Direct Randomized Benchmarking (DRB)

• DRB is a protocol that directly tests native gate layers, bypassing the need for Clifford compilation and reducing physical-depth challenges.
• It estimates an Ω-weighted average error rate using an exponential decay model, providing a clear metric for gate performance under stochastic noise.
• DRB is versatile, applied across multiple quantum platforms for both holistic system benchmarking and detailed native gate fidelity evaluations.

Searching arXiv for recent and foundational papers on Direct Randomized Benchmarking. Direct Randomized Benchmarking (DRB) is a randomized benchmarking protocol for directly benchmarking a customizable gate set—typically native gates or native circuit layers—rather than compiled $n$-qubit Clifford gates. Its central purpose is to retain the practical advantages associated with randomized benchmarking, especially SPAM robustness and simple exponential-decay fitting, while avoiding the compilation bottleneck that makes standard Clifford randomized benchmarking increasingly impractical on larger systems. In the literature, DRB is also described as direct RB and, in related theoretical discussions, sometimes called generator RB; it is used to estimate an $\Omega$-weighted average error rate for the layers actually sampled in the experiment rather than an error per compiled Clifford (Proctor et al., 2018, Polloreno et al., 2023, Heinrich et al., 2022).

1. Historical motivation and conceptual scope

DRB was introduced to address a specific limitation of standard Clifford randomized benchmarking. In Clifford RB, the random benchmark objects are $n$-qubit Clifford gates, but those gates are generally not native to hardware and must be compiled into many one- and two-qubit operations. As emphasized in the foundational multiqubit DRB paper, this creates both a physical-depth problem and an interpretability problem: the measured quantity is an error per compiled Clifford, and rescaling that number into an error per native gate is an unreliable extrapolation (Proctor et al., 2018).

The multiqubit DRB literature therefore frames the protocol as a method for benchmarking native circuit layers directly. In the original formulation, the random central circuit is built from a user-chosen distribution $\Omega$ over native $n$-qubit layers, and the reported quantity is intended to track the $\Omega$-averaged error probability of those layers under stochastic Pauli noise (Proctor et al., 2018). The later theory paper sharpens this target by defining

$$\epsilon_{\Omega} = \sum_{G \in \mathbb{G}} \Omega(G)\,\epsilon\!\left[\tilde{\mathcal{G}}(G),\mathcal{G}(G)\right],$$

namely the $\Omega$-weighted average entanglement infidelity of the benchmarked gates (Polloreno et al., 2023).

This “directness” has two consequences. First, the protocol can be tailored to benchmark architecture-specific layer families, such as native one-qubit layers, layers with tunable two-qubit-gate density, or pair-specific native entangling gates. Second, the sampling distribution becomes part of the benchmark definition. DRB therefore reports a sampling-dependent quantity, but that dependence is explicit and controllable rather than hidden in a Clifford compiler (Proctor et al., 2018).

Within broader randomized-benchmarking theory, DRB is typically classified as a non-uniform or subset-RB protocol. The general-framework literature explicitly places direct RB in the subset-RB class, with constant-depth native layers as the sampled subset, and treats it as part of the larger family of scalable random-circuit benchmarking protocols that depart from exact group-uniform sampling (Helsen et al., 2020).

2. Canonical protocol structure

The canonical DRB experiment combines three ingredients: randomized state preparation, a random native-gate core, and a final mapping that converts the ideal output into a computational-basis success test. In the original multiqubit formulation, a depth-$m$ DRB circuit is constructed by sampling a random $n$-qubit stabilizer state $\Omega$0, sampling a random $\Omega$1-layer circuit

$\Omega$2

preparing $\Omega$3 from $\Omega$4, applying $\Omega$5, applying a circuit that maps the ideal output $\Omega$6 to a known computational-basis state $\Omega$7, and then measuring in the computational basis, recording success when the observed bit string equals $\Omega$8 (Proctor et al., 2018).

The 2023 theory paper presents the same protocol in a more implementation-oriented way. For each depth $\Omega$9, one chooses a target output bit string $n$0, samples a random state-preparation group element $n$1, finds a circuit $n$2 such that

$n$3

samples $n$4 core gates $n$5 independently from $n$6, and then finds a measurement-preparation circuit $n$7 satisfying

$n$8

The sampled DRB circuit is therefore

$n$9

which ideally yields the designated success string with probability $\Omega$0 (Polloreno et al., 2023).

A notable implementation detail is conditional compilation. The state-preparation and measurement-preparation circuits need only prepare or unprepare the correct state on the specific input $\Omega$1; they do not need to realize the corresponding full unitary on arbitrary inputs. This is one of the reasons DRB is more feasible than Clifford RB at larger $\Omega$2 (Polloreno et al., 2023).

The single-qubit neutral-atom realization illustrates the same structure in a minimal setting. There, a DRB circuit prepares a stabilizer state from $\Omega$3, applies $\Omega$4 random layers of native single-qubit gates, applies a final operation mapping the target stabilizer to the computational basis, and measures in the computational basis. The six single-qubit stabilizer states are prepared using one gate from

$\Omega$5

and the random layers are sampled from native rotations sufficient to generate the single-qubit Clifford group (Rozanov et al., 8 Sep 2025).

3. Signal model, fitted parameters, and estimated quantities

The hallmark DRB observable is a success or survival probability as a function of sequence depth. In the foundational and theoretical DRB papers, the sequence-averaged success probability is fit to

$\Omega$6

or equivalently

$\Omega$7

where the nuisance parameters $\Omega$8 and $\Omega$9 absorb SPAM-like effects and $n$0 is the decay parameter of interest (Proctor et al., 2018, Polloreno et al., 2023).

The original multiqubit DRB convention reports the error rate

$n$1

and under stochastic Pauli errors with suitable scrambling this is approximately

$n$2

so the fitted decay directly estimates the $n$3-averaged native-layer error probability (Proctor et al., 2018).

The later theory paper keeps the same convention and defines

$n$4

because its target quantity is an average entanglement infidelity rather than the more common average gate infidelity (Polloreno et al., 2023). In that framework, DRB is explicitly an estimator for $n$5, the $n$6-weighted average entanglement infidelity of the benchmarked gate set.

Platform-specific implementations sometimes use a different but compatible parameterization. In the neutral-atom single-qubit study, the fitted model is

$n$7

and the decay parameter is converted to an average gate fidelity through

$n$8

with $n$9 for one qubit, so

$\Omega$0

That work also uses separate fits for $\Omega$1- and $\Omega$2-targeting circuits to infer readout asymmetry via

$\Omega$3

(Rozanov et al., 8 Sep 2025).

This suggests that DRB literature contains both entanglement-infidelity and average-fidelity parameterizations, depending on the protocol framing. The common feature is the same RB-style exponential decay with SPAM-dominated intercept terms and a decay parameter interpreted as an average quality metric for the sampled layer ensemble.

4. Reliability theory and its assumptions

The dedicated DRB theory develops two complementary explanations for why DRB can work. The first is an error-scrambling theory for stochastic Pauli noise. In that picture, a depth-$\Omega$4 random circuit succeeds for exactly three reasons: no errors occur, multiple errors occur but cancel, or errors occur and do not cancel but the final stabilizer measurement fails to detect them. Writing these contributions as $\Omega$5, $\Omega$6, and $\Omega$7, the average success probability decomposes as

$\Omega$8

with

$\Omega$9

If the cancellation term $$\epsilon_{\Omega} = \sum_{G \in \mathbb{G}} \Omega(G)\,\epsilon\!\left[\tilde{\mathcal{G}}(G),\mathcal{G}(G)\right],$$0 is negligible, then

$$\epsilon_{\Omega} = \sum_{G \in \mathbb{G}} \Omega(G)\,\epsilon\!\left[\tilde{\mathcal{G}}(G),\mathcal{G}(G)\right],$$1

so the decay is a single exponential with $$\epsilon_{\Omega} = \sum_{G \in \mathbb{G}} \Omega(G)\,\epsilon\!\left[\tilde{\mathcal{G}}(G),\mathcal{G}(G)\right],$$2 and $$\epsilon_{\Omega} = \sum_{G \in \mathbb{G}} \Omega(G)\,\epsilon\!\left[\tilde{\mathcal{G}}(G),\mathcal{G}(G)\right],$$3 (Polloreno et al., 2023).

The same paper makes the negligible-cancellation condition concrete for a class of random local Clifford-plus-CNOT layer ensembles. There it defines a delocalization scale $$\epsilon_{\Omega} = \sum_{G \in \mathbb{G}} \Omega(G)\,\epsilon\!\left[\tilde{\mathcal{G}}(G),\mathcal{G}(G)\right],$$4 and gives the sufficient condition

$$\epsilon_{\Omega} = \sum_{G \in \mathbb{G}} \Omega(G)\,\epsilon\!\left[\tilde{\mathcal{G}}(G),\mathcal{G}(G)\right],$$5

or, under local Pauli noise, the corresponding per-qubit condition

$$\epsilon_{\Omega} = \sum_{G \in \mathbb{G}} \Omega(G)\,\epsilon\!\left[\tilde{\mathcal{G}}(G),\mathcal{G}(G)\right],$$6

The physical claim is that random circuit layers spread low-weight Pauli errors quickly enough that later cancellation becomes extremely unlikely (Polloreno et al., 2023).

The second theory treats general gate-dependent Markovian noise. Its central structural condition is that $$\epsilon_{\Omega} = \sum_{G \in \mathbb{G}} \Omega(G)\,\epsilon\!\left[\tilde{\mathcal{G}}(G),\mathcal{G}(G)\right],$$7 form a sequence-asymptotic unitary 2-design, meaning that long random sequences converge in second moments to Haar-random unitaries: $$\epsilon_{\Omega} = \sum_{G \in \mathbb{G}} \Omega(G)\,\epsilon\!\left[\tilde{\mathcal{G}}(G),\mathcal{G}(G)\right],$$8 Under low-error, benchmarking-compatible implementations, the resulting $$\epsilon_{\Omega} = \sum_{G \in \mathbb{G}} \Omega(G)\,\epsilon\!\left[\tilde{\mathcal{G}}(G),\mathcal{G}(G)\right],$$9-matrix has two dominant modes associated with the depolarizing subspace, yielding

$\Omega$0

with $\Omega$1 equal to a gauge-invariant $\Omega$2-weighted mean infidelity of the benchmarked gates in a suitable gauge (Polloreno et al., 2023).

Broader RB theory places DRB in a larger class of non-uniform random-circuit protocols. The 2020 general framework treats DRB as subset RB and shows that, after sufficient mixing, such protocols are described by linear combinations of matrix exponential decays rather than necessarily a single exponential (Helsen et al., 2020). The beyond-groups framework reformulates DRB-like native-layer protocols as universal RB or twirling schemes and proves approximate single-exponential behavior when the ensemble twirling map is sufficiently close to the Haar twirling map in induced diamond norm (Chen et al., 2022). Filtered-RB theory further frames direct RB as part of the non-uniform random-circuit regime and emphasizes spectral-gap control of the dominant benchmarking decay versus the subdominant circuit-mixing decay (Heinrich et al., 2022).

5. Experimental realizations across hardware platforms

DRB has been implemented as both a holistic multiqubit benchmark and a component-level native-gate benchmark. The original experimental demonstration used IBMQX5 and benchmarked 2, 3, 4, and 5 qubits. That work reported clear exponential decay of the measured success probabilities, used two sampling distributions with $\Omega$3 and $\Omega$4, and showed how comparing those two datasets enables inference of separate average error rates for layers containing a CNOT versus all-single-qubit layers (Proctor et al., 2018).

A large trapped-ion realization used DRB as the primary component-level benchmark on the 30-qubit IonQ Forte system. Single-qubit DRB was performed on each qubit, and pairwise two-qubit DRB was performed on all

$\Omega$5

qubit pairs. The reported 1Q DRB distribution had median $\Omega$6, 10th percentile $\Omega$7, and 90th percentile $\Omega$8. The 2Q DRB distribution had median $\Omega$9, 10th percentile $m$0, and 90th percentile $m$1, with best observed infidelity $m$2, worst pair up to $m$3, and no statistically significant correlation between 2Q DRB error and ion separation, with fitted slope $m$4 (Chen et al., 2023).

A neutral-atom realization benchmarked single-qubit gates on a neutral-atom quantum processor and then extended the protocol to a 25-qubit array under global single-qubit control. After DRB-based recalibration, the reported single-qubit result was

$m$5

at 95% confidence level. Applied across the 25-qubit array, the study reported

$m$6

That work also used DRB not only as a passive benchmark but as an active coherent-calibration tool, fitting a two-parameter model of overrotation and axis misalignment to full per-circuit DRB data (Rozanov et al., 8 Sep 2025).

Platform	DRB scope	Reported findings
IBMQX5	Holistic 2–5 qubit DRB	Clear exponential decay; separate average CNOT-related error estimates from two sampling distributions (Proctor et al., 2018)
IonQ Forte	30-qubit 1Q and pairwise 2Q DRB	$m$7 pairs benchmarked; 1Q median $m$8; 2Q median $m$9; no significant ion-distance dependence (Chen et al., 2023)
Neutral-atom processor	Single-qubit DRB and 25-qubit spatial DRB	$n$0 single-qubit average fidelity; $n$1 array average; DRB-guided coherent calibration (Rozanov et al., 8 Sep 2025)

These implementations span superconducting, trapped-ion, and neutral-atom platforms, and collectively show that DRB can be used as a native-layer benchmark, a pairwise entangling-gate survey, a spatial map across an array, and a calibration diagnostic.

6. Relation to other benchmarking and characterization methods

The most direct comparison is with standard Clifford randomized benchmarking. Clifford RB samples uniformly from an $n$2-qubit 2-design and measures an error per compiled Clifford; DRB instead samples random native-gate layers directly and reports a benchmark tied to the actual sampled layer ensemble. The foundational DRB papers therefore present DRB as both more scalable in practice and more directly informative about native-gate performance (Proctor et al., 2018, Polloreno et al., 2023).

The neutral-atom study gives a second important comparison, between DRB and gate set tomography (GST). There DRB is characterized as fast, scalable, and SPAM-robust, suited to average fidelity estimation, whereas GST reconstructs the full gate set, input states, and measurements, is more experimentally and computationally intensive, and requires gauge fixing for meaningful fidelity comparisons. In that work, GST and DRB were reported as consistent at the level of overall high single-qubit fidelity, while GST additionally revealed gate-specific structure such as lower $n$3 fidelity (Rozanov et al., 8 Sep 2025).

A third comparison comes from the superconducting-noise study that evaluated DRB alongside MRB and CRB on specially designed random circuits. In that comparison framework, MRB, DRB, and CRB sequentially overestimate the average error rate under $n$4 and $n$5 noise relative to a tomography-based reference, and DRB provides a more stable estimate at a specific precision while being more resource-consuming. The same study found that DRB, MRB, and CRB exhibit almost the same level of sensitivity to coherent error (Chai et al., 2023).

DRB has also been used as an input to system-level modeling rather than only as a reporting metric. The trapped-ion 30-qubit study constructed a simple depolarizing simulation model from median 1Q and 2Q DRB values and found that the resulting predictions correlated with observed application-benchmark behavior but systematically overestimated application fidelity, indicating substantial out-of-model errors (Chen et al., 2023).

7. Limitations, failure modes, and interpretive cautions

Several limitations recur across the literature. First, DRB’s interpretation is simplest under stochastic or effectively Pauli-like noise. The neutral-atom single-qubit study explicitly states that its fidelity interpretation is most naturally justified under a stochastic Pauli noise model and reports that stronger-noise simulations begin to marginally exceed the descriptive scope of DRB (Rozanov et al., 8 Sep 2025). The dedicated theory paper similarly separates a stochastic-Pauli scrambling theory from a more formal low-error Markovian theory, rather than claiming one universal argument for all regimes (Polloreno et al., 2023).

Second, coherent and gate-dependent errors can distort the meaning of a fitted average-fidelity-like number. In the neutral-atom work, a pre-calibration DRB fit yielded

$n$6

at 95% confidence level, yet many individual circuits had success probabilities below $n$7, which the authors interpreted as evidence of coherent miscalibration. The same study emphasizes that large circuit-to-circuit spread at fixed depth signals nontrivial coherent or gate-dependent error mechanisms even when the averaged decay remains fit-able (Rozanov et al., 8 Sep 2025).

Third, DRB can fail under unfavorable noise processes. In the superconducting comparative study, DRB loses reliability as $n$8 grows, with the proposed mechanism being damage to the initial stabilizer-state preparation. There the authors argue that strong $n$9 scrambles the prepared initial state into a mixed state so severely that the final survival measurement ceases to reflect the intended circuit noise of interest (Chai et al., 2023).

Fourth, DRB is not a full process-characterization method. It does not reconstruct gates, states, and measurements individually, and it does not by itself resolve coherent versus stochastic structure or context dependence. The trapped-ion 30-qubit study makes this limitation explicit by showing that a simple DRB-derived depolarizing model predicts application benchmarks only qualitatively; observed algorithmic performance was closer to an effective error rate of about $\Omega$00 than to the DRB-derived $\Omega$01, with the discrepancy attributed to drift, crosstalk, context dependence, and likely non-Markovian stochastic spin-phase noise (Chen et al., 2023).

Finally, the benchmark is intrinsically ensemble-dependent. The reported DRB quantity depends on the chosen gate set $\Omega$02 and sampling distribution $\Omega$03. This is a feature rather than a defect, because it allows DRB to target native layers of direct operational interest; however, it also means that DRB numbers obtained under different $\Omega$04 choices are not interchangeable without specifying the sampled ensemble (Proctor et al., 2018, Polloreno et al., 2023).