Randomized benchmarking

Randomized benchmarking estimates average gate performance by observing how random gate sequences degrade a prepared quantum state. A typical experiment prepares a reference state, applies m randomly selected elements of the Clifford group, appends an operation intended to invert their ideal product, and measures whether the initial state survives. Repeating this for several sequence lengths and random realizations produces an average survival curve.¹

Under simplified, gate-independent noise, the curve is modeled as S(m) = A pm + B. The parameter p captures the decay caused by the gates, while A and B absorb preparation and readout effects. For a qubit, the associated depolarizing error estimate is commonly r = (1 − p)/2; this is an average metric, not an error probability for every individual gate.²

The main strength of randomized benchmarking is its relative insensitivity to SPAM errors, because those errors primarily alter the fitted offset and amplitude rather than the long-run decay rate.² The method therefore provides a practical comparison of processors, pulse calibrations, and compiled gate sets without requiring perfectly characterized input states or detectors.

The estimate is an average over the sampled gate set and sequence distribution. It can include control errors, decoherence, crosstalk, and other effects that occur during the sequence, but it does not by itself identify their physical causes. Randomized benchmarking is consequently complementary to quantum process tomography and gate set tomography, which can reveal more detailed error structure but generally demand more measurements and stronger calibration assumptions.³

Variants adapt the protocol to more specific questions. In interleaved randomized benchmarking, a target gate is inserted between random reference gates; comparing the two decay rates estimates that gate's average error, subject to assumptions about noise and statistical uncertainty.³ Simultaneous benchmarking applies sequences on multiple qubits or subsystems to expose crosstalk and context-dependent degradation.

Modern analyses allow gate-dependent and time-dependent noise rather than treating all errors as identical. Such work shows that the fitted decay can remain useful under broader conditions, but its interpretation as a precise average infidelity requires care.⁴ Confidence intervals should account for finite numbers of random sequences, finite sampling of measurement outcomes, and uncertainty in nonlinear curve fitting.

Leakage outside the computational subspace is a major edge case because leaked population may not obey the single-exponential model. A sequence can therefore show a decay with additional components or a misleadingly favorable asymptote unless leakage-sensitive measurements or extended models are used. Coherent, systematic errors can also be disguised by randomization: their effects may appear as an average decay rather than as a directly visible calibration bias.

The protocol's random gates are usually implemented through a smaller native gate set, so compilation errors and compiler choices become part of what is measured. Different Clifford decompositions can consequently yield different reported performance. Randomized benchmarking is also used alongside error-correcting codes to characterize physical operations relevant to fault-tolerant thresholds, while its aggregate nature means that it cannot replace targeted tests for crosstalk, leakage, or particular error mechanisms.⁴

Clifford group

The set of quantum operations that maps Pauli operators to Pauli operators under conjugation; it is efficiently representable and commonly used for randomized sequences.

Survival probability

The measured probability that the final state agrees with the intended reference outcome after a randomized sequence.

SPAM errors

Errors in state preparation and measurement, abbreviated from state preparation and measurement.

Interleaved randomized benchmarking

A variant that inserts a fixed target gate into otherwise random sequences to estimate that gate's average error.

Depolarizing channel

A noise model that replaces part of a quantum state with the maximally mixed state, providing a convenient interpretation of a fitted decay parameter.