Bivariate bicycle (BB) code[1]
Description
One of several Abelian 2BGA codes which admit time-optimal syndrome measurement circuits that can be implemented in a two-layer architecture, a generalization of the square-lattice architecture optimal for the surface codes. Codes can be classified by the weight of their checks, e.g., by BB\(w\) where \(w\) is the check weight.
The qubit connectivity graph is not quite a 2D grid and is instead decomposable into two planar subgraphs of degree three; there exists an optimized layout minimizing Euclidean communication distance for check operators [2]. There are \(n\) \(X\) and \(Z\) check operators, each of weight \(|A|+|B|\). The weight-six BB codes of Ref. [1] are on par with the surface code in terms of threshold and admit a much higher ancilla-added encoding rate. The price is check operators that are not geometrically local in a two-layer nearest-neighbor layout [1].
Multivariate (e.g., trivariate) bicycle codes generalize BB codes to Abelian 2BGA codes with polynomials in more than two variables [3]; other variants and generalizations exist [4]. Versions with open boundary conditions are tile codes [5], obtained by pruning [6] or by condensing boundary anyons [7,8]. There exist qudit BB codes that achieve \(kd^2 / n = 20\) [9].
BB codes as lattice codes on twisted tori
A BB code with polynomials \(A\) and \(B\) is the quotient by a torus of a 2D translation-invariant CSS code with two qubits per unit cell [10]. In the polynomial representation, the stabilizer generators of that code are \((A,B|0,0)\) and \((0,0|\overline{B},\overline{A})\) and their translations [10]. The bar denotes the antipode map \(x^iy^j\mapsto x^{-i}y^{-j}\). The torus is specified by two lattice vectors \((0,\alpha)\) and \((\beta,\gamma)\), which impose \(y^{\alpha}=1\) and \(x^{\beta}y^{\gamma}=1\) on \(n=2\alpha\beta\) qubits [10]. Every torus can be brought to this form by a unimodular change of basis, and the case \(\gamma\neq 0\) is called a twisted torus [10; Sec. II.B]. The polynomials of the gross code on the twisted torus with lattice vectors \((0,30)\) and \((6,6)\) yield a \([[360,12,d\leq 24]]\) code whose checks have range three [10; Sec. III.B and Table IV].
A BB code satisfies the topological order condition iff \(A\) and \(B\) are coprime as Laurent polynomials [10,11]. Its \(m\)-type anyons are then the elements of the quotient ring \(\mathbb{F}_2[x^{\pm1},y^{\pm1}]/\langle A,B\rangle\) [10]. The number of logical qubits on any torus is at most \(k_{\max}=2\dim\mathbb{F}_2[x^{\pm1},y^{\pm1}]/\langle A,B\rangle\), twice the number of independent \(m\)-type anyons [10; Thm. 1]. On the torus with lattice vectors \((0,\alpha)\) and \((\beta,\gamma)\), the number of logical qubits is [10; Thm. 3] \begin{align} k=2\dim\frac{\mathbb{F}_2[x^{\pm1},y^{\pm1}]}{\langle A,B,y^{\alpha}-1,x^{\beta}y^{\gamma}-1\rangle}~. \tag*{(1)}\end{align} The maximum \(k=k_{\max}\) is attained iff \(y^{\alpha}-1\) and \(x^{\beta}y^{\gamma}-1\) lie in the ideal \(\langle A,B\rangle\) [10; Corr. 2.2]. Otherwise, \(k<k_{\max}\) [10,12].
Weight-six BB codes whose polynomials \(A=1+x+x^ay^b\) and \(B=1+y+x^cy^d\) extend the toric-code pair \(1+x\), \(1+y\) by one monomial each include the checks of the 6.6.6 color code and of the gross code [10; Exams. 2 and 3].
Protection
A BB code with \(A=B\) and \(k>0\) has distance two regardless of check weight [13; Thm. 1].Rate
When ancilla qubit overhead is included, the encoding rate of the weight-six codes of Ref. [1] surpasses that of the surface code. A general \([[n,k,d]]\) bivariate bicycle code requires \(n\) ancilla qubits for encoding, meaning that its ancilla-added encoding rate is \(k/2n\). BB codes have zero design rate, so all of their logical qubits come from dependent stabilizer generators [14].Transversal and Permutation-Based Gates
Logical Pauli operators and fold-transversal gates studied in Ref. [15,16].Gates
Certain bivariate bicycle codes admit a cup product structure and can thus have logical gates in the Clifford hierarchy implemented by constant-depth Clifford circuits [17].Decoding
BP-OSD decoder [18] has been extended [1] to account for measurement errors (i.e., the circuit-based noise model [19]).Decoding under circuit-level noise has been studied for the BP, BP+OSD, and AutDEC decoders [20].Transformer-based neural-network decoder [21].Matching decoder [22].Fault Tolerance
Fault-tolerant state initialization using lattice surgery techniques [23,24] and an ancillary surface code [1].Syndrome extraction circuit requires seven layers of CNOT gates regardless of code length [1].The depth-7 syndrome extraction schedules studied in Ref. [1] are not generally distance-preserving; for the \([[144,12,12]]\) code, all \(936\) depth-7 variants obtained by reordering the CNOT layers satisfy \(d_{\mathrm{circ}}\leq 10 < d\).Some long-range check operators can be measured less frequently than others [25].Syndrome extraction circuits called morphing circuits [26], generalizing circuits for the color code [27].Code Capacity Threshold
Fully postselected failure rates of weight-six BB codes under bit-flip noise cross near \(p=1/(2+\sqrt{2})\approx 0.2929\) [28]. This value is the self-dual critical point of zero-rate PSD code families with a unique threshold [28]. Without postselection, the \(k=12\) weight-six family of Ref. [29] has a threshold of \(8.38(6)\%\) under BP-OSD decoding [28].Threshold
Admits a \(0.8\%\) pseudo-threshold for circuit-level noise under BP-OSD decoder [1] (cf. [19]).Notes
A database of bivariate bicycle codes is available in QECDB [30].A catalogue of BB codes found by LLM-guided search is available at qcode-discovery [13].See Ref. [10; Tables I-IV] for weight-six BB codes of this form on twisted tori with \(n\leq 400\), and Ref. [11; Tables I and II] for weight-eight self-dual BB codes with \(n\leq 200\).See Ref. [31] for further self-dual BB codes.Cousins
- Honeycomb (6.6.6) color code— Certain bivariate bicycle codes are equivalent to a family of 6.6.6 color codes [6; Exam. 1]. The 6.6.6 color code on a torus is the self-dual BB code with \(A=1+x+y\) [11; Exam. 2].
- Self-dual CSS code— BB codes with \(B=\overline{A}\), where the bar denotes the antipode map \(x^iy^j\mapsto x^{-i}y^{-j}\), have \(H_X=H_Z\) and are self-dual CSS codes [11]. Weight-six ones, \(A=1+x^ay^b+x^cy^d\), have \(k_{\max}=4|ad-bc|\) and reduce to \(|ad-bc|\) decoupled copies of the 6.6.6 color code [11; Thm. 1 and Exam. 2]. Weight-eight ones are doubly even, and \(A=1+x+y+y^{-1}\) yields the \([[16,4,4]]\) and \([[64,8,8]]\) codes [11; Table I].
- Abelian topological code— BB codes have been investigated in terms of their anyons and topological order [10,12]. A BB code satisfies the topological order condition iff its polynomials are coprime [10]. Its \(m\)-type anyons are then the elements of \(\mathbb{F}_2[x^{\pm1},y^{\pm1}]/\langle A,B\rangle\), and the number of logical qubits on any torus is at most twice the dimension of this ring [10; Thm. 1].
- Generalized bicycle (GB) code— GB codes (BB codes) are 2BGA codes over the cyclic group \(\mathbb{Z}_{\ell}\) (Abelian group \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\)). The two codes are the same when \(r\) and \(s\) are relatively prime due to the isomorphism \(\mathbb{Z}_{r} \times \mathbb{Z}_{s} \cong \mathbb{Z}_{\ell = rs}\). BB codes on twisted tori with lattice vectors \((0,\ell)\) and \((1,\gamma)\) reduce to GB codes upon substituting \(x=y^{-\gamma}\) [10; Sec. III.C].
- \([[27,3,3]]\) bivariate tricycle code— The \([[18,2,3]]\) BB code with \(a=x^2y+x^2y^2\) and \(b=1+xy^2\) is the 2D component code of the \([[27,3,3]]\) bivariate tricycle code [32].
- \([[81,3,5]]\) bivariate tricycle code— The \([[54,2,6]]\) BB code with \(a=xy^3+x^2y\) and \(b=1+xy^8\) is the 2D component code of the \([[81,3,5]]\) bivariate tricycle code [32].
- Tile quantum code— Tile codes are open-boundary (planar) analogues of bivariate bicycle (BB) codes, which arise by tiling a torus with stabilizer tiles [5]. Tile codes instead have open boundaries and thus retain true \(O(1)\)-locality on a planar 2D lattice [5]. Sharing the same bulk stabilizer tiles does not yield the same code. The planar \([[288,8,12]]\) tile code has the same bulk stabilizers as the \((-2,2)\)-BB code. That BB code requires a \(217\times 217\) torus to attain its maximal logical dimension \(k=16\) and yields a \([[98,6,12]]\) code on a \(7\times 7\) torus [8]. The logical dimension of a BB code is determined by the period of the same cellular automaton that generates tile-code logical operators [33].
Member of code lists
- 2D stabilizer codes
- Lattice qubit stabilizer codes
- Quantum codes
- Quantum codes based on homological products
- 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 or permutation-based gates
- Qubit CSS codes
Primary Hierarchy
Generalized homological-product qubit CSS codeQLDPC Qubit CSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum
Parents
Abelian two-block group-algebra codeCSS Generalized homological-product Stabilizer Hamiltonian-based QECC Quantum
Bivariate bicycle codes are Abelian 2BGA (equivalently, two-variable multivariate bicycle) codes over groups of the form \(\mathbb{Z}_{r} \times \mathbb{Z}_{s}\).
Bivariate bicycle codes are permutationally self-dual. Their standard \(XZ\)-duality exchanges the two blocks of qubits while sending \(x\mapsto x^{-1}\) and \(y\mapsto y^{-1}\). This involution yields a Hadamard-type fold-transversal gate [15; Secs. 3 and 4.3]. BB codes on \(\mathbb{Z}_{\ell}\times\mathbb{Z}_{\ell}\) whose two polynomials are exchanged by swapping \(x\) and \(y\) admit a second \(XZ\)-duality. It sends \(x\mapsto y^{-1}\) and \(y\mapsto x^{-1}\) within each block and yields a phase-type fold-transversal gate [15; Sec. 4.3].
BB codes are PBB codes with \(C=D=0\) [13].
BB codes are mirror codes up to qubit permutations and Hadamard gates [34; Prop. 3.21].
A BB code with polynomials \(A\) and \(B\) is the quotient by a torus \(\mathbb{Z}^2/\Lambda\) of a 2D translation-invariant CSS code with two qubits per unit cell [10,11]. The generators of that code are \((A,B|0,0)\) and \((0,0|\overline{B},\overline{A})\), with range given by the degrees of the Laurent representatives of \(A\) and \(B\) [10]. Holding the polynomials fixed while enlarging the torus, including twisted tori, yields a 2D lattice stabilizer code family with checks of bounded range [7,10].
Bivariate bicycle (BB) code
Children
The \([[16,4,4]]\) code is the weight-eight self-dual bivariate bicycle (BB) code with \(A=1+x+y+y^{-1}\) and \(B=\overline{A}\) on the twisted torus with lattice vectors \((0,4)\) and \((2,2)\) [11; Table I].
The \([[18,4,4]]\) color code is simultaneously the bivariate bicycle code with \(\ell=m=3\), \(A=x+1+y^2\) and \(B=y+1+x^2\) [35], i.e., an Abelian 2BGA code over \(\mathbb{Z}_3\times\mathbb{Z}_3\). It is equivalently the \(\ell=m=3\) member of the bivariate bicycle family \(A=1+x+xy\), \(B=1+y+xy\), which yields 6.6.6 color codes whenever \(\ell\) and \(m\) are multiples of three [6; Exam. 1]. It is also the weight-six self-dual BB code with \(A=1+x+y\) on that torus [11; Exam. 2].
The \([[6,4,2]]\) error-detecting code is the weight-six self-dual BB code with the 6.6.6 color-code polynomial \(A=1+x+y\) on the twisted torus with lattice vectors \((3,0)\) and \((1,1)\) [11; Exam. 2].
The three-quarters-gross code is the BB code with \(A=1+x+x^{-1}y^3\) and \(B=1+y+x^3y^{-1}\) on the torus with lattice vectors \((0,9)\) and \((6,0)\) [10; Table I].
The half-gross code places the checks of the gross code on the \(6\times 6\) torus [1; Table 3].
Directional codes form a subfamily of BB codes. Writing the support of an \(X\)-type stabilizer generator on horizontal and vertical edges as bivariate polynomials \(f\) and \(g\), the parity-check matrices are \(H_X=[B(f)\mid B(g)]\) and \(H_Z=[B(\overline{g})\mid B(\overline{f})]\), where \(B\) denotes the binary representation of the quotient ring and the bar denotes the reciprocal polynomial [36]. Every stabilizer generator then has weight \(w\), and every data qubit participates in exactly \(w\) checks, so the Tanner graph is \((w,w)\)-regular [36]. Directional codes are 2D lattice stabilizer codes whose generators are supported on translates of a fixed bounded string of the square lattice [37].
The gross code is the BB code with \(A=1+x+x^{-1}y^3\) and \(B=1+y+x^3y^{-1}\) on the torus with lattice vectors \((0,12)\) and \((6,0)\) [10; Exam. 3 and Table II].
The toric code on any torus \(\mathbb{Z}^2/\Lambda\), including twisted tori, is the BB code with \(A=1+x\) and \(B=1+y\) [10; Exam. 1 and Appx. B].
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Cite as:
“Bivariate bicycle (BB) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/qcga, arXiv:2606.11484