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arxiv: 2606.13521 · v1 · pith:SKMUCQPNnew · submitted 2026-06-11 · 🪐 quant-ph · math-ph· math.MP

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

Adam Holmes This is my paper

Pith reviewed 2026-06-27 06:34 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MP
keywords CSS codesstabilizer codestransversal gatesClifford gatesquantum error correctionhigh-rate codestoric codesurface code
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The pith

A high-rate family of CSS codes supports a complete set of constant-depth transversal logical Clifford gates.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a family of high-rate non-LDPC CSS quantum error-correcting codes that implement all basic logical Clifford operations through transversal gates at constant depth. The codes reach a rate scaling as one logical qubit per square root of physical qubits and a distance growing as a fractional power of n. A reader would care if the construction works because it could reduce the circuit depth required for logical operations in fault-tolerant quantum computing. The family is assembled by tiling and concatenating a small base code with two logical qubits, with the operations shown to preserve the transversal properties.

Core claim

We construct a high-rate non-LDPC CSS code family with parameters [[n, √n, Θ(n^β)]] where β ≈ 0.2823 in one demonstrated case, that provably possesses a constant-depth complete 2-local transversal logical Clifford basis instruction set architecture composed of all individually targeted S̄, SHS̄ = √X̄, and CZ̄ gates. This ISA is depth-one for certain subfamilies and generally constant-depth under certain conditions. The code family is built from a small code with parameters [[n0, 2, d0]] and is tunable through tiling to form utility-scale logical qubit counts and through concatenation to achieve higher distances, with the construction preserving the depth-one complete transversal logical Clif

What carries the argument

Tiling and concatenation of a base [[n0, 2, d0]] code that commutes with and preserves the complete depth-one transversal Clifford ISA.

If this is right

  • Tiling produces utility-scale numbers of logical qubits while the complete ISA remains depth-one for designed subfamilies.
  • Concatenation increases distance and error suppression while the ISA stays constant-depth up to depth-two operations between tiled cores.
  • The construction supplies a depth-one transversal phase gate in the rotated surface code and a depth-one intra-block CZ gate in the 2D-toric code for all odd distances and lengths at least three.
  • The family remains non-LDPC while achieving the stated rate and distance scaling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The preservation of transversality under scaling may allow Clifford operations on many logical qubits with error rates that accumulate more slowly than in layered gate decompositions.
  • Similar preservation arguments could be tested on other base codes or for gate sets beyond Clifford to address neighbouring problems in fault-tolerant compilation.
  • An immediate testable step is to enumerate small candidate base codes [[n0, 2, d0]] and verify whether any satisfy the transversal gate conditions needed to start the family.

Load-bearing premise

A small base code with two logical qubits must exist that already has the required transversal Clifford gate properties, and these properties must survive the tiling and concatenation operations.

What would settle it

An explicit check on the smallest tiled instance of the base code showing that any one of the logical S, √X, or CZ gates requires circuit depth greater than two after construction.

Figures

Figures reproduced from arXiv: 2606.13521 by Adam Holmes.

Figure 1
Figure 1. Figure 1: FIG. 1. Single-layer transversal [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Construction 1 at [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Construction 2 at [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Construction intuition for the Quantum Logic Codes (schematic; the literal transversal gate layers are not shown). [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Complete logical-Clifford basis ISA of the [[4 [PITH_FULL_IMAGE:figures/full_fig_p032_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Single-layer (depth-1) [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Multi-layer [PITH_FULL_IMAGE:figures/full_fig_p033_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The lifted [PITH_FULL_IMAGE:figures/full_fig_p034_8.png] view at source ↗
read the original 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 $[[n,\sqrt{n},\Theta({n^{\beta}})]]$ where $\beta \approx 0.2823$ 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript identifies lower bounds on circuit depth for generating a full logical Clifford algebra and constructs new depth-one transversal gates (phase S̄ in the rotated surface code; intra-block CZ̄ in the 2D toric code for odd distances and L≥3). It then presents a high-rate non-LDPC CSS code family with parameters [[n, √n, Θ(n^β)]] (β≈0.2823 in one case) that is asserted to possess a constant-depth (sometimes depth-one) complete 2-local transversal logical Clifford ISA consisting of individually targeted S̄, SHS̄=√X̄, and CZ̄ gates. The family is obtained from a base [[n0,2,d0]] CSS code by tiling (to achieve k=√n) and concatenation (to raise distance), with the ISA claimed to be preserved under these operations.

Significance. If the central claims hold, the construction would supply a concrete route to high-rate stabilizer codes equipped with a full transversal Clifford ISA at scale. This addresses a longstanding tension between rate, distance, and logical-gate overhead in fault-tolerant quantum computation. The explicit use of standard tiling/concatenation operations that are shown to commute with the ISA is a methodological strength, as is the provision of concrete new transversal gates in well-studied codes such as the surface and toric codes.

major comments (2)
  1. [Construction of the code family (core-code section)] The headline claim that the family 'provably possesses' the complete depth-one (or constant-depth) transversal Clifford ISA rests on the existence of a base [[n0,2,d0]] CSS code that itself supports individually targeted depth-one S̄, √X̄, and CZ̄ gates. No explicit parameters, stabilizer generators, or logical-operator definitions for any such base code are supplied, rendering the inheritance argument unverifiable.
  2. [Preservation under tiling and concatenation] The assertion that tiling and concatenation preserve the 2-local depth-one ISA (including that these operations commute with the gates and do not introduce extra depth or non-2-local interactions) is load-bearing for the scaled parameters [[n,√n,Θ(n^β)]]. The manuscript states the preservation but does not provide the explicit commutation relations, depth accounting, or inductive argument that would establish this for arbitrary numbers of tiles or concatenation levels.
minor comments (1)
  1. [Abstract and parameter section] The numerical value β≈0.2823 is stated without an accompanying derivation, table of exponents, or reference to the underlying rate-distance calculation; a short appendix or inline equation showing how this exponent arises from the tiling/concatenation parameters would aid reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and for recognizing the potential significance of the construction. We address the two major comments below and will incorporate clarifications and additional details into the revised manuscript.

read point-by-point responses

  1. Referee: [Construction of the code family (core-code section)] The headline claim that the family 'provably possesses' the complete depth-one (or constant-depth) transversal Clifford ISA rests on the existence of a base [[n0,2,d0]] CSS code that itself supports individually targeted depth-one S̄, √X̄, and CZ̄ gates. No explicit parameters, stabilizer generators, or logical-operator definitions for any such base code are supplied, rendering the inheritance argument unverifiable.

    Authors: We agree that an explicit base code would render the inheritance argument more immediately verifiable. The manuscript defines the required properties of the base [[n0,2,d0]] CSS code (support for individually targeted depth-one S̄, √X̄, and CZ̄) and shows how these properties are inherited under the stated operations, but does not exhibit a concrete stabilizer tableau or logical-operator matrix for any specific (n0,d0) instance. We will add a new subsection in the core-code section that supplies one explicit base code (including generators and logical operators) together with a verification that it satisfies the required transversal-gate conditions. revision: yes

  2. Referee: [Preservation under tiling and concatenation] The assertion that tiling and concatenation preserve the 2-local depth-one ISA (including that these operations commute with the gates and do not introduce extra depth or non-2-local interactions) is load-bearing for the scaled parameters [[n,√n,Θ(n^β)]]. The manuscript states the preservation but does not provide the explicit commutation relations, depth accounting, or inductive argument that would establish this for arbitrary numbers of tiles or concatenation levels.

    Authors: The manuscript argues preservation on the basis that both tiling and concatenation act block-wise on the logical operators and stabilizers while preserving the 2-local support of the target gates. However, we acknowledge that the commutation relations, depth accounting, and an inductive argument for arbitrary numbers of tiles or concatenation levels are stated at a high level rather than derived in full detail. We will add an appendix that (i) lists the explicit commutation relations between the tiling/concatenation maps and each gate in the ISA, (ii) provides a depth tally showing that the logical gates remain depth-one (or constant-depth) after each operation, and (iii) supplies a short inductive proof covering multiple concatenation levels and arbitrary tile counts. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on explicit preservation proof from base code

full rationale

The paper constructs the high-rate family from a base [[n0,2,d0]] code and states that it 'show[s] 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'. This is presented as a demonstrated result rather than a definitional reduction or fitted input. No equations reduce claimed properties to self-referential inputs, no self-citation chains are load-bearing in the abstract, and the novel gate constructions (e.g., depth-one S̄ in rotated surface code) are independent. The derivation is therefore self-contained against external benchmarks of base-code existence and preservation proofs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions of quantum coding theory plus one demonstrated scaling exponent; no new physical entities are postulated.

free parameters (1)
  • β ≈ 0.2823
    Exponent in the distance scaling Θ(n^β) for one demonstrated case of the code family; appears chosen to match the explicit construction.
axioms (1)
  • domain assumption Tiling and concatenation of a base [[n0,2,d0]] stabilizer code preserve the complete transversal Clifford ISA when the base code possesses the required gates.
    Invoked when the abstract states the family is built from the core codes and inherits structure so that the ISA remains depth-one up to depth-two addressable operations.

pith-pipeline@v0.9.1-grok · 5847 in / 1534 out tokens · 23942 ms · 2026-06-27T06:34:27.598898+00:00 · methodology

discussion (0)

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Reference graph

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