Randomized benchmarking¶
This experiment measures the average single-qubit gate fidelity using randomized benchmarking (RB).
Description¶
High-fidelity quantum gates are crucial for robust quantum computation. It's therefore necessary to benchmark their performance accurately1.
Randomized benchmarking (RB) uses random circuits of varying length to quantify the average error rate per gate within a gateset. From this, the corresponding average gate fidelity can be determined. Unlike characterisation techniques such as quantum process tomography, RB yields a fidelity metric that is largely insensitive to state preparation and measurement (SPAM) errors.
There are many types of RB protocol, each with its own strengths and limitations2. We encourage interested readers to check out the references below to find out more about the different types of RB, but in this experiment we focus on Clifford-group-based RB.
Clifford gates
A Clifford gate is a quantum gate that maps each Pauli operator (X, Y, Z) to another Pauli operator under conjugation. The main Clifford gates are the Hadamard (\(\text{H}\)), Phase (\(\text{S}\)), and \(\text{CNOT}\) gates. They’re important in randomized benchmarking because they’re easy to simulate and preserve stabiliser states.
A Clifford-group-based RB sequence might look like this:
where \(\mathrm{G_1}\), \(\mathrm{G_2}\), and \(\mathrm{G_3}\) are different Clifford gates, and \(\mathrm{G_{recovery}}\) is chosen such that (in the absence of errors) the qubit returns to its initial state at the end of the sequence. The probability that this happens in practice is known as the survival probability, \(P_\text{surv}\), and is given by
where \(A\), \(p\), and \(B\) are fit parameters, and \(x\) is the gate sequence length. For a single qubit, the average gate fidelity, \(\alpha\), can be obtained from \(p\) using
The fit parameters \(A\) and \(B\) capture SPAM errors, as setting \(x=0\) gives the fidelity when no gates are applied:
Experiment steps¶
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\(N\) different sequences of the same length, \(x\), are defined, each consisting of gates from the set \(R_x(\frac{\pi}{2})\), \(R_x(-\frac{\pi}{2})\), \(R_y(\frac{\pi}{2})\), \(R_y(-\frac{\pi}{2})\), and \(I\).
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The RB sequences are applied to the qubit, which begins each sequence in its ground state \(|0\rangle\).
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The resonator transmission is measured at the end of each of the \(N\) sequences.
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Steps 1 to 3 are repeated for varying lengths of the gate sequence (\(x\)).
Two-qubit RB
The above instructions are for single-qubit RB. When performing two-qubit RB, the only relevant change is the choice of the gateset. Along with the single-qubit gates described here, we also include the \(\text{CNOT}\) gate when generating the RB sequences.
Analysis steps¶
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The qubit state is determined from the \(I\)-\(Q\) data by applying the discriminator trained in the Readout discriminator training experiment.
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The survival probability, \(P_{\text{surv}}(x)\), is calculated by averaging over the \(N\) different sequences.
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\(P_{\text{surv}}(x)\) is plotted against the sequence length, \(x\) (see figure below), and an exponential curve is fit to the data.
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The fit parameter \(p\) is used to calculate the average gate fidelity, \(\alpha\).
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The fit parameters \(A\) and \(B\) are used to calculate the SPAM error.
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Matthew James Baldwin. Randomized benchmarking simulations of quantum gate sequences with z-gate virtualization. 2021. URL: https://hdl.handle.net/1721.1/139448. ↩
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Akel Hashim, Long B. Nguyen, Noah Goss, Brian Marinelli, Ravi K. Naik, Trevor Chistolini, Jordan Hines, J. P. Marceaux, Yosep Kim, Pranav Gokhale, Teague Tomesh, Senrui Chen, Liang Jiang, Samuele Ferracin, Kenneth Rudinger, Timothy Proctor, Kevin C. Young, Robin Blume-Kohout, and Irfan Siddiqi. A practical introduction to benchmarking and characterization of quantum computers. 2024. arXiv:2408.12064. ↩