Description
Subsystem CSS code defined on an \(m_1 \times m_2\) lattice of qubits that generalizes the \([[9,1,3]]\) (subspace) Shor code. It is said to be symmetric when \(m_1=m_2\), and asymmetric otherwise.
The \(X\)-type and \(Z\)-type stabilizers defined as \(X\) and \(Z\) operators acting on all qubits on adjacent columns and rows, respectively. Let \(O_{i,j}\) denote an operator acting on the qubit at a position \((i,j)\) on the lattice, with \(i\in\{0,1,\ldots ,m_1-1\}\) and \(j\in\{0,1,\ldots,m_2-1\}\). The code's stabilizer group is \begin{align} \mathsf{S}=\langle X_{i,*}X_{i+1,*},Z_{*,j}Z_{*,j+1}\rangle~, \tag*{(1)}\end{align} with generators expressed as products of nearest-neightbour 2-qubit gauge operators, \begin{align} \begin{split} X_{i,*}X_{i+1,*}= \bigotimes_{k=0}^{m_2-1} X_{i,k}X_{i+1,k} \\ Z_{*,j}Z_{*,j+1}=\bigotimes_{k=0}^{m_1-1} Z_{k,j}Z_{k,j+1}~. \end{split} \tag*{(2)}\end{align} Syndrome extraction can be done by measuring these gauge operators, which are on fewer qubits and local.
A Floquet version of the Bacon-Shor code admits a period-four measurement sequence that utilizes its gauge degrees of freedom as defects evolving across measurement rounds. This Floquet-Bacon-Shor code saturates the subsystem BT bound. Applying a period-four measurement schedule to the original Bacon-Shor code yields a numerical threshold under circuit-level noise [4].
Protection
The \([[m_1 m_2,1,min(m_1,m_2)]]\) variant has distance \(d=min(m_1,m_2)\). In a symmetric 3-dimensional case (defined on a cubic lattice) with \(L^3\) qubits, the code has the parameters \([[L^3,1,L]]\). Bacon-Shor code parameteres can be optimized by changing the block geometry, yielding good performance against biased noise [5].Rate
A non-LDPC family of Bacon-Shor codes achieves a distance of order \(\Omega(n^{1-\epsilon})\) with sparse gauge operators.Transversal Gates
Logical Hadamard is transversal in symmetric Bacon-Shor codes up to a qubit permutation [6] and can be implemented with teleportation [7].Bacon-Shor codes on an \(m \times m^k\) lattice admit transversal \(k\)-qubit-controlled \(Z\) gates [8].Gates
Pieceably fault-tolerant circuits can be employed to construct non-transversal gates effectively [9].Subsystem lattice surgery [10].Measurement-free deformation protocol realizing the \(CCZ\) gate [11].Decoding
Both Steane error correction and Shor error correction can be used for syndrome extraction, with the former outperforming the latter [12].Utilizing the mapping of the effect of the noise to a statistical mechanical model [13,14] yields several copies of the 1D Ising model [15; Sec. V.B].While check operators are few-body, stabilizer weights scale with the number of qubits, and stabilizer expectation values are obtained by taking products of gauge-operator expectation values. It is thus not clear how to extract stabilizer values in a fault-tolerant manner [16,17].Autonomous QEC [18].Applying a period-four measurement schedule to the original Bacon-Shor code yields a numerical threshold under circuit-level noise [4].Fault Tolerance
Fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes that is effective against highly biased noise [19].Pieceably fault-tolerant circuits can be employed to construct non-transversal gates effectively [9].Code Capacity Threshold
The number of check operators scales sublinearly with system size, so the Bacon-Shor codes alone do not exhibit a topological threshold in the \(m_1,m_2 \to \infty\) limit [20]. However, a threshold can be obtained from concatenated Bacon-Shor codes that are further restricted to planar geometries, whose recovery circuit is a subset of a circuit used by a larger bona-fide Bacon-Shor code [21]. This threshold differs from a concatenated threshold in that there are no long-range connectivity requirements.Lower bounds for the concatenated threshold of various small Bacon-Shor codes are tabulated in [6; Table I].Threshold
Numerical study of concatenated thresholds of logical CNOT gates for various codes against depolarizing noise [22].The Bacon-Shor code has a measurement threshold of zero [23].Applying a period-four measurement schedule to the original Bacon-Shor code yields a numerical threshold under circuit-level noise [4].Realizations
Superconducting qubits: The Floquet-Bacon-Shor code has been realized on a 3-by-3 lattice [24].Notes
See [25; Sec. III.C1] for an exposition.Cousins
- Hamiltonian-based code— The 2D Bacon-Shor gauge-group Hamiltonian is the compass model [26–28].
- Hybrid stabilizer code— There are several ways to convert Bacon-Shor codes to hybrid qubit stabilizer codes [29,30]
- Hastings-Haah Floquet code— A Floquet version of the Bacon-Shor code admits a period-four measurement sequence that utilizes its gauge degrees of freedom as defects evolving across measurement rounds. This Floquet-Bacon-Shor code saturates the subsystem BT bound. Applying a period-four measurement schedule to the original Bacon-Shor code yields a numerical threshold under circuit-level noise [4].
- Asymmetric quantum code— Bacon-Shor code parameters against bit- and phase-noise can be optimized by changing the block geometry, yielding good performance against biased noise [5]. A fault-tolerant teleportation-based computation scheme for asymmetric Bacon-Shor codes is effective against highly biased noise [19].
- Majorana subsystem stabilizer code— Bacon-Shor codes can be fermionized into fermionic subsystem codes with two-body terms [31].
- GNU PI code— GNU codes of length \((2t+1)^2\) result from projecting Bacon-Shor codes into the PI qubit subspace [32].
- Quantum parity code (QPC)— Bacon-Shor codes reduce to QPCs when all \(X\)-type gauge generators are fixed [33; pg. 6].
- Heavy-hexagon code— Bacon-Shor stabilizers are used to measure the X-type stabilizers of the code.
Member of code lists
- Asymmetric quantum codes
- Hamiltonian-based codes
- Lattice subsystem stabilizer codes
- Quantum codes
- Quantum codes with a rate
- Quantum codes with code capacity thresholds
- Quantum codes with fault-tolerant gadgets
- Quantum codes with notable decoders
- Quantum codes with other thresholds
- Quantum codes with transversal gates
- Quantum CSS codes
- Realized quantum codes
- Subsystem codes
Primary Hierarchy
Parents
A compass code on a fully non-colored lattice reduces to the Bacon-Shor code.
Bacon-Shor code
Children
The four-qubit subsystem code is the shortest error-detecting Bacon-Shor code.
The nine-qubit Bacon-Shor code is the shortest error-correcting Bacon-Shor code.
References
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Page edit log
- Victor V. Albert (2022-07-31) — most recent
- Mazin Karjikar (2022-06-28)
- Victor V. Albert (2022-03-15)
- Srilekha Gandhari (2022-01-20)
- Victor V. Albert (2021-12-03)
Cite as:
“Bacon-Shor code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/bacon_shor