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arXiv:2610.20591

Logical CCZ state cultivation

Yuta Hirasaki, Jong Yeon Lee

quant-phSubmitted Oct 14, 2026• Announced Oct 15, 2026• 0 citations
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Abstract

Logical CCZ states enable Toffoli gates, a central building block of fault-tolerant quantum arithmetic. We introduce an end-to-end protocol for direct CCZ-state preparation on bivariate tricycle and bicycle (BT/BB) codes. Our factory combines the cup-product CCZ gates native to BT codes with two novel ingredients, both enabled by engineering the logical coupling between BT and BB codes through projection-preserving lifts. Index-2 lifted BB ancillas enable selective $Z$-basis initialization, allowing the native cup-product gates to directly prepare two disjoint logical CCZ states. The prepared CCZ states are faithfully transferred to index-3 lifted BB memories with high code distance. We use decoder-based postselection to obtain high-fidelity output states with low overhead. We perform exact simulations of our factory under a circuit-level noise model motivated by neutral-atom hardware, obtaining joint infidelities of $1.84\times10^{-5}$ at $21.94\%$ acceptance and $4.10\times10^{-7}$ at $8.89\%$ acceptance for single-qubit, CZ, and CCZ error rates of $0.01\%$, $0.1\%$, and $0.5\%$, respectively. At the output-infidelity target of our smallest factory, its estimated spacetime cost per CCZ state is lower than that of $T$-state cultivation followed by CCZ conversion, providing an efficient route to logical CCZ states for fault-tolerant quantum computation.

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Logical CCZ states enable Toffoli gates, a central building block of fault-tolerant quantum arithmetic. We introduce an end-to-end protocol for direct CCZ-state preparation on bivariate tricycle and bicycle (BT/BB) codes. Our factory combines the cup-product CCZ gates native to BT codes with two novel ingredients, both enabled by engineering the logical coupling between BT and BB codes through projection-preserving lifts. Index-2 lifted BB ancillas enable selective ZZZ-basis initialization, allowing the native cup-product gates to directly prepare two disjoint logical CCZ states. The prepared CCZ states are faithfully transferred to index-3 lifted BB memories with high code distance. We use decoder-based postselection to obtain high-fidelity output states with low overhead. We perform exact simulations of our factory under a circuit-level noise model motivated by neutral-atom hardware, obtaining joint infidelities of 1.84×10−51.84\times10^{-5}1.84×10−5 at 21.94%21.94\%21.94% acceptance and 4.10×10−74.10\times10^{-7}4.10×10−7 at 8.89%8.89\%8.89% acceptance for single-qubit, CZ, and CCZ error rates of 0.01%0.01\%0.01%, 0.1%0.1\%0.1%, and 0.5%0.5\%0.5%, respectively. At the output-infidelity target of our smallest factory, its estimated spacetime cost per CCZ state is lower than that of TTT-state cultivation followed by CCZ conversion, providing an efficient route to logical CCZ states for fault-tolerant quantum computation.
The multi-controlled Toffoli (MCT) gate is a fundamental primitive in quantum circuit design, with applications in quantum arithmetic, cryptanalysis, and algorithmic implementations. Being a high-level logical operation, the efficient decomposition of MCT gates into lower-level netlists has remained a major optimization challenge for decades. While emerging quantum error-correcting codes such as bivariate bicycle (BB) codes drastically reduce fault-tolerance overhead, realizing non-Clifford circuits on modular BB-code architectures introduces complex compilation bottlenecks governed by inter-module routing, factory density, and layout. Consequently, the mapping of MCT gates onto BB-code architectures remains relatively unexplored. In this paper, we overcome these challenges by mapping optimal-Toffoli-depth MCT decompositions (Dutta et al., PRA, 2025) onto BB-code-based fault-tolerant architectures via direct ∣CCZ⟩\lvert \mathrm{CCZ} \rangle∣CCZ⟩ state injection from an external magic state factory. We introduce a targeted placement strategy that exploits the binary-tree structure of MCT decompositions to co-locate interacting subtrees. This approach reduces inter-module instruction counts by up to 16.02%\mathbf{16.02}\%16.02% compared to a naive sequential first-fit placement. We also evaluate the impact of factory placement across different topologies, demonstrating that grid-based layouts yield up to a 23.7%\mathbf{23.7}\%23.7% reduction in inter-module instructions relative to linear architectures (Yoder et al., arXiv, 2025). Finally, we validate the practical viability of our compiled circuits by analyzing aggregate execution errors and logical failure probabilities using the bicycle-ISA error estimator bicycle_numerics provided by the Qiskit community, https://github.com/qiskit-community/bicycle-architecture-compiler.
We introduce transversal dimension jump, a code-switching protocol for lifted product (LP) quantum low-density parity-check (qLDPC) codes across different chain-complex dimensions, enabling universal fault-tolerant quantum computation with low overhead. The construction leverages the product structure of LP codes to implement one-way transversal CNOTs between a 3D code and its 2D component codes, enabling teleportation-based switching with geometrically nonlocal gates. Combined with constant-depth CCZ gates in 3D LP codes and low-overhead transversal Clifford gates in 2D LP codes, this yields universal, high-rate quantum logical computation with high thresholds and low space-time costs. Beyond asymptotic schemes, we identify explicit 3D-2D LP code pairs supporting cup-product CCZ gates, including bivariate tricycle-bicycle families such as the [[81,3,5]][[81,3,5]][[81,3,5]]-[[54,2,6]][[54,2,6]][[54,2,6]] pair, where the 3D tricycle codes admit depth-2 CCZ, weight-6 stabilizers, and pseudo-thresholds ≳0.4%\gtrsim 0.4\%≳0.4%. As a byproduct, we show that the 3D codes enable highly efficient magic-state preparation: a single round of stabilizer measurements followed by depth-2 CCZ and postselection produces states with error <10−9<10^{-9}<10−9 and success probability ∼35%\sim 35\%∼35%. Our results establish a native integration of qLDPC codes with complementary transversal gates-covering nearly all practically relevant families known so far-and open a broad design space for scalable, low-overhead universal quantum computation.
Quantum low-density parity-check (qLDPC) codes can substantially reduce the qubit overhead of fault-tolerant quantum computation, but any useful architecture based on such codes also requires an efficient mechanism to implement non-Clifford gates in a fault-tolerant manner. We introduce a space-time-efficient magic state factory for high-rate qLDPC codes by combining the transversal non-Clifford gates of recently developed tricycle codes with single-shot four-dimensional balanced product quadcycle codes. The key ingredient is an efficient "escape" mechanism that teleports postselected logical magic states from tricycle codes to larger distance quadcycle codes. Under a standard circuit-level noise model at physical error rate 10−310^{-3}10−3, our factories produce CCZ\mathrm{CCZ}CCZ states at logical error rates as low as 10−910^{-9}10−9 with 10-40 times lower space-time cost than state-of-the-art magic state generation methods. The resulting states are directly encoded in high-rate blocks suitable for computation. We identify the quadcycle codes by an AI-assisted search and show that they can serve as excellent memories capable of single-shot error correction. Finally, we present a concrete implementation of the entire factory with highly efficient atom transport on reconfigurable neutral-atom quantum computers. Together, these results provide an end-to-end, practical protocol for producing and utilizing high-fidelity magic states within a high-rate qLDPC architecture, with the lowest space-time cost to date.
Magic State Distillation is considered to be one of the promising methods for supplying the non-Clifford resources required to achieve universal fault tolerance. Conventional MSD protocols implemented in surface codes often require multiple code blocks and lattice surgery rounds, resulting in substantial qubit overhead, especially at low target error rates. In this work, we present practical magic state distillation factories on Bivariate Bicycle (BB) codes that execute Pauli-measurement-based Clifford circuits inside a single BB code block. We formulate distillation circuit design as a joint optimization of logical qubit mapping, gate scheduling, measurement nativization, and protocol compression via qubit recycling. Based on detailed resource analysis and simulations, our BB factories have space-time volume comparable to that of leading distillation factories while delivering lower target error at a smaller qubit footprint, and are particularly compelling as second-round distillers following magic state cultivations.
Fault-tolerant quantum computation architectures are frequently bottlenecked by the overhead of producing high-fidelity magic states. In this work, we use algebraic geometric techniques to construct codes over binary extension fields F2s\mathbb{F}_{2^s}F2s​, thus discovering new protocols for the distillation of qubit magic states, where our focus is on the regime of practical qubit-based quantum computing architectures. To do this, we show that multi-qubit gates of interest such as CS\text{CS}CS, CCZ\text{CCZ}CCZ, and TOF#=CCZ123CCZ345\text{TOF}\# = \text{CCZ}_{123}\text{CCZ}_{345}TOF#=CCZ123​CCZ345​, can be packaged into simple gates over the larger fields, and we derive simple algebraic conditions in the extension fields allowing the distillation of these gates. Because they are derived from Galois qudits, the corresponding qubits codes naturally handle the correlated errors present on such multi-qubit states. Moreover, the protocols we discover are extremely compact; for example, we show that 4 CS\text{CS}CS states can be distilled to 1 CS\text{CS}CS state at distance 2, using only 4 logical qubits. For a case study, we consider the distillation of CS\text{CS}CS and CCZ\text{CCZ}CCZ states from injected T\text{T}T and CS\text{CS}CS states. When optimized for magic state production per unit time, or logical spacetime volume, we find that our protocols outperform the state-of-the-art in almost every situation, both at input error rates 10−310^{-3}10−3 (direct injection), and 10−610^{-6}10−6 (allowing some cultivation pre-injection).
Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer-level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general generator weight bound of ⌈d/2⌉\lceil d/2\rceil⌈d/2⌉ for distance-ddd codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle (BB) and non-CSS Mirror qLDPC codes, the framework gives explicit low-weight Clifford and controlled-phase gates. Our implementations yield new low-weight gates, including an addressed TTT gate on the [[16,4,4]][[16,4,4]][[16,4,4]] BB code, TTT-type rotations on the [[64,8,8]][[64,8,8]][[64,8,8]] BB code, and CCZ-, CS-, and TTT-type operations on the [[36,6,6]][[36,6,6]][[36,6,6]] Mirror code. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and provide a systematic route to logical addressability. We use this approach to analytically construct addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed--Muller codes yields Cr{}^rrZ gates that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code, the logical computations it can directly support, and the fault-tolerance properties of their physical realizations.