Sound certification of memory-bounded quantum computers

Abstract

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.

Figures (4)

• Figure 1: The considered scenario: a quantum system is initialized, undergoes a series of transformations, specified by classical instructions $x_1,x_2,\dots,x_m$, chosen from a finite set $\mathrm{X}$, and is measured producing an outcome $a\in \mathrm{A}$.
• Figure 2: Results of the numerical investigation for $\mathcal{S}_2$ (left), $\mathcal{S}_3$ (center), and $\mathcal{S}_4$ (right). The coefficients for the soundness (upper borderline) are approx. $2.6447,2.1993$, and $1.9274$, respectively, and the coefficients for the completeness (lower borderline) are approx. $0.2448,0.2217$, and $0.2096$, respectively. Each of the plots was created from sampling $10^7$ random noisy models.
• Figure 3: Numerics for robustness for the $\mathcal{S}_2$-model. Infidelity between the implemented and target models is plotted against the probability of the former failing a single repetition of \ref{['protocol']}. In the region with higher infidelity values, SPAM errors are set to $0$. In the region with lower infidelity values, initial state and measurement are affected by depolarizing noise and statistical errors, respectively, both up to $5\%$. Each region contains $\approx 2\times 10^6$ randomly generated models.
• Figure 4: Numerics for robustness for the $\mathcal{C}l_2$-model. The worst average gate infidelity among the implemented gates is plotted against the probability of the corresponding noisy model failing a single repetition of \ref{['protocol']}. SPAM errors are set to $0$. The larger region corresponds to coherent noise applied to all the gates, and contains $\approx 2\times 10^6$ random models, while the smaller region represents the case of coherent noise affecting only the $C_\mathrm{h} X$ gate and contains $\approx 7\times 10^6$ randomly drawn models. The dashed line represents the case where only the $C_\mathrm{h} X$ gate is affected by depolarizing noise.

Theorems & Definitions (37)

• Definition 1
• Definition 2
• Definition 3
• Theorem 4
• Definition 5
• proof : Proof sketch
• Definition 6
• Theorem 7
• proof
• Corollary 8