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arXiv:2608.17005

Sound and Efficient Certification of High-Quality Qubit Operations: Theory and Experiment

Nikolai Miklin, Jan Nöller, José Martínez, Lucas B. Vieira, Ulrich Poschinger, Ferdinand Schmidt-Kaler, Mariami Gachechiladze

quant-phSubmitted Aug 17, 2026• Announced Aug 19, 2026• 0 citations
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Abstract

Can a high-quality quantum gate be certified when uncharacterized state-preparation and measurement errors are dominant? Can this be achieved with low experimental overhead? Here, we introduce a sound black-box certification protocol for a single-qubit gate based on a small set of fixed, deterministic sequences. From the data, the protocol derives finite-sample bounds on the gate's rotation eigenvalue, a gauge-invariant property. Its phase reveals the accuracy of the rotation angle, while its modulus quantifies the loss of coherence under repeated gate applications. We implement the protocol on a $^{40}\mathrm{Ca}^{+}$ trapped-ion processor and certify the $\sqrt{\mathrm{X}}$-gate rotation eigenvalue using $22\,000$ circuit executions, and demonstrate the robustness of certification to state-preparation and measurement errors by deliberately degrading the readout. Finally, we prove that these spectral constraints imply, up to a physically meaningful unitary change of basis, a rigorous average gate-fidelity lower bound for every time-independent qubit model compatible with the data. In both readout settings, the spectral bounds yield the same fidelity certificate of $99.94(3)\%$ with $99\%$ confidence. Our results establish a new standard for quantum-gate certification by combining soundness and experimental efficiency without requiring trusted reference operations, randomized circuits, or model fitting.

Comments
27 pages, 4 figures, comments welcome

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Can a high-quality quantum gate be certified when uncharacterized state-preparation and measurement errors are dominant? Can this be achieved with low experimental overhead? Here, we introduce a sound black-box certification protocol for a single-qubit gate based on a small set of fixed, deterministic sequences. From the data, the protocol derives finite-sample bounds on the gate's rotation eigenvalue, a gauge-invariant property. Its phase reveals the accuracy of the rotation angle, while its modulus quantifies the loss of coherence under repeated gate applications. We implement the protocol on a 40Ca+^{40}\mathrm{Ca}^{+}40Ca+ trapped-ion processor and certify the X\sqrt{\mathrm{X}}X​-gate rotation eigenvalue using 22 00022\,00022000 circuit executions, and demonstrate the robustness of certification to state-preparation and measurement errors by deliberately degrading the readout. Finally, we prove that these spectral constraints imply, up to a physically meaningful unitary change of basis, a rigorous average gate-fidelity lower bound for every time-independent qubit model compatible with the data. In both readout settings, the spectral bounds yield the same fidelity certificate of 99.94(3)%99.94(3)\%99.94(3)% with 99%99\%99% confidence. Our results establish a new standard for quantum-gate certification by combining soundness and experimental efficiency without requiring trusted reference operations, randomized circuits, or model fitting.
The rapid advancement of quantum hardware calls for the development of reliable methods to certify its correct functioning. However, existing certification tests often fall short: they either rely on flawless state preparation and measurement or lack soundness guarantees, meaning that they do not rule out incorrect implementations of the target operations by a quantum device. We introduce an approach, which we call quantum system quizzing, for the certification of quantum gates in a practical server-user scenario, where a classical user tests the results of quantum computation performed by a quantum server by checking its responses to a set of predesigned small-sized computational problems. Importantly, this approach does not require trusted state preparation and measurement and is thus inherently free from the associated systematic errors. For a wide range of relevant gate sets, including a universal one, we prove our certification protocol to be sound; i.e., it is guaranteed to reject any incorrect gate implementation, under the assumptions of a known Hilbert space dimension and context independence of error. A major technical challenge that we are first to resolve is recovering the tensor product structure of a multi-qubit system in the memory-bounded single-device setup. Finally, we prove the robustness of our protocol and validate its sample and computational efficiency through extensive numerical experiments. Our protocol is platform-agnostic and introduces a new paradigm for benchmarking and comparing diverse quantum architectures.
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.
Gate fidelity -- an average fidelity over all possible input states -- is the workhorse metric for benchmarking quantum gates or circuits, yet fault-tolerant quantum computing ultimately depends on the worst-case behavior, typically quantifiable by so-called the diamond distance. In the low-error regime, the coherent errors can inflate the worst-case error even when the reported gate fidelity is high, making the gate fidelity alone an unreliable proxy for fault-tolerance readiness. To capture the missing information, we introduce a companion observable -- what we dub the fidelity deviation -- that quantifies how strongly the state-dependent fidelities fluctuate across input states. Adopting such fluctuations in assessing the fault-tolerance is physically natural because some input directions are nearly unaffected while others form narrow "valleys" that dominate adversarial circuit behavior. For coherent (unitary) gate errors on two or more qubits, we show that the gate fidelity together with the fidelity deviation constrains the relevant spectral moments of the error unitary, enabling an explicit and tight certificate of the worst-case error. Both quantities are estimated directly from the same randomized input-measurement experiment, without full process tomography. We show that the fidelity and its deviation can provide an economical, operationally meaningful, and accurate standard for assessing the fault tolerance of the engineered quantum gates and circuits.
A central task in quantum information science is state certification: testing whether an unknown state is ε1ε_1ε1​-close to a fixed target state, or ε2ε_2ε2​-far. Recent work has shown that surprisingly simple measurement protocols--comprising only single-qubit measurements--suffice to certify arbitrary nnn-qubit states [Huang, Preskill, Soleimanifar '25; Gupta, He, O'Donnell '25]. However, these certification protocols are not robust: rather than allowing constant ε1ε_1ε1​, they can only positively certify states within ε1=O(1/n)ε_1=O(1/n)ε1​=O(1/n) trace distance of the target. In many experimental settings, the appropriate error tolerance is constant as the system size grows, so this lack of robustness renders existing tests inapplicable at scale, no matter how many times the test is repeated. Here we present robust certification protocols based on few-qubit measurements that apply to all but a O(2−n)O(2^{-n})O(2−n)-fraction of pure target states. Our first protocol achieves constant robustness, i.e. ε1=Θ(1)ε_1=Θ(1)ε1​=Θ(1), using a single O(log⁡n)O(\log n)O(logn)-qubit measurement along with single-qubit measurements in the ZZZ or XXX basis on the other qubits. As a corollary of its robustness, this protocol also achieves constant (in nnn) copy complexity, which is optimal. Our second protocol uses exclusively single-qubit measurements and is nearly robust: ε1=Ω(1/log⁡n)ε_1=Ω(1/\log n)ε1​=Ω(1/logn). Our tests are based on a new uncertainty principle for conditional fidelities, which may be of independent interest.
Quantum computers have the potential to bring about a new age of technology, but in order for them to become a reality, the elephant in the room must be addressed, quantum errors. The complexity of quantum errors grows exponentially as the system size is increased; scalable characterization protocols which work under physically realistic noise are therefore needed to assess and optimize control. \n \nRandomized Benchmarking (RB) is a family of characterization protocols with the requisite qualities. RB protocols are independent of state preparation and measurement (SPAM) errors, and because they extract partial information about the error profiles in the form of figures of merit, they are scalable in the number of qubits. In this thesis, we study these figures of merit under various error models, gate-sets, and sample distributions, to further our understanding of their physical meaning, and to fine-tune the best practice for diagnosing errors in basic gate-sets. \n \nThe first study presented in this thesis looks at the physical meaning of the unitarity under physically realistic gate-dependent noise. Numerically, we demonstrate that under these realistic noise models, the unitarity obtained via Extended RB (XRB) overlaps strongly with the average of the unitarity of the errors over the set of individual operations. \n \nThe second study numerically and analytically proves that the Standard RB (SRB) decay parameter p coincides with the decay parameter of the novel figure of merit gate-set circuit fidelity. We also prove that, in the limit of high-fidelity single-qubit experiments, the disconnect between the average gate fidelity and SRB experimental results is caused by a basis mismatch between the gates and the state-preparation and measurement procedures. \n \nIn the third part of this thesis, we compare the results of two variations of the SRB protocol, one that samples uniformly (SRB) and one that samples non-uniformly (NIST RB) from the Clifford group. We prove that NIST RB produces an exponential fidelity curve that gives different infidelity results than SRB. \n \nThe fourth and final study gives a standard for assessing basic gate-sets. Experimental results are given from the assessment run on super-conducting and ion-trap devices. We also experimentally show how the systematic error bounds can be tightened by accounting for the error's degree of coherence, as well as using weaker randomizing sets.
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.