Small high-rate non-CSS stabilizer codes provide compact platforms for encoded quantum computation, but mixed-Pauli checks and limited native transversal logical gates complicate fault-tolerant dynamics. Block-level constructions offer an alternative by mapping an entire logical block to a physical circuit rather than compiling separately protected logical gates. We investigate this approach using the high-rate [[8,3,3]] non-CSS code and logical Trotter circuits as a testbed. We construct flagged syndrome-extraction circuits and establish a circuit-level memory pseudo-threshold near (1.5\times10^{-3}). We then apply our symplectic-transvection construction, which maps a logical Trotter circuit to a physical circuit with the same block pattern for any stabilizer code. Although this mapping preserves the intended unitary algebraically, encoded Trotter circuits exhibit asymmetry between logical-(X) and logical-(Z) failure channels. Single-fault analysis identifies the mechanism: a fault on the shared parity ancilla can propagate through the uncomputation network into an undetectable logical operator, reducing the effective circuit distance in the affected sector. We evaluate flag-conditioned recovery, biased-noise decoding, CliNR resource verification, flag postselection, and asymmetric gate-noise models. These methods suppress propagated faults but do not simultaneously suppress both logical sectors in the realistic configurations studied. A diagnostic protected limit removing the identified malignant first-order locations restores pseudo-threshold behavior in both sectors, approaching memory performance. These results demonstrate the potential of block-level logical constructions for non-CSS codes without rich native transversal gate sets and the joint protection of the parity network, analog rotation, and recovery required to preserve fault-tolerant distance.
The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group
G and two subsets of
G whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find
[[60,4,10]],[[36,6,6]],[[48,8,6]], and
[[85,8,9]] codes, all of which are not CSS; we also find several weight-7 codes with
kd>n.
Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of
0.2%, approximately matching that of the
[[144,12,12]] BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.
Conventional approaches to fault-tolerant quantum computing realize logical circuits gate-by-gate, synthesizing each gate independently on one or more code blocks. This incurs excess overhead and doesn't leverage common structures in quantum algorithms. In contrast, we propose a framework that enables the execution of entire logical circuit blocks at once, preserving their global structure. This whole-block approach allows for the direct implementation of logical Trotter circuits—of arbitrary rotation angles—on any stabilizer code, providing a powerful new method for fault tolerant Hamiltonian simulation within a single code block. At the heart of our approach lies a deep structural correspondence between symplectic transvections and Trotter circuits. This connection enables both logical and physical circuits to share the Trotter structure while preserving stabilizer centralization and circuit symmetry even in the presence of non-Clifford rotations. We discuss potential approaches to fault tolerance via biased noise and code concatenation. While we illustrate the key principles using a [[8], [3], [3]] code, our simulations show that the framework applies to Hamiltonian simulation on even good quantum LDPC codes. These results open the door to new algorithm-tailored, block-level strategies for fault tolerant circuit design, especially in quantum simulation.
Topological codes have many desirable properties that allow fault-tolerant quantum computation with relatively low overhead. A core challenge for these codes, however, is to achieve a low-overhead universal gate set with limited connectivity. In this work, we explore a non-Pauli stabilizer code that can be used to complete a universal gate set on topological toric and surface codes in strictly two dimensions. Fault-tolerant syndrome extraction for the non-Pauli code requires mid-circuit
X corrections, a key difference to conventional Pauli codes. We construct and benchmark a just-in-time (JIT) matching decoder to reliably decide these corrections. Under a phenomenological error model with equally likely physical and measurement errors, we find a high threshold of
≈2.5%, close to the
≈2.9% of a decoder with access to the full syndrome history. We also perform a finite-size scaling analysis to estimate how the logical error rate scales below threshold and verify an exponential suppression in both physical error rate and in the system size. A second global decoding step for
Z errors is required and the non-Clifford gates in the circuit reduce the threshold from
≈2.9% to
≈1.8% with a naive decoder. We show how
Z decoding can be improved using knowledge of the
X corrections, pushing the threshold to
≈2.2%. Our results suggest non-Clifford logic in 2D codes could perform comparably to 2D quantum memory. Our formalism for efficient benchmarking and decoding directly generalizes to a broader family of CSS codes whose
X stabilizers are twisted by diagonal Clifford operators, and spacetime versions thereof, defined by CSS-like circuits enriched by
CCZ,
CS, and
T gates.
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.
We develop an operator-algebraic framework for analyzing fault tolerance in quantum circuits and establish necessary and sufficient conditions for exact recovery of the ideal computation under a prescribed noise model. For prescribed circuit families and noise models, we derive a semidefinite program (SDP) based test whose infeasibility certifies that there does not exist any recovery map restoring its intended operation. For non-adaptive stabilizer circuits, analyzing the stabilizer symmetries of its components with symplectic geometry, we obtain closed-form detection and logical-effect matrices that characterize circuit distance algebraically. Applied to a finite Hastings--Haah honeycomb Floquet-code circuit, these matrices certify the dynamical encoding of two logical qubits and distance four under Pauli noise including measurement errors. Using these matrices, we then derive MacWilliams identities for circuit weight enumerators and obtain linear-programming (LP) upper bounds on circuit distance. Output-code constraints give additional Singleton-like bounds. Under independent and identically distributed (i.i.d.) depolarizing noise, the weight enumerators determine decoder failure probabilities and yield upper bounds on circuit pseudothresholds. Finally, we derive LP distance bounds for families of flag syndrome-extraction circuits specified by their CNOT orderings, by incorporating the
t-flag criterion as additional linear constraints to the distance-bounding LP. An analytical CSS Hamming-code benchmark yields a tight distance-three bound attained by the Chao--Reichardt one-flag construction.
We present a fault-tolerant [[8, 1, 3]] non-CSS quantum error correcting code and study its logical error rates. We modify Gottesman's encoding procedure for stabilizer codes to suit a class of non-CSS quantum codes. We adopt the bare ancilla method presented by Brown et al. to reorder the measurement sequence in the syndrome extraction step and upgrade it to obtain higher pseudo-thresholds and lower leading order terms of logical error rates under the standard depolarizing and anisotropic noise models.