\([[6,2,2]]\) \(C_6\) code

\([[6,2,2]]\) \(C_6\) code[1]

Description

Error-detecting normal self-dual CSS code on three qubit pairs that encodes a logical qubit pair and detects any error acting on one pair [1]. In Knill’s \(C_4/C_6\) architecture, this code is used at the second and higher concatenation levels. A choice of check operators used in that construction is \(XIIXXX\), \(XXXIIX\), \(ZIIZZZ\), and \(ZZZIIZ\), with logical operators \(X_L = IXXIII\), \(Z_L = IIZZIZ\), \(X_S = XIXXII\), and \(Z_S = IIIZZI\) [1][2; ID 126].

Protection

As a distance-two code, the \(C_6\) code detects any single-qubit error. In the qubit-pair grouping used by Knill, it detects any error acting on one of the three pairs, and therefore can correct a pair error when its location is already known [1].

Magic

Various magic-state distillation protocols exist for the \([[4,2,2]]\) qubit code and the \(C_6\) code in what are known as Meier-Eastin-Knill (MEK) protocols [3]. For example, the magic-state yield parameter is \(\gamma = \log_2 5 \approx 2.322\) for a protocol using the \([[10,2,2]]\) code [4; Box 2]; see also [5; Table IV].

Transversal Gates

Physical Hadamard gates implement logical Hadamard gates, making this a normal self-dual qubit CSS code [6].

Gates

Fault-tolerant magic-state preparation [7].

Fault Tolerance

Knill’s \(C_4/C_6\) architecture uses the \([[4,2,2]]\) code at the first level and the \(C_6\) code at higher levels, together with error-correcting teleportation [1]. That paper gives evidence for postselected thresholds above \(0.03\) and extrapolates to \(0.06\), while the error-correcting architecture has evidence for a threshold above \(0.01\). Later work refined the postselected-threshold analysis [8,9] (see also Ref. [10]).Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [11]. In the optimized protocol of Ref. [11], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively.Fault-tolerant magic-state preparation [7].One of the code’s logical qubits can be relaxed to a gauge qubit to yield a \([[6,1,1,2]]\) subsystem qubit stabilizer code with a particular set of transversal gates. This code admits a fault-tolerant circuit relevant to magic-state preparation [2].

Realizations

Trapped ions: fault-tolerant magic-state preparation demonstrated on a 20-qubit H1-1 device by Quantinuum [7].

Cousins

\([[6,1,2]]\) semi-self-dual CSS code— Fixing one logical qubit of the \([[6,2,2]]\) \(C_6\) code to \(|Y^{-}\rangle_L\) yields this \([[6,1,2]]\) code [12][2; ID 59].
Concatenated qubit code— Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [1] admitting a post-selected threshold [8,9] (see also Ref. [10]) and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [3]. Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [11]. In the optimized protocol of Ref. [11], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively.
Concatenated GKP code— Recursively concatenating the \(C_6\) and \([[4,2,2]]\) codes with GKP codes achieves the hashing bound of the displacement channel [13].
Ladder Floquet code— The ISG of the ladder code includes \(Z\)-type Pauli products around squares of the qubit ladder. These are also included in the checks of the \(C_6\) code.
\([[12,2,4]]\) carbon code— The carbon code is a concatenation of the \([[4,2,2]]\) code and the \(C_6\) code.
\([[4,2,2]]\) Four-qubit code— Concatenations of \([[4,2,2]]\) and \(C_6\) codes yield fault-tolerant quantum computation schemes [1] admitting a post-selected threshold [8,9] (see also Ref. [10]) and the Meier-Eastin-Knill (MEK) magic-state distillation protocols [3]. Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [11]. In the optimized protocol of Ref. [11], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively.
\([[7,1,3]]\) Steane code— In Knill’s \(C_4/C_6\) architecture, noisy \(\ket{\pi/8}\) states are injected using \(C_4/C_6\) logical Bell pairs and then purified by encoding them into the Steane code; Knill also proposed using the Steane code as a final concatenation level for the \(C_4/C_6\) scheme [1].
\([[2^r-1, 2^r-2r-1, 3]]\) quantum Hamming code— Concatenating quantum Hamming codes on top of the \([[4,2,2]]\) and \(C_6\) codes yields fault-tolerant quantum computation with constant space and quasi-polylogarithmic time overheads [11]. In the optimized protocol of Ref. [11], a level-five \(C_4/C_6\) code underlies concatenated quantum Hamming codes \(\mathcal{Q}_5,\mathcal{Q}_6,\mathcal{Q}_7,\mathcal{Q}_7\), yielding a \(2.5\%\) threshold and space overheads \(162\) and \(373\) physical qubits per logical qubit at physical error rate \(0.1\%\) for logical CNOT error rates \(10^{-10}\) and \(10^{-24}\), respectively.

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

Qubit QLDPC codeStabilizer Hamiltonian-based QECC Quantum

Color codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

2D color codeTwist-defect color QLDPC Qubit Generalized homological-product Lattice stabilizer CSS Stabilizer Abelian topological Topological Hamiltonian-based QECC Quantum

Parents

2D color code

The \(C_6\) code is a color code on a ladder with three rungs and periodic boundary conditions, (a.k.a. a triangular prism with no top and bottom faces) [14]. Purely \(Z\)- or \(X\)-type stabilizers lie on the three square faces of the ladder.

\([[k+4,k,2]]\) H codeCSS Stabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

The \([[k+4,k,2]]\) H code for \(k=2\) is the \(C_6\) code.

\([[6r,2r,2]]\) Ganti-Onunkwo-Young codeCSS Stabilizer Hamiltonian-based Qubit Small-distance block quantum QECC Quantum

The Ganti-Onunkwo-Young code for \(r=1\) is the \(C_6\) code.

Khesin-Lu-Shor codeCSS Stabilizer Hamiltonian-based Qubit QECC Quantum

The Khesin-Lu-Shor code for \(r=2\) and \(m=2^r - 1 = 3\) is the \(C_6\) code.

Hermitian qubit codeStabilizer Hamiltonian-based Qubit QECC Quantum

The \(C_6\) code is Hermitian [2; Table 6].