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

Children

Cousins

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, Combinatorics, Physics and their Interactions 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'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]
I. Marvian and D. A. Lidar, “Quantum Error Suppression with Commuting Hamiltonians: Two Local is Too Local”, Physical Review Letters 113, (2014) arXiv:1410.5487 DOI
[11]
Y. Cao, S. Liu, H. Deng, Z. Xia, X. Wu, and Y.-X. Wang, “Robust analog quantum simulators by quantum error-detecting codes”, (2024) arXiv:2412.07764
[12]
Z. Jiang and E. G. Rieffel, “Non-commuting two-local Hamiltonians for quantum error suppression”, Quantum Information Processing 16, (2017) arXiv:1511.01997 DOI
[13]
M. Marvian and D. A. Lidar, “Error Suppression for Hamiltonian-Based Quantum Computation Using Subsystem Codes”, Physical Review Letters 118, (2017) arXiv:1606.03795 DOI
[14]
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
[15]
B. Shaw, M. M. Wilde, O. Oreshkov, I. Kremsky, and D. A. Lidar, “Encoding one logical qubit into six physical qubits”, Physical Review A 78, (2008) arXiv:0803.1495 DOI
[16]
A. Nemec, “Quantum Data-Syndrome Codes: Subsystem and Impure Code Constructions”, (2023) arXiv:2302.01527
[17]
O. Novak and N. Rengaswamy, “GNarsil: Splitting Stabilizers into Gauges”, (2024) arXiv:2404.18302
[18]
N. P. Breuckmann and J. N. Eberhardt, “Balanced Product Quantum Codes”, IEEE Transactions on Information Theory 67, 6653 (2021) arXiv:2012.09271 DOI
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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} }
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“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.