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arXiv:2608.05688

Beyond transversality: structure of Clifford circuits for CSS codes

Victor V. Albert

quant-phcs.ITmath.COmath.GRmath.RTSubmitted Aug 6, 2026• Announced Aug 7, 2026• 0 citations
v1 · Aug 6, 2026v2 · Oct 1, 2026
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Abstract

We characterize four groups of Clifford circuits for Calderbank--Shor--Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice. We further show that every code-preserving automorphism circuit, consisting of single-qubit Clifford gates and permutations, has a normal form comprising a Hadamard layer, a permutation, and two diagonal circuits. We also define a two-fold automorphism group, in which a depth-one two-local circuit may be code-preserving up to a permutation, and show that its logical image can be larger than that of the two-fold transversal group. For 136 CSS codes, we provide explicit generators and determine the logical image of the two-fold transversal group. We find 78 codes whose full logical Clifford group is generated by two-fold-transversal circuits, including codes of distances 3, 4, 5, 6, 8, and 12, with respective rates $2/5$, $3/4$, $1/9$, $1/5$, $2/5$, and $3/56$. We construct three families of CSS codes from bipartite grids, cut-complements, and quadrics, many of which realize the full logical Clifford group in this way. More generally, the induced logical group can be large even when it is not full logical Clifford group: it has order at least $460\,800$ for the gross code and roughly $10^{26}$ for a clustered-cyclic code.

Comments
18 + 37 pages, 3 figures, and 14 tables

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We characterize four groups of Clifford circuits for Calderbank--Shor--Steane (CSS) codes that are relevant to fault-tolerant logical operations. First, we show that every code-preserving Clifford circuit is a product of Z-diagonal circuits, composed of S and CZ gates, and their X-basis analogues. Second, we define the two-fold transversal group, generated by depth-one two-local code-preserving circuits, and show that each of its elements can be expressed as a product of layers consisting of either Z-diagonal, X-diagonal, or CNOT gates. As a corollary, every transversal gate is a product of three transversal diagonal circuits; for connected non-self-dual codes, two such circuits suffice.
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Storing quantum information in a quantum error correction code can protect it from errors, but the ability to transform the stored quantum information in a fault tolerant way is equally important. Logical Pauli group operators can be implemented on Calderbank-Shor-Steane (CSS) codes, a commonly-studied category of codes, by applying a series of physical Pauli X and Z gates. Logical operators of this form are fault-tolerant because each qubit is acted upon by at most one gate, limiting the spread of errors, and are referred to as transversal logical operators. Identifying transversal logical operators outside the Pauli group is less well understood. Pauli operators are the first level of the Clifford hierarchy which is deeply connected to fault-tolerance and universality. In this work, we study transversal logical operators composed of single- and multi-qubit diagonal Clifford hierarchy gates. We demonstrate algorithms for identifying all transversal diagonal logical operators on a CSS code that are more general or have lower computational complexity than previous methods. We also show a method for constructing CSS codes that have a desired diagonal logical Clifford hierarchy operator implemented using single qubit phase gates. Our methods rely on representing operators composed of diagonal Clifford hierarchy gates as diagonal XP operators and this technique may have broader applications.
Realizing Logical Diagonal Gates via Transversal Physical ZZZ-Rotations in CSS Codes
Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes C2⊆C1C_2 \subseteq C_1C2​⊆C1​, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs (C1,C2)(C_1, C_2)(C1​,C2​) whose resulting CSS codes realize a target logical diagonal gate via transversal physical ZZZ-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit ZZZ-rotations and multi-qubit controlled ZZZ-rotations via transversal physical ZZZ-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an [[n′,k′]][[n',k']][[n′,k′]] CSS code Q′Q'Q′ and a target logical ZZZ-rotation (single-qubit or multi-controlled) ULU_LUL​, and extends Q′Q'Q′ by systematically appending n′′n''n′′ physical qubits to obtain an [[n,k]][[n,k]][[n,k]] CSS code QQQ with n=n′+n′′n = n'+n''n=n′+n′′ and k=k′k=k'k=k′. The target logical gate ULU_LUL​ is realized in QQQ by applying a well-chosen physical transversal ZZZ-rotation to the n′′n''n′′ appended physical qubits. Moreover, any logical gate realized via transversal physical ZZZ-rotations in the input code Q′Q'Q′ remains transversally realizable in the extended code QQQ. The CSS code QQQ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code Q′Q'Q′ to obtain a CSS code QQQ that supports fault-tolerant implementations of multiple desired logical ZZZ-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled ZZZ-rotations.
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 S‾\mathrm{\overline{S}}S gate in the rotated surface code and a depth-one intra-block CZ‾\mathrm{\overline{CZ}}CZ gate in the 2D-toric code that generalizes to all odd distances and all lengths L≥3L\ge3L≥3, respectively. Finally, we construct a high-rate non-LDPC CSS code family with parameters [[n,n,Θ(nβ)]][[n,\sqrt{n},Θ({n^β})]][[n,n​,Θ(nβ)]] where β≈0.2823β\approx 0.2823β≈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 S‾\mathrm{\overline{S}}S, SHS‾=X\mathrm{\overline{SHS}} = \sqrt{X}SHS=X​, and CZ‾\mathrm{\overline{CZ}}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 [[n0,2,d0]][[n_0, 2, d_0]][[n0​,2,d0​]], 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.