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 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. 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 \( N(\mathsf{S}) \setminus \mathsf{G}\).

There is an upper bound of \(d = O(L^{D-1})\) on the distance [2] of geometrically local subsystem stabilizer codes arranged in a \(D\)-dimensional lattice of length \(L\) with \(n=L^D\). More generally, there is a tradeoff theorem [3] stating that, for any logical operator, there is an equivalent logical operator with weight \(\tilde{d}\) such that \(\tilde{d}d^{1/(D-1)}=O(L^{D})\).

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

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

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]
S. Bravyi and B. Terhal, “A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes”, New Journal of Physics 11, 043029 (2009) arXiv:0810.1983 DOI
[3]
J. Haah and J. Preskill, “Logical-operator tradeoff for local quantum codes”, Physical Review A 86, (2012) arXiv:1011.3529 DOI
[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]
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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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.