Gross-code parameter and provenance ledger

This page derives every static number attached to the shipped gross instance and labels what the repository recomputes versus what remains a literature obligation.

Inputs and matrix sizes

The instance records

input	value	role
$$l$$	12	order of the $$x$$ translation
$$m$$	6	order of the $$y$$ translation
$$A$$	$$x^3+y+y^2$$	left half of each X check
$$B$$	$$y^3+x+x^2$$	right half of each X check

The torus algebra has binary dimension

N=lm=12\cdot6=72.

Both $$A$$ and $$B$$ are $72\times72$ . Concatenating two blocks gives $H_X,H_Z\in\mathbb F_2^{72\times144}$ , hence

\[ n=2N=144. \]

This is the data-qubit count. It excludes syndrome ancillas, couplers, routing workspace, spare modules, and decoder hardware.

Check weight and qubit degree

Each gross polynomial contains three distinct reduced monomials. A monomial is a permutation matrix, so each contributes exactly one one per row and column. Therefore:

every row and column of $$A$$ has weight three;
every row and column of $$B$$ has weight three;
every row of $$H_X=[A|B]$$ has weight six;
every row of $$H_Z=[B^T|A^T]$$ has weight six;
every data qubit belongs to three X checks and three Z checks.

The last statement is sometimes compressed to “degree six.” Keeping the Pauli split visible matters for a circuit: three X-type and three Z-type interactions have different ancilla preparations and fault propagation.

reproduce.py checks every row and column, rather than inferring the result from the input term count.

Rank and encoded dimension

Gaussian elimination over $\mathbb F_2$ gives

\operatorname{rank}H_X=66,\qquad \operatorname{rank}H_Z=66.

The CSS stabilizer rank is their sum because pure-X and pure-Z binary symplectic rows cannot cancel each other. Thus

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

There is a useful quotient reading. The map

[A|B]:R^2\longrightarrow R,\qquad (u,v)\mapsto Au+Bv

has image equal to the ideal $$(A,B)$$ . Its rank is 66, so its cokernel $$R/(A,B)$$ has dimension

\dim_{\mathbb F_2}R/(A,B)=72-66=6.

For the displayed gross complex, the dual side has the same six-dimensional defect, leaving twelve logical dimensions in the middle. This is a computed fact for these periods and polynomials. Three terms in $$A$$ and three in $$B$$ do not determine the rank: changing their exponents changes the common ideal and can change $$k$$ .

The quotient perspective is the route developed further in postema2025existence. It turns code search into a tension:

a larger common quotient can increase $$k$$ ;
the same shared structure may introduce short nontrivial cycles and reduce distance;
locality constraints restrict the permitted monomial offsets.

Distance: two certificates, not one label

The gross instance records

lower: {value: 12}
upper: {value: 12, method: rand-lowweight-v1, seed: 5, iters: 2000}
origin: {kind: bounded, ref: bravyi2024bicycle}

The two sides mean different things.

Lower bound. bravyi2024bicycle supplies the claim that no nontrivial logical Pauli has weight below 12. The Atlas does not currently replay that lower-bound certificate. The citation is therefore part of the structured origin, not decorative bibliography.

Upper bound. The repository constructs the centralizer nullspaces, samples deterministic binary combinations with NumPy seed 5, rejects stabilizer classes, greedily reduces representatives by same-type stabilizers, and finds weight 12 within 2,000 iterations. This is a rerunnable existence witness for a logical of weight at most 12.

Because the cited lower bound and machine upper bound meet, $$d=12$$ . The origin is bounded, not verified: the machine has not exhaustively excluded weights one through eleven at $$n=144$$ .

The seed is not tuned data. A seed-zero run with 2,000 iterations reaches only weight 14, while a longer seed-zero run reaches 12. The stored seed-five search is a compact deterministic witness. If a future engine finds a smaller logical, verification reports an improvement or mismatch; the expected distance must then be diagnosed against the literature rather than adjusted silently.

The finite-block overhead comparison

The data-only ratio is

\frac{n}{k}=\frac{144}{12}=12

data qubits per encoded qubit at distance 12. A distance-12 rotated surface-code patch uses $$d^2=144$$ data qubits for one logical qubit. This is the twelve-fold finite-block comparison emphasized by bravyi2024bicycle.

It is not a full-machine ratio. A BB cycle also needs syndrome ancillas, degree-six interactions, a seven-CNOT-layer schedule in the original proposal, measurement/reset, classical detector processing, and BP-OSD. Architectures may add movement, communication, spare blocks, or surface-code ancillas for logical operations. Comparisons must put the same resources in both denominators.

Static versus circuit distance

The value 12 belongs to the static stabilizer code. A syndrome circuit defines a spacetime fault code whose minimum undetectable fault set can be smaller. CNOT ordering controls how an ancilla error spreads across data qubits; the detector construction controls which histories are distinguishable.

Consequently, none of these follows from the static ledger alone:

a distance-12 fault circuit;
a universal threshold;
a decoder-independent logical error rate;
a logical-gate error rate.

Those quantities move to the syndrome-circuit ledger or the operations ledger.

Recomputed summary

For the exact shipped row and qubit order, reproduce.py asserts:

$$N=72$$ , $$H_X,H_Z$$ shapes $72\times144$ ;
rank 66 for both check matrices and stabilizer rank 132;
[[144,12]] CSS parameters;
weight six for every check;
three X checks and three Z checks incident on every data qubit;
quotient codimension six;
a deterministic weight-12 upper witness.

Only the distance lower bound remains sourced rather than replayed.

input	value	role
$\(l\)$	12	order of the $\(x\)$ translation
$\(m\)$	6	order of the $\(y\)$ translation
$\(A\)$	$\(x^3+y+y^2\)$	left half of each X check
$\(B\)$	$\(y^3+x+x^2\)$	right half of each X check