Bivariate bicycle code

Essence

A bivariate bicycle (BB) code starts with two sparse binary polynomials on a finite torus. Replacing each monomial by a cyclic translation gives two commuting matrices $$A$$ and $$B$$ . The CSS checks

H_X=[A\mid B],\qquad H_Z=[B^T\mid A^T]

then commute for an algebraic reason: $$H_XH_Z^T=AB+BA=0$$ over $\mathbb F_2$ . Instead of searching over stabilizer matrices and separately repairing anticommutation, the construction chooses an abelian algebra where commutation is automatic and spends the search budget on rate, distance, and hardware layout.

The reference gross block uses a $12\times6$ torus and two three-term polynomials. The constructor recomputes 144 data qubits, 12 logical qubits, weight-six checks, and qubit degree six. Its static distance is 12 with split provenance: Bravyi et al. supply the lower bound, while the Atlas reruns a fixed-seed weight-12 upper witness. The full calculation is in the gross parameter ledger.

Design problem

The target is not an asymptotically good two-dimensional phase. It is a useful finite memory block: pack several logical qubits behind one syndrome circuit without giving up bounded-weight checks. A distance-12 rotated surface-code patch uses 144 data qubits for one logical qubit; the gross block uses the same data-qubit count for twelve. That twelve-fold data-only comparison is the finite-block attraction reported by bravyi2024bicycle, not a claim that ancillas, routing, decoding, or logical operations are free.

The price is connectivity. Each check touches three qubits in each of two data layers, and the Tanner graph needs two planar degree-three coupling layers rather than one nearest-neighbor square grid. The syndrome circuit, CNOT ordering, detector model, and decoder are part of the memory. A static [[n,k,d]] check matrix alone does not certify circuit distance.

There is also an asymptotic boundary. The quotient-ring analysis of postema2025existence finds the studied BB subclass asymptotically bad. Moderate blocks may still have excellent constants, but increasing $$l,m$$ does not silently turn the gross construction into a constant-rate, growing-relative-distance qLDPC family.

Derivation chain

Make two translations commute. Let $$S_l$$ and $$S_m$$ be cyclic shifts and set $x=S_l\otimes I_m$ , $y=I_l\otimes S_m$ . They commute because they act on different tensor factors.
Turn sparse polynomials into sparse matrices. A monomial $$x^a y^b$$ is a permutation matrix, so a polynomial with $$w$$ distinct monomials has row and column weight $$w$$ , unless repeated monomials cancel over $\mathbb F_2$ .
Use the swapped dual. Put $$A,B$$ in $$H_X=[A|B]$$ , then put their swapped transposes in $$H_Z=[B^T|A^T]$$ . The swap makes the overlap $$AB+BA$$ , rather than the generally nonzero $$A^2+B^2$$ .
Read it as a chain complex. The maps $R\xrightarrow{[B;A]}R^2\xrightarrow{[A|B]}R$ compose to zero. Logical Paulis are middle homology classes. The complete convention and the toric-code anchor are derived in the polynomial chain-complex page.
Search the quotient, not arbitrary checks. The rank deficit of the ideal generated by $$A,B$$ determines how many logical qubits remain. Sparse terms constrain check weight and connectivity; common algebraic structure raises $$k$$ , but too much common structure can create low-weight logicals.

The card.yaml replay recipe binds each instance's l, m, a_terms, and b_terms into two bivariate_polynomial@1 factors, then applies bivariate_bicycle@1. The verifier requires the resulting stabilizer array to match the ordinary constructor exactly, including polynomial, check, and qubit order, for both shipped instances.

Multiple pictures

Polynomial picture. Monomials are translations and transpose is translation reversal. This is the shortest route from four input parameters to the exact matrices.
Group-algebra picture. The ring is the regular representation of $\mathbb Z_l\times\mathbb Z_m$ . It explains why BB is an abelian two-block group-algebra specialization, not a synonym for every 2BGA or lifted-product code.
Homological picture. Checks are consecutive boundary maps and logicals are the middle quotient. This separates “commutes with all checks” from “is itself a product of checks.”
Tanner/circuit picture. Every polynomial term is an incidence edge. This exposes degree, routing layers, hook propagation, and why a syndrome schedule cannot be inferred from a static matrix.
Quotient-module picture. Common factors and roots predict rank defects, finite rate, and boundary sensitivity.
Anyon/Koszul picture. Polynomial ideals become excitation types and mobility constraints. The topological-frustration result prevents the overstatement that all finite BB codes are merely decoupled toric copies.
Toric-cylinder decoder picture. Selected symmetries expose matching subproblems on cylinders. That is a decoder reduction, not a global equivalence of phases.

The representation dictionary states what each picture preserves and where it stops.

Why these parameters

For the gross instance, $$lm=72$$ . There are two qubits per torus site, so $$n=2lm=144$$ . Each of $$A=x^3+y+y^2$$ and $$B=y^3+x+x^2$$ is a sum of three permutation matrices. Every X or Z check therefore has weight $$3+3=6$$ , and every data qubit participates in three checks of each Pauli type.

The constructor finds

\operatorname{rank}H_X=\operatorname{rank}H_Z=66, \qquad k=144-66-66=12.

Equivalently, the image ideal has codimension six and the CSS middle homology contains twice that many logical dimensions. This is a machine result for the displayed matrices, not a promise that all three-term BB polynomials encode twelve qubits.

Distance has two different obligations. The literature lower bound excludes all nontrivial logicals below weight 12. Independently, rand-lowweight-v1 with seed 5 and 2,000 iterations finds a weight-12 centralizer class outside the stabilizer row space. The bounds meet, so the recorded distance is 12 with bounded provenance. The Atlas has not reproduced the literature's lower-bound certificate; that remaining division of trust is explicit.

The second shipped instance uses $$A=1+x$$ , $$B=1+y$$ on a $3\times3$ torus. Exact enumeration gives [[18,2,3]], providing a small toric anchor that checks the conventions without claiming every BB block decomposes into toric codes.

Failure modes

Static distance can overstate a circuit. One ancilla fault may propagate through later CNOTs into a correlated data error. The seven-layer schedule of bravyi2024bicycle is a specific circuit; other orderings can have a smaller circuit distance. See syndrome circuits and decoding.
Connectivity is a resource. The two planar coupling layers still require long-range or reconfigurable couplers, crossings between layers, or motion. A one-layer nearest-neighbor chip does not obtain gross-code overhead by relabeling qubits.
BP is not a magic black box. Short cycles and degeneracy can stall belief propagation; OSD improves robustness but adds global, data-dependent latency. A reported threshold belongs to the full family-circuit-decoder experiment.
Finite constants do not imply asymptotic goodness. The gross point is a strong finite block. It is not evidence that fixed sparse polynomials evade two-dimensional tradeoffs as the torus grows.
Packed logicals need addressability. Twelve logical qubits in one block are useful only if an architecture can prepare, measure, couple, and route the desired logical combinations without erasing the memory advantage.
Variants change the object. Pruning, open boundaries, morphing, self-duality, extra variables, and qudits change rank, distance, circuits, or alphabet. The variant and boundary map keeps those claims separate.

Operations story

An aligned physical CNOT between two identical CSS blocks is transversal: control X checks acquire the matching target X check, while target Z checks acquire the matching control Z check. This preserves the two-block stabilizer group and induces the corresponding logical CNOT layer. Physical alignment, module transport, and correlated decoding remain architecture costs.

Torus translations also preserve the static gross code. If $$P$$ is an $$x$$ - or $$y$$ -shift, then $$A$$ and $$B$$ commute with $$P$$ , giving $H_X\operatorname{diag}(P,P)=PH_X$ , and likewise for $$H_Z$$ . This proves a physical permutation is a code automorphism. It does not identify its action on the twelve logical qubits. An automorphism becomes a useful gate only after its logical symplectic action and physical routing are resolved.

The literature supplies several non-universal certificates: fold-transversal Cliffords on selected symmetric examples, automorphism-derived Cliffords, surface-ancilla surgery in modular proposals, cup-product operations on qualified blocks, and transversal Cliffords in an extra self-dual subfamily. None is a universal transversal gate set for the original gross block. Non-Clifford completion still needs magic states, switching, surgery, or another certified resource. The applicability boundaries are derived in logical operations.

Litmus questions

Q1: Why does swapping $$A^T$$ and $$B^T$$ matter? A: With the swap, $$[A|B][B^T|A^T]^T=AB+BA=0$$ . Without it, the product is $$A^2+B^2$$ , which need not vanish. The swap turns CSS commutation into the commutator of two commuting algebra elements.

Q2: Does the Atlas machine-verify the gross code's complete distance proof? A: No. It reruns a weight-12 upper witness, while the exclusion of weights below 12 remains sourced to bravyi2024bicycle. The matching bounds justify value 12 with bounded, not verified, provenance.

Q3: Is 0.8% an intrinsic threshold of every BB code? A: No. Bravyi et al. estimate a circuit-level threshold for their BB memory family under the standard circuit-noise model, depth-seven extraction, detector construction, and BP-OSD configuration. It is a protocol-qualified family threshold, not a one-block pseudo-threshold or a property of an arbitrary BB check matrix. Changing the family, circuit, noise normalization, or decoder defines a different threshold experiment.

Q4: Why does $$k=12$$ not follow from having three terms in each polynomial? A: Term count fixes sparsity, not rank. Here both CSS check matrices have rank 66, leaving twelve logical dimensions. Different three-term polynomials can have a different common quotient, hence different $$k$$ and distance.

Q5: If translations preserve the stabilizers, are they already twelve addressable logical gates? A: No. Stabilizer preservation proves only that the code space maps to itself. One must compute the induced action on a logical Pauli basis; it may be trivial, may mix several logical qubits, and may be expensive to route. The card records only operations with a cited certificate and explicit applicability.