# Subsystem qubit stabilizer code[1]

## Description

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 chosen 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.

There is the following analogue of the Knill-Laflamme conditions for subsystem qubit stabilizer codes. 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 \( N(\mathsf{S}) \setminus \mathsf{G}\).

Entropic conditions have been formulated for random projective measurement noise [2].

## Encoding

## Gates

## Fault Tolerance

## Code Capacity Threshold

## Notes

## Parents

- Subsystem qubit code
- Operator-algebra qubit stabilizer code
- Subsystem modular-qudit stabilizer code — Subsystem modular-qudit stabilizer codes reduce to subsystem qubit stabilizer codes for qudit dimension \(q=2\).
- Subsystem Galois-qudit stabilizer code — Subsystem Galois-qudit stabilizer codes reduce to subsystem qubit stabilizer codes for qudit dimension \(q=2\).

## Children

- Subsystem hypergraph code
- Subsystem CSS code — Subsystem CSS codes are subsystem stabilizer codes whose gauge groups admit a generating set of pure-\(X\) and pure-\(Z\) Pauli strings. Additionally, any \([[n,k,r,d]]_{\mathbb{Z}_q}\) subsystem stabilizer code can be mapped onto a \([[2n,2k,2r,\geq d]]_{\mathbb{Z}_q}\) subsystem CSS code, with the mapping preserving geometric locality of a code up to a constant factor [7]. Every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints [7].
- Subsystem spacetime circuit code
- Three-fermion (3F) subsystem code

## Cousins

- Qubit stabilizer code — Subsystem stabilizer codes reduce to stabilizer codes when there are no gauge qubits.
- Bose–Chaudhuri–Hocquenghem (BCH) code — BCH codes yield subsystem stabilizer codes via the subsystem extension of the Hermitian construction to subsystem codes [8; Exam. 1].
- Reed-Solomon (RS) code — Primitive RS codes yield subsystem stabilizer codes via the subsystem extension of the Hermitian construction to subsystem codes [8; Exam. 3].
- Hastings-Haah Floquet code — This code can be viewed as a subsystem stabilizer code, albeit one with less logical qubits.
- Spacetime circuit code — Spacetime circuit codes can be upgraded to subsystem codes by gauging a subgroup of the logical Pauli group which causes trivial faults in the corresponding Clifford circuit.
- Balanced product (BP) code — Distance balancing is used to form balanced-product subsystem codes [9].
- Distance-balanced code

## References

- [1]
- D. Poulin, “Stabilizer Formalism for Operator Quantum Error Correction”, Physical Review Letters 95, (2005) arXiv:quant-ph/0508131 DOI
- [2]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
- [3]
- P. K. Sarvepalli and A. Klappenecker, “Encoding Subsystem Codes”, (2008) arXiv:0806.4954
- [4]
- D. Banfield and A. Kay, “Implementing Logical Operators using Code Rewiring”, (2023) arXiv:2210.14074
- [5]
- 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) arXiv:1809.10704 DOI
- [6]
- G. M. Crosswhite and D. Bacon, “Automated searching for quantum subsystem codes”, Physical Review A 83, (2011) arXiv:1009.2203 DOI
- [7]
- M. L. Liu, N. Tantivasadakarn, and V. V. Albert, “Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat’s Lemma”, (2023) arXiv:2311.18003
- [8]
- S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “Subsystem Codes”, (2006) arXiv:quant-ph/0610153
- [9]
- N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI

## Page edit log

- Victor V. Albert (2022-03-17) — most recent
- Victor V. Albert (2021-12-16)
- Eric Kubischta (2021-12-14)

## Cite as:

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