We study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase $\mathrm{\overline{S}}$ gate in the rotated surface code and a depth-one intra-block $\mathrm{\overline{CZ}}$ gate in the 2D-toric code that generalizes to all odd distances and all lengths $L\ge3$ , respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters $[[n,\sqrt{n},Θ({n^β})]]$ where $β\approx 0.2823$ in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted $\mathrm{\overline{S}}$ , $\mathrm{\overline{SHS}} = \sqrt{X}$ , and $\mathrm{\overline{CZ}}$ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters $[[n_0, 2, d_0]]$ , and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.We study the structure and transversal logical capabilities of stabilizer quantum error correcting codes. Among our results, we identify universal lower bounds on circuit depth to generate a full logical Clifford algebra, and develop novel constructions of logical transversal gates including a new depth-one transversal phase $\mathrm{\overline{S}}$ gate in the rotated surface code and a depth-one intra-block $\mathrm{\overline{CZ}}$ gate in the 2D-toric code that generalizes to all odd distances and all lengths $L\ge3$, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters $[[n,\sqrt{n},Θ({n^β})]]$ where $β\approx 0.2823$ in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted $\mathrm{\overline{S}}$, $\mathrm{\overline{SHS}} = \sqrt{X}$, and $\mathrm{\overline{CZ}}$ gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters $[[n_0, 2, d_0]]$, and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.

quant-pharXiv:2606.13521Jun 12, 20260 citations

From Syndromes to Computation: A General Framework for Logical Operations in Quantum Error-Correcting Codes

Hunter Nelson, Arshag Danageozian, ChunJun Cao +2 more

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 $\lceil d/2\rceil$ 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 ${}^r$ Z 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.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 $\lceil d/2\rceil$ 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${}^r$Z 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.

quant-pharXiv:2610.17612Oct 14, 20260 citations

Quantum logic gates

Александр Алексеевич Горбацевич, Н. М. Шубин

Uspekhi Fizicheskih NaukDec 1, 20173 citations

Efficiently estimating failure rates of fault-tolerant logical non-Clifford blocks

Julio C. Magdalena de la Fuente, Thomas R. Scruby

Useful quantum computation requires the fault-tolerant implementations of a universal gate set which are generically not efficiently simulable. We devise an algebraic framework that allows for efficient sampling from the measurement distribution of fault-tolerant circuits that implement diagonal logic gates in the third level of the Clifford hierarchy. Upon successful sampling we also provide sufficient conditions for decoding success. The resulting estimate for the logical failure rate is an overestimate, which becomes more accurate for blocks that are fault tolerant against arbitrary local errors. The non-Clifford simulation overhead is independent of the number of logical qubits, making the method particularly attractive for large-scale simulations. The framework is based on viewing the non-Clifford gates in the circuit as a cohomology invariant of an underlying spacetime fault complex. The method can also be interfaced with Clifford simulators to simulate larger fault-tolerant circuits and algorithmic subroutines.Useful quantum computation requires the fault-tolerant implementations of a universal gate set which are generically not efficiently simulable. We devise an algebraic framework that allows for efficient sampling from the measurement distribution of fault-tolerant circuits that implement diagonal logic gates in the third level of the Clifford hierarchy. Upon successful sampling we also provide sufficient conditions for decoding success. The resulting estimate for the logical failure rate is an overestimate, which becomes more accurate for blocks that are fault tolerant against arbitrary local errors. The non-Clifford simulation overhead is independent of the number of logical qubits, making the method particularly attractive for large-scale simulations. The framework is based on viewing the non-Clifford gates in the circuit as a cohomology invariant of an underlying spacetime fault complex. The method can also be interfaced with Clifford simulators to simulate larger fault-tolerant circuits and algorithmic subroutines.

quant-pharXiv:2609.19485Sep 18, 20260 citations

\mathrm{\overline{S}}

gate in the rotated surface code and a depth-one intra-block

\mathrm{\overline{CZ}}

gate in the 2D-toric code that generalizes to all odd distances and all lengths

L\ge3

, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters

[[n,\sqrt{n},Θ({n^β})]]

where

β\approx 0.2823

in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture (ISA) composed of all individually targeted

\mathrm{\overline{S}}

\mathrm{\overline{SHS}} = \sqrt{X}

, and

\mathrm{\overline{CZ}}

gates. This ISA is depth-one for certain subfamilies that we design and generally constant-depth under certain conditions. The code family is built from a small code with parameters

[[n_0, 2, d_0]]

, and is tunable in the standard way: it tiles out to form utility-scale logical qubit counts, and it scales up through concatenation to achieve higher distances and error suppression. We show that this construction preserves the depth-one complete transversal logical Clifford basis ISA when composed with these commuting construction actions, inheriting structure from the core codes so that at scale the complete logical Clifford basis ISA remains depth-one up to depth-two addressable operations between tiled cores. We call these Quantum Logic Codes.

\lceil d/2\rceil

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

{}^r

Z 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.