Interleaved RB

Standard randomized benchmarking gives you the average error rate over the whole Clifford group. But often you want the error of one specific gate — say, the controlled-NOT, or a particular single-qubit rotation you suspect is mis-calibrated. Interleaved randomized benchmarking (IRB) isolates exactly that.

The idea: run RB twice

IRB is a difference measurement. You run two RB experiments and compare their decay rates.

Reference experiment. Ordinary RB: random Cliffords, then the inverting Clifford. Fit the decay $\bar F(m) = A\,p^{\,m} + B$ to extract the reference depolarizing parameter $p_{\text{ref}}$ .
Interleaved experiment. The same, but after every random Clifford you insert the target gate $\mathcal{G}$ you want to characterize. The sequence becomes $C_1,\, \mathcal{G},\, C_2,\, \mathcal{G},\, \dots,\, C_m,\, \mathcal{G}$ , followed by the single Clifford that inverts the whole thing. Fit this decay to extract $p_{\text{gate}}$ .

Because the interleaved sequence carries the same random Cliffords plus one copy of $\mathcal{G}$ per step, the extra decay must come from $\mathcal{G}$ .

Extracting the gate's error

The ratio of the two depolarizing parameters isolates the target gate. For dimension $d = 2^n$ , the interleaved gate error is estimated as

r_{\mathcal{G}} = \frac{(d-1)}{d}\left(1 - \frac{p_{\text{gate}}}{p_{\text{ref}}}\right).

Dividing $p_{\text{gate}}$ by $p_{\text{ref}}$ cancels the contribution of the random Clifford "background", leaving the per-application error of $\mathcal{G}$ alone. The reference run plays the role of a calibration baseline, much as a blank does in a chemistry assay.

Caveats worth knowing

IRB is widely used but it is an estimate, not an exact measurement, and the literature is careful about its limits:

Systematic bounds. The ratio formula assumes the target gate's noise twirls into a depolarizing channel just like the random Cliffords. When that assumption is imperfect, $r_{\mathcal{G}}$ carries a known systematic uncertainty; Magesan et al. derive rigorous upper and lower bounds on the true error around the IRB estimate.
Coherent errors. If $\mathcal{G}$ has a coherent (unitary) error — a small over-rotation — it can partially cancel or compound with the random Cliffords, biasing the estimate. RB-type protocols see the average effect, not the worst case.
The gate must be a Clifford (or be compiled into one) so that the inverting gate at the end remains a single Clifford and the twirl still applies.

Where IRB fits

Standard RB answers "how good are my gates on average?"; IRB answers "how good is this gate?". The two together let an engineer rank a device's operations, find the weakest link, and target calibration effort where it matters. The next lessons broaden the lens again — from individual gates to whole-device figures of merit such as cross-entropy benchmarking and quantum volume.