Accurate benchmarking of quantum gates is crucial for understanding and enhancing the performance of quantum hardware. A standard method for this is interleaved benchmarking, a technique which estimates the error on an interleaved target gate by comparing cumulative error rates of randomized sequences implemented with the interleaved gate and without it. In this work, we show both numerically and experimentally that the standard approach of interleaved randomized benchmarking (IRB), which uses the multi-qubit Clifford group for randomization, can produce highly inaccurate and even physically impossible estimates for the error on the interleaved gate in the presence of coherent errors. Fortunately we also show that interleaved benchmarking performed with cycle benchmarking, which randomizes with single qubit Pauli gates, provides dramatically reduced systematic uncertainty relative to standard IRB, and further provides as host of additional benefits including data reusability. We support our conclusions with a theoretical framework for bounding systematic errors, extensive numerical results comparing a range of interleaved protocols under fixed resource costs, and experimental demonstrations on three quantum computing platforms.
We describe a scalable experimental protocol for estimating the average error of individual quantum computational gates. This protocol consists of interleaving random Clifford gates between the gate of interest and provides an estimate as well as theoretical bounds for the average error of the gate under test, so long as the average noise variation over all Clifford gates is small. This technique takes into account both state preparation and measurement errors and is scalable in the number of qubits. We apply this protocol to a superconducting qubit system and find a bounded average error of 0.003 [0,0.016] for the single-qubit gates X(π/2) and Y(π/2). These bounded values provide better estimates of the average error than those extracted via quantum process tomography.
We demonstrate a new technique that adapts single-qubit randomized benchmarking to two-qubit Mølmer-Sørensen gates. We use the controllable gate phase to generate Cliffords that act on a two-state subspace, enabling benchmarking of two-qubit gates without single-qubit operations. In addition to quantifying the gate infidelity, the protocol provides valuable information about the type of error by distinguishing between those that conserve the two-state subspace and those that result in leakage out of it. We demonstrate the protocol for calibrating and validating all-electronic maximally entangling gates in a trapped-ion quantum computer, achieving a two-qubit gate error of
2.6(2)×10−4.
Interleaved randomized benchmarking (IRB) provides a scalable estimate of a gate's error rate, but its standard guarantees require the interleaved gate to be Clifford~\cite{Magesan2012Interleaved,magesan2012characterizing}. In superconducting processors, many non-Clifford phase gates in compiled circuits are implemented virtually as software-defined frame updates rather than as additional control pulses~\cite{mckay2017efficient}. This raises the question of whether inserting or removing such virtual phases measurably changes IRB error estimates.
We introduce \emph{phase-altered interleaved randomized benchmarking} (PA-IRB), a paired-IRB diagnostic protocol comparing phase-stripped and phase-dressed Clifford interleaving gates derived from the same compiled implementation. PA-IRB reports
Δr=rd−rs with combined uncertainty to test whether virtual phase gates affect the extracted IRB decay beyond statistical error.
As a case study, we apply PA-IRB to a compiled Toffoli gate executed on IBM superconducting processors, where the constituent
T/T† gates are implemented as virtual
Z rotations. Across tested calibration runs,
Δr is consistent with zero within uncertainty, indicating that virtual phase addition or removal does not measurably alter the IRB-derived error estimate under the employed compilation and execution stack. More generally, PA-IRB provides a lightweight, abstraction-aware diagnostic for benchmarking workflows involving software-defined phase operations. The same paired comparison can also be used to place operational bounds on the contribution of non-Clifford components to the compiled gate error, even when those components are physically executed rather than implemented virtually.
Randomised benchmarking is a widely used experimental technique to characterise the average error of quantum operations. Benchmarking procedures that scale to enable the characterisation of n-qubit circuits rely on efficient procedures for manipulating those circuits and, as such, have been limited to subgroups of the Clifford group. However, universal quantum computers require additional, non-Clifford gates to approximate arbitrary unitary transformations. We define a scalable randomised benchmarking procedure over n-qubit unitary matrices that correspond to protected non-Clifford gates for a class of stabiliser codes. We present efficient methods for representing and composing group elements, sampling them uniformly and synthesising corresponding poly(n)-sized circuits. The procedure provides experimental access to two independent parameters that together characterise the average gate fidelity of a group element. A scalable procedure to determine the error of quantum operations has been developed by researchers in the United States. Andrew Cross and colleagues at the IBM T.J. Watson Research Center have expanded previously used protocols to a broader range of quantum circuits, thereby enabling error benchmarking for all common logical operations used in leading quantum computing protocols. The standard technique for error characterization in these devices is randomized benchmarking, which provides benefits such as insensitivity to measurement errors, but on the other hand only functions in a limited number of circumstances. In their theoretical study, the researchers demonstrate how randomized benchmarking approaches can be expanded to a much broader range of logical operations. The procedure can be readily implemented experimentally, and provides access to important experimental noise parameters in quantum computing.
We implement a complete randomized benchmarking protocol on a system of two superconducting qubits. The protocol consists of randomizing over gates in the Clifford group, which experimentally are generated via an improved two-qubit cross-resonance gate implementation and single-qubit unitaries. From this we extract an optimal average error per Clifford operation of
0.0936. We also perform an interleaved experiment, alternating our optimal two-qubit gate with random two-qubit Clifford gates, to obtain a two-qubit gate error of
0.0653. We compare these values with a two-qubit gate error of $\ensuremath{\sim}0.12$ obtained from quantum process tomography, which is likely limited by state preparation and measurement errors.