A key appeal of quantum low-density parity check (qLDPC) codes is their ability to suppress stochastic Pauli noise below nonzero thresholds. Coherent errors are fundamentally different: they produce superpositions of error patterns whose amplitudes can interfere even after syndrome measurement. Rigorous understanding of coherent errors remains limited. Here we show that general qLDPC codes admit a nonzero code capacity threshold against local coherent noise and more generally local channel noise. For any family of qLDPC codes with distance
d=Ω(logn), we show that there is a constant noise strength below which the logical recovery error in diamond distance decays exponentially with the code distance. The result is established for optimal recovery as well as the minimum-weight decoder. The key technical ingredient is what we call a \emph{cluster resummation}: rather than bounding superposed error configurations one by one, we isolate a large connected error cluster in the channel expansion and exactly resum all errors disconnected from it before taking norms. Standard cluster counting then yields exponential suppression. This work resolves a longstanding challenge in fault tolerance theory, providing general robustness guarantees for qLDPC codes against coherent noise and laying a rigorous foundation for future studies of fault-tolerant quantum technologies.
Threshold analyses of quantum error-correcting codes are well established for stochastic error models, in which errors occur randomly with given probabilities. However, errors in actual devices can also be coherent, such as unwanted
Z rotations due to imperfect control, which are not captured by stochastic error models. For the surface code, numerical studies have indicated threshold behavior even under coherent errors, but a rigorous proof of threshold existence is lacking. Here we prove that a positive threshold for coherent
Z-rotation errors exists for quantum low-density parity-check codes with a bounded number of logical qubits, including the surface code and other topological codes. Specifically, we show that the maximum-likelihood Pauli recovery suppresses the entanglement infidelity exponentially in the code distance up to a prefactor linear in the number of physical qubits whenever the rotation angles lie below a constant that is independent of the code size. The proof combines Fourier analysis to retain the interference among the amplitudes of coherent errors with the cluster expansion of abstract polymer models. Our results expand the theoretical foundation of quantum error correction and offer a statistical-mechanical description of quantum error correction beyond stochastic errors.
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⌉ for distance-
d 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
T gate on the
[[16,4,4]] BB code,
T-type rotations on the
[[64,8,8]] BB code, and CCZ-, CS-, and
T-type operations on the
[[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 C
rZ 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.
In this work, we analyze a framework for constructing fault-tolerant measurement schedules of varying lengths by combining stabilizer generators, and prove results about the distance of such schedules by combining according to classical codes. Using this framework, we produce explicit measurement schedules sufficient for fault-tolerant error correction of quantum codes of distance
d with
r independent stabilizer generators using only
O(dlogr) measurements if the code is LDPC, and
O(dlogdlogr) measurements if the code is produced via concatenating a smaller code with itself
O(logd) times. In both of these cases the number of measurements can be asymptotically fewer than the number of stabilizer generators which define the code. Although optimizing our construction to use the fewest measurements produces high-weight stabilizers, we also show that we can reduce the number of measurements used for specific examples while maintaining low-weight stabilizer measurements. We numerically examine the performance of our construction on the surface code under several noise models and demonstrate the exponential error suppression with increasing distance which is characteristic of weak fault tolerance.
Non-Clifford gates are essential for universal quantum computation, yet implementing them fault-tolerantly remains a central challenge for stabilizer codes. Here, we show how a syndrome degree of freedom can mediate a logical non-Clifford gate. Releasing one stabilizer check makes an additional logical qubit available within the encoded data block. Two Pauli rotations, followed by syndrome measurement and Clifford feed-forward, then implement a deterministic logical
T gate on every stabilizer code of distance at least two, in a suitable logical basis. During the ideal rotations, the state remains in an intermediate stabilizer code whose distance we determine exactly. For pure codes with a balanced factorization, this distance grows with the original code distance, whereas the maximum weight of the stabilizer generators bounds the intermediate distance from above. We realize the mechanism in two circuits that tolerate a single fault under local stochastic circuit noise: a fixed 22-qubit construction admitting recursive error suppression and a direct Golay-code gate protected by measuring a stabilizer check transported through the non-Clifford rotation, which can recover the unknown encoded state even after rejection. Serial implementations with ancilla reuse, reset, and flexible two-qubit connectivity require at most 33 and 32 physical qubits, respectively. These results establish a general mechanism for logical non-Clifford gates and demonstrate how syndrome measurements and recovery can protect the intermediate evolution of encoded information.
Quantum low-density parity-check (LDPC) codes are an important class of quantum error correcting codes. In such codes, each qubit only affects a constant number of syndrome bits, and each syndrome bit only relies on some constant number of qubits. Constructing quantum LDPC codes is challenging. It is an open problem to understand if there exist good quantum LDPC codes, i.e. with constant rate and relative distance. Furthermore, techniques to perform fault-tolerant gates are poorly understood. We present a unified way to address these problems. Our main results are a) a bound on the distance, b) a bound on the code dimension and c) limitations on certain fault-tolerant gates that can be applied to quantum LDPC codes. All three of these bounds are cast as a function of the graph separator of the connectivity graph representation of the quantum code. We find that unless the connectivity graph contains an expander, the code is severely limited. This implies a necessary, but not sufficient, condition to construct good codes. This is the first bound that studies the limitations of quantum LDPC codes that does not rely on locality. As an application, we present novel bounds on quantum LDPC codes associated with local graphs in D -dimensional hyperbolic space.