Subsystem qubit stabilizer code[1]

Description

Also called a gauge stabilizer code. A stabilizer code with some of its logical qubits denoted as gauge qubits and not used for storage of logical information. Note that this doesnt lead to new codes but does lead to new error correction and fault tolerance procedures. Subsystem codes are denoted by \([[n,k,r,d]]\), similar to stabilizer codes, but with an extra parameter \(r\) denoting the number of gauge qubits.

To create these codes proceed as follows. Choose \(2n\) operators \(\{ \tilde{X}_j,\tilde{Z}_j\}_{j=1}^n\) from \(\mathsf{P}_n\), the Pauli group on \(n\) qubits, such that they obey the same commutation relations as the regular \(n\)-qubit Pauli generators \( \{X_j,Z_j\}_{j=1}^n \) (the subscript on these latter operators indicates the single qubit the Pauli matrix acts on). The tilde operators might act on more than one physical (or bare) qubit but they behave as if they acted only on a single qubit. WLOG we can choose a stabilizer group as \( \mathsf{S} = \langle Z_1,\dots,Z_s \rangle \). It follows that the normalizer of \(\mathsf{S} \) is \( N(\mathsf{S}) = \langle i, \tilde{Z}_1,\dots, \tilde{Z}_n, \tilde{X}_{s+1},\dots, \tilde{X}_n \rangle \). We now choose a gauge group as \( \mathsf{G} = \langle i, \tilde{Z}_1,\dots, \tilde{Z}_s, \tilde{X}_{s+1}, \tilde{Z}_{s+1}, \dots, \tilde{X}_{s+r}, \tilde{Z}_{s+r} \rangle \) with \( s + r \leq n \). The logical group is choosen as \( \mathsf{L} = N(\mathsf{S})/\mathsf{G} \simeq \langle \tilde{X}_{s+r+1},\tilde{Z}_{s+r+1}, \dots, \tilde{X}_n,\tilde{Z}_n \rangle \). Now the codespace \( C \) is as usual the \(+1\) eigenspace of the stabilizer \( \mathsf{S} \). But the gauge and logical groups have further decomposed this space into \( C = A \otimes B \simeq (\mathbb{C}^2)^{\otimes k} \otimes (\mathbb{C}^2)^{\otimes r} \). Thus the Hilbert space is partitioned into 3 sets; \(k\) logical qubits, \(r\) gauge qubits, and \(s\) stabilizer qubits, with \(s+r+k=n\).

Protection

Detects errors on \(d-1\) qubits, corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qubits. More generally, a set of errors \( \{ E_a \} \) is correctable iff \( E_aE_b \not\in N(\mathsf{S}) \setminus \mathsf{G} \) for all pairs \(a,b\). The distance of the code is the minimal weight of operators in \( N(\mathsf{S}) \setminus \mathsf{G}\).

Decoding

Syndrome measurements are obtained by first measuring gauge operators of the code and taking their products, which give the stabilizer measurement outcomes.

Code Capacity Threshold

For correlated Pauli noise, bounds can be obtained by mapping the effect of noise on the code to a statistical mechanical model [2].

Notes

When the gauge group \( \mathsf{G} \) is abelian, the above is reduced to the standard stabilizer formalism.

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Zoo code information

Internal code ID: subsystem_stabilizer

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Zoo Code ID: subsystem_stabilizer

Cite as:
“Subsystem qubit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/subsystem_stabilizer
BibTeX:
@incollection{eczoo_subsystem_stabilizer, title={Subsystem qubit stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/subsystem_stabilizer} }
Permanent link:
https://errorcorrectionzoo.org/c/subsystem_stabilizer

References

[1]
D. Poulin, “Stabilizer Formalism for Operator Quantum Error Correction”, Physical Review Letters 95, (2005). DOI; quant-ph/0508131
[2]
C. T. Chubb and S. T. Flammia, “Statistical mechanical models for quantum codes with correlated noise”, Annales de l’Institut Henri Poincaré D 8, 269 (2021). DOI; 1809.10704
[3]
N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021). DOI; 2012.09271

Cite as:

“Subsystem qubit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/subsystem_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/subsystem/subsystem_stabilizer.yml.