Certifying Quantum Gates via Automata Advantage

TL;DR

The paper argues that promise problems from finite-automata theory offer a natural, sound framework for testing and certifying quantum gate quality on near-term devices. It demonstrates exponential separations between quantum finite automata and classical pvDFA for several promise problems, establishing a memory-based criterion for quantumness in gate-testing tasks. By extending to generalized and restricted promise problems and applying them to a QSQ certification protocol, the authors analyze robustness to noise and provide both analytic and numerical evidence that quantum models outperform classical ones under realistic conditions. The work bridges automata theory and quantum information, highlighting how automata minimality and uniqueness inform soundness guarantees and suggesting future directions, including the potential use of entanglement-breaking channels to reveal quantum advantage.

Abstract

There is growing interest in developing rigorous tests of quantumness that are feasible even before practical quantum advantages become a reality. Such tests not only aim to certify the quantum nature of a system but also serve as benchmarks for precise quantum control. In this work, we argue that promise problems, studied in the theory of finite automata, provide a natural framework for designing sound tests of quantum gate quality. Soundness, the property that only implementations of sufficiently high quality can pass the test, is a central requirement for meaningful certification. We study several promise problems relevant to quantum gate testing and establish separations between the memory resources required by quantum and classical finite automata to solve them. These separations form the theoretical basis for using promise problems as tests of quantumness. Finally, we show how results from automata theory, in particular the minimality of automata, can be used to derive soundness guarantees.

Figures (5)

• Figure 1: (Left) A qubit QFA solving $\mathtt{EO}^1$. (Right) The state diagram of a pvDFA solving the same promise problem, which requires $4$ states. The initial state is marked by an incoming arrow, the accept state by a double circle, and the reject state by a filled circle.
• Figure 2: (Left) A qubit QFA solving $\mathtt{EO}^2$. (Right) The state diagram of a pvDFA solving the same promise problem, which requires $8$ states. Red dashed lines indicate an alternative transition function for the same set of states.
• Figure 3: The state diagram of a pvDFA solving $\mathtt{DIOF}^{2}$ with $\Sigma=\{\mathrm{s},\mathrm{t}\}$. The solid arrows indicate transitions for $\mathrm{t}$, while the dashed arrows represent transitions for $\mathrm{s}$.
• Figure 4: (Left) A qubit QFA solving $\mathtt{Cl}$. (Right) The state diagram of a pvDFA solving $\mathtt{Cl}$, which needs $6$ states.
• Figure 5: Numerical survey of qubit channels for the promise problems $\mathtt{EO}^1_3$ (left) and $\mathtt{EO}^1_5$ (right). Blue regions each depict around $100$ million qubit channels, positioned according to their probability of failure and infidelity, with the noiseless $\sqrt{X}$ gate at the origin. The soundness guarantee of QSQ can be seen as a sharp upper boundary near the origin. Blue dashed line shows a conservative worst-case infidelity upper bound, with slope $\alpha$ given by the optimal classical strategy with $p_\mathrm{fail} = p_C$ and $\mathrm{inFid} = 1/2$. Curves show the behavior of the $\sqrt{X}$ gate under different noise models, with tick marks every 10% increment for the noise parameter $t$. Black dots depict noisy $U_s$ channels for every available qubit of every simulated backend in IBM's quantum platform, placed according to their reported noise models.

Theorems & Definitions (26)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Example 1
• Example 2
• proof
• Example 3
• Example 4