Technical Benchmarking¶
Technical Benchmarking refers to tests measuring fidelities, success probabilities, or other noise measures for specific sets of gates and qubits. The Classiq package supports, for example, randomized benchmarking, a test measuring the average error per Clifford gate on a specific, usually narrow, set of qubits.
Randomized Benchmarking¶
Randomized Benchmarking (RB) is a test meant to measure the average Clifford error or fidelity on a specific set of qubits. The test is performed by applying a series of random Clifford gates, then their inverse, which is another Clifford gate, precomputed in advance. Several analytical results allow to extract the fidelity by fitting the RB experiment results. More mathematically inclined readers can see the next subsection for a summary. Following it is a usage example of RB on the Classiq platform.
Theory of Randomized Benchmarking¶
The Clifford group forms a two-design; namely, a set on which all degree two-polynomial integrals may be evaluated by a discrete sum. Random samples from the Clifford group approximately have the same property [1]. A classical paper by Nielsen [2] shows that when averaging over all unitary gates according to the Haar measure, the average noise channel is a depolarizing channel. This process is commonly referred to as "twirling". Direct calculation of the survival probability (the chance not to be "depolarized"), yields an exponentially decreasing success probability, with the rate given by the average Clifford fidelity \(f\).
Here, \(m\) is the number of Clifford gates, and \(A\) and \(B\) are constants that depend on state preparation and measurement (SPAM) errors. This is the basic RB scheme, which may be extended. See, for example, [3].
Usage¶
References¶
[1] C. Dankert, R. Cleve, J. Emerson, and E. Livine, “Exact and Approximate Unitary 2-Designs and their Application to Fidelity Estimation”. https://arxiv.org/pdf/quant-ph/0606161.pdf.
[2] M. A. Nielsen, "A simple formula for the average fidelity of a quantum dynamical operation". https://arxiv.org/pdf/quant-ph/0205035.pdf.
[3] J. Helsen, X. Xue, L. M. L Vandersypen, S. Wehner "A new class of efficient randomized benchmarking protocols". https://www.nature.com/articles/s41534-019-0182-7.