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 $d = 11$ surface code at $1\%(3\%)$ dropout rate for qubits and couplers, these optimizations allow for a total improvement of $14.5\%(23.6\%)$ over $4d$ round of syndrome extraction using the SI1000 noise model at $0.1\%$ noise.
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=11 surface code at 1%(3%) dropout rate for qubits and couplers, these optimizations allow for a total improvement of 14.5%(23.6%) over 4d round of syndrome extraction using the SI1000 noise model at 0.1% noise.
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.
Fault-tolerant Quantum Computing (FTQC) relies on Quantum Error Correction (QEC) codes that encode logical qubits across many physical qubits to detect and correct errors. The surface code is among the most widely studied codes due to its high error threshold, the existence of efficient decoders, and hardware-friendly properties: a planar, two-dimensional layout with nearest-neighbor connectivity. In practice, however, the fabrication of solid-state quantum processors introduces hardware defects, resulting in defective qubits and couplers that must be discarded. Adapting the surface code to these defects often requires measuring the X- and Z-type checks in separate rounds rather than simultaneously. In this work, we investigate the optimal X-to-Z checks round-scheduling ratio under biased noise systems. Our results characterize how key architectural parameters, such as noise bias, code distance, and defect rate, impact the logical error rate. We provide insights into how to determine the optimal scheduling ratio directly from device calibration data, enabling manufacturers to maximize performance without extensive simulations. Our approach reduces the logical error rate by up to 4.25× at a 1% defect rate and up to 8.46× at a 2% defect rate for a distance-13 surface code under moderately biased noise. Furthermore, we demonstrate that the benefits of round-scheduling extend beyond the defective-hardware setting. In biased-noise architectures subject to CNOT crosstalk, separating X and Z measurement rounds yields up to 4.5× reduction in logical error rate.
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× (no overhead) and 1.5× relative to the depth of defect-free circuits. The LUCI algorithm, the best prior art, yielded a 2× overhead, so ACID offers a significant time saving. The yield of surface code chips with a logical error rate at most 10× the dropout-free baseline is up to 3× 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× (no overhead) and 2.5× 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.
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.
We present a quantum code implementable on a regular 2D hex grid with an estimated encoding rate up to 4.5× of that of a rotated surface code patch using circuit-level noise in a one- and two-qubit 10−3 error uniform depolarizing model. Our approach is based on yoking a dense packing of surface code twist defects, enabled by new stabilizer measurement cycles with an optimal four layers of nearest-neighbor two-qubit gates, almost no distance-reducing hook errors, and efficient decoding. We demonstrate a space-efficient architecture for computing on densely packed logical qubits, including new padding-free lattice surgery protocols in an optimal bounding box of 2d2 data and measurement qubits per patch. Assuming a 1μs surface code cycle time and a 10μs reaction time, these developments enable chemically accurate ground state phase estimation of a broad class of `utility-scale' electronic structure simulation problems such as the 108 spin-orbital FeMoco-based nitrogen fixation catalyst in under a month with 89k noisy superconducting qubits. We elucidate a Pareto frontier of space-time trade-offs and find a minimum physical quantum volume of 1.3 mega-qubit-hours. These correspond to a 36× space and 6.6× spacetime improvement, respectively, over our previous state-of-the-art minimum-Toffoli resource estimates (Phys. Rev. X 15, 041016).
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 X or Z 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) for logical X errors and 1.93(12) for logical Z errors, whereas the standard approach yields $ 1.58(13)$ and 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.