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arXiv:2610.27260

Optimal four-copy detection of the three-tangle and its fault-tolerant cost

Federico Tomás B. Pérez, Berta Casas, I-Chi Chen, Su Yeon Chang, Philipp Schleich, M. Cerezo

quant-phSubmitted Oct 19, 2026• Announced Oct 21, 2026• 0 citations
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Abstract

The three-tangle, $τ_3$, 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 $τ_3^2/48$, while previous four-copy constructions achieve $τ_3^2/256$ 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 $τ_3^2/96$. 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.

Comments
8+12 pages, 3+1 figures, 2 tables

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To achieve large-scale fault-tolerant quantum computation, it may be easier to combine many small sets of qubits than to construct a single large set. For example via quantum error correction with block codes, or distributed quantum processors utilizing shared entanglement. In these regimes, the time or error budget of the overall quantum computation may be dominated by non-local operations. Hence, it is worthwhile to minimize the number of these operations. We consider the case where both non-local and local connectivity may be arbitrarily restricted, and give an asymptotically optimal synthesis method for distributed CNOT and Clifford circuits, based on block-matrix Gaussian elimination. We extend this to all Clifford+RZ circuits by generalizing the Pauli exponential circuit representation; this naturally integrates with existing methods for optimizing T-count. As an application, we show how to implement CNOT circuits in a CSS code encoding n logical qubits in k blocks using O(nk) inter-block transversal CNOTs and intra-block Pauli measurements.
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Fault-tolerant (FT) quantum computation offers a pathway to quantum advantage, but its realization is hindered by its significant overhead. The partially FT "STAR" architecture reduces the cost of universal computation at the price of relaxing FT requirements for non-Clifford operations, increasing exposure to hardware noise. Therefore, maximizing its performance and applicability demands careful co-design with the underlying hardware. In this work we investigate the co-design of the STAR architecture with neutral-atom platforms, connecting the underlying atomic physics to logical error modeling and application-level performance. At the physical level, we identify a favorable noise bias in 171Yb^{171}\mathrm{Yb}171Yb Rydberg gates, and show how it can be enhanced through error-selective pulse optimization. At the logical level, we identify an intrinsic error suppression mechanism, yielding vanishing logical coherent noise in leading order. Detailed circuit-level simulations of surface code implementations show more than an order of magnitude reduction in logical error relative to previous estimates. As a concrete target application, we focus on the Fermi-Hubbard model with periodic boundaries, for which we design a simulation circuit particularly suited to early implementations. We quantify the resource tradeoff of partially and fully FT approaches to gate synthesis, with the former showing approximately an order of magnitude reduction in state-injection depth, and an even larger reduction in Clifford resources. Comparing with classical and fully FT approaches, we identify regimes where partial fault tolerance provides both lower overheads and lower error rates. In our estimates, this is accessible with moderate error rates p∼10−3p\sim 10^{-3}p∼10−3 and a few thousand qubits, underlining that for these types of problems, partial FT approaches provide a compelling pathway to early quantum advantage.