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arXiv:1409.8320 · Quantum Information and Computation

Classification of transversal gates in qubit stabilizer codes

Jonas T. Anderson, Tomas Jochym-O’Connor

Published Jul 1, 2016• 11 citations
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Abstract

This work classifies the set of diagonal gates that can implement a single or two-qubit transversal logical gate for qubit stabilizer codes. We show that individual physical diagonal gates on the underlying qubits that compose the code are restricted to have entries of the form e iπc/2 k along their diagonal, resulting in a similarly restricted class of logical gates that can be implemented in this manner. As such, we show that all diagonal logical gates that can be implemented transversally by individual physical diagonal gates must belong to the Clifford hierarchy. Moreover, we show that for a given stabilizer code, the two-qubit diagonal transversal gates must belong to the same level of Clifford hierarchy as the single-qubit diagonal transversal gates available for the given code. We use this result to prove a conjecture about arbitrary transversal gates made by Zeng et al. in 2007.

DOI
10.26421/qic16.9-10-3

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arXiv:1910.09333•IEEE Journal on Selected Areas in Information Theory•Jul 30, 2020•56 citations
This work classifies the set of diagonal gates that can implement a single or two-qubit transversal logical gate for qubit stabilizer codes. We show that individual physical diagonal gates on the underlying qubits that compose the code are restricted to have entries of the form e iπc/2 k along their diagonal, resulting in a similarly restricted class of logical gates that can be implemented in this manner. As such, we show that all diagonal logical gates that can be implemented transversally by individual physical diagonal gates must belong to the Clifford hierarchy. Moreover, we show that for a given stabilizer code, the two-qubit diagonal transversal gates must belong to the same level of Clifford hierarchy as the single-qubit diagonal transversal gates available for the given code. We use this result to prove a conjecture about arbitrary transversal gates made by Zeng et al. in 2007.
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.
Abstract 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.
We establish that transversal diagonal gates on exact qubit codes of distance at least two can be exhaustively classified within a finite search space. Although code-preserving physical gates can have continuous phases, any projective logical action induced by a transversal diagonal gate can always be realized by a physical gate of finite order. Crucially, we prove that this physical order has an upper bound determined solely by the code parameters. This order bound enables the subset-sum-linear-programming (SSLP) framework introduced in arXiv:2504.20847 to exhaustively enumerate all candidate discrete symmetries, after which the Knill-Laflamme equations determine the remaining continuous code parameters. Applying this method to the nontrivial diagonal sector of exact ((6,2,3))((6,2,3))((6,2,3)) codes, SSLP yields 666666 candidate symmetry classes, of which exactly 151515 are realizable, supporting full projective diagonal logical cyclic groups of orders 222, 333, and 555. In particular, for order 555, the code is unique up to local unitaries and qubit permutations, equivalent to the code constructed in arXiv:2504.20847. This code also attains the minimum length n=6n=6n=6 among exact ((n,2,≥3))((n, 2, \ge 3))((n,2,≥3)) codes whose full projective transversal group cannot be conjugated into the projective Clifford group.
This work classifies stabilizer codes by the set of diagonal Clifford gates that can be implemented transversally on them. We show that, for any stabilizer code, its group of diagonal transversal Clifford gates on ℓ\ellℓ code blocks must be one of six distinct families of matrix groups. We further develop the theory of classifying stabilizer codes by via matrix algebras of endomorphisms first introduced by Rains, and give a complete classification of the diagonal Clifford symmetries of ℓ\ellℓ code blocks. A number of corollaries are given in the final section.
Universal quantum computation requires non-Clifford gates, which stabilizer codes can provide transversally only up to a finite level of the Clifford hierarchy. Non-stabilizer codes are not subject to this bound and are the natural candidates to fill the gap. However, it is difficult to efficiently evaluate the nonstabilizerness (or magic) of non-stabilizer codes. Here, we resolve this problem for codeword-stabilized (CWS) codes by showing that the nonstabilizerness of a CWS code is determined by the difference structure of its codewords. Whenever a Sidon set of the required size exists, the most magical codes are related to the Sidon sets, whose pairwise differences are all distinct. Structural analysis demonstrates that nonstabilizerness is invariant under coset closure, giving rise to non-stabilizer codes with arbitrarily many logical qubits and constant nonstabilizerness. Nonstabilizerness of whole code families becomes exactly computable, and we obtain closed-form values for the Kerdock codes. Moreover, we reveal a tradeoff between nonstabilizerness and transversal gates. For a code with kkk logical qubits, a diagonal transversal gate that is non-Clifford on ttt coordinates forces the nonstabilizerness benchmark to be at most 2(k−t)2(k-t)2(k−t). These results reduce the search for codes with high nonstabilizerness and for transversal non-Clifford gates to classical counting problems, to which the standard tools of additive combinatorics apply.
In order to perform universal fault-tolerant quantum computation, one needs to implement a logical non-Clifford gate. Consequently, it is important to understand codes that implement such gates transversally. In this paper, we adopt an algebraic approach to characterize all stabilizer codes for which transversal T and T†gates preserve the codespace. Our Heisenberg perspective reduces this question to a finite geometry problem that translates to the design of certain classical codes. We prove three corollaries of this result: (a) For any non-degenerate [n, k, d] stabilizer code supporting a physical transversal T (which might not be logical T), there exists an [n, k, d]] CSS code with the same property; (b) Triorthogonal codes form the most general family of CSS codes that realize logical transversal T via physical transversal T; (c) Triorthogonality is necessary for physical transversal T on a CSS code to realize the logical identity. The main tool we use is a recent characterization of a particular family of diagonal gates in the Clifford hierarchy that are efficiently described by symmetric matrices over rings of integers [N. Rengaswamy et al., Phys. Rev. A 100, 022304]. We refer to these operations as Quadratic Form Diagonal (QFD) gates. Our framework generalizes all existing stabilizer code constructions that realize logical gates via transversal T. We provide several examples of codes and briefly discuss connections to decreasing monomial codes, pin codes, generalized triorthogonality and quasitransversality. We partially extend these results towards characterizing all stabilizer codes that support transversal π/2ℓZ-rotations. In particular, using Ax's theorem on residue weights of polynomials, we provide an alternate characterization of logical gates induced by transversal π/2ℓZ-rotations on a family of quantum Reed-Muller codes. We also briefly discuss a general approach to analyze QFD gates that might lead to a characterization of all stabilizer codes that support any given physical transversal 1- or 2-local diagonal gate.