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Characterizing large-scale quantum computers via cycle benchmarking
Alexander Erhard, Joel James Wallman, Lukas Postler, Michael Meth, Roman Stricker, Esteban Adrian Martinez, Philipp Schindler, Thomas Monz, Joseph Emerson, Rainer Blatt
TL;DR
Large-scale quantum processors require scalable characterization that accounts for global and local errors while remaining robust to SPAM effects. The paper develops cycle benchmarking and proves an estimator for process fidelity under stated assumptions. Its analysis supports efficient estimation with variance independent of qubit count, while experiments assess fidelity drift and finite-sampling behavior.
Problem
Large-scale quantum-process characterization must handle SPAM errors and remain practical as the number of qubits grows.
Source: A. State preparation and measurement… · pp. 7–8Method
Cycle benchmarking uses Pauli eigenstate preparations, Pauli measurements, and sequence-length comparisons to estimate process fidelity.
Source: A. State preparation and measurement… · p. 7, C. Estimating the process fidelity · p. 9Results
The estimator’s variance is independent of the number of qubits, supporting efficient estimation of 1 − ˆF to multiplicative precision.
Source: D. Finite sampling effects · p. 10Takeaways & Limitations
Cycle benchmarking provides a practical route to characterizing multi-qubit processes under the stated experimental and noise assumptions.
Source: A. State preparation and measurement… · p. 8, C. Estimating the process fidelity · p. 9
Abstract
from arXiv · showhide
Quantum computers promise to solve certain problems more efficiently than their digital counterparts. A major challenge towards practically useful quantum computing is characterizing and reducing the various errors that accumulate during an algorithm running on large-scale processors. Current characterization techniques are unable to adequately account for the exponentially large set of potential errors, including cross-talk and other correlated noise sources. Here we develop cycle benchmarking, a rigorous and practically scalable protocol for characterizing local and global errors across multi-qubit quantum processors. We experimentally demonstrate its practicality by quantifying such errors in non-entangling and entangling operations on an ion-trap quantum computer with up to 10 qubits, with total process fidelities for multi-qubit entangling gates ranging from 99.6(1)% for 2 qubits to 86(2)% for 10 qubits. Furthermore, cycle benchmarking data validates that the error rate per single-qubit gate and per two-qubit coupling does not increase with increasing system size.
I. THEORETICAL METHODS
The appendix establishes notation for state preparation, measurement, and noisy quantum processes using density matrices and quantum channels.
A. State preparation and measurement procedures
The protocol prepares Pauli eigenstates and measures Pauli expectations through local basis changes and computational-basis readout, while explicitly accounting for SPAM errors.
State preparation and measurement procedures
- The protocol prepares eigenstates of Pauli matrices P and measures expectation values of Pauli matrices C(P).For the investigated processes, C(P) is always an N-qubit Pauli matrix.Source: A. State preparation and measurement… · p. 7
- Local Clifford operators construct the basis changes to maximize SPAM coefficients and reduce statistical uncertainty.Source: A. State preparation and measurement… · p. 7
- Expectation values are obtained by rotating to the computational basis, measuring outcomes, and weighting probabilities using ideal quantities.Source: A. State preparation and measurement… · p. 7
- The number of measurements needed for fixed additive precision is independent of the qubit count N.This follows from averaging relative frequencies over all outcomes.Source: A. State preparation and measurement… · p. 7
- State preparation and measurement include initialization, readout, and basis-change errors on each qubit.These SPAM errors motivate a protocol robust to SPAM.Source: A. State preparation and measurement… · pp. 7–8
B. Modelling the decay as a function of the sequence length
Cycle benchmarking models repeated noisy cycles through randomized Pauli operations, producing a decay whose sequence-length dependence isolates the process under study under stated noise assumptions.
Modelling the decay
- The decay model assumes a Clifford cycle with Markovian noise and gate-independent noise on the random Pauli operations.The gate of interest may undergo an arbitrary Markovian process.Source: B. Modelling the decay as a function… · p. 8
- The effective error process is E = G† ˜GA, with a scalar β depending only on P and G^m(P).β = 1 in the absence of SPAM errors.Source: B. Modelling the decay as a function… · p. 8
- Random Pauli relabelling converts the sequence average into a Pauli twirl of the effective error process.The relabelled operations remain Pauli processes because G is a Clifford process.Source: B. Modelling the decay as a function… · p. 8
- Under Pauli covariance, each Pauli operator is an eigenoperator of the effective error channel.Source: B. Modelling the decay as a function… · p. 8
- The protocol compares outcomes at sequence lengths m1 and m2 whose ideal Pauli outputs agree, allowing SPAM factors to cancel in a ratio.Source: B. Modelling the decay as a function… · pp. 8–9
C. Estimating the process fidelity
The appendix proves that the cycle-benchmarking estimator provides a conservative approximation to process fidelity under the protocol’s assumptions.
Estimating the process fidelity
- The cycle-benchmarking estimator is an accurate, conservative estimate of process fidelity under the stated assumptions.Source: C. Estimating the process fidelity · p. 9
- The estimator satisfies ˆF ≤ F_RC(˜G, G).Source: C. Estimating the process fidelity · p. 9
- For a unitary target U, process fidelity reduces to fidelity with the identity after applying U† to the noisy process.Source: C. Estimating the process fidelity · p. 9
- The estimator’s approximation error is second order in the process infidelity.The appendix expresses this scaling as O([1 − F(E, I)]^2).Source: C. Estimating the process fidelity · p. 9
- The bound uses Pauli-channel covariance and the fact that Pauli expectation values lie in the unit disc.Source: C. Estimating the process fidelity · pp. 9–10
D. Finite sampling effects
Finite-sampling analysis separates uncertainty from random sequences and measurements from uncertainty due to sampling Pauli matrices. The variance from Pauli sampling is independent of qubit number and scales favorably when sequence-length differences are chosen appropriately.
Formal qualification
- Finite-sample statements can be made rigorous using concentration results and the union bound, but with additional notation and less favorable pessimistic constants.This qualification concerns the formal treatment of the approximately normal statements.Source: D. Finite sampling effects · p. 10
Finite sampling sources
- Finite random sequences and measurements produce approximately normally distributed errors in estimated expectation values.The resulting error scale depends on the sequence-length difference used in the estimator.Source: D. Finite sampling effects · p. 10
Pauli-matrix sampling
- The variance from sampling K Pauli matrices is independent of the number of qubits.Pauli matrices are sampled uniformly at random with replacement and their fidelity estimates are averaged.Source: D. Finite sampling effects · p. 10
- Choosing sequence-length differences proportional to 1/(1 − F̂) makes the variance proportional to (1 − F̂)^2.This allows efficient estimation of 1 − F̂ to multiplicative precision.Source: D. Finite sampling effects · p. 10
Experimental convergence
- σ = 0.0135(3)/√K describes the fitted decrease in standard deviation as the number of sampled subspaces increases.The observed standard deviation is larger than the quantum projection-noise lower bound σproj = 0.00151(2)/√K.Source: D. Finite sampling effects · p. 10
E. Correction operators for the MS gate
The MS-gate correction analysis uses the gate’s periodicity and its action on Pauli operators to simplify cycle-benchmarking sequences. For even N, the MS gate is related to an X⊗N operation at half-period.
Gate periodicity
- MS^4 = I allows sequence lengths for the MS gate to be restricted to integral multiples of 4.Because MS^2 ∝ X⊗N, even sequence lengths are also possible if the Pauli-dependent sign is tracked.Source: E. Correction operators for the MS gate · p. 10
Pauli propagation
- For even N, MS ∝ (I − iX⊗N)/2, which determines how arbitrary Pauli operators propagate through the gate.This propagation is needed to compute the expectation value of C(P).Source: E. Correction operators for the MS gate · p. 10
II. EXPERIMENTAL METHODS
The experiments implement cycle benchmarking with sequences of single-qubit rotations and N-qubit Mølmer–Sørensen gates, compiled into the ion-trap machine language. The implementation shares collective basis changes across qubit operations, while addressed z-rotations are expected to limit local-operation fidelity.
Sequence construction
- CB experiments use sequences containing single-qubit rotations and N-qubit MS gates.The sequences are defined using N-qubit Clifford gates according to the experimental protocol.Source: §II · pp. 10–11
- A rotation R(θ)j = exp(iθpj/2) uses a single-qubit Pauli operation pj ∈ [X, Y, Z].This defines the elementary rotation notation used in the experimental sequences.Source: §II · p. 11
Machine implementation
- Compilation represents an elementary single-qubit operation as one addressed z-rotation between two collective x- or y-rotations.Collective rotations act as shared basis changes across the register and can therefore be reused by individual qubit operations.Source: §II · p. 11
Error considerations
- Addressed z-rotations are expected to have larger infidelity than collective rotations because of greater intensity sensitivity.The AC-Stark implementation is quadratically more sensitive to intensity fluctuations than resonant x- and y-rotations.Source: §II · p. 11
Experimental parameters
- Table II summarizes the experimental parameters used to estimate local CB and dressed MS fidelities.Source: §II · p. 11, Table II · p. 11
A. Testing the dependence of the estimator on the sequence length
The estimator’s dependence on sequence length is tested on a six-qubit register using multiple sequence-length pairs. The measured process fidelities agree within half a standard deviation, supporting the assumed sequence-length independence.
Sequence-length test
- The Markovian-noise model predicts process-fidelity estimates independent of sequence lengths m1 and m2 up to O([1 − FRC(G̃, G)]^2).The experiment tests this prediction at three sequence lengths for six qubits.Source: A. Testing the dependence of the… · p. 11
Experimental result
- Three length pairs—4-8, 4-12, and 8-12—produce measured fidelities agreeing within half a standard deviation.This supports the validity of the assumptions for the experimental apparatus.Source: A. Testing the dependence of the… · p. 11, Table III · p. 11
B. Analyzing fidelity drift
Fidelity loss over time was attributed primarily to temperature-driven laser-beam misalignment, and modeled as a linear drift. The dressed MS gate drifted faster than local gates, with recalibration every two hours limiting expected loss to 1%.
- Temperature-driven laser-beam misalignment changes the addressing beam position, causing Rabi-frequency miscalibration and increased intensity fluctuations.Source: B. Analyzing fidelity drift · pp. 11–12
- 3.3(5)·10^-3 h^-1 was the average fidelity loss rate for local gates.Source: B. Analyzing fidelity drift · p. 12
- 5.4(8)·10^-3 h^-1 was the fidelity loss rate for the dressed MS gate.Source: B. Analyzing fidelity drift · p. 12
- 1 % was the expected maximum fidelity loss when recalibrating the apparatus every two hours.Source: B. Analyzing fidelity drift · p. 12
- The drift rates were obtained from estimated linear slopes of the 4-qubit fidelity measurements in Fig. 5.Source: Table IV · p. 12, Figure 5 · p. 13