Classification of small triorthogonal codes

Nezami, Sepehr; Haah, Jeongwan

doi:10.1103/physreva.106.012437

Cited by 8 publications

(3 citation statements)

References 25 publications

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“…In the linked Jupyter notebook, we apply the method of Section 3.4 to find generating sets of diagonal logical operators for the 38 triorthogonal code classes in Table II of Ref. [27]. In this example, we consider codes with k = 3 logical qubits (this choice can be modified by the user).…”

Section: Example 36 (Triorthogonal Codes)mentioning

confidence: 99%

Transversal Diagonal Logical Operators for Stabiliser Codes

Webster¹,

Quintavalle²,

Bartlett³

2023

Preprint

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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.

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Section: Example 36 (Triorthogonal Codes)mentioning

confidence: 99%

Transversal Diagonal Logical Operators for Stabiliser Codes

Webster¹,

Quintavalle²,

Bartlett³

2023

Preprint

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“…For triorthogonal codes [10], there is always a logical operator of form T ⊗k := UT ⊗n where U is a product of CZ and S operators and k is the number of logical qubits of the code. In the linked Jupyter notebook, we apply the method of section 3.4 to find generating sets of diagonal logical operators for the 38 triorthogonal code classes in table II of [27]. In this example, we consider codes with k = 3 logical qubits (this choice can be modified by the user).…”

Section: Example 36 (Triorthogonal Codes)mentioning

confidence: 99%

Transversal diagonal logical operators for stabiliser codes

Webster,

Quintavalle,

Bartlett

2023

New J. Phys.

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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.

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“…(Generalized) triorthogonal codes [9,22] are Calderbank-Shor-Steane (CSS) codes [12,37] designed to implement a non-Clifford logical gate (up to some diagonal Clifford logical gates). Hamming weights in the classical codes that determine the CSS codes are required to satisfy certain divisibility properties [13,21,26,31,39]. Many examples employ Reed-Muller (RM) codes.…”

Section: Introduction and Reviewmentioning

confidence: 99%

Designing the Quantum Channels Induced by Diagonal Gates

Liang

Calderbank

2022

Quantum

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The challenge of quantum computing is to combine error resilience with universal computation. Diagonal gates such as the transversal T gate play an important role in implementing a universal set of quantum operations. This paper introduces a framework that describes the process of preparing a code state, applying a diagonal physical gate, measuring a code syndrome, and applying a Pauli correction that may depend on the measured syndrome (the average logical channel induced by an arbitrary diagonal gate). It focuses on CSS codes, and describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. The interaction of code states and diagonal gates depends very strongly on the signs of Z-stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. The paper derives necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provides an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate (introduced by Rengaswamy et al.), the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find application in magic state distillation and elsewhere. When all the signs are positive, the paper characterizes all possible CSS codes, invariant under transversal Z-rotation through π/2l, that are constructed from classical Reed-Muller codes by deriving the necessary and sufficient constraints on l. The generator coefficient framework extends to arbitrary stabilizer codes but there is nothing to be gained by considering the more general class of non-degenerate stabilizer codes.

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Classification of small triorthogonal codes

Cited by 8 publications

References 25 publications

Transversal Diagonal Logical Operators for Stabiliser Codes

Transversal Diagonal Logical Operators for Stabiliser Codes

Transversal diagonal logical operators for stabiliser codes

Designing the Quantum Channels Induced by Diagonal Gates

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