideal channels use calligraphic letters, noisy implementations use a tilde, and channel composition is written as multiplication.The normalized identity is denoted by Id-hat, and the identity channel corresponding to a unitary U maps ρ to UρU†.Source: §2 · pp. 2–3
- The uniform average over a set X is written as E_x∈X[f(x)], and groups and sets use blackboard font.The unital component of a trace-preserving map is defined by applying G after removing the input's trace component.Source: §2 · p. 3
- Channels are represented as matrices in a trace-orthonormal operator basis, with Hermitian basis elements and the normalized identity as the first element.For Hermiticity-preserving maps, this convention makes the matrix representation real-valued.Source: §2 · p. 3
3 Randomized benchmarking protocol
The protocol samples random sequences from a unitary two-design, measures their survival probabilities, and averages over repetitions.
3 Randomized benchmarking protocol
- Choose a sequence length m and sample m operations uniformly at random from the group, then append the inverse of their product as the final operation.The group is a unitary two-design, such as the n-qubit Clifford group.Source: §3 · p. 4
- Prepare a d-level system in state ρ, apply the m sampled operations and final inverse, and estimate the observable Q's expectation as the sequence survival probability.Source: §3 · p. 4
- Repeat the sequence and measurement steps to estimate the average survival probability over random sequences to the desired precision.Source: §3 · p. 4
5. Repeat steps 1–4 for different values of m and fit to the decay model
The decay model uses the twirling properties of unitary two-designs, and the paper extends the gate-independent form to gate-dependent noise with a perturbation.
5. Repeat steps 1–4 for different values of m and fit to the decay model
- Best-practice choices for sequence lengths and the number of sequences per length remain unknown, though the paper points to guidance in Refs..Source: §5 · p. 4
- A unitary two-design reproduces the Haar measure's first and second moments, and its twirl maps a general CPTP channel to a depolarizing channel.The twirl of a channel C is its uniform group average under G†CG.Source: §5 · p. 4
- The twirled linear map has a depolarizing component and a uniform trace-decreasing component, with p and t computed from a lemma [30].Source: §5 · p. 4
- For time-independent Markovian gate-dependent noise, the RB decay model holds up to a small, exponentially decaying perturbation.Source: §5 · p. 5
4 Representing gate-dependent noise
The paper represents gate-dependent implementations using a decomposition into left and right maps around ideal gates, chosen to make the residual deviation small. This construction supports a gauge-invariant description of the noise between gates.
4 Representing gate-dependent noise
- The maps L and R decompose each implementation around its ideal gate, and for gate-independent noise they recover the shared noise factors.For gate-dependent noise, the construction instead chooses an approximate decomposition with a residual deviation.Source: §4 · p. 5
- The decomposition's parameters p and t describe the fidelity and trace loss of the average noise between ideal gates.When R is not invertible, an arbitrarily small perturbation can make it invertible, so the stated relation holds to arbitrary precision.Source: §4 · p. 5
- Choosing the largest eigenvalues in the stated averages identifies the maps used in the decomposition and can minimize the residual deviation.Theorem 3 bounds the deviation after choosing L = I, in terms of the implementation's distance from a depolarizing form.Source: §4 · pp. 6–7
- The noise between ideal gates, RL, is gauge invariant even though a gauge transformation changes L and R individually.The paper contrasts this with the original RB analysis, which it describes as not gauge invariant.Source: §4 · p. 6
5 Analyzing RB with arbitrarily gate-dependent noise
For gate-dependent noise, the average RB survival probability has the form of exponential decay with a perturbation term. The decay parameters retain an interpretation in terms of the average noise between ideal gates, while the perturbation decreases exponentially with sequence length.
5 Analyzing RB with arbitrarily gate-dependent noise
- The perturbation is generally negligible for noise close to identity when m ≥ 3, while m ≈ 1 can show larger, potentially oscillating deviations.For δ2 ⪅ 1/2, the decay is essentially a single exponential for m ≥ 10.Source: §5 · p. 8
- The average survival probability has a two-exponential form plus a perturbation, with p and t describing the fidelity and trace loss of the average noise between ideal gates.Theorem 4 states this for implementations that are Hermiticity-preserving maps.Source: §5 · p. 8
- When every implementation is CPTP, the model reduces to a standard single-exponential decay plus an exponentially decreasing perturbation.The passage attributes this form to the eigenvalues of the relevant CPTP averaged maps lying in the unit disc.Source: §5 · p. 8
- The numerical figure compares fitted decay-parameter confidence intervals that neglect δ1δm with theorem 4 predictions, and reports that the perturbation scales as δ1δm.The caption identifies the intervals as 90% confidence intervals and the predictions as p values.Source: §5 · p. 10, Figure 1 · p. 10
6 Numerics
The numerical section simulates single-qubit RB with independently sampled random unitary noise and compares fitted decay parameters with theorem 4 predictions. It uses repeated random sequences over a range of lengths to fit the decay.
6 Numerics
- The simulations compare decay parameters fitted from RB experiments with theorem 4 predictions and calculate δ1 and δ2 to assess the perturbation.The analysis plots the perturbation quantities in the gauge where L = I.Source: §6 · p. 10
- The simulated operation set is a subgroup of the single-qubit Clifford group that is also a unitary two-design.The noisy implementation uses independently sampled unitaries U and V with fixed infidelity r.Source: §6 · p. 10
- For each m ∈ {4, 8, 16, ..., 2048}, the authors average 100 random sequences and fit the results to a single-exponential model.They use variance-weighted fitting and set A = B = 0.5 because of the SPAM choice and unital noise.Source: §6 · p. 11
7 Alternative analysis of RB with gate-dependent noise
The section argues that alternative analyses do not rigorously justify single-exponential fits in practical regimes because their perturbation bounds can produce large systematic uncertainty. The problems include restrictive sequence-length conditions, a mismatch between the bounded noise measure and the RB decay parameter, and regimes where the resulting bounds are uninformative.
7 Alternative analysis of RB with gate-dependent noise
- The initial analysis's higher-order terms can make the systematic uncertainty dominate the infidelity even when fidelity is dominated by stochastic errors.The section contrasts this with the tighter bound developed in the present paper.Source: §7 · p. 11, §7.1 · p. 12, §7.2 · p. 13
7.1 Initial RB Analysis
- Higher-order perturbations in the initial RB analysis are negligible only when mγ ≪ 1, a condition that some noise models do not satisfy.This creates tension between gate-dependent noise and the sequence lengths needed for a useful fit.Source: §7 · p. 11, §7.1 · p. 12
- For the hypothetical single-qubit model, keeping the zeroth-order estimate within a factor of two requires γ ≤ r(E), or |θ| ≲ 1 − ν.The estimate can under-report r by 9r when ν = 0.99 and gate-dependent errors contribute 10% of infidelity, or when ν = 0.999 and they contribute 1%.Source: §7.1 · pp. 12–13
7.2 Subsequent analyses of RB
- Ref. 's analysis gives no operational interpretation of its decay parameter, while its fixed-decomposition noise measure can differ from rRB by orders of magnitude.Refs. and bound a perturbation around (1 − rRB)^m using a diamond norm tied to a fixed noise decomposition, leaving a potentially much looser bound.Source: §7.2 · p. 13
- At rRB ≈ 0.01, the best perturbation bound is approximately √rRB = 0.1, making sequence lengths with survival probabilities above 0.9 or below 0.6 effectively useless when A = B = 0.5.In the linear part of the decay, the perturbation's systematic uncertainty can dominate the decay.Source: §7.2 · p. 13
- The subsequent analysis requires m2 − m1 > 1000 at 1 − p ≈ 10^-6 only if 1 − p(RL) ≈ r(E), an unjustified assumption that leaves the needed spacing unknown.The authors therefore call the analysis impractical and say it does not explain RB's empirical success or justify a single-exponential fit without further analysis.Source: §7.2 · p. 14
8 Conclusion
The conclusion states that RB under gate-dependent noise has a single-exponential decay with negligible perturbation, and that its decay parameter is the fidelity of the average noise between ideal gates. The authors connect this result to diagnosing non-Markovian noise, discuss short-sequence deviations, and identify unresolved limits and future work.
8 Conclusion
- The decay parameter is the average fidelity of the noise between ideal gates, and the quantity p = E_G(RGL) is gauge invariant.The paper states that gauge transformations change L and R individually but leave RL invariant.Source: §8 · p. 14
- Significant revivals in average survival probability as sequence length increases can be attributed directly to non-Markovian noise.The conclusion distinguishes this from time-dependent noise, which can only alter the decay rate.Source: §8 · p. 14
- For m = 1, gate-dependent fluctuations may reach 0.1, and for m = 2 they may reach 0.01; these lengths should be omitted from exponential fits.Such deviations may also indicate gate-dependent noise.Source: §8 · p. 14
- Applying the analysis to other RB protocols that assume gate-independent noise is left for future work.Source: §8 · p. 14
- The solutions L and R in theorem 2 are not guaranteed to be CPTP maps, even after rescaling.Source: §8 · p. 14