\([[15,1,3]]\) quantum Reed-Muller code

\([[15,1,3]]\) quantum Reed-Muller code[1–3]

Alternative names: Tetrahedral code.

Description

\([[15,1,3]]\) CSS code that is most easily thought of as a tetrahedral 3D color code.

This code contains 15 qubits, represented by four vertices, four face centers, six edge centers, and one body center. The tetrahedron is cellulated into four identical polyhedron cells by connecting the body center to all four face centers, where each face center is then connected by three adjacent edge centers. Each colored cell corresponds to a weight-eight \(X\)-check, and each face corresponds to a weight-4 \(Z\)-check. A logical \(Z\) is any weight-3 \(Z\)-string along an edge of the entire tetrahedron. The logical \(X\) is any weight-7 \(X\)-face of the entire tetrahedron.

The code can also be thought of as a code on all corners of a tesseract except one [4].

Magic

Magic-state yield parameter \( \gamma= \log_d (n/k)\approx 2.47\) [6][5; Box 2].

Encoding

Fault-tolerant logical zero and logical plus state preparation using reinforcement learning [7].

Transversal Gates

A transversal logical \(T\) is implemented by applying a \(T^\dagger\) gate on every qubit [1,5,8]. This is the smallest qubit stabilizer code with a (strongly) transversal gate outside of the Clifford group [9].A subsystem version yields a transversal \(CCZ\) gate [10].

Gates

Code is often used in magic-state distillation protocols because of its transversal \(T\) gate [3].

Decoding

Decoding has been studied for the BP, BP+OSD, and AutDEC decoders [11].

Fault Tolerance

A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [8,10,12,13].Fault-tolerant logical zero and logical plus state preparation using reinforcement learning [7].

Threshold

Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [14].

Cousins

Doubled color code— The \([[15,1,3]]\) code can be viewed as a (gauge-fixed) doubled color code obtained from the Steane code via the doubling transformation [15].
\([[7,1,3]]\) Steane code— The \([[15,1,3]]\) code can be viewed as a (gauge-fixed) doubled color code obtained from the Steane code via the doubling transformation [15]. A fault-tolerant universal gate set can be done via code switching between the Steane code and the \([[15,1,3]]\) code [8,10,12,13,16]. An \([[105,1,3]]\) alternative concatenation of the \([[15,1,3]]\) and Steane codes allows for a universal gate set consisting of gates that are close to transversal [17,18].
Concatenated Steane code— The \([[105,1]]\) concatenation of the \([[15,1,3]]\) and Steane codes allows for a universal gate set consisting of gates that are transversal w.r.t. to two different partitions [17,18].
Concatenated qubit code— The concatenated \([[15,1,3]]\) code has a measurement threshold less than one [19].
Tensor-network code— The quantum Lego framework yields an \([[8,1,2]]\) stabilizer code admits a transversal logical \(T\) gate that originates from that of a trivial (distance-one) \([[7,1]]\) code. This code, in turn, is obtained from the \([[15,1,3]]\) code [20].
\([[10,1,2]]\) CSS code— The \([[10,1,2]]\) code can be obtained by morphing the \([[15,1,3]]\) code [21].
\([[8,3,2]]\) Smallest interesting color code— The \([[8,3,2]]\) code can be obtained from a subset of physical qubits of the \([[15,1,3]]\) code [21].
Subset-Sum-Linear-Programming (SS-LP) code— The \(((7,2,3))\) SS-LP code realizes the \(T\) gate transversally, but requires fewer qubits than the \([[15,1,3]]\) quantum Reed-Muller code.
\([[15, 7, 3]]\) quantum Hamming code— Gauging out six of the seven logical qubits of the \([[15,7,3]]\) code yields the \([[15,1,3]]\) code [22].

Member of code lists

Primary Hierarchy

Quantum Domain

Qubit Kingdom

Qubit codeQECC Quantum

Union stabilizer (USt) codeQECC Quantum

Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum

Coherent-parity-check (CPC) code

Qubit CSS codeCSS Stabilizer Qubit Hamiltonian-based QECC Quantum

Generalized homological-product qubit CSS codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum

Quantum rainbow code

Quantum pin codeStabilizer Hamiltonian-based Qubit QECC Quantum

Color code

3D color codeQubit Generalized homological-product Lattice stabilizer QLDPC CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

Tetrahedral color code

Parents

Tetrahedral color code

The \([[15,1,3]]\) code is a tetrahedral color code defined on a single tetrahedron.

\([[2^r-1,1,3]]\) simplex codeColor Quantum Reed-Muller Generalized homological-product QLDPC CSS Stabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

XS stabilizer codeQubit QECC Quantum

The \([[15,1,3]]\) code can be viewed as an XS stabilizer code [23; Exam. 6.4].

Triorthogonal codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

The \([[15, 1, 3]]\) code is a triorthogonal code [24].