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arXiv:2410.14891 · Quantum

LUCI in the Surface Code with Dropouts

Dripto M. Debroy, Matt McEwen, Craig Gidney, Noah Shutty, Adam Zalcman

Published Dec 11, 2025• 9 citations
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Abstract

Recently, usage of detecting regions facilitated the discovery of new circuits for fault-tolerantly implementing the surface code. Building on these ideas, we present LUCI, a framework for constructing fault-tolerant circuits flexible enough to construct aperiodic and anisotropic circuits, making it a clear step towards quantum error correction beyond static codes. We show that LUCI can be used to adapt surface code circuits to lattices with imperfect qubit and coupler yield, a key challenge for fault-tolerant quantum computers using solid-state architectures. These circuits preserve spacelike distance for isolated broken couplers or isolated broken measure qubits in exchange for halving timelike distance, substantially reducing the penalty for dropout compared to the state of the art and creating opportunities in device architecture design. For qubit and coupler dropout rates of 1% and a patch diameter of 15, LUCI achieves an average spacelike distance of 13.1, compared to 9.1 for the best method in the literature. For a SI1000(0.001) circuit noise model, this translates to a 36x improvement in median logical error rate per round, a factor which increases with device performance. At these dropout and error rates, LUCI requires roughly 25% fewer physical qubits to reach algorithmically relevant one-in-a-trillion logical codeblock error rates.

DOI
10.22331/q-2025-12-11-1936

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Recently, usage of detecting regions facilitated the discovery of new circuits for fault-tolerantly implementing the surface code. Building on these ideas, we present LUCI, a framework for constructing fault-tolerant circuits flexible enough to construct aperiodic and anisotropic circuits, making it a clear step towards quantum error correction beyond static codes. We show that LUCI can be used to adapt surface code circuits to lattices with imperfect qubit and coupler yield, a key challenge for fault-tolerant quantum computers using solid-state architectures. These circuits preserve spacelike distance for isolated broken couplers or isolated broken measure qubits in exchange for halving timelike distance, substantially reducing the penalty for dropout compared to the state of the art and creating opportunities in device architecture design. For qubit and coupler dropout rates of 1% and a patch diameter of 15, LUCI achieves an average spacelike distance of 13.1, compared to 9.1 for the best method in the literature. For a SI1000(0.001) circuit noise model, this translates to a 36x improvement in median logical error rate per round, a factor which increases with device performance. At these dropout and error rates, LUCI requires roughly 25% fewer physical qubits to reach algorithmically relevant one-in-a-trillion logical codeblock error rates.
Long-lived logical qubits are essential for fault-tolerant quantum computation. However, the practical performance of traditional error correction protocols relies on performing specific syndrome circuits, causing vulnerability to hardware defects and imposing rigid connectivity constraints. Recent theoretical findings have proposed that flexible subroutine circuits within the LUCI framework can maintain space-time distance in the presence of isolated or broken components, albeit at the expense of temporal distance. However, these approaches have solely targeted defect avoidance and have not yet been demonstrated to suppress errors with reduced temporal distances on physical hardware. In this work, we propose a reset-free scenario for the LUCI framework and experimentally benchmark it on IBM quantum hardware. By asymmetrically scaling the XXX or ZZZ distance, we compare our reset-free approach against the standard surface code and successfully demonstrate error suppression ratios for targeted logical Pauli errors. Remarkably, despite a nearly halved syndrome density in time, which requires two subroutine rounds for full syndrome extraction, the LUCI framework remains competitive with the rotated surface code implementation. In the LUCI framework, we observe error suppression of 1.75(10)1.75(10)1.75(10) for logical XXX errors and 1.93(12)1.93(12)1.93(12) for logical ZZZ errors, whereas the standard approach yields $ 1.58(13)$ and 2.44(7)2.44(7)2.44(7), respectively. These results demonstrate that dynamic codes outperform standard methods by avoiding highly noisy components, even without physical defects, while preserving logical boundaries. Our findings challenge the conventional dependency on static fault-tolerant architectures by verifying the feasibility and efficacy of the LUCI framework on physical hardware and pave the way for hybrid, hardware-compatible code designs in quantum computing.
Recent work has shown that a hexagonal grid qubit layout, with only three couplers per qubit, is sufficient to implement the surface code with performance comparable to that of a traditional four-coupler layout [McEwen et al., 2023]. In this work we propose a method for handling broken qubits and couplers even in hex-grid surface code architectures, using an extension of the LUCI framework [Debroy et al., 2024]. We show that for isolated broken qubits, the circuit distance drops by one, while for isolated broken couplers, the distance drops by one in one or both bases. By providing a viable dropout strategy, we have removed a critical roadblock to the implementation of hexagonal qubit grids in hardware for large-scale quantum error correction.
Recent work has shown that fabrication defects can be well-handled using a strategy relying on the mid-error-correction-cycle state. In this work we present two improvements to the original prescription. First, we quantify the impact of the choice of a more complete set of gauge operators originally proposed for the hex-grid surface code on the standard square-grid surface code, as well as a new method for excising effectively unused qubits. Second, we leverage the expressivity of the LUCI framework as an intermediate representation, using integer linear programming to find performant physical circuits from the large space of valid LUCI circuits. We show that on the d=11d = 11d=11 surface code at 1%(3%)1\%(3\%)1%(3%) dropout rate for qubits and couplers, these optimizations allow for a total improvement of 14.5%(23.6%)14.5\%(23.6\%)14.5%(23.6%) over 4d4d4d round of syndrome extraction using the SI1000 noise model at 0.1%0.1\%0.1% noise.
Utility-scale solid-state quantum devices will need to fabricate quantum devices at scale using imperfect processes. By introducing tolerance to fabrication defects into the design of the quantum devices, we can improve the yield of usable quantum chips and lower the cost of useful systems. Automated Compilation Including Dropouts (ACID) is a framework that works in the ancilla-free (or `middle-out') paradigm, to generate syndrome extraction circuits for general stabiliser codes in the presence of defective couplers or qubits. In the ancilla-free paradigm, we do not designate particular qubits as measurement ancillas, instead measuring stabilisers using any of the data qubits in their support. This approach leads to a great deal of flexibility in how syndrome extraction circuits can be implemented. ACID works by constructing and solving an optimisation problem within the ancilla-free paradigm to find a short syndrome extraction circuit. Applied to the surface code, ACID produces syndrome-extraction circuits of depth between 1×1\times1× (no overhead) and 1.5×1.5\times1.5× relative to the depth of defect-free circuits. The LUCI algorithm, the best prior art, yielded a 2×2 \times2× overhead, so ACID offers a significant time saving. The yield of surface code chips with a logical error rate at most 10×10\times10× the dropout-free baseline is up to 3×3\times3× higher using ACID than using LUCI. I demonstrate the broad applicability of ACID by compiling syndrome extraction circuits for bivariate bicycle codes and the colour code. For these circuits, we incur a circuit-depth overhead of between 1×1\times1× (no overhead) and 2.5×2.5\times2.5× relative to defect-free circuits. I believe this work is the first to simulate both of these families of codes in the presence of fabrication defects.
Quantum computation holds the promise of solving certain complex problems exponentially faster than classical computers. However, the high prevalent noise in current quantum devices impedes the accurate execution of even basic algorithms. This can be remedied by protecting quantum information with a quantum error-correcting code, where the logical information of an algorithmic qubit is spread across multiple physical qubits. Individual quantum errors are then located and corrected by the fault-tolerant measurement of multi-qubit stabilizer operators (parity checks). Unfortunately, error correction and fault tolerance both impose large demands on the qubit overhead: hundreds to thousands of physical qubits per logical qubit. We reduce the spacetime cost of fault tolerance by redesigning key building blocks of an error-corrected quantum computer. First, we develop a combinatorial proof with flag fault tolerance that exponentially reduces the extra qubits needed to measure a stabilizer of any size, while tolerating one fault. We leverage these proofs to then design state preparation circuits for the Steane and Golay codes with 100% yield. Next, we improve error correction on a planar layout by showing that a distance-four code encoding six logical qubits protects information as well as the distance-five surface code, using one-tenth as many physical qubits. Finally, we optimize the time overhead of logical gates in surface code quantum computers by protecting measurement results with a classical code, cutting computation time by a factor of two to six. Our hardware-agnostic optimizations of fault tolerance overheads thus suggest new routes to advance the timeline of error-free quantum computing.
In fault-tolerant quantum computation, the preparation of logical states is a ubiquitous subroutine, yet significant challenges persist even for the simplest states required. In the present work, we present a unitary, scalable, distance-preserving encoding scheme for preparing Pauli eigenstates in surface codes. Unlike previous unitary approaches whose fault-distance remains constant with increasing code distance, our scheme ensures that the protection offered by the code is preserved during state preparation. Building on strategies discovered by reinforcement learning for the surface-17 code, we generalize the construction to arbitrary code distances and both rotated and unrotated surface codes. The proposed encoding relies only on geometrically local gates, and is therefore fully compatible with planar 2D qubit connectivity, and it achieves circuit depth scaling as O(d)\mathcal{O}(d)O(d), consistent with fundamental entanglement-generation bounds. We design explicit stabilizer-expanding circuits with and without ancilla-mediated connectivity and analyze their error-propagation behavior. Numerical simulations under depolarizing noise show that our unitary encoding without ancillas outperforms standard stabilizer-measurement-based schemes, reducing logical error rates by up to an order of magnitude. These results make the scheme particularly relevant for platforms such as trapped ions and neutral atoms, where measurements are costly relative to gates and idling noise is considerably weaker than gate noise. Our work bridges the gap between measurement-based and unitary encodings of surface-code states and opens new directions for distance-preserving state preparation in fault-tolerant quantum computation.