Quantum Logic Codes: Complete Transversal Logical Clifford Instruction Sets for High-Rate Stabilizer Quantum Error Correcting Codes

Abstract: 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.

• The paper introduces a complete transversal Clifford instruction set architecture enabling constant-depth logical operations in high-rate CSS quantum codes.
• It establishes rigorous lower bounds on circuit depth using Pauli spread and symplectic entropy metrics for synthesizing the full logical Clifford group.
• Explicit transversal gate constructions, such as depth-1 S and CZ gates, are validated via SAT-solver searches, ensuring fault-tolerance and code distance preservation.

Quantum Logic Codes: Complete Transversal Logical Clifford Instruction Sets for High-Rate Stabilizer Quantum Error Correcting Codes

Introduction and Framework

The paper "Quantum Logic Codes: Complete Transversal Logical Clifford Instruction Sets for High-Rate Stabilizer Quantum Error Correcting Codes" (2606.13521) addresses the construction and characterization of transversal Clifford logic in high-rate stabilizer quantum error correcting codes, with a focus on CSS codes. Stabilizer codes encode logical qubits into physical qubit blocks, enabling robust quantum computation by fault-tolerant error correction. The work formulates general lower bounds on circuit depth for synthesizing the full logical Clifford group, and develops a systematic theory of codespace-preserving transversal gates, including multi-layer synthesis protocols and code automorphism techniques. A primary objective is to realize a complete, constant-depth transversal instruction set architecture (ISA) for logical Clifford operations in high-rate, non-LDPC CSS code families.

Circuit Depth Lower Bounds for Clifford Synthesis

A rigorous analysis is presented regarding universal lower bounds on the circuit depth required to synthesize the full logical Clifford group in stabilizer codes using codespace-preserving $W$-local transversal gate layers. The dominant terms controlling these bounds arise from (i) information transfer limitations—expressed as the growth in Pauli operator weight required for logical Clifford transformations, and (ii) Clifford group entropy—accounting for the logarithmic scaling in the size of the projective symplectic group $|\text{Sp}(2k, \mathbb{F}_2)|$ relative to the space of available physical gate layers.

For codes with $k = O(1)$, the information spread term dominates. For codes with $k = \Theta(n)$, the entropy term supersedes. The precise lower bound is:

$$D \geq \max \left\{ \left\lceil \log_W \frac{w}{d} \right\rceil, \left\lceil \log_{N_{n,W}} \frac{|\text{Sp}(2k, \mathbb{F}_2)|}{|\rho(\text{Aut}_\text{perm}(Q))|} \right\rceil \right\}$$

where $w$ is the Pauli radius, $d$ the code distance, $N_{n,W}$ the number of codespace-preserving $W$-local gate layers, and $|\rho(\text{Aut}_\text{perm}(Q))|$ the logical automorphism group action.

For Quantum Reed-Muller codes and related high-rate families, these scaling laws generalize earlier results (e.g., Tansuwannont et al.) that connect logical density to transversal circuit depth.

Transversality Theory: Logical Reach and Gate Layer Characterization

The paper develops a comprehensive theory of logical reach in CSS codes under codespace-preserving $|\text{Sp}(2k, \mathbb{F}_2)|$0-local Clifford gates. The logical actions possible within single transversal layers are constrained, and the reach is characterized by (i) the symplectic algebra of per-block gates and (ii) code automorphism-induced basis changes.

Key structural results include:

• 1-local gates: Logical reach restricted to diagonal gates $|\text{Sp}(2k, \mathbb{F}_2)|$1, $|\text{Sp}(2k, \mathbb{F}_2)|$2; entanglement (e.g., logical CZ) is permitted for certain codes.
• 2-local gates: The reach expands to phase/CZ sets composed of symmetric blocks, bounded by support and confinement hypotheses; no global sector exchange or arbitrary basis changes are realizable in a single layer.
• W-local gates ($|\text{Sp}(2k, \mathbb{F}_2)|$3): Increasing $|\text{Sp}(2k, \mathbb{F}_2)|$4 allows more physical circuits to target the same logical operation set but does not enlarge logical reach beyond the 2-local case, except for $|\text{Sp}(2k, \mathbb{F}_2)|$5.
• Automorphism augmentation: Logical reach can be expanded via permutation automorphisms, yielding a closure under basis changes in $|\text{Sp}(2k, \mathbb{F}_2)|$6.
• Multi-layer closure: Sequential composition of 2-local gate layers generates the full logical Clifford group, i.e., $|\text{Sp}(2k, \mathbb{F}_2)|$7.

The formulation is rigorous, relying on symplectic matrix block structure, stabilizer preservation, and explicit conditions for addressability and codespace preservation.

Novel Transversal Gate Constructions

Two explicit transversal Clifford gate constructions are introduced:

1. Rotated Surface Code Transversal S Gate: A depth-1 2-local diagonal circuit, composed solely of $|\text{Sp}(2k, \mathbb{F}_2)|$8 and $|\text{Sp}(2k, \mathbb{F}_2)|$9 gates, acts transversally on data qubits, realizing the logical $k = O(1)$0 gate for all odd code distances $k = O(1)$1. The construction is CSS-preserving, distance-preserving, and single-fault-tolerant for $k = O(1)$2. This gadget generalizes transversal Clifford gates for non-folded surface code architectures.
2. 2D-Toric Code Intra-block CZ Gate: A depth-1 2-local circuit enables an intra-block logical $k = O(1)$3 gate in the $k = O(1)$4 toric code for $k = O(1)$5. The logical reach map is validated via explicit combinatorial constructions, and the circuit preserves the code distance and stabilizer structure.

Both constructions were validated via SAT-solver search and symbolic computation.

Quantum Logic Codes: High-Rate CSS Family with Complete Clifford ISA

A family of high-rate non-LDPC CSS codes, termed Quantum Logic Codes, is constructed. These codes possess a complete transversal logical Clifford basis ISA: every generator ($k = O(1)$6, $k = O(1)$7, $k = O(1)$8, $k = O(1)$9, code automorphisms) is realized via codespace-preserving, constant-depth, W-local circuits. The code parameters scale as $k = \Theta(n)$0 with $k = \Theta(n)$1 (e.g., $k = \Theta(n)$2 for certain tiling/concatenation strategies).

Construction involves:

• Self-dual core tiling: Replicating small group-algebra CSS codes increases $k = \Theta(n)$3 at fixed $k = \Theta(n)$4.
• Concatenation with inner doubly-even codes: Using Steane-type $k = \Theta(n)$5 codes multiplies the distance without increasing circuit depth.
• Automorphism augmentations: Logical code automorphisms and inter-block aggregate CZs are included as depth-1 generators.

The compositional operations (tiling, concatenation) are proven to preserve both code distance and the circuit depth of logical Clifford gate layers. The per-generator depth is constant, independent of code scaling parameters.

Strong Numerical Results and Claims

• Complete transversal Clifford ISA: For two of three core codes, every logical Clifford group generator is realized at depth-1; for the third, depth-2 is optimal via SAT-infeasibility.
• Fault tolerance: Rotated surface code S gate and 2D-toric code CZ gate constructions are single-fault-tolerant and distance-preserving.
• Scaling parameters: The family achieves $k = \Theta(n)$6 (asymptotic form), with rate $k = \Theta(n)$7 tunable via composition.
• Instruction set completeness: Any logical Clifford operation on $k = \Theta(n)$8 qubits is synthesized in $k = \Theta(n)$9 circuit layers (constant depth per generator), performing full symplectic group operations.

Practical and Theoretical Implications

The results have high impact for designing scalable, fault-tolerant quantum computing architectures:

• Reduction of gate overhead: Complete transversal ISAs enable logical operations without relying on ancilla-based gadgets, reducing quantum circuit complexity and latency.
• Hardware-friendly constructions: Depth-1 and depth-2 transversal gates can be implemented efficiently on architectures with parallel/qubit locality constraints.
• High-rate code applicability: Non-LDPC, high-rate codes with constant-depth logical ISAs are candidates for quantum LDPC applications where logical operation accessibility is typically a bottleneck.
• System design: The compositional nature of the Quantum Logic Codes offers systematic approaches to building utility-scale systems with full transversal logic, directly supporting maximal concurrency in logical qubit manipulation.

Future Directions

These results suggest avenues for:

• Extending transversal logic to higher levels of the Clifford hierarchy and non-Clifford gates.
• Generalizing the compositional framework to subsystem codes and hypergraph product codes.
• Optimized hardware mapping of transversal ISAs for atomic, superconducting, and trapped ion systems.
• Numerical classification of codes with optimal transversal logical circuits via SAT-based and algebraic methods.

Conclusion

This work advances the theoretical foundation and practical synthesis of transversal Clifford gate sets in high-rate stabilizer codes, culminating in the construction of Quantum Logic Codes—CSS codes with complete, constant-depth transversal logical Clifford ISAs (2606.13521). The analytical framework rigorously quantifies circuit depth lower bounds, characterizes logical reach of transversal gates, and demonstrates explicit constructions of novel logical Clifford gadgets. The compositional construction paradigms and depth-efficient circuit design directly contribute to scalable, fault-tolerant quantum computation, supporting robust logical unit architectures for future quantum processors.