Subsystem qubit stabilizer code

Subsystem qubit stabilizer code[1]

Also known as Gauge qubit stabilizer code.

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 doesn't lead to new codes but does lead to new error correction and fault tolerance procedures. Subsystem codes are denoted by \([[n,k,g,d]]\), similar to stabilizer codes, but with an extra parameter \(g\) denoting the number of gauge qubits.

Subsystem qubit stabilizer codes are defined by a gauge group and a stabilizer group that, up to phase, forms the centralizer of the gauge group. The table below summarizes the relevant groups and their sizes for a subsystem qubit stabilizer code.

purpose	symbol	size
gauge group	\(\mathsf{G}\)	\(4\cdot 2^{n-k+g}\)
stabilizer group	\(\mathsf{S}\)	\(2^{n-k-g}\)
code-preserving Paulis	\(\mathsf{N}(\mathsf{S})\)	\(4\cdot 2^{n+k+g}\)
logical Paulis	\(\mathsf{N}(\mathsf{S})/\mathsf{G}\)	\(4^{k}\)
gauge Paulis	\(\frac{\mathsf{N}(\mathsf{S})}{\mathsf{N}(\mathsf{S})/\mathsf{G}\times\left\langle i,\mathsf{S}\right\rangle }=\mathsf{G}/\left\langle i,\mathsf{S}\right\rangle\)	\(4^{g}\)
gauge-preserving Paulis	\(\mathsf{N}(\mathsf{G})\)	\(4\cdot 2^{n+k-g}\)
bare logicals	\(\mathsf{N}(\mathsf{G})/\left\langle i,\mathsf{S}\right\rangle\)	\(4^{k}\)

Table I: Groups relevant to subsystem qubit stabilizer codes. The normalizer \(\mathsf{N}(\mathsf{S})\) (technically, the centralizer, but these are equivalent for this case) is the group formed by all elements of the \(n\)-qubit Pauli group that commute with all elements in \(\mathsf{S}\). The gauge group and the normalizer are defined so as to include \(i\) and its powers as elements, while the stabilizer group is not.

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 \( \mathsf{S} = \tilde{Z}_1,\dots, \tilde{Z}_s \rangle \). It follows that the normalizer of \(\mathsf{S} \) is \( \mathsf{N}(\mathsf{S}) = \langle i, \tilde{Z}_1,\dots, \tilde{Z}_n, \tilde{X}_{s+1},\dots, \tilde{X}_n \rangle \). We now choose a gauge group as \( \mathsf{G} = \langle i, \tilde{Z}_1,\dots, \tilde{Z}_s, \tilde{X}_{s+1}, \tilde{Z}_{s+1}, \dots, \tilde{X}_{s+g}, \tilde{Z}_{s+g} \rangle \) with \( s + g \leq n \). The logical group is chosen as \( \mathsf{L} = \mathsf{N}(\mathsf{S})/\mathsf{G} \simeq \langle \tilde{X}_{s+g+1},\tilde{Z}_{s+g+1}, \dots, \tilde{X}_n,\tilde{Z}_n \rangle \). 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 \( C = A \otimes B \simeq (\mathbb{C}^2)^{\otimes k} \otimes (\mathbb{C}^2)^{\otimes g} \). Thus the Hilbert space is partitioned into 3 sets; \(k\) logical qubits, \(g\) gauge qubits, and \(s\) error-syndrome qubits, with \(s+g+k=n\).

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 \( E_aE_b \not\in \mathsf{N}(\mathsf{S}) \setminus \mathsf{G} \) for all pairs \(a,b\). The distance of the code is the minimal weight of operators in \( \mathsf{N}(\mathsf{S}) \setminus \mathsf{G}\).

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

Encoding

A subsystem codeword can be encoded with the Clifford circuits of the stabilizer code corresponding to treating all gauge qubits as logical qubits [3].

Gates

Logical Clifford gates can be implemented fault-tolerantly for subsystem codes of distance at least three [4].

Fault Tolerance

Logical Clifford gates can be implemented fault-tolerantly for subsystem codes of distance at least three [4].

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 [5].

Notes

See Ref. [6] for algorithms and lists of possible tilings of particular subsystem codes.

Parents

Subsystem qubit 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 \(q=2\).
Operator-algebra (OA) qubit stabilizer code — An OA qubit stabilizer code storing no classical information but retaining gauge qubits for its quantum code is a subsystem qubit stabilizer code.

Children

Majorana subsystem stabilizer code — Subsystem qubit stabilizer codes have been formulated in terms of Majorana operators [7].
Holographic hybrid code — The holographic hybrid code is constructed out of alternating isometries of the five-qubit and \([[4,1,1,2]]\) Bacon-Shor codes.
Sarvepalli-Brown subsystem 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. Any \([[n,k,r,d]]\) subsystem stabilizer code can be mapped onto a \([[2n,2k,2r,\geq d]]\) subsystem CSS code via symplectic doubling, which preserves geometric locality of a code up to a constant factor. Every subsystem stabilizer code can be constructed from two nested subsystem CSS codes satisfying certain constraints [8].
Subsystem spacetime circuit code
Three-fermion (3F) subsystem code

Cousins

Qubit stabilizer code — Subsystem qubit stabilizer codes reduce to qubit 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 [9; Exam. 1].
Reed-Solomon (RS) code — Primitive RS codes yield subsystem stabilizer codes via the subsystem extension of the Hermitian construction to subsystem codes [9; 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 out a subgroup of the logical Pauli group which causes trivial faults in the corresponding Clifford circuit.
Hybrid stabilizer code — Hybrid stabilizer codes can be constructed from qubit subsystem stabilizer codes by using the gauge qubits of the latter to store classical information [10; Thm. 4].
\([[6,1,3]]\) Six-qubit stabilizer code — The \([[6,1,3]]\) six-qubit code can be converted into a \([[6,1,1,3]]\) subsystem code that saturates the subsystem Singleton bound [11].
Quantum data-syndrome (QDS) code — The DS construction can be extended to qubit subsystem qubit stabilizer codes [12].
Qubit CSS code — Qubit CSS "seed" codes can be used to produce subsystem stabilizer codes [13].
Balanced product (BP) code — Distance balancing is used to form balanced-product subsystem codes [14].
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]: A. Chapman, S. T. Flammia, and A. J. Kollár, “Free-Fermion Subsystem Codes”, (2022) arXiv:2201.07254
[8]: M. L. Liu, N. Tantivasadakarn, and V. V. Albert, “Subsystem CSS codes, a tighter stabilizer-to-CSS mapping, and Goursat&apos;s Lemma”, Quantum 8, 1403 (2024) arXiv:2311.18003 DOI
[9]: S. A. Aly, A. Klappenecker, and P. K. Sarvepalli, “Subsystem Codes”, (2006) arXiv:quant-ph/0610153
[10]: A. Nemec and A. Klappenecker, “Encoding classical information in gauge subsystems of quantum codes”, International Journal of Quantum Information 20, (2022) arXiv:2012.05896 DOI
[11]: B. Shaw et al., “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI
[12]: A. Nemec, “Quantum Data-Syndrome Codes: Subsystem and Impure Code Constructions”, (2023) arXiv:2302.01527
[13]: O. Novak and N. Rengaswamy, “GNarsil: Splitting Stabilizers into Gauges”, (2024) arXiv:2404.18302
[14]: 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

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