# Extended Randomized Benchmarking (XRB)¶

See make_xrb() for API documentation.

XRB is a protocol for estimating how the probability of error decomposed into stochastic errors vs coherent/calibration errors that can typically be corrected through improved control [7].

Terms that may be returned from the analysis are as follows. These descriptions are also available via mouse-overs when running True-Q™ in a Jupyter or Colab notebook.

## Estimated Parameters¶

$${e}_{S}$$ -

The probability of a stochastic error for an error channel $$\mathcal{A}$$ is defined to be

$e_S = 1 - \sqrt{\Phi(\mathcal{A}^\dagger \mathcal{A}, \mathcal{I})}$

where $$\Phi(\cdot, \cdot)$$ is the process fidelity between two channels (see eq. 13 in [8]). By looking at the process fidelity of $$\mathcal{A}^\dagger \mathcal{A}$$, we remove any coherent errors. The square root corrects for the residual stochastic errors being applied twice.

$${e}_{U}$$ -

We define this quantity to be $$e_U = e_F - e_S$$, so that it will be $$O(e_S^2)$$ if there are no residual unitary errors. Formally, the quantity $$e_U$$ is equivalent to the process infidelity of the unitary correction required to make the noise channel decoherent up to corrections of $$O(e_F^2)$$ (see arxiv 1904.08897, eq. 65).

$${A}$$ -

This parameter has an ideal value of $$A=1$$ and only includes state preparation and and measurement (SPAM) errors, up to statistical noise.

This is one of two parameters in the decay curve $$y=Au^m$$ where $$m$$ is the sequence length, i.e. the number of random gates in a particular circuit, or the circuit depth before compiling into native gates. At least two distinct values of $$m$$ must be used to distinguish $$A$$ from $$u$$. The value of $$y$$ at a particular value of $$m$$ is the average purity after $$m$$ random gates, including the loss of purity due to SPAM errors. Many random circuits must be used per sequence length $$m$$ to gain an accurate estimate of the average purity. If the raw decay is plotted, the average value $$y$$ is shown, as well as individual estimates of final state purity for each random circuit.

As a technical aside, given a final density matrix $$\rho$$ expressed in the Pauli basis as $$\rho=\sum_P r_P P/d$$ where $$d$$ is the Hilbert space dimension, the purity is given by $$\operatorname{Tr}\rho^2=\sum_P r_P^2/d$$. However, purity values discussed above omit the component $$r_I$$ which allows the decay of $$y$$ to asymptote to 0 making the fit simpler. This would upper bound any $$y$$ value to the value $$(d-1)/d$$ rather than the nominal value of $$1$$. Therefore, for aesthetic reasons, $$y$$ is an estimate of $$\sum_{P\neq I} r_P^2/(d-1)$$ instead of $$\operatorname{Tr}\rho^2$$. Note that this scaling of $$A$$ does not affect the interpretation of $$u$$ or derived quantities such as $$e_U$$.

$${u}$$ -

The unitarity of the noise, that is, the average decrease in the purity of an initial state.

This is one of two parameters in the decay curve $$y=Au^m$$ where $$m$$ is the sequence length, i.e. the number of random gates in a particular circuit, or the circuit depth before compiling into native gates. At least two distinct values of $$m$$ must be used to distinguish $$A$$ from $$u$$. The value of $$y$$ at a particular value of $$m$$ is the average purity after $$m$$ random gates, including the loss of purity due to SPAM errors. Many random circuits must be used per sequence length $$m$$ to gain an accurate estimate of the average purity. If the raw decay is plotted, the average value $$y$$ is shown, as well as individual estimates of final state purity for each random circuit.

As a technical aside, given a final density matrix $$\rho$$ expressed in the Pauli basis as $$\rho=\sum_P r_P P/d$$ where $$d$$ is the Hilbert space dimension, the purity is given by $$\operatorname{Tr}\rho^2=\sum_P r_P^2/d$$. However, purity values discussed above omit the component $$r_I$$ which allows the decay of $$y$$ to asymptote to 0 making the fit simpler. This would upper bound any $$y$$ value to the value $$(d-1)/d$$ rather than the nominal value of $$1$$. Therefore, for aesthetic reasons, $$y$$ is an estimate of $$\sum_{P\neq I} r_P^2/(d-1)$$ instead of $$\operatorname{Tr}\rho^2$$. Note that this scaling of $$A$$ does not affect the interpretation of $$u$$ or derived quantities such as $$e_U$$.

## Examples¶

 Extended Randomized Benchmarking (XRB)¶ Comparing Infidelities with SRB and XRB¶