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Efficient learning of quantum noise
Robin Harper, Steven T. Flammia, Joel J. Wallman
TL;DR
Characterizing quantum noise requires efficient estimation as the full error-rate distributions grow exponentially with qubit number. The protocol reconstructs reduced, locally averaged noise information and reveals correlations, including strong long-range correlations involving qubit 3 in a 14-qubit architecture.
Problem
Full distributions of observed or Pauli error rates grow exponentially with the number of qubits, motivating scalable noise characterization.
Method
The protocol uses independent measurements and single-qubit Clifford twirls to reconstruct relevant averaged noise parameters while preserving information about many-body correlations.
Results
Qubit 3 became strongly correlated with all other machine qubits across three two-qubit-gate activation patterns, with JSD values 0.216(1), 0.218(3) and 0.212(2).
Takeaways & Limitations
Covariances and correlations between qubits can be determined scalably, with Gibbs random field decompositions motivated by the device's physical layout.
Takeaways & Limitations
The best covariance-matrix estimator for learning SPAM-free observed or Pauli error rates remains an open question.
Abstract
from arXiv · showhide
Noise is the central obstacle to building large-scale quantum computers. Quantum systems with sufficiently uncorrelated and weak noise could be used to solve computational problems that are intractable with current digital computers. There has been substantial progress towards engineering such systems. However, continued progress depends on the ability to characterize quantum noise reliably and efficiently with high precision. Here we describe such a protocol and report its experimental implementation on a 14-qubit superconducting quantum architecture. The method returns an estimate of the effective noise and can detect correlations within arbitrary sets of qubits. We show how to construct a quantum noise correlation matrix allowing the easy visualization of correlations between all pairs of qubits, enabling the discovery of long-range two-qubit correlations in the 14 qubit device that had not previously been detected. Our results are the first implementation of a provably rigorous and comprehensive diagnostic protocol capable of being run on state of the art devices and beyond. These results pave the way for noise metrology in next-generation quantum devices, calibration in the presence of crosstalk, bespoke quantum error-correcting codes, and customized fault-tolerance protocols that can greatly reduce the overhead in a quantum computation.
I. METHODS
The methods define probability-distribution metrics, correlation measures, and channel representations used to analyze observed quantum errors. They also describe the experimental implementation on a 14-qubit device.
- Mutual information quantifies dependence between two error variables by measuring information gained about one from observing the other.
- The covariance matrix treats observed qubit errors as 0/1 variables and is computed from their centered second moments.
- Pearson correlations are obtained by normalizing covariances by the variables’ standard deviations, producing the correlation matrix plotted in the paper.
- The Jensen-Shannon distance provides a finite, symmetric metric when relative entropy may be undefined because q assigns zero probability where p does not.
- The experiment was conducted on IBM’s 14-qubit Quantum Experience Melbourne device using separate single-qubit and two-qubit protocol runs.
- The channel’s unital block contains the diagonal information needed to extract Pauli noise, including channel or subspace fidelity.
A. Randomized Benchmarking
The protocol extends randomized benchmarking from estimating a dominant decay parameter to reconstructing many observed error rates simultaneously. Clifford twirls, probability transforms, decay fitting, and marginalization support single- and two-qubit characterization.
- Randomized benchmarking uses random Clifford sequences of varying lengths, measurements, sequence averaging, and exponential-decay fitting to estimate noise parameters robust to SPAM errors.
- Simultaneous single-qubit twirls reconstruct 2^n decay parameters from the probability distribution of independent measurements across n qubits.
- The protocol applies a Walsh-Hadamard transform to averaged measurement probabilities, fits decay models, and transforms the estimates back into a probability distribution.
- Two-qubit characterization uses randomized benchmarking on qubit pairs, where errors can spread between coupled qubits and appear as correlations.
- Full two-qubit Clifford twirls can produce decay rates too large to measure accurately when all qubits are active simultaneously.
D. Extracting the fidelity
The protocol’s decay estimates can be converted into fidelity measures and reduced to subsets of qubits through probability marginalization. This supports fidelity extraction without necessarily running separate experiment types.
- Fidelity can be reconstructed from the decay estimates by using the average of the non-identity diagonal elements of the noise superoperator.
- Using an alternative twirling group can extract 2^n parameters per run, and real Clifford twirling can place all decay parameters on selected λ{I,Z} terms for fidelity calculation.
- The simulated six-qubit reconstruction recovered 64 averaged noise-matrix elements with relative error below 2% using 50 sequences across 11 gate lengths.
- For selected qubit subsets, decay parameters are obtained by converting to probabilities, marginalizing, and converting back.
III. MEANING OF “OBSERVED ERROR RATE”
The observed error rates are a reduced distribution obtained after local Clifford averaging, retaining the information needed to describe many-body noise correlations.
- The protocol uses a 2^n reduced distribution called the observed error rates.
- Local Clifford averaging reduces the reconstructed parameters from 4^n to 2^n.
- Despite this reduction, the observed error rates can describe all many-body correlations in the noise.
2. Are the observed error rates a meaningful quantity?
The section frames whether the reduced observed error-rate distribution remains sufficient to represent the noise correlations of interest.
- The key question is whether observed error rates remain sufficient to describe all many-body noise correlations.
A. What is the definition of the observed error rates?
Observed error rates are joint probabilities for error patterns after Clifford averaging, rather than probabilities for individual qubit errors or the full Pauli-error distribution.
- Observed error rates form a joint distribution giving the chance of seeing each error pattern across the qubits.
- A Walsh-Hadamard transform of the reduced 2^n eigenvalues produces the observed error rates.
- For a single qubit, Clifford averaging combines the X, Y, and Z outcomes into averaged observed-error probabilities.
- Figure 7 organizes two-qubit protocols through gate-activation schematics, mutual-information diagrams, and covariance matrices.
- The observed error rates differ from the full Pauli-error rates, which separately distinguish no error and X, Y, and Z errors.
B. Are the observed error rates a meaningful quantity?
Observed error rates have an operational interpretation as probabilities of observed error patterns and can retain enough information to reconstruct the averaged Pauli noise description under Clifford twirling.
- A Walsh-Hadamard transform maps the observed distribution to the averaged Pauli noise matrix needed by the reconstruction procedure.
- The reduced distribution represents the chance of observing an error pattern in the computational basis.
- Clifford averaging makes Pauli errors observable only probabilistically, producing the observed distribution from averaged detection events.
- Under the assumption of a correct Clifford twirl, the observed distribution allows reconstruction of the full averaged Pauli noise matrix.
- The observed error rates are informationally interchangeable with the averaged global Pauli distribution.
C. Are the observed error rates are still capable of describing all many-body correlations in the noise?
The reduced observed error-rate distribution remains compact while preserving the correlations present in two-qubit Pauli noise. A two-qubit example shows that Clifford twirling and reduction retain independent-error structure.
- Protocol tests: The demonstration tests the two-qubit protocol by comparing the full and reduced algorithms and checking whether qubit error rates remain uncorrelated.It also introduces an arbitrary two-Pauli correlation to test whether the averaging procedure can detect it.
- Independent errors: Two-qubit Pauli errors are first constructed independently and then transformed through Clifford twirling to obtain a reduced representation.The reduced representation is described as substantially more compact than the full probability distribution.
- Independent errors: The reduced algorithm produces data consistent with the independently generated, Clifford-twirled two-qubit errors.The corresponding tables compare the independent Pauli errors, their Clifford-twirled form, and the reduced result.
- Independent errors: The mini-correlation matrix has no correlations, matching the independent-error construction.The original probability distribution likewise indicates that the errors are independent.
TABLE IV – continued from previous page
Introducing an XY correlation changes the two-qubit Pauli error distribution, and the reconstructed correlation matrix detects the resulting dependence between qubits.
- Correlated errors: The protocol reconstructs the altered qubit eigenvalues, observations, fitted eigenvalues, and observed error distribution from the correlated-noise example.These quantities are organized in Table VI for the final Walsh-Hadamard transform.
- Correlated errors: The resulting correlation matrix contains non-zero off-diagonal elements, showing that the introduced correlation is detected.The displayed off-diagonal value is 0.0639497067.
D. Converting between twirled Pauli eigenvalues/Pauli error rates and qubit eigenvalues
The reduced observed error rates can be converted to and from twirled Pauli eigenvalues and error rates using Walsh-Hadamard-transform relationships. Correlation patterns are preserved across the two representations, although their magnitudes differ.
- Conversion relationships: Pauli eigenvalues and Pauli error rates are organized as vectors, with reduced eigenvalues obtained by removing eigenvalues associated with X and Y components.The reduced vector supports reconstruction of the full Pauli error-rate representation.
- Conversion relationships: The averaged error rates are recovered from the reduced representation through the stated transform relationships.These conversions make the procedures of Ref. [28] applicable after obtaining the reduced data.
- Conversion relationships: The final conversion step relates averaged error rates and observed error rates through H⊗nµo = λΣ.This relationship connects the reduced eigenvalue vector to the observed error-rate representation.
- Correlation comparison: The Pauli-error correlation magnitudes are larger than the observed-error magnitudes, but both correlation matrices have the same pattern.Thus the reduced observed error rates preserve which qubit pairs are correlated while changing absolute correlation strength.
IV. SCALABLE ESTIMATIONS
Full probability-distribution reconstruction becomes infeasible as qubit number grows, motivating scalable covariance and Gibbs random-field descriptions. Marginalization and transforms enable scalable correlation analysis, while covariance estimation remains an open methodological issue.
- Scalability: The full observed- or Pauli-error distribution grows exponentially with qubit number, whereas covariance matrices and GRF decompositions remain polynomial-time targets.The Walsh-Hadamard transform commutes with marginalization, allowing experiment-level marginalization before fitting and transformation back.
- Scalability: All qubit covariances and correlations can be determined scalably, and GRF decompositions can be motivated by the device’s physical layout.This provides a structured description without reconstructing the entire global distribution.
- Scalability: For the demonstrated 14-qubit-scale setting, the observed-error-rate values were manageable on current computers, but larger systems require consistent estimators from marginalized covariance matrices.The text notes that brute-force manipulation is not generally available when the system becomes larger.
- Correlation comparison: Correlation matrices from observed and averaged Pauli error rates preserve the same pattern, while observed-error correlations have reduced magnitude.Figure 8 compares the two matrices and supports marginal analysis of pairwise differences.
- Open limitations: The best covariance-estimation method for SPAM-free observed or Pauli error rates remains an open question.The text also leaves open whether matrix-product-state-like methods can be generalized provably to quantum channels.
- Simulation: Simulation studies examine six-qubit single- and two-qubit noise models, including arbitrary high-fidelity single-qubit noise and controlled-qubit-rotation noise.The single-qubit simulation extracts 2^6 averaged λ values and reconstructs distributions over 2^6 measurement outcomes.
VI. TWO-QUBIT PROTOCOL ON THE 14-QUBIT ARCHITECTURE
The two-qubit protocol was tested under three gate-activation patterns, revealing strong correlations involving qubit 3 and discrepancies in empirically observed decay rates.
- Three two-qubit gate-activation patterns were tested with the protocol.
- Qubit 3 became strongly correlated with all other machine qubits in every activation pattern.
- The correlation matrices revealed empirically distinct decay rates for qubit pairs that should have identical rates under a two-qubit twirl.The protocol highlighted this discrepancy, although its cause was left for future work.
A. Further experiments relating to the two-qubit protocol
Further experiments used conditional mutual information and wait-time controls to examine the correlations observed during the two-qubit protocol. Most long-range correlations also appeared under a comparable wait time, while the experiments comprised separate single- and two-qubit runs.
- Further experiments relating to the two-qubit protocol: Conditional mutual information measured residual information between two qubits after conditioning on all other qubits.Independence corresponds to 0, while complete dependence corresponds to 1.
- Further experiments relating to the two-qubit protocol: Most long-range correlations seen during the two-qubit protocol also arose when a comparable wait time was imposed.The authors did not speculate about the cause because their access to and knowledge of the machine were limited.
- Experimental runs: The single-qubit experiment used 1000 runs, whereas the two-qubit experiment retrieved 973 jobs.
- Figure 10: Figure 10 plotted each qubit's conditional mutual information with every other qubit, using bar color for physical Manhattan distance and bar length for mutual-information value.