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arXiv:2608.20676

Continuous-angle logical rotations in the Steane code

Eric Huang, Daiwei Zhu, Matteo Ippoliti, Christopher Monroe, Michael J. Gullans

quant-phSubmitted Aug 21, 2026• Announced Aug 24, 2026• 0 citations
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Abstract

We experimentally demonstrate continuous-angle logical $Z$ rotations in the $[[7,1,3]]$ Steane code on the IonQ Forte trapped-ion processor. A round of the protocol applies a transversal physical $Z$ rotation by $θ$, followed by Steane syndrome extraction and decoding, which induces a syndrome-dependent logical $Z$ rotation. We analytically derive the effect of dephasing noise on the logical rotation angle and logical dephasing rate. Using logical Ramsey interferometry, we observe coherent syndrome-dependent logical rotations from a single round of the protocol. We find that the logical channel reconstructed from process tomography is a noisy logical $Z$ rotation well explained by a dephasing model. We further implement a two-round protocol applying physical rotations $+θ$ and $-θ$, and observe cancellation of the total logical angle with low logical dephasing for repeated trivial syndromes. This constitutes a proof-of-principle demonstration of continuously tunable non-Clifford logical gates by transversal rotations and standard error correction in a small quantum code.

Comments
12 pages, 14 figures

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We experimentally demonstrate continuous-angle logical ZZZ rotations in the [[7,1,3]][[7,1,3]][[7,1,3]] Steane code on the IonQ Forte trapped-ion processor. A round of the protocol applies a transversal physical ZZZ rotation by θθθ, followed by Steane syndrome extraction and decoding, which induces a syndrome-dependent logical ZZZ rotation. We analytically derive the effect of dephasing noise on the logical rotation angle and logical dephasing rate. Using logical Ramsey interferometry, we observe coherent syndrome-dependent logical rotations from a single round of the protocol. We find that the logical channel reconstructed from process tomography is a noisy logical ZZZ rotation well explained by a dephasing model. We further implement a two-round protocol applying physical rotations +θ+θ+θ and −θ-θ−θ, and observe cancellation of the total logical angle with low logical dephasing for repeated trivial syndromes. This constitutes a proof-of-principle demonstration of continuously tunable non-Clifford logical gates by transversal rotations and standard error correction in a small quantum code.
We propose hardware-efficient schemes for implementing logical h and s gates transversally on the rotated surface code with reconfigurable neutral atom arrays. Our protocol for logical s gates utilizes the time dynamics of the data and ancilla qubits during syndrome extraction (SE). In particular, we move beyond traditional schemes where transversal logical gates take place between two SE rounds and instead embed our fold-transversal logical operation within a single SE round, leveraging the fact that data and ancilla qubits can be morphed into an unrotated surface code state halfway through an SE round. Under circuit noise, we observe that the performance of our s gate protocol is comparable to that of the quantum memory. For logical h gates, we first develop a simple strategy to rotate code patches efficiently with two sets of 2D acousto-optic deflectors (2D-AOD). We also propose a new transversal h gate protocol, based on the same principle as our s gate protocol, to remove the need for an additional set of 2D-AOD. Together with transversal logical cnot gates, our protocols complete a transversal logical Clifford gate set on the rotated surface code and allow for efficient implementation on neutral atom array platforms.
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Quantum computing platforms based on arrays of tweezer-confined neutral atoms have recently emerged as a competitive modality thanks to a direct path toward high qubit count, rapidly advancing operation fidelities, and their ability to execute circuits with arbitrary qubit connectivity. These features will enable the use of efficient error correction schemes with high encoding-rates, time-efficient decoding, and resource-efficient architectures based on transversal gates. With these goals in mind, recent state of the art neutral atom demonstrations focus on the transition from the use of physical qubits to error-corrected logical qubits, but to date there has been no demonstration of repeated error correction scalable to arbitrary depth. Here, we demonstrate many cycles of syndrome extraction in a toric quantum error correcting code, using mid-circuit measurement and replacement of lost qubits, including reloading of a qubit reservoir for indefinite coherent operation. We characterize the logical error rate after up to 90 cycles, showing that logical information can be preserved through multiple rounds of qubit reloading. Comparing two distances of the code up to 8 rounds of syndrome extraction shows a lower absolute logical error rate for the larger distance code.
We implement logically encoded three-qubit circuits for the quantum Fourier transform (QFT), using the [[7,1,3]] Steane code, and benchmark the circuits on the Quantinuum H2-1 trapped-ion quantum computer. The circuits require multiple logical two-qubit gates, which are implemented transversally, as well as logical non-Clifford single-qubit rotations, which are performed by non-fault-tolerant state preparation followed by a teleportation gadget. First, we benchmark individual logical components using randomized benchmarking for the logical two-qubit gate, and a Ramsey-type experiment for the logical TTT gate. We then implement the full QFT circuit, using two different methods for performing a logical control-TTT, and benchmark the circuits by applying it to each basis state in a set of bases that is sufficient to lower bound the process fidelity. We compare the logical QFT benchmark results to predictions based on the logical component benchmarks.
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.