Randomized Benchmarking Protocol Research Articles

Quantification and characterization of leakage errors

We present a general framework for the quantification and characterization of leakage errors that result when a quantum system is encoded in the subspace of a larger system. To do this we introduce new metrics for quantifying the coherent and incoherent properties of the resulting errors, and we illustrate this framework with several examples relevant to superconducting qubits. In particular, we propose two quantities: the leakage and seepage rates, which together with average gate fidelity allow for characterizing the average performance of quantum gates in the presence of leakage and show how the randomized benchmarking protocol can be modified to enable the robust estimation of all three quantities for a Clifford gate set.

Randomized benchmarking in measurement-based quantum computing

Randomized benchmarking is routinely used as an efficient method for characterizing the performance of sets of elementary logic gates in small quantum devices. In the measurement-based model of quantum computation, logic gates are implemented via single-site measurements on a fixed universal resource state. Here we adapt the randomized benchmarking protocol for a single qubit to a linear cluster state computation, which provides partial, yet efficient characterization of the noise associated with the target gate set. Applying randomized benchmarking to measurement-based quantum computation exhibits an interesting interplay between the inherent randomness associated with logic gates in the measurement-based model and the random gate sequences used in benchmarking. We consider two different approaches: the first makes use of the standard single-qubit Clifford group, while the second uses recently introduced (non-Clifford) measurement-based 2-designs, which harness inherent randomness to implement gate sequences.

Randomized benchmarking of quantum gates implemented by electron spin resonance

Spin systems controlled and probed by magnetic resonance have been valuable for testing the ideas of quantum control and quantum error correction. This paper introduces an X-band pulsed electron spin resonance spectrometer designed for high-fidelity coherent control of electron spins, including a loop-gap resonator for sub-millimeter sized samples with a control bandwidth ∼40MHz. Universal control is achieved by a single-sideband upconversion technique with an I–Q modulator and a 1.2GS/s arbitrary waveform generator. A single qubit randomized benchmarking protocol quantifies the average errors of Clifford gates implemented by simple Gaussian pulses, using a sample of gamma-irradiated quartz. Improvements in unitary gate fidelity are achieved through phase transient correction and hardware optimization. A preparation pulse sequence that selects spin packets in a narrowed distribution of static fields confirms that inhomogeneous dephasing (1/T2∗) is the dominant source of gate error. The best average fidelity over the Clifford gates obtained here is 99.2%, which serves as a benchmark to compare with other technologies.

Robust characterization of leakage errors

Leakage errors arise when the quantum state leaks out of some subspace of interest, for example, the two-level subspace of a multi-level system defining a computational ‘qubit’, the logical code space of a quantum error-correcting code, or a decoherence-free subspace. Leakage errors pose a distinct challenge to quantum control relative to the more well-studied decoherence errors and can be a limiting factor to achieving fault-tolerant quantum computation. Here we present a scalable and robust randomized benchmarking protocol for quickly estimating the leakage rate due to an arbitrary Markovian noise process on a larger system. We illustrate the reliability of the protocol through numerical simulations.

Effect of noise correlations on randomized benchmarking

Among the most popular and well studied quantum characterization, verification and validation techniques is randomized benchmarking (RB), an important statistical tool used to characterize the performance of physical logic operations useful in quantum information processing. In this work we provide a detailed mathematical treatment of the effect of temporal noise correlations on the outcomes of RB protocols. We provide a fully analytic framework capturing the accumulation of error in RB expressed in terms of a three-dimensional random walk in "Pauli space." Using this framework we derive the probability density function describing RB outcomes (averaged over noise) for both Markovian and correlated errors, which we show is generally described by a gamma distribution with shape and scale parameters depending on the correlation structure. Long temporal correlations impart large nonvanishing variance and skew in the distribution towards high-fidelity outcomes -- consistent with existing experimental data -- highlighting potential finite-sampling pitfalls and the divergence of the mean RB outcome from worst-case errors in the presence of noise correlations. We use the Filter-transfer function formalism to reveal the underlying reason for these differences in terms of effective coherent averaging of correlated errors in certain random sequences. We conclude by commenting on the impact of these calculations on the utility of single-metric approaches to quantum characterization, verification, and validation.

Error compensation of single-qubit gates in a surface-electrode ion trap using composite pulses

The fidelity of laser-driven quantum logic operations on trapped ion qubits tend to be lower than microwave-driven logic operations due to the difficulty of stabilizing the driving fields at the ion location. Through stabilization of the driving optical fields and use of composite pulse sequences, we demonstrate high fidelity single-qubit gates for the hyperfine qubit of a $^{171}\text{Yb}^+$ ion trapped in a microfabricated surface electrode ion trap. Gate error is characterized using a randomized benchmarking protocol, and an average error per randomized Clifford group gate of $3.6(3)\times10^{-4}$ is measured. We also report experimental realization of palindromic pulse sequences that scale efficiently in sequence length.

Investigating the limits of randomized benchmarking protocols

In this paper, we analyze the performance of randomized benchmarking protocols on gate sets under a variety of realistic error models that include systematic rotations, amplitude damping, leakage to higher levels, and $1/f$ noise. We find that, in almost all cases, benchmarking provides better than a factor-of-2 estimate of average error rate, suggesting that randomized benchmarking protocols are a valuable tool for verification and validation of quantum operations. In addition, we derive models for fidelity decay curves under certain types of non-Markovian noise models such as $1/f$ and leakage errors. We also show that, provided the standard error of the fidelity measurements is small, only a small number of trials are required for high-confidence estimation of gate errors.

Optimization of Clifford circuits

We study optimal synthesis of Clifford circuits, and apply the results to peep-hole optimization of quantum circuits. We report optimal circuits for all Clifford operations with up to four inputs. We perform peep-hole optimization of Clifford circuits with up to 40 inputs found in the literature, and demonstrate the reduction in the number of gates by about 50%. We extend our methods to the optimal synthesis of linear reversible circuits, partially specified Clifford functions, and optimal Clifford circuits with five inputs up to input/output permutation. The results find their application in randomized benchmarking protocols, quantum error correction, and quantum circuit optimization.

Process verification of two-qubit quantum gates by randomized benchmarking

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 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 ~0.12 obtained from quantum process tomography, which is likely limited by state preparation and measurement errors.

Characterizing quantum gates via randomized benchmarking

We describe and expand upon the scalable randomized benchmarking protocol proposed in Phys. Rev. Lett. 106, 180504 (2011) which provides a method for benchmarking quantum gates and estimating the gate-dependence of the noise. The protocol allows the noise to have weak time and gate-dependence, and we provide a sufficient condition for the applicability of the protocol in terms of the average variation of the noise. We discuss how state preparation and measurement errors are taken into account and provide a complete proof of the scalability of the protocol. We establish a connection in special cases between the error rate provided by this protocol and the error strength measured using the diamond norm distance.

Scalable and Robust Randomized Benchmarking of Quantum Processes

In this Letter we propose a fully scalable randomized benchmarking protocol for quantum information processors. We prove that the protocol provides an efficient and reliable estimate of the average error-rate for a set operations (gates) under a very general noise model that allows for both time and gate-dependent errors. In particular we obtain a sequence of fitting models for the observable fidelity decay as a function of a (convergent) perturbative expansion of the gate errors about the mean error. We illustrate the protocol through numerical examples.

Randomized benchmarking of single- and multi-qubit control in liquid-state NMR quantum information processing

Being able to quantify the level of coherent control in a proposed device implementing a quantum information processor (QIP) is an important task for both comparing different devices and assessing a device's prospects with regards to achieving fault-tolerant quantum control. We implement in a liquid-state nuclear magnetic resonance QIP the randomized benchmarking protocol presented by Knill et al (2008 Phys. Rev. A 77 012307). We report an error per randomized π/2 pulse of 1.3±0.1×10−4 with a single-qubit QIP and show an experimentally relevant error model where the randomized benchmarking gives a signature fidelity decay which is not possible to interpret as a single error per gate. We explore and experimentally investigate multi-qubit extensions of this protocol and report an average error rate for one- and two-qubit gates of 4.7±0.3×10−3 for a three-qubit QIP. We estimate that these error rates are still not decoherence limited and thus can be improved with modifications to the control hardware and software.