# Techniques: Restricted Randomized Benchmarking with Universal Gates of Fixed Sequence Length

## What the paper contributes (one paragraph)

The paper introduces a variant of randomized benchmarking (RB) that estimates the average gate fidelity of a quantum device using directly accessible native universal gates compiled into sequences of equal length, rather than relying on a unitary t-design or an approximate one. The authors give an explicit method to generate exact Haar-random single- and two-qubit unitaries from a fixed-depth native gate set (Rx, Rz, and CZ or CNOT), by sampling the circuit parameters from suitable marginal and joint distributions, using the known result that any two-qubit unitary needs only three two-qubit entangling gates. Running this on Rigetti's Aspen-M-3 processor, they show that standard and t-design RB overestimate average gate fidelity once compiler optimisation is enabled, and quantify that overestimation. A supplementary noise analysis argues the observed exponential fidelity decay is consistent with predominantly depolarizing noise, suggesting the protocol can support accurate experimental noise models.

## Techniques offered

- **Restricted randomized benchmarking (fixed-length native-gate RB)** - benchmarks the average gate fidelity of a device by implementing every Haar-random operation with native gate decompositions of equal sequence length, removing the depth disparity that biases standard RB.
  - guarantee: statistical estimate of the average gate fidelity (SPAM-free) via exponential decay fit, and a quantified comparison showing standard and t-design RB overestimate device performance.
  - quote: "In this work, we devised a new version of the protocol, which we call {\it Restricted Randomized Benchmarking} where for each Haar-random operation we use universal gate decomposition of equal sequence length."

- **Fixed-depth Haar-random unitary generation from a native gate set** - a constructive method to produce exact Haar-random single- and two-qubit unitaries using fixed-depth native gate circuits by sampling the gate parameters from derived marginal and joint distributions, exploiting that three two-qubit gates suffice for any two-qubit unitary.
  - guarantee: provable Haar-uniformity of the generated ensemble (verified against uniform Bloch-sphere coverage and circular-unitary-ensemble eigenvalue spacing), not relying on an approximate t-design.
  - quote: "For single qubits, any random operations can be expressed using 4 sequences of universal gates while for the two-qubits operations, results from Ref.\ \cite{PhysRevA.69.032315, PhysRevA.69.010301} guarantee that only three C-NOTs together with a few single qubit rotations are enough."

- **Noise characterisation for experimental noise modelling** - uses the restricted RB fidelity-decay data, plus a twirl analysis of coherent and incoherent (amplitude-damping and depolarizing) noise components, to infer the device noise profile and support accurate noise models.
  - guarantee: none (an empirical, non-exclusive explanatory model; the authors note the explanation may not be unique).
  - quote: "We augment our findings with a noise analysis which demonstrates that our method could be an effective tool for building accurate models of experimental noise."

## Where it could apply

- Target primitive(s): sampling (Haar-random circuit sampling and randomized benchmarking sequences); applicable to any gate-based primitive whose average gate quality is benchmarked (VQE, QAOA, Trotterised simulation, QPE).
- Target application group(s): platforms_tooling; error_correction (noise modelling input towards mitigation and error correction).
- Code family / hardware assumptions: superconducting native gate set (Rx with theta = +/- pi, +/- pi/2; Rz arbitrary; CZ or CNOT), demonstrated on Rigetti Aspen-M-3; restricted to small qubit numbers (single- and two-qubit demonstrations), no surface code or qLDPC dependence.

## Caveats

The protocol is demonstrated and proven optimal only for single- and two-qubit operations; the authors state that generalisation beyond two qubits is not straightforward because the optimal fixed-depth construction of an n-qubit unitary is not known, and any fixed-length extension would not be guaranteed optimal. The noise channels of native gates do not commute with the gate operations, so unitary invariance of the Haar twirl is not guaranteed; the observed exponential decay and depolarizing-noise explanation are empirical and admittedly non-exclusive. Results are practical only for few-qubit systems where generating Haar randomness remains resource-manageable.