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arXiv:2107.02365 · Physical Review Letters

Efficient Experimental Verification of Quantum Gates with Local Operations

Rui-Qi Zhang, Zhibo Hou, Jun-Feng Tang, Jiangwei Shang, Huangjun Zhu, Guo‐Yong Xiang, Chuan-Feng Li, Guang-Can Guo

Published Jan 14, 2022• 7 citations
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Abstract

Verifying the correct functioning of quantum gates is a crucial step toward reliable quantum information processing, but it becomes an overwhelming challenge as the system size grows due to the dimensionality curse. Recent theoretical breakthroughs show that it is possible to verify various important quantum gates with the optimal sample complexity of O(1/ε) using local operations only, where ε is the estimation precision. In this Letter, we propose a variant of quantum gate verification (QGV) that is robust to practical gate imperfections and experimentally realize efficient QGV on a 2-qubit controlled-not gate and a 3-qubit Toffoli gate using only local state preparations and measurements. The experimental results show that, by using only 1600 and 2600 measurements on average, we can verify with 95% confidence level that the implemented controlled-not gate and Toffoli gate have fidelities of at least 99% and 97%, respectively. Demonstrating the superior low sample complexity and experimental feasibility of QGV, our work promises a solution to the dimensionality curse in verifying large quantum devices in the quantum era.

DOI
10.1103/physrevlett.128.020502

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Verifying the correct functioning of quantum gates is a crucial step toward reliable quantum information processing, but it becomes an overwhelming challenge as the system size grows due to the dimensionality curse. Recent theoretical breakthroughs show that it is possible to verify various important quantum gates with the optimal sample complexity of O(1/ε) using local operations only, where ε is the estimation precision. In this Letter, we propose a variant of quantum gate verification (QGV) that is robust to practical gate imperfections and experimentally realize efficient QGV on a 2-qubit controlled-not gate and a 3-qubit Toffoli gate using only local state preparations and measurements. The experimental results show that, by using only 1600 and 2600 measurements on average, we can verify with 95% confidence level that the implemented controlled-not gate and Toffoli gate have fidelities of at least 99% and 97%, respectively. Demonstrating the superior low sample complexity and experimental feasibility of QGV, our work promises a solution to the dimensionality curse in verifying large quantum devices in the quantum era.
To employ a quantum device, the performance of the quantum gates in the device needs to be evaluated first. Since the dimensionality of a quantum gate grows exponentially with the number of qubits, evaluating the performance of a quantum gate is a challenging task. Recently, a scheme called quantum gate verification (QGV) has been proposed, which can verifies quantum gates with near-optimal efficiency. In this paper, we implement a proof-of-principle optical experiment to demonstrate this QGV scheme. We show that for a single-qubit quantum gate, only $\ensuremath{\sim}300$ samples are needed to confirm the fidelity of the quantum gate to be, at least, 97% with a 99% confidence level using the QGV method, whereas, at least, $\ensuremath{\sim}3000$ samples are needed to achieve the same result using the standard quantum process tomography method. The QGV method validated by this paper has the potential to be widely used for the evaluation of quantum devices in various quantum information applications.
Efficient verification of the functioning of quantum devices is a key to the development of quantum technologies, but is a daunting task as the system size increases. Here we propose a simple and general framework for verifying unitary transformations that can be applied to both individual quantum gates and gate sets, including quantum circuits. This framework enables efficient verification of many important unitary transformations, including but not limited to all bipartite unitaries, Clifford unitaries, generalized controlled-ZZZ gates, generalized controlled-not gates, the controlled-swap gate, and permutation transformations. For all these unitaries, the sample complexity increases at most linearly with the system size and is often independent of the system size. Moreover, little overhead is incurred even if one can only prepare Pauli eigenstates and perform local measurements. Our approach is applicable in many scenarios in which randomized benchmarking (RB) does not apply and is thus instrumental to quantum computation and many other applications in quantum information processing.
The initialization of a quantum system into a certain state is a crucial aspect of quantum information science. While a variety of measurement strategies have been developed to characterize how well the system is initialized, for a given one, there is in general a trade-off between its efficiency and the accessible information of the quantum state. Conventional quantum state tomography can characterize unknown states while requiring exponentially expensive time-consuming postprocessing. Alternatively, recent theoretical breakthroughs show that quantum state verification provides a technique to quantify the prepared state with significantly fewer samples, especially for multipartite entangled states. In this Letter, we modify the original proposal to be robust to practical imperfections, and experimentally implement a scalable quantum state verification on two-qubit and four-qubit entangled states with nonadaptive local measurements. For all the tested states, the estimated infidelity is inversely proportional to the number of samples, which illustrates the power to characterize a quantum state with a small number of samples. Compared to the globally optimal strategy which requires nonlocal measurements, the efficiency in our experiment is only worse by a small constant factor (<2.5). We compare the performance difference between quantum state verification and quantum state tomography in an experiment to characterize a four-photon Greenberger-Horne-Zeilinger state, and the results indicate the advantage of quantum state verification in both the achieved efficiency and precision.
Most near-term quantum information processing devices will not be capable of implementing quantum error correction and the associated logical quantum gate set. Instead, quantum circuits will be implemented directly using the physical native gate set of the device. These native gates often have a parameterization (e.g., rotation angles) which provide the ability to perform a continuous range of operations. Verification of the correct operation of these gates across the allowable range of parameters is important for gaining confidence in the reliability of these devices. In this work, we demonstrate a procedure for sample-efficient verification of continuously-parameterized quantum gates for small quantum processors of up to approximately 10 qubits. This procedure involves generating random sequences of randomly-parameterized layers of gates chosen from the native gate set of the device, and then stochastically compiling an approximate inverse to this sequence such that executing the full sequence on the device should leave the system near its initial state. We show that fidelity estimates made via this technique have a lower variance than fidelity estimates made via cross-entropy benchmarking. This provides an experimentally-relevant advantage in sample efficiency when estimating the fidelity loss to some desired precision. We describe the experimental realization of this technique using continuously-parameterized quantum gate sets on a trapped-ion quantum processor from Sandia QSCOUT and a superconducting quantum processor from IBM Q, and we demonstrate the sample efficiency advantage of this technique both numerically and experimentally.
The rapid advancement of quantum hardware necessitates the development of reliable methods to certify its correct functioning. However, existing certification tests fall short, as they either suffer from systematic errors or do not guarantee that only a correctly functioning quantum device can pass the test. We introduce a certification method for quantum gates tailored for a practical server-user scenario, where a classical user tests the results of exact quantum computations performed by a quantum server. This method is free from the systematic state preparation and measurement (SPAM) errors. For single-qubit gates, including those that form a universal set for single-qubit quantum computation, we demonstrate that our approach offers soundness guarantees based solely on the dimension assumption. Additionally, for a highly-relevant phase gate – which corresponds experimentally to a π/2 -pulse – we prove that the method's sample complexity scales as O(ε−1) relative to the average gate infidelity ε . By combining the SPAM-error-free and sound notion of certification with practical applicability, our approach paves the way for promising research into efficient and reliable certification methods for full-scale quantum computation.
Efficient verification of quantum states and gates is crucial to the development of quantum technologies. Although the sample complexities of quantum state verification and quantum gate verification have been studied by many researchers, the number of experimental settings has received little attention and is poorly understood. In this work we study systematically quantum state verification and quantum gate verification with a focus on the number of experimental settings. We show that any bipartite pure state can be verified by only two measurement settings based on local projective measurements. Any bipartite unitary in dimension ddd can be verified by 2d2d2d experimental settings based on local operations. In addition, we introduce the concept of entanglement-free verification and clarify its connection with minimal-setting verification. Finally, we show that any two-qubit unitary can be verified with at most five experimental settings; moreover, a generic two-qubit unitary (except for a set of measure zero) can be verified by an entanglement-free protocol based on four settings. In the course of study we clarify the properties of Schmidt coefficients of two-qubit unitaries, which are of independent interest.