Source-linked AI summary
Estimating the Coherence of Noise
Joel J. Wallman, Christopher Granade, Robin Harper, Steven T. Flammia
TL;DR
The paper addresses how to quantify coherence in quantum noise beyond average error rates. It defines coherence through unitarity, presents an efficient randomized-benchmarking estimation protocol, and establishes connections to unitary noise and achievable gate infidelity.
Problem
The paper asks how the coherence of quantum noise can be quantified using its effect on output purity.
Method
The paper defines unitarity through average purity change over pure-state inputs and estimates it using a protocol based on randomized benchmarking.
Results
The unitarity equals 1 if and only if the noise is unitary, provides a tight lower bound in terms of infidelity, and lower-bounds the best achievable gate infidelity under perfect unitary control.
Takeaways & Limitations
Unitarity quantifies the intermediate regime between fully incoherent and unitary errors and may improve bounds on worst-case error.
Takeaways & Limitations
The unitarity is not generally a monotone for trace-decreasing noise, and necessary and sufficient monotonicity conditions remain open.
Abstract
from arXiv · showhide
Noise mechanisms in quantum systems can be broadly characterized as either coherent (i.e., unitary) or incoherent. For a given fixed average error rate, coherent noise mechanisms will generally lead to a larger worst-case error than incoherent noise. We show that the coherence of a noise source can be quantified by the unitarity, which we relate to the average change in purity averaged over input pure states. We then show that the unitarity can be efficiently estimated using a protocol based on randomized benchmarking that is efficient and robust to state-preparation and measurement errors. We also show that the unitarity provides a lower bound on the optimal achievable gate infidelity under a given noisy process.
I. DEFINING UNITARITY
The paper defines unitarity to measure the coherence preserved by a noise channel through average output purity, after removing identity-related contributions. This correction avoids assigning unitary-like coherence to channels that do not preserve coherent superpositions.
- Definition: A naive output-purity average would assign the same unitarity to a nonunital state-preparation channel and a unitary channel.This occurs even though the state-preparation channel does not preserve coherent superpositions.
- Definition: A filtering channel similarly fails to preserve coherent superpositions and should match a completely depolarizing channel in unitarity.Both problematic cases arise when the identity is mapped to coherent terms or vice versa.
- Definition: Unitarity is based on the average purity of output states from pure inputs, with identity components subtracted.The subtraction accounts for trace-decreasing and non-unital channels.
- Definition: Equivalently, unitarity is the average squared length of the generalized Bloch vector after identity-related components are removed.The Bloch-vector formulation uses traceless, trace-orthonormal operator components.
II. THE ESTIMATION PROTOCOL
The estimation protocol adapts randomized benchmarking to estimate noise unitarity under a unitary 2-design model. Averaging squared observable expectations over random sequences yields an exponential decay whose rate identifies the unitarity while remaining robust to SPAM.
- Protocol assumptions: Each implemented 2-design operation is modeled as ideal unitary conjugation followed by a fixed CPTP noise channel independent of the ideal operation.The assumption can be relaxed without dramatically affecting randomized-benchmarking results.
- Protocol: The protocol randomly samples length-m sequences and estimates an observable expectation after applying each sequence.Many independent trials are performed for multiple sequence lengths.
- Estimation: For trace-preserving noise, the decay model contains the unitarity u(E), which equals 1 if and only if the noise channel is unitary.The parameter is constrained to u(E) ∈ [0, 1].
- Estimation: Fitting the averaged squared expectation to the protocol’s decay model provides an efficient and robust estimator of unitarity.The decay model incorporates constants accounting for SPAM and noise nonunitality.
A. Estimators
The estimator can measure purity through observables rather than single-outcome probabilities, using either two-copy SWAP measurements or one-copy operator-basis averaging. The one-copy approach reduces sequence-to-sequence uncertainty but is not scalable as written.
- Purity estimators: The protocol averages squared expectations of observables, allowing multiple observables to be evaluated with the same randomized sequence.Averaging over non-identity Pauli operators can simulate a two-state SWAP measurement.
- Purity estimators: For a single qubit in the Pauli basis, the shifted purity quantity is Pj = ⟨X⟩2 + ⟨Y⟩2 + ⟨Z⟩2.For physical states, this quantity lies in the interval [0, 1].
- Purity estimators: Purity can be measured directly with two parallel copies and a SWAP gate immediately before measurement.A one-copy alternative sums measurements over an orthonormal operator basis for identical sequences.
- Estimator behavior: Averaging over the operator basis reduces the between-sequence contribution to uncertainty in the squared-expectation estimates.The reduction follows because approximately unitary noise leaves relatively pure states that overlap with many non-identity Paulis.
- Limitations: The one-copy Pauli summation is not scalable with qubit number because n-qubit systems have exponentially many Pauli operators.Optimizations and analysis of a scalable two-copy protocol remain open problems.
- Protocol design: Unlike standard randomized benchmarking, the protocol does not require the 2-design to be a group or require inverse operations.The set G also need not be closed under composition.
B. Trace-decreasing noise
The framework extends to trace-decreasing noise by incorporating average survival rates and generalized decay curves. Its randomized-benchmarking structure preserves robustness to SPAM while changing the post-processing and interpretation of the decay.
- Trace-decreasing noise: Average survival rate measures the trace of states surviving the error channel, averaged over Haar-distributed pure inputs.When the channel is the average noise over G, the loss rate can be estimated experimentally.
- Trace-decreasing noise: The estimated loss rate includes a constant determined by state-preparation and measurement errors.This retains an explicit SPAM contribution in the trace-decreasing setting.
- Trace-decreasing noise: For trace-decreasing noise, the standard decay curve is generalized with additional constants and survival-related terms.The generalized model retains an exponential-decay fitting structure.
- Protocol relation: The protocol estimates an exponential decay rate analogously to randomized benchmarking and remains robust to SPAM.It is a variation of randomized benchmarking and resembles the protocol for estimating loss.
- Protocol relation: Compared with the loss protocol, this method squares individual-sequence survival probabilities before averaging and uses different preparation and measurement procedures.These differences change the analysis and interpretation of the resulting decay curves.
III. NUMERICAL SIMULATIONS
Numerical simulations test the unitarity model and estimation protocol across extreme unitary, composite non-unital, and random-channel noise. The results support accurate unitarity estimation and show that unitarity reflects channel structure while providing information distinct from average gate fidelity.
- Protocol estimation: The protocol estimates unitarity from purity-decay fits for non-unital amplitude damping composed with Haar-random or gate-dependent rotation noise.Simulations include SPAM on prepared states and measurements, with the single-qubit Clifford group used for randomized sequences.
- Model validation: The model agrees with simulated purity data for both Haar-random fixed unitaries and a 0.1-radian X-axis rotation.These simulations test the model under extreme unitary noise and show insensitivity to unitary noise.
- Statistical considerations: Statistical fluctuations arise from between-sequence sampling and within-sequence observable-estimation variation, with simulations using 30 sequences and 150 measurements.The purity observable reduces between-sequence variation, while squared-expectation values are estimated without bias.
- Protocol estimation: Estimated unitarities of 0.994, 0.993, and 0.978 are consistent with theoretical values of 0.994, 0.994, and 0.981, respectively.The estimates come from slopes of fits to the purity-decay model; confidence intervals assume Gaussian noise.
- Random-channel structure: Random single-qubit channels with higher Kraus rank tend toward smaller unitarity, indicating that unitarity carries information about channel structure.The channels are drawn from the Bruzda et al. random ensemble and evaluated using QuTiP.
- Random-channel structure: Unitarity and average gate fidelity are correlated but nonredundant, providing different information about random single-qubit noise channels.Figure 4 compares unitarity with fidelity to the identity channel.
IV. DERIVATION OF THE FIT MODELS
The derivation expresses the randomized-benchmarking signal in the Liouville representation, where the averaged operator reduces to a two-dimensional invariant subspace. Its decay model separates SPAM-dependent constants from exponential terms governed by the unitarity, with trace-preserving noise giving λ− = u(E).
- The Liouville representation converts channels into matrices, so channel composition becomes matrix multiplication and unitary conjugations form a unitary representation.
- The channel decomposes into state-dependent leakage, nonunital, and unital blocks, with the unital block Eu connected to the unitarity.
- Averaging the doubled representation identifies a two-dimensional invariant subspace spanned by the identity and SWAP operators.
- The averaged operator’s matrix element satisfies M22 = (d2 −1)−1∥Eu∥2 F = u(E), directly linking the fit operator to the unitarity.
- For time- and gate-independent noise, the expected squared signal follows decay equations whose constants depend only on SPAM errors and the unitary diagonalizing M.
- For trace-preserving noise, λ+ = 1 and λ− = u(E), while neglecting the unitarity dependence of the diagonalizing unitary makes the fit slightly less sensitive to u.
V. PROPERTIES OF THE UNITARITY
The unitarity has structural properties that make it a coherence measure and relate it to achievable gate infidelity. It is bounded by one, invariant under unitary composition, and can oscillate under general channel composition.
- For trace-preserving noise, the nonunital block obeys ∥En∥2 ≤ (d −1)[1 −u(E)], bounding nonunital behavior using the unitarity.
- For any channel, u(E) ≤ 1 with equality if and only if E is unitary, and unitary pre- and post-composition leaves u(E) unchanged.
- The unitarity and average gate infidelity satisfy inequalities that connect coherence to the best infidelity achievable with perfect unitary control.
- The inequality chain is saturated for depolarizing noise and depolarizing noise composed with amplitude damping, while its first bound reaches one for unitary noise.
- The unitarity is closely related to the purity of the Jamiołkowski state associated with the noise channel.
VI. CONCLUSION
The paper introduces unitarity as a measure of noise coherence, estimates it efficiently for average noise in a unitary 2-design, and links it to infidelity and worst-case-error bounds. It also identifies limitations concerning monotonicity, individual-gate errors, and scalability.
- The protocol efficiently estimates the unitarity of average noise in a unitary 2-design.
- Unitarity equals 1 if and only if the noise source is unitary, and it has a tight lower bound in terms of infidelity.
- Combining unitarity with infidelity can quantify intermediate noise coherence and potentially improve worst-case-error bounds.
- Smaller unitarity reduces sequence variance and allows more precise estimation of average incoherent survival probability for a fixed number of experiments.
- Open problems include monotonicity for trace-decreasing noise and bounding the unitarity of errors in individual gates.
- The Pauli-based purity measurement is not scalable beyond a handful of qubits because of exponential growth in the required measurements.