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arXiv:1203.3733 · Physical Review Letters

Randomized Benchmarking of Multiqubit Gates

John Gaebler, Adam M. Meier, Ting Rei Tan, R. Bowler, Yiheng Lin, David Hanneke, John D. Jost, Jonathan Home, Emanuel H. Knill, Dietrich G. Leibfried, D. J. Wineland

Published Jun 26, 2012• 125 citations
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Abstract

We describe an extension of single-qubit gate randomized benchmarking that measures the error of multiqubit gates in a quantum information processor. This platform-independent protocol evaluates the performance of Clifford unitaries, which form a basis of fault-tolerant quantum computing. We implemented the benchmarking protocol with trapped ions and found an error per random two-qubit Clifford unitary of 0.162±0.008, thus setting the first benchmark for such unitaries. By implementing a second set of sequences with an extra two-qubit phase gate inserted after each step, we extracted an error per phase gate of 0.069±0.017. We conducted these experiments with transported, sympathetically cooled ions in a multizone Paul trap-a system that can in principle be scaled to larger numbers of ions.

DOI
10.1103/physrevlett.108.260503

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We describe an extension of single-qubit gate randomized benchmarking that measures the error of multiqubit gates in a quantum information processor. This platform-independent protocol evaluates the performance of Clifford unitaries, which form a basis of fault-tolerant quantum computing. We implemented the benchmarking protocol with trapped ions and found an error per random two-qubit Clifford unitary of 0.162±0.008, thus setting the first benchmark for such unitaries. By implementing a second set of sequences with an extra two-qubit phase gate inserted after each step, we extracted an error per phase gate of 0.069±0.017. We conducted these experiments with transported, sympathetically cooled ions in a multizone Paul trap-a system that can in principle be scaled to larger numbers of ions.
Benchmarking methods that can be adapted to multiqubit systems are essential for assessing the overall or "holistic" performance of nascent quantum processors. The current industry standard is Clifford randomized benchmarking (RB), which measures a single error rate that quantifies overall performance. But, scaling Clifford RB to many qubits is surprisingly hard. It has only been performed on one, two, and three qubits as of this writing. This reflects a fundamental inefficiency in Clifford RB: the n-qubit Clifford gates at its core have to be compiled into large circuits over the one- and two-qubit gates native to a device. As n grows, the quality of these Clifford gates quickly degrades, making Clifford RB impractical at relatively low n. In this Letter, we propose a direct RB protocol that mostly avoids compiling. Instead, it uses random circuits over the native gates in a device, which are seeded by an initial layer of Clifford-like randomization. We demonstrate this protocol experimentally on two to five qubits using the publicly available ibmqx5. We believe this to be the greatest number of qubits holistically benchmarked, and this was achieved on a freely available device without any special tuning up. Our protocol retains the simplicity and convenient properties of Clifford RB: it estimates an error rate from an exponential decay. But, it can be extended to processors with more qubits-we present simulations on 10+ qubits-and it reports a more directly informative and flexible error rate than the one reported by Clifford RB. We show how to use this flexibility to measure separate error rates for distinct sets of gates, and we use this method to estimate the average error rate of a set of cnot gates.
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.
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 0.09360.09360.0936. 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 0.06530.06530.0653. 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.
Randomized benchmarking (RB) protocols are the most widely used methods for assessing the performance of quantum gates.However, the existing RB methods either do not scale to many qubits or cannot benchmark a universal gate set.Here, we introduce and demonstrate a technique for scalable RB of many universal and continuously parametrized gate sets, using a class of circuits called randomized mirror circuits.Our technique can be applied to a gate set containing an entangling Clifford gate and the set of arbitrary single-qubit gates, as well as gate sets containing controlled rotations about the Pauli axes.We use our technique to benchmark universal gate sets on four qubits of the Advanced Quantum Testbed, including a gate set containing a controlled-S gate and its inverse, and we investigate how the observed error rate is impacted by the inclusion of non-Clifford gates.Finally, we demonstrate that our technique scales to many qubits with experiments on a 27-qubit IBM Q processor.We use our technique to quantify the impact of crosstalk on this 27-qubit device, and we find that it contributes approximately 2=3 of the total error per gate in random many-qubit circuit layers.
Randomized benchmarking allows one to efficiently and scalably characterize the average error of a unitary 2-design such as the Clifford group C\mathcal{C}C on a physical candidate for quantum computation, as long as there are no noncomputational leakage levels in the system. We investigate the effect of leakage errors on randomized benchmarking induced from an additional level per physical qubit and provide a modified protocol that allows us to derive reliable estimates for the error per gate in their presence. We assess the variance of the sequence fidelity corresponding to the number of random sequences needed for valid fidelity estimation. Our protocol allows for gate-dependent error channels without being restricted to perturbations. We show that our protocol is compatible with interleaved randomized benchmarking and expand to benchmarking of arbitrary gates. This setting is relevant for superconducting transmon qubits, among other systems.
We describe a practical experimental protocol for robustly characterizing the error rates of non-Clifford gates associated with dihedral groups, including small single-qubit rotations. Our dihedral benchmarking protocol is a generalization of randomized benchmarking that relaxes the usual unitary 2-design condition. Combining this protocol with existing randomized benchmarking schemes enables practical universal gate sets for quantum information processing to be characterized in a way that is robust against state-preparation and measurement errors. In particular, our protocol enables direct benchmarking of the $\ensuremath{\pi}/8$ gate even under the gate-dependent error model that is expected in leading approaches to fault-tolerant quantum computation.