Quickly extracting fidelity decay rates in random circuit benchmarking experiments

AI summary

Overview

Research area: Quantum computing hardware characterization — randomized benchmarking (RB) and cross-entropy benchmarking protocols for measuring gate fidelity on near-term devices.

Technical level: Advanced. The core idea is explainable in plain terms, but the paper's guarantees rest on channel-twirling theory, Pauli/Clifford symmetry, Weingarten calculus, and tensor-network calculations.

Scope: The paper introduces a one-parameter family of classical post-processing "filter functions" for random-circuit benchmarking — most importantly a "trace filter function" (α = 3) — that removes finite-depth mixing bias so that gate fidelities can be extracted from short circuits, and validates the method on up to 12 qubits of a Rigetti superconducting processor.

What This Paper Is About

Randomized benchmarking with structured circuits (such as one-dimensional brickwork circuits, or the circuits used in linear cross-entropy benchmarking, LinXEB) only produces a clean single-exponential fidelity decay after the circuit ensemble has "mixed" — i.e., after some minimum circuit depth. Below that depth, extra transient decay modes contaminate the signal and bias the extracted error rate, and waiting for them to die out can push the signal below the noise floor. The authors show that this bias can be removed purely in classical post-processing, without changing the circuits or normalizing by an ideal reference simulation, by reweighting measurement outcomes according to the symmetries of the random circuit ensemble.

Key Contributions

1. A one-parameter family of filter functions. The authors define f_α(x, g) = (2^n + 1)[⟨x| g ξ(α) g† |x⟩ − (1+α)^(−n)] with ξ(α) = (I + αZ)^(⊗n)/(1+α)^n and α ∈ [1, 3]. At α = 1 this reproduces the standard LinXEB-type filter function; at α = 3 it gives the new "trace filter function."

2. A provable de-biasing result (Proposition 1, informal). If the circuit distribution ν is right-invariant under local Clifford rotations and the noise is time-stationary, Markovian, gate-dependent, and does not break that local Clifford symmetry of the ν-twirls, then the expected trace-filter signal takes the form Tr[P T^m(M̃)] with P the noiseless fixed point of the self-adjoint twirl T — so the subdominant transient decays are suppressed and the mixing length becomes zero in the noiseless limit.

3. Variance and complexity analysis. Proposition 2 bounds the noiseless long-sequence variance as O([1 + ((α−1)/(α+1))^2]^n), an exponential base ranging from 1 to 5/4 over α ∈ [1, 3]; the paper also analyzes the short-depth variance, the bias–variance trade-off, and the classical evaluation cost (including a #P-hardness result for worst-case evaluation of α > 1 with Clifford circuits, plus efficient regimes).

4. An experimental demonstration on superconducting hardware. The protocol is implemented on Rigetti's 36-qubit Cepheus-1-36Q processor with FPGA-embedded random gate sampling, benchmarking 1D brickwork circuits at 6, 10, and 12 qubits with tetrahedral (12-element), Clifford (24-element), and icosahedral (60-element) single-qubit gate ensembles.

Main Findings

• Transients are the obstacle, and they are removable. For α < 3 the RB signal deviates from a single exponential for up to 30 layers in exact 30-qubit simulations; the deviation shrinks as α grows, and the α = 3 trace filter has a mixing length "indistinguishable from zero."

• Short-depth extraction at 12 qubits. In the experiments, the standard estimator requires depth beyond a 40-layer experimental window at 12 qubits, whereas the trace filter function yields consistent estimates below 10 layers — the paper's headline result.

• Variance grows with α, creating an optimal intermediate α. In exact 30-qubit simulations (open 1D brickwork, two-qubit Clifford bricks, single-qubit depolarizing noise with p = 5·10⁻³), the optimal choice was α = 1.5, reducing decay-rate fitting uncertainty by a factor of 4 relative to α = 1. As a rule of thumb, the authors report that higher noise rates push the optimal α closer to 3.

• Short-depth variance inverts. At minimal circuit depth the variance ordering reverses and is largest for α = 1; computing the signal-to-noise ratio restores the expected monotonicity in α, because the α < 3 signal is exponentially larger at short depth.

• Optimistic fitting is biased; cautious fitting is not. In a 6-qubit Qiskit IBM Guadalupe emulator study (1.2·10⁵ circuits per length for lengths {1, …, 50}, 100 shots each), single-exponential regression over the full dataset gives a substantial bias at α = 1 that decreases with α and vanishes near α = 3, while cautious fits agree across all α.

• Mixing-length estimates grow with system size. Model-based tensor-network estimation gives a minimal mixing length of 18 for the 6-qubit data (so cautious fitting is possible for all α), but at n = 12 the estimated mixing lengths for α near 1 approach or exceed the 40-layer window — e.g. 36 and 40 for the icosahedral ensemble.

• All three single-qubit groups agree. Tetrahedral, Clifford, and icosahedral ensembles yield comparable infidelities, making them qualitatively interchangeable benchmarks of this device.

• Measured infidelities (trace filter, α = 3, 68% intervals). Table 1 reports values of 0.045±0.001 and 0.047±0.001 at 6 qubits, 0.08±0.01 and 0.082±0.003 at 10 qubits, and 0.10±0.02, 0.103±0.007, and 0.14±0.03 at 12 qubits.

• Long-depth second moments match theory qualitatively. Experimental converged second moments behave similarly to the idealized predictions of Proposition 2 even for non-Clifford groups, with a constant offset that mainly affects small α.

• Classical cost is regime-dependent. Worst-case evaluation of f_α for α > 1 is #P-hard even for Clifford circuits (it contains the weight-enumerator problem for binary linear codes), but direct evaluation for uniformly random global Cliffords takes O(1) time in expectation, the short-depth local-Clifford case factorizes and takes O(n), and 1D circuits of O(log n) depth admit efficient tensor-network evaluation. A stabilizer-based implementation with O(n²) memory, using Stim and Galois, proved scalable in practice; non-Clifford unitaries were simulated with qsim.

Methodology in Plain English

The protocol removes the final inversion gate of standard RB and instead does the work classically. For each of many random circuits, the device is prepared in |0⟩, a random depth-m circuit is run, and a bit string x is measured together with a record of which random gates g were applied. A "filter function" f(x, g) scores each shot, and the average over shots at each depth is fitted to an exponential; the decay rate gives the infidelity.

The trick is in how the scoring is done. The standard score (α = 1) asks how large the Born probability |⟨x|g|0⟩|² is compared to uniform. The new score (α = 3, and interpolating values in between) additionally reweights outcomes according to the local Clifford symmetry of the circuit ensemble — mathematically, it computes a trace-like inner product ⟨x| g ξ(α) g† |x⟩ with ξ(α) built from tensor products of (I + αZ). This reweighting makes the signal sensitive only to the dominant, interpretable decay mode, so the transient modes that normally require deep circuits are suppressed from the first layer. The price is higher statistical variance, which is why the family is parameterized: α tunes the trade-off between bias (mixing length) and variance.

Because the method needs many distinct random circuits with few shots each, the authors push randomness onto the control hardware: a 48-bit linear-feedback-shift-register pseudo-random generator (masked to 16 bits) on the FPGA picks gate indices from a small lookup table of RZ angles, so millions of distinct circuits can be sampled without recompiling or reloading programs. To handle finite-statistics data, they compare "optimistic fitting" (fit everything) with "cautious fitting" (fit only beyond an empirically estimated mixing length), and evaluate three mixing-length heuristics: residual χ² statistics, the number of nontrivial singular values of a Hankel matrix, and a tensor-network denoising heuristic — the last being judged the most reliable, though the least formally justified.

Why This Matters

Benchmarks that are experimentally convenient (shallow, geometrically local, hardware-native circuits) are exactly the ones whose ensembles mix slowly, so conventional estimators either report biased numbers or require depths where the signal has already died. Trace-filtered RB lets experimenters keep the convenient circuit ensemble and still get an interpretable decay rate, which matters for any device where 1D or otherwise restricted connectivity is the thing being characterized.

Real-world applications:

• Device calibration and fabrication feedback: faster, lower-depth fidelity estimates shorten the loop between measurement and hardware tuning on multi-qubit processors.
• Benchmarking restricted-topology and native-gate architectures: architectures whose entangling pattern is deliberately constrained can be benchmarked directly rather than approximated by a mixing-friendly surrogate.
• Error-budget inputs for error mitigation and error correction: reliable per-layer infidelities feed resource estimates and mitigation tuning.
• Validating digital models of many-body dynamics and local scrambling, where 1D brickwork circuits are the object of interest rather than a stand-in.

Industry relevance: The work is a collaboration including the Technology Innovation Institute and Rigetti Computing, and the experiment ran on Rigetti's Cepheus-1-36Q processor, a 36-qubit device made of four 9-qubit chiplets (one qubit was not operational). The FPGA-integrated random sampling achieves 10⁶ distinct circuits in a few minutes, limited mainly by a 200 μs passive reset time — a throughput argument that speaks directly to production benchmarking workflows, where per-circuit compilation overhead otherwise dominates.

Future Directions

• Extending beyond 1D brickwork. The paper validates one-dimensional circuits up to 12 qubits; whether the trace filter's zero-mixing-length guarantee remains practical for two-dimensional or all-to-all architectures, and at larger n where variance grows exponentially in n, is left open.

• Predicting the optimal α. The authors state that a closed-form a priori prediction of the optimal α is difficult, especially with noisy data; a practical prescription mapping noise rate, architecture, and shot budget to an α would remove the remaining tuning step.

• Better mixing-length estimation. The most reliable heuristic in their study (tensor-network denoising) has the least formal justification, and reliable determination of the mixing length became challenging at larger n and higher noise — a clear target for rigorous methods.

• Scaling the classical post-processing. Beyond the efficient long-depth O(1), short-depth O(n), and O(log n)-depth tensor-network regimes, the intermediate regime and non-Clifford gates remain costly; the authors' evidence is described as suggesting a formally efficient short-depth algorithm might exist, which is an open question.

Target Audience

Experimental quantum-computing groups running benchmarking on superconducting or trapped-ion hardware; benchmarking and characterization theorists working on RB, LinXEB, and non-uniform circuit ensembles; engineers building FPGA-based control and sampling infrastructure; and readers who need to interpret reported fidelity numbers and want to know why a shallow-circuit benchmark can disagree with a deep-circuit one. Familiarity with quantum channels and Clifford/Pauli formalism is needed to follow the proofs, but the experimental results and the bias-versus-variance message are accessible to intermediate readers.

Authors’ abstract

Randomized benchmarking (RB) with structured circuits is experimentally scalable, but gives biased short-depth fidelity estimates because incomplete ensemble mixing introduces transient modes. Waiting for these transients to decay in deeper circuits can push the signal below the noise floor. This is a well-established obstacle, e.g. in linear cross-entropy benchmarking (LinXEB). Here we overcome this obstacle solely through classical post-processing. We introduce a one-parameter family of estimators that interpolate between the standard LinXEB-type post-processing and a new post-processing technique we call the trace filter function. Under practical assumptions, the trace filter function provably suppresses transient modes, exposing an interpretable fidelity decay at short depth, with theoretical guarantees comparable to standard RB, at the cost of an increased variance. We bound the variance, analyze the evaluation complexity of the new estimators, and develop practical post-processing algorithms. Experiments on a Rigetti superconducting processor validate the method for one-dimensional circuits of up to 12 qubits: where the standard estimator requires depth beyond a 40-layer experimental window, the trace filter function yields consistent estimates below 10 layers. Trace-filtered randomized benchmarking therefore extends reliable fidelity characterization to relevant structured circuit ensembles and short-depth regimes that are inaccessible to conventional estimators.