---
title: Interleaved Randomized Benchmarking
url: https://www.emergentmind.com/topics/interleaved-randomized-benchmarking
type: topic
---

# Interleaved Randomized Benchmarking

Interleaved Randomized Benchmarking (IRB) is a quantum characterization protocol designed to estimate the average error rate of a specific quantum gate or operation by alternating—or “interleaving”—its application within sequences of randomly sampled gates from a unitary 2-design (typically the Clifford group). IRB provides an operationally meaningful, SPAM-robust, and scalable method to isolate and quantify the fidelity of individual operations, including Clifford and non-Clifford gates, mid-circuit measurements, resets, and general dynamic circuit primitives. The method and its rigorous guarantees underlie much of the modern literature on quantum benchmarking and have been extended to address time-dependent, gate-dependent, leakage, and coherent error mechanisms.

## 1. Principles and Protocol Structure

Interleaved Randomized Benchmarking builds on the theory of randomized benchmarking (RB), which leverages random gate sequences from a unitary 2-design to twirl general errors into (approximately) depolarizing channels. Standard RB provides an average gate error rate for a reference gate set (e.g., the Clifford group) by fitting the decay of the survival probability in sequences of increasing length:

\[
F_{\rm seq}(m) = A p^m + B
\]

Here $p$ is the depolarizing decay parameter, and $A,B$ absorb state preparation and measurement (SPAM) errors.

Interleaved RB modifies this protocol to insert a specific gate or block $\mathcal{C}$ (or, more generally, a dynamic circuit primitive $\mathcal{F}$) between each random gate. The procedure is as follows [1203.4550, 2408.07677]:

- For each sequence length $m$, generate multiple random sequences of $m$ gates $G_1, \ldots, G_m$ from the reference group (e.g., Clifford).
- Build two sets of sequences:
  - **Reference**: $G_1, G_2, ..., G_m$, followed by the unique group inverse.
  - **Interleaved**: $G_1, \mathcal{C}, G_2, \mathcal{C}, ..., G_m, \mathcal{C}$, followed by the inversion of the total ideal sequence.
- Implement each sequence on the device, measure the survival probability, and average over many randomizations.

The interleaved protocol yields two decay curves:

\[
F_{\mathrm{seq}}^{\mathrm{ref}}(m) = A p^m + B \qquad
F_{\mathrm{seq}}^{\mathrm{int}}(m) = A' \tilde{p}^m + B'
\]

where $p$ and $\tilde{p}$ are the reference and interleaved decay rates, respectively.

The estimated average error rate of $\mathcal{C}$ (for $d$-dimensional Hilbert space) is then [1203.4550, 1404.6025]:

\[
r_\mathcal{C}^{\mathrm{est}} = \frac{d-1}{d}\left(1 - \frac{\tilde{p}}{p}\right)
\]

Confidence intervals and error bounds are rigorously derived, enabling quantitative fidelity statements even in the presence of gate-dependent and time-dependent Markovian noise [1404.6025, 2010.07974].

## 2. Mathematical Foundations and Fidelity Extraction

The mathematical basis for IRB is the unitary twirl, which ensures that under the action of a 2-design (such as the Clifford group), arbitrary noise channels are depolarized. This process equates exponential decay rates with average gate fidelity metrics. IRB relies on the following key relationships [1203.4550, 1806.02048, 2010.07974]:

- **Depolarizing Model:** Averaged noise over a 2-design reduces to a single-parameter depolarizing channel:
  \[
  \mathcal{E}_{\mathrm{dep}}(\rho) = p \rho + (1-p)\frac{I}{d}
  \]
- **Gate Fidelity:** The average fidelity of $\mathcal{C}$ is obtained as
  \[
  F_{\mathrm{avg}}(\mathcal{C}) = \frac{(d-1)\tilde{p}/p + 1}{d}
  \]
- **Robustness to SPAM:** SPAM errors enter only as offsets $A$, $B$ in the fits and do not affect the decay constants $p$, $\tilde{p}$.
- **Rigorous Bounds:** The true gate infidelity lies within an interval $[r_\mathcal{C}^{\mathrm{est}}-E,\, r_\mathcal{C}^{\mathrm{est}}+E]$, where $E$ is a computable bound depending on the twirling group and sequence parameters [1404.6025].

When multiple nontrivial irreducible representations or non-depolarizing noise are present, the protocol generalizes to multi-exponential fits or filtered postprocessing, but the gate fidelity is still extractable using representation-theoretic tools [2010.07974, 1806.02048].

## 3. Extensions: Non-Clifford Gates, Dynamic Circuits, Leakage, and Measurement

IRB has been systematically extended along several axes:

### Non-Clifford and Arbitrary Gates

Variants including 2-for-1 RB, CNOT-dihedral RB, and hybrid Monte-Carlo RB enable benchmarking of non-Clifford gates (e.g., T, CS gates) using arbitrary reference groups, as long as sufficient randomness and invertibility are maintained [1608.02943, 2007.08532, 1806.02048, 1606.03927]. These methods preserve SPAM independence and enable estimation of average fidelity even when the target gate does not belong to the reference gate set.

### Dynamic-Circuit Elements

Recent protocols interleave entire dynamic-circuit operation blocks, such as measurement, reset, and feedforward primitives, within RB sequences. By constructing identity blocks (e.g., $H$-CNOT, conditional Pauli operations with measurement and feedforward), one quantifies the error budgets for mid-circuit measurement, feedforward, and coherence effects [2408.07677]. Error sources, including readout assignment, measurement-induced phase, and decoherence during idle windows, can be isolated and mitigated via dynamical decoupling techniques.

### Leakage-Aware IRB

For systems with leakage (e.g., transmons with $|2\rangle$ states), IRB is modified by adding phase randomization layers to destroy residual coherences and by fitting multi-exponential models to capture population flow into and out of the computational subspace [1505.00580]. The protocol robustly estimates both average gate error and leakage/seepage rates under mild assumptions.

### Measurement-Based and Bias-Optimized IRB

Variants have also been demonstrated for measurement-based quantum computation (using cluster state measurements to generate unitary 2-designs) [2203.14995], and for scenarios with biased noise (e.g., dephasing-favored architectures). In the latter, character-weighted survival probabilities and Z-group twirls replace full Pauli twirling to accommodate hardware constraints [2206.00009].

## 4. Statistical Confidence and Practical Implementation

IRB protocols exhibit exponential decay in survival probability, with finite-sampling variance that grows only polynomially in the sequence length and error rate. Explicit sample complexity formulas guarantee sub-percent estimation precision with moderate numbers of sequences and shots [1404.6025]. The median-of-means estimator, gate-set shadow tomography, and filtered RB post-processing further improve statistical robustness and minimize resource requirements [2510.18453].

Modern implementations routinely use the following practical parameters [2408.07677, 1203.4550]:

- Sequence length: up to $m=30$–$100$ gates.
- Number of sequences per $m$: $20$–$50$.
- Number of shots per sequence: $>300$.
- Fitting: least-squares or subspace signal processing (e.g., MUSIC, ESPRIT).
- Device: multi-qubit superconducting architectures up to $127$ qubits.

Coherent errors and gate-dependent cross-talk can induce systematic bias in the IRB estimator if not properly mitigated; recent work demonstrates the superiority of single-qubit Pauli-twirl protocols (“cycle benchmarking”) in suppressing such bias compared to multi-qubit Clifford twirls [2512.24744].

## 5. Applications and Notable Results

Interleaved RB is the standard for high-fidelity benchmarking of critical quantum operations, including:

- Single- and two-qubit gate errors in superconducting, ion-trap, and silicon devices [1203.4550, 2007.08532].
- Mid-circuit measurement, reset, and feedforward primitives in dynamic circuits, isolating assignment error, $T_1/T_2$ decoherence, and cross-talk [2408.07677].
- Non-Clifford gate benchmarking (T, CS, controlled-S, Toffoli) for fault-tolerant gate sets [1608.02943, 2007.08532].
- Measurement-based gate fidelities on large-scale cluster-state hardware [2203.14995].
- Leakage and bias characterization in advanced device candidates [1505.00580, 2206.00009].
- Benchmarking protocols adapted for NISQ hardware and circuits with stabilizer verification or native-gate synthesis [2102.13044].

An illustrative experimental result: On the IBM 127-qubit “Eagle” processor, interleaved RB for dynamic-circuit primitives revealed ε_F of ≈$10^{-2}$ dominated by measurement-assignment error and idling, which could be mitigated to the expected incoherent sum via feedforward-aware dynamical decoupling [2408.07677]. Non-Clifford two-qubit gates (CS) were benchmarked to error rates below standard CNOTs and approaching $T_2$-limited performance [2007.08532]. Cycle benchmarking yielded order-of-magnitude lower systematic uncertainty under coherent error drift compared to Clifford-twirl IRB [2512.24744].

## 6. Limitations, Advanced Techniques, and Future Directions

Standard IRB protocols assume gate-independent, Markovian noise and may exhibit systematic bias in the presence of strong coherent or gate-dependent errors. Refined bounds and protocol variants, including hybrid (Monte Carlo) IRB, subspace signal-processing, cycle benchmarking, and character-filtered RB, address these limitations by:

- Explicitly quantifying systematic uncertainty due to incomplete twirling [2512.24744].
- Filtering multi-exponential decay data to extract target gate contributions [2010.07974, 1806.02048].
- Utilizing gate-set shadow tomography and median-of-means estimators for optimal finite-sample confidence and reduced group sizes [2510.18453].
- Implementing minimal or hardware-tailored gate sets for efficient on-device benchmarking [2510.18453, 2206.00009, 2408.07677].

Application of interleaved RB is expanding to multi-qubit correlated noise, dynamic circuit benchmarking, quantum error-correcting code performance, and adapted tomography-free fidelity estimation [2510.18453, 2408.07677]. Systematic study and minimization of error bias, leakage corrections, and crosstalk remain active topics of research.

## 7. Summary Table: Core Elements of Interleaved RB

| Element                                | Purpose / Action                                                    | Key Formula or Concept                                   |
|-----------------------------------------|---------------------------------------------------------------------|----------------------------------------------------------|
| Reference RB Sequence                   | Baseline decay, depolarization parameter $p$                        | $F_{\rm seq}(m) = A p^m + B$                            |
| Interleaved RB Sequence                 | Alternate random gates and target $\mathcal{C}$, extract $\tilde{p}$| $F_{\rm seq}^{\rm int}(m) = A' \tilde{p}^m + B'$         |
| Error Rate Estimate                     | Quantifies additional error due to $\mathcal{C}$                    | $r_\mathcal{C} = \frac{d-1}{d}\left(1 - \frac{\tilde{p}}{p}\right)$ |
| SPAM Robustness                         | Fitting offsets absorb SPAM, does not bias decay constants          | $A, B$ (reference), $A', B'$ (interleaved)               |
| Confidence Interval / Bound             | Rigorous error bar on estimate                                      | See explicit $E$ in [1404.6025]                          |
| Extensions (Non-Clifford, Leakage, etc) | Embedded via sequence, fitting, or representation theory            | Multi-exponential or filtered single-exponential fits     |

Interleaved Randomized Benchmarking thus provides a robust, scalable, and theoretically principled methodology for isolating and quantifying the fidelity of individual quantum operations—Clifford, non-Clifford, measurement, reset, or dynamic circuit—in experimental platforms, with rigorous confidence even in nonideal devices [2408.07677, 1203.4550, 1806.02048, 2010.07974, 2512.24744].

Source: https://www.emergentmind.com/topics/interleaved-randomized-benchmarking