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This paper presents the statistically optimal four-copy detector for three-tangle in pure three-qubit states, significantly improving experimental efficiency by reducing repetition counts compared to prior constructions. The study rigorously quantifies the tradeoff between statistical gain and implementation cost, analyzing fault-tolerant resource overheads for both non-Clifford synthesis and early implementations using $T$ gates and STAR rotations. These findings provide critical benchmarks for optimizing entanglement verification protocols in scalable quantum computing architectures.
Optimal four-copy detection of the three-tangle and its fault-tolerant cost
arXiv:2610.27260v1 Announce Type: new Abstract: The three-tangle, $\tau_3$, is a measure of genuine tripartite entanglement that distinguishes GHZ-type from W-type entanglement in pure three-qubit states. Here, we identify the statistically optimal one-sided four-copy detector of the three-tangle and study its implementation cost. The optimal detector has click probability $\tau_3^2/48$, while previous four-copy constructions achieve $\tau_3^2/256$ and therefore require asymptotically $16/3$ times as many experimental repetitions. Using the representation-theoretic structure of the problem, we construct an ancilla-free implementation of the optimal detector using 15 CNOT gates, together with an 11-CNOT detector with click probability $\tau_3^2/96$. These results reveal a tradeoff between statistical efficiency and implementation cost, since reducing the number of repetitions requires increasingly expensive circuits. We then show that this tradeoff persists once the circuits are encoded. In a fully fault-tolerant setting, the increased click probability comes with a substantial non-Clifford synthesis cost, while an early fault-tolerant implementation based on $T$ gates and STAR rotations reduces this overhead at the expense of additional encoded resources. We quantify these costs using both the Steane and $[[8,3,2]]$ codes, providing explicit resource estimates for implementing the optimal detector across different computational regimes.
Abstract:The three-tangle, $\tau_3$, is a measure of genuine tripartite entanglement that distinguishes GHZ-type from W-type entanglement in pure three-qubit states. Here, we identify the statistically optimal one-sided four-copy detector of the three-tangle and study its implementation cost. The optimal detector has click probability $\tau_3^2/48$, while previous four-copy constructions achieve $\tau_3^2/256$ and therefore require asymptotically $16/3$ times as many experimental repetitions. Using the representation-theoretic structure of the problem, we construct an ancilla-free implementation of the optimal detector using 15 CNOT gates, together with an 11-CNOT detector with click probability $\tau_3^2/96$. These results reveal a tradeoff between statistical efficiency and implementation cost, since reducing the number of repetitions requires increasingly expensive circuits. We then show that this tradeoff persists once the circuits are encoded. In a fully fault-tolerant setting, the increased click probability comes with a substantial non-Clifford synthesis cost, while an early fault-tolerant implementation based on $T$ gates and STAR rotations reduces this overhead at the expense of additional encoded resources. We quantify these costs using both the Steane and $[[8,3,2]]$ codes, providing explicit resource estimates for implementing the optimal detector across different computational regimes.
arXiv:2610.27260 [quant-ph]
(or arXiv:2610.27260v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2610.27260 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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