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 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 | \(\mathsf{N}(\mathsf{G})/\left\langle i,\mathsf{S}\right\rangle\) | \(4^{k}\) |
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.Subsystem qubit stabilizer codes can be used to estimate logical Pauli noise for any correctable physical noise [7].Cousins
- Qubit stabilizer code— Subsystem qubit stabilizer codes reduce to qubit stabilizer codes when there are no gauge qubits. An \([[n,k,d]]\) qubit stabilizer code can be converted into an order \([[O(\ell \delta n),k,\Omega(d/w)]]\) subsystem qubit stabilizer code with weight-three gauge operators via the wire-code mapping [8], which uses weight reduction. Here, \(w\) and \(\delta\) are the weight and degree of the input code's Tanner graph, while \(\ell\) is the length of the longest edge of a particular embedding of that graph.
- 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].
- Commuting-projector Hamiltonian code— Ground-state spaces of commuting-projector Hamiltonians with weight-two (two-body) terms cannot be used to suppress errors in adiabatic quantum computation [10], but this can be circumvented with excited-state subspaces [11] or ground-state subspaces of subsystem code Hamiltonians, e.g., using BBS codes [12,13].
- 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 [14; 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 [15].
- Quantum data-syndrome (QDS) code— The DS construction can be extended to qubit subsystem qubit stabilizer codes [16].
- Qubit CSS code— Qubit CSS "seed" codes can be used to produce subsystem stabilizer codes [17].
- Balanced product (BP) code— Distance balancing is used to form balanced-product subsystem codes [18].
- Distance-balanced code
Primary Hierarchy
References
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- A. Nemec, “Quantum Data-Syndrome Codes: Subsystem and Impure Code Constructions”, (2023) arXiv:2302.01527
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- O. Novak and N. Rengaswamy, “GNarsil: Splitting Stabilizers into Gauges”, (2024) arXiv:2404.18302
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- 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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- A. Chapman, S. T. Flammia, and A. J. Kollár, “Free-Fermion Subsystem Codes”, (2022) arXiv:2201.07254
- [20]
- 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
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