Large Language Model-Guided Discovery of Weight-Five Bivariate Bicycle Codes

Juan Cruz-Benito

quant-phOct 5, 2026

Large Language Model-Guided Discovery of Weight-Five Bivariate Bicycle Codes

Authors: Juan Cruz-Benito

Organizations: IBM Research

Abstract

Building on our earlier program-evolution workflow guided by large language models (LLMs), we study weight-five bivariate bicycle (BB) and perturbed bivariate bicycle (PBB) codes. The resulting catalogue contains 1,142 distinct code proposals, including 1,081 nonbaseline proposals attributable to LLM-generated programs. Across the catalogue, we certify connected Calderbank--Shor--Steane (CSS) realizations [[96,4,10]], [[140,6,10]], and [[180,4,14]]. A post-search comparison certifies seven imported Lin--Pryadko archive constructions. For leading parameter triples also represented in that archive, we provide exact distance evidence, explicit bivariate presentations, and verified component reductions. A basis-independent connectivity analysis identifies 409 of the 1,142 catalogue entries as disconnected and shows that 73.1% of the classes with exact distance certificates contain repeated connected components. Algebraic analysis organizes the connected CSS classes into order-3, order-7, and order-15 cyclotomic-kernel strata. The strongest exact connected PBB parameter point is [[216,4,10]], attained by two distinct component classes. Among the 936 distinct CSS proposals from the LLM-guided campaign with a recorded positive distance, 816 (87.18%) are certified at $d\geq5$ . For comparison, three random-search controls each sample 6,444 CSS code proposals uniformly without replacement, using the same per-lattice and encoded-dimension sample counts as the LLM-guided campaign. In these controls, 4,672--4,785 proposals (72.50--74.26%) meet the same criterion. The LLM-guided campaign has the higher certified yield, while the random controls cover more connected classes. Together, these results extend LLM-guided discovery to a more constrained code family and provide a reproducible structural and exact-distance account of its strongest candidates.

Figures & tables

Campaign	Targets	Exact	Nonexact
CSS-small	16	12	4
CSS-large	78	38	40
PBB-small	30	30	0
PBB-large	51	26	25
Total	175	106	69

Table 1: Certification outcomes for the targeted follow-up after excluding proposals with

d\leq 2

. “Nonexact” combines a certified lower bound with either a rigorous upper bound or a decoder estimate.

Parameters	Role in this study	Prior-art relationship
$\left[\!\left[96,4,10\right]\!\right]$	Connected realization of a generated CSS parent	Equivalent to archive construction over $C_{48}$
$\left[\!\left[140,6,10\right]\!\right]$	Connected realization of a generated CSS parent	Archive parameter match
$\left[\!\left[180,4,14\right]\!\right]$	Generated directly as a connected CSS presentation	Archive parameter match
$\left[\!\left[132,4,12\right]\!\right]$	Post-search archive construction	Distance certified here
$\left[\!\left[216,4,10\right]\!\right]$	Two connected PBB component classes	—

Table 2: Provenance of selected exact results. “Parameter match” records a shared

(n,k,d)

; equivalence is listed where an explicit certificate is available.

Parameters	$(\ell,m)$	$A(x,y)$	$B(x,y)$	$\mathrm{FOM}$	Evidence
$\left[\!\left[90,4,9\right]\!\right]$	$(5,9)$	$1+x^{2}y^{8}+x^{3}y^{7}$	$1+x^{2}y^{6}$	3.60	M+component
$\left[\!\left[96,4,10\right]\!\right]$	$(16,3)$	$1+y+x^{2}y^{2}$	$1+x^{3}$	4.17	M+component
$\left[\!\left[120,4,10\right]\!\right]$	$(6,10)$	$1+xy+x^{5}y^{5}$	$1+y$	3.33	M
$\left[\!\left[140,6,10\right]\!\right]$	$(5,14)$	$1+y^{13}+x^{4}y^{11}$	$1+xy^{7}$	4.29	M+component
$\left[\!\left[180,4,13\right]\!\right]$	$(10,9)$	$1+x^{2}y^{3}$	$1+x^{5}y+x^{7}y^{8}$	3.76	M
$\left[\!\left[180,4,14\right]\!\right]$	$(10,9)$	$1+x^{2}y^{2}+x^{5}y$	$1+x^{3}y^{3}$	4.356	M

Table 3: Selected connected weight-five CSS BB presentations. Exact distances are certified; interval rows combine a certified lower bound with a supported upper endpoint, so the corresponding displayed FOMs are upper bounds. M denotes exact MILP certification over all logical sectors of the displayed presentation; M+class transfers such a certificate from an isomorphic presentation in the same presentation class; M+component denotes a connected BB realization shown by BLISS canonicalization to be isomorphic to a component of an MILP-certified parent. B denotes a decoder estimate whose operator was not retained, I a nonoptimal MILP incumbent, W an independently reconstructed and verified logical operator, and L4 exhaustive exclusion through weight four. Each row specifies a presentation through its lattice and generators; matching

(n,k)

values do not imply equivalence.

Parameters	Ambient group	Cyclic form $(a(z),b(z))$	Complete common-kernel sector
$\left[\!\left[90,4,9\right]\!\right]$	$C_{45}$	$(1+z^{17}+z^{43},\ 1+z^{42})$	$h=z^{2}+z+1$ (order 3)
$\left[\!\left[96,4,10\right]\!\right]$	$C_{48}$	$(1+z^{2}+z^{16},\ 1+z^{3})$	$h=z^{2}+z+1$ (order 3)
$\left[\!\left[120,4,10\right]\!\right]$	$C_{6}\times C_{10}$	—	$\phi(i,j)=i\pmod{3}$
$\left[\!\left[140,6,10\right]\!\right]$	$C_{70}$	$(1+z^{39}+z^{55},\ 1+z^{21})$	$h=z^{3}+z^{2}+1$ (order 7)
$\left[\!\left[180,4,13\right]\!\right]$	$C_{90}$	$(1+z^{12},\ 1+z^{17}+z^{55})$	$h=z^{2}+z+1$ (order 3)
$\left[\!\left[180,4,14\right]\!\right]$	$C_{90}$	$(1+z^{2}+z^{55},\ 1+z^{3})$	$h=z^{2}+z+1$ (order 3)

Table 4: Algebraic classification of the exact connected CSS constructions. The cyclic forms use a CRT generator

z

;

C_{r}

denotes the cyclic group of order

r

. The ordering of

a

and

b

matches the corresponding presentation in Table 3 , which need not be the representative selected for the supplemental class summary. For noncyclic ambient groups, the last column gives an explicit quotient character.

Figure 1: Exact-distance Pareto frontiers within the compiled dataset, grouped by maximum stabilizer weight after excluding

d\leq 2

. Panel (a) shows the nondominated rate–distance frontier; panel (b) shows the idealized data-qubit ratio

kd^{2}/n

relative to

k

independent rotated surface-code patches as block length increases. Stars and diamonds denote component-class points originating from CSS and PBB codes in this study, respectively; circles denote prior constructions and Challenge entries; color denotes maximum stabilizer weight. The cohorts are noncumulative, and the compiled dataset is not an exhaustive census of the corresponding design spaces. The

\left[\!\left[12,2,3\right]\!\right]

label marks the first available point in the compiled weight-five series; shorter lengths are currently unsampled. Lines are guides to the eye. Every current or prior campaign entry with stored polynomial supports is reduced to one translation-connected component before frontier selection, and all Challenge entries receive the basis-independent connectivity test.

Weight	Panel	Pareto points, in plot order
4	(a)	$\left[\!\left[16,2,4\right]\!\right]^{\mathrm{Q_{c}}}$ , $\left[\!\left[84,2,9\right]\!\right]^{\mathrm{Q}}$
4	(b)	$\left[\!\left[16,2,4\right]\!\right]^{\mathrm{Q_{c}}}$
5	(a)	$\left[\!\left[180,4,14\right]\!\right]^{\mathrm{O}}$ , $\left[\!\left[132,4,12\right]\!\right]^{\mathrm{LP}}$ , $\left[\!\left[140,6,10\right]\!\right]^{\mathrm{O_{c}}}$ , $\left[\!\left[78,4,9\right]\!\right]^{\mathrm{LP}}$ , $\left[\!\left[60,4,8\right]\!\right]^{\mathrm{LP}}$ , $\left[\!\left[48,4,7\right]\!\right]^{\mathrm{LP}}$ , $\left[\!\left[36,4,6\right]\!\right]^{\mathrm{K}}$ , $\left[\!\left[30,4,5\right]\!\right]^{\mathrm{V}}$ , $\left[\!\left[40,10,4\right]\!\right]^{\mathrm{Q}}$
5	(b)	$\left[\!\left[12,2,3\right]\!\right]^{\mathrm{O_{p}}}$ , $\left[\!\left[30,4,5\right]\!\right]^{\mathrm{V}}$ , $\left[\!\left[36,4,6\right]\!\right]^{\mathrm{K}}$ , $\left[\!\left[48,4,7\right]\!\right]^{\mathrm{LP}}$ , $\left[\!\left[60,4,8\right]\!\right]^{\mathrm{LP}}$ , $\left[\!\left[132,4,12\right]\!\right]^{\mathrm{LP}}$
6	(a)	$\left[\!\left[336,12,20\right]\!\right]^{\mathrm{QL}}$ , $\left[\!\left[340,16,18\right]\!\right]^{\mathrm{L}}$ , $\left[\!\left[254,14,16\right]\!\right]^{\mathrm{L}}$ , $\left[\!\left[144,12,12\right]\!\right]^{\mathrm{B}}$ , $\left[\!\left[90,8,10\right]\!\right]^{\mathrm{B}}$ , $\left[\!\left[72,8,8\right]\!\right]^{\mathrm{P}}$ , $\left[\!\left[72,12,6\right]\!\right]^{\mathrm{B}}$ , $\left[\!\left[18,4,4\right]\!\right]^{\mathrm{Q}}$
6	(b)	$\left[\!\left[18,4,4\right]\!\right]^{\mathrm{Q}}$ , $\left[\!\left[30,4,6\right]\!\right]^{\mathrm{Q}}$ , $\left[\!\left[48,4,8\right]\!\right]^{\mathrm{Q}}$ , $\left[\!\left[60,4,10\right]\!\right]^{\mathrm{K}}$ , $\left[\!\left[72,8,8\right]\!\right]^{\mathrm{P}}$ , $\left[\!\left[84,6,10\right]\!\right]^{\mathrm{Q}}$ , $\left[\!\left[90,8,10\right]\!\right]^{\mathrm{B}}$ , $\left[\!\left[120,8,12\right]\!\right]^{\mathrm{L}}$ , $\left[\!\left[144,12,12\right]\!\right]^{\mathrm{B}}$ , $\left[\!\left[254,14,16\right]\!\right]^{\mathrm{L}}$ , $\left[\!\left[336,12,20\right]\!\right]^{\mathrm{QL}}$ , $\left[\!\left[340,16,18\right]\!\right]^{\mathrm{L}}$

Table 5: Exact-distance points defining the per-weight frontiers in the compiled dataset shown in Fig. 1 , listed in plot order. Source superscripts are O (this work, connected CSS catalogue proposal), O c (this work, CSS component-class point, with a separately supplied explicit polynomial realization when available), O p (this work, PBB component-class point), Q (connected QEC Challenge entry), Q c (QEC Challenge component), V (Voss et al. [ 25 ] ), K (Khesin–Lu [ 11 ] ), B (Bravyi et al. [ 2 ] ), LP (Lin–Pryadko archive construction, with distance independently certified here), L (Liang et al. [ 13 ] ), P (our previous work [ 5 ] ), QL (Qian–Li [ 19 ] ), and S (Symons et al. [ 22 ] ).

Parameters	$(\ell,m)$	$(A,B)$	$(C,D)$	$\mathrm{FOM}$	Structure	Evidence
$\left[\!\left[144,4,6\right]\!\right]$	$(9,8)$	$(1+x^{6}y,\ 1+x^{2}y^{4}+x^{4})$	$(0,\ x^{2}y^{4})$	1.00	conn.	Exh.+M
$\left[\!\left[144,4,8\right]\!\right]$	$(18,4)$	$(1+x^{3}y,\ 1+xy^{2}+x^{2})$	$(1+x^{3}y,\ xy^{2})$	1.78	conn.	M+component
$\left[\!\left[180,2,11\right]\!\right]$	$(15,6)$	$(1+x^{2}y^{2},\ 1+x^{2}y^{3}+x^{4})$	$(1+x^{2}y^{2},\ 1+x^{4})$	1.34	conn.	M+component
$\left[\!\left[216,2,8\right]\!\right]$	$(12,9)$	$(1+x+x^{2},\ 1+y)$	$(x,\ 1+y)$	0.59	conn.	M; baseline
$\left[\!\left[216,4,10\right]\!\right]$	$(18,6)$	$(1+x^{2}y,\ 1+xy^{3}+x^{2})$	$(1+x^{2}y,\ 1+x^{2})$	1.85	conn.	M+component
$\left[\!\left[216,4,10\right]\!\right]$	$(18,6)$	$(1+x^{4}y,\ 1+xy^{3}+x^{2})$	$(0,\ 1+x^{2})$	1.85	conn.	M+component

Table 6: Selected weight-five PBB presentations. Exact rows are certified. Interval rows combine exclusion through weight four with a supported upper endpoint, so their FOMs are upper bounds. Evidence labels are Exh.+M (exhaustive search corroborated by all-logical MILP), M (all-logical MILP), M+component (a connected presentation matched by colored-BLISS to a component of a homogeneous M-certified parent), and L4+I+B (certified

d\geq 5

plus both a nonoptimal MILP incumbent and a decoder estimate). Baseline rows identify fixed-family controls supplied by the campaign evaluators independently of model output. Each row specifies a particular polynomial presentation, and repeated parameter triples need not denote the same class. In particular, the two

\left[\!\left[216,4,10\right]\!\right]

rows belong to distinct component classes, and the

\left[\!\left[144,4,6\right]\!\right]

component class underlying the displayed

\left[\!\left[288,8,6\right]\!\right]

parent is distinct from that of the displayed connected

\left[\!\left[144,4,6\right]\!\right]

row. “conn.” denotes connected.

Endpoint	LLM-guided output	Stratified-uniform control
Records/evaluations	6,444 logged positive-distance observations	6,444 completed selected evaluations
Distinct proposals	936	6,444
Distinct certified- $d\geq 5$ proposals	816/936 (87.18%)	4,672–4,785/6,444 (72.50–74.26%)
Connected certified- $d\geq 5$ classes	311	566–572
Highest-FOM campaign-certified class sampled	$\left[\!\left[180,4,14\right]\!\right]$ ; 4.356	$\left[\!\left[180,4,14\right]\!\right]$ ; 4.356

Table 7: Weight-five transfer control. Control entries are ranges over three stratified-uniform replicates.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Campaign	Logged obs.	Kept obs.	All	Kept	Classes	Removed	Exact	Unresolved
CSS-small	5,610	5,269	382	365	164	17	99	266
CSS-large	834	824	554	547	371	7	94	453
PBB-small	4,890	3,625	257	151	39	106	151	0
PBB-large	2,589	2,318	104	79	56	25	54	25
Total	13,923	12,036	1,297	1,142	630	155	398	744

Appendix

Table S1: Catalogue accounting after filtering. “Logged obs.” counts all observations with a positive reported distance; “Kept obs.” restricts this count to retained proposals. “All” and “Kept” give proposal counts before and after filtering, and “Classes” gives the number of retained presentation classes. Exact and unresolved counts are proposal-level statuses before the final class merge and can include low-weight evidence transferred from a verified representative. Unresolved rows pair a certified lower bound with either a rigorous upper bound or a decoder estimate.

Code	$(\ell,m)$	$A$	$B$	FOM	Certified parent
$\left[\!\left[90,4,9\right]\!\right]$	$(5,9)$	$1+x^{2}y^{8}+x^{3}y^{7}$	$1+x^{2}y^{6}$	3.60	$\left[\!\left[180,8,9\right]\!\right]$
$\left[\!\left[96,4,10\right]\!\right]$	$(16,3)$	$1+y+x^{2}y^{2}$	$1+x^{3}$	4.17	$\left[\!\left[384,16,10\right]\!\right]$
$\left[\!\left[140,6,10\right]\!\right]$	$(5,14)$	$1+y^{13}+x^{4}y^{11}$	$1+xy^{7}$	4.29	$\left[\!\left[420,18,10\right]\!\right]$

Appendix

Table S2: Explicit connected BB presentations of components of certified disconnected parents. Canonical labelling verifies that the corresponding presentation graphs are isomorphic, transferring the component’s certified distance to each presentation.

Code	$(\ell,m)$	$(A,B)$	$(C,D)$	FOM	Certified parent
$\left[\!\left[144,4,8\right]\!\right]$	$(18,4)$	$(1+x^{3}y,\ 1+xy^{2}+x^{2})$	$(1+x^{3}y,\ xy^{2})$	1.78	$\left[\!\left[432,12,8\right]\!\right]$
$\left[\!\left[180,2,11\right]\!\right]$	$(15,6)$	$(1+x^{2}y^{2},\ 1+x^{2}y^{3}+x^{4})$	$(1+x^{2}y^{2},\ 1+x^{4})$	1.34	$\left[\!\left[360,4,11\right]\!\right]$
$\left[\!\left[216,4,10\right]\!\right]$	$(18,6)$	$(1+x^{2}y,\ 1+xy^{3}+x^{2})$	$(1+x^{2}y,\ 1+x^{2})$	1.85	$\left[\!\left[432,8,10\right]\!\right]$
$\left[\!\left[216,4,10\right]\!\right]$	$(18,6)$	$(1+x^{4}y,\ 1+xy^{3}+x^{2})$	$(0,\ 1+x^{2})$	1.85	$\left[\!\left[432,8,10\right]\!\right]$

Appendix

Table S3: Explicit connected PBB presentations of components of certified disconnected parents. Canonical labelling verifies that the corresponding presentation graphs are isomorphic, transferring the component’s certified distance to each presentation.

Fam.	Code	Rep.	$(\ell,m)$	$(A,B)$	$(C,D)$	Decomp.	FOM	Props.	Cls.	Conn./disc.
CSS	$\left[\!\left[120,4,4\right]\!\right]$	CS-d74f23a3	$(6,10)$	$(1+y^{3},\,1+x+x^{2})$	–	conn.	0.53	1	1	1/0
CSS	$\left[\!\left[120,4,10\right]\!\right]$	CS-07d0fcb6	$(6,10)$	$(1+xy+x^{5}y^{5},\,1+y)$	–	conn.	3.33	6	1	1/0
CSS	$\left[\!\left[120,8,3\right]\!\right]$	CS-183fa7f4	$(6,10)$	$(1+xy^{2}+x^{2}y^{4},\,1+x^{3}y^{6})$	–	$2\times\left[\!\left[60,4\right]\!\right]$	0.60	2	1	0/1
CSS	$\left[\!\left[120,8,4\right]\!\right]$	CS-ab5fa0e9	$(6,10)$	$(1+x^{2}y^{2}+x^{4}y^{4},\,1+x^{3}y^{8})$	–	$2\times\left[\!\left[60,4\right]\!\right]$	1.07	5	3	0/3
CSS	$\left[\!\left[120,8,5\right]\!\right]$	CS-999229aa	$(6,10)$	$(1+y^{2},\,1+x+x^{5}y^{3})$	–	conn. *	1.67	17	9	3/6
CSS	$\left[\!\left[120,16,3\right]\!\right]$	CS-c600e4d1	$(6,10)$	$(1+x^{2}y^{2}+x^{4}y^{6},\,1+y^{2})$	–	$4\times\left[\!\left[30,4\right]\!\right]$	1.20	1	1	0/1

Appendix

Table S4 : Certified parameter summary after combining evidence within presentation classes. For each exact parameter triple, Props. and Cls. respectively count retained proposals and presentation classes; Conn./disc. splits the classes into connected and disconnected. The displayed distance and FOM reflect class-level evidence. Rep. is chosen deterministically, preferring an exact proposal-level status and then a connected presentation. The representative columns give its lattice, generators (with – in

(C,D)

for CSS), and component decomposition. The selected representative need not supply the decisive certificate. An asterisk marks a row containing both connected and disconnected classes; in such a row Decomp. still describes only the representative. The seven explicit connected-component presentations in Tables S2 and S3 are excluded from these counts.

ID/class	$(\ell,m)$	$\left[\!\left[n,k\right]\!\right]$	$d$	Own evidence	Class effect	FOM	$A$	$B$	Evid.	Obs.	Attr.	Decomp.
CS-435681c1/C040	$(6,10)$	$\left[\!\left[120,16\right]\!\right]$	$5$	Exact	–	$3.33$	$1+x^{2}y^{2}+x^{4}$	$1+y^{6}$	M	9	G	$4\times\left[\!\left[30,4\right]\!\right]$
CS-9dfccec1/C038	$(6,10)$	$\left[\!\left[120,16\right]\!\right]$	$4$	Exact	–	$2.13$	$1+y^{4}$	$1+x^{2}+x^{4}y^{4}$	Exh.	1	G	$4\times\left[\!\left[30,4\right]\!\right]$
CS-8146fb41/C038	$(6,10)$	$\left[\!\left[120,16\right]\!\right]$	$4$	Exact	–	$2.13$	$1+x^{2}y^{4}+x^{4}$	$1+y^{6}$	Exh.	1	G	$4\times\left[\!\left[30,4\right]\!\right]$
CS-c600e4d1/C039	$(6,10)$	$\left[\!\left[120,16\right]\!\right]$	$3$	Exact	–	$1.20$	$1+x^{2}y^{2}+x^{4}y^{6}$	$1+y^{2}$	Exh.	121	G/O	$4\times\left[\!\left[30,4\right]\!\right]$
CS-999229aa/C022	$(6,10)$	$\left[\!\left[120,8\right]\!\right]$	$5$	Exact	–	$1.67$	$1+y^{2}$	$1+x+x^{5}y^{3}$	L4+W+B	2	G	conn.
CS-fc4dc8ca/C025	$(6,10)$	$\left[\!\left[120,8\right]\!\right]$	$5$	Exact	–	$1.67$	$1+y^{2}$	$1+xy^{2}+x^{2}y^{6}$	L4+W+B	1	G	$2\times\left[\!\left[60,4\right]\!\right]$

Appendix

Table S5 : CSS-small compact catalogue (224 displayed of 365 retained proposals; 164 presentation classes). The table includes every proposal in an exact class and one deterministic representative of every nonexact class. Own evidence and Evid. describe the proposal before class merging; Class effect reports when merging makes the displayed

d

and FOM exact, tighter, or rigorously supported. Column abbreviations, filtering, and evidence conventions are defined above.

ID/class	$(\ell,m)$	$\left[\!\left[n,k\right]\!\right]$	$d$	Own evidence	Class effect	FOM	$A$	$B$	Evid.	Obs.	Attr.	Decomp.
CL-929c2a70/C248	$(16,12)$	$\left[\!\left[384,64\right]\!\right]$	$3$	Exact	–	$1.50$	$1+x^{4}y^{4}+x^{12}y^{8}$	$1+x^{4}$	H c	1	–	$16\times\left[\!\left[24,4\right]\!\right]$
CL-a25b2baf/C247	$(16,12)$	$\left[\!\left[384,32\right]\!\right]$	$6$	Exact	–	$3.00$	$1+x^{4}y^{2}+x^{4}y^{10}$	$1+x^{2}y^{9}$	H c	1	–	$8\times\left[\!\left[48,4\right]\!\right]$
CL-83dd7b5d/C245	$(16,12)$	$\left[\!\left[384,32\right]\!\right]$	$5$	Exact	–	$2.08$	$1+x^{2}y^{6}$	$1+x^{2}y^{2}+x^{4}y^{4}$	M	2	G	$8\times\left[\!\left[48,4\right]\!\right]$
CL-a94de289/C244	$(16,12)$	$\left[\!\left[384,32\right]\!\right]$	$4$	Exact	–	$1.33$	$1+x^{2}y^{5}+x^{4}y^{10}$	$1+x^{4}y^{6}$	Exh.+I	4	G/O	$8\times\left[\!\left[48,4\right]\!\right]$
CL-02bb1787/C243	$(16,12)$	$\left[\!\left[384,32\right]\!\right]$	$4$	Exact	–	$1.33$	$1+x^{4}$	$1+y^{2}+y^{4}$	Exh.	2	G/N	$8\times\left[\!\left[48,4\right]\!\right]$
CL-d9f65a7e/C246	$(16,12)$	$\left[\!\left[384,32\right]\!\right]$	$3$	Exact	–	$0.75$	$1+x^{4}y^{2}+x^{10}y^{7}$	$1+x^{2}y^{3}$	H c	1	–	$8\times\left[\!\left[48,4\right]\!\right]$

Appendix

Table S6 : CSS-large compact catalogue (391 displayed of 547 retained proposals; 371 presentation classes). The table includes every proposal in an exact class and one deterministic representative of every nonexact class. Own evidence and Evid. describe the proposal before class merging; Class effect reports when merging makes the displayed

d

and FOM exact, tighter, or rigorously supported. Column abbreviations, filtering, and evidence conventions are defined above.

ID/class	$(\ell,m)$	$\left[\!\left[n,k\right]\!\right]$	$d$	Own evidence	Class effect	FOM	$A$	$B$	$C$	$D$	Evid.	Obs.	Attr.	Decomp.
PS-5feb6ebb/P013	$(6,6)$	$\left[\!\left[72,12\right]\!\right]$	$3$	Exact	–	$1.50$	$1+x^{3}y^{3}$	$1+xy^{3}+x^{2}$	$0$	$1+x^{2}$	Exh.	1	G	$6\times\left[\!\left[12,2\right]\!\right]$
PS-64d64fd9/P012	$(6,6)$	$\left[\!\left[72,12\right]\!\right]$	$3$	Exact	–	$1.50$	$1+x^{3}y^{3}$	$1+xy^{3}+x^{2}$	$0$	$xy^{3}$	Exh.	8	G/O	$6\times\left[\!\left[12,2\right]\!\right]$
PS-106b8669/P014	$(6,6)$	$\left[\!\left[72,12\right]\!\right]$	$3$	Exact	–	$1.50$	$1+x^{3}y^{3}$	$1+xy^{3}+x^{2}$	$1+x^{3}y^{3}$	$1+x^{2}$	Exh.	2	G/N	$6\times\left[\!\left[12,2\right]\!\right]$
PS-f4956d3c/P011	$(6,6)$	$\left[\!\left[72,12\right]\!\right]$	$3$	Exact	–	$1.50$	$1+x^{3}y^{3}$	$1+xy^{3}+x^{2}$	$1+x^{3}y^{3}$	$xy^{3}$	Exh.	1	G	$6\times\left[\!\left[12,2\right]\!\right]$
PS-edb2da66/P013	$(6,6)$	$\left[\!\left[72,12\right]\!\right]$	$3$	Exact	–	$1.50$	$1+x^{3}y^{3}$	$1+x^{4}+x^{5}y^{3}$	$0$	$1+x^{4}$	Exh.	1	G	$6\times\left[\!\left[12,2\right]\!\right]$
PS-1a20f6c1/P012	$(6,6)$	$\left[\!\left[72,12\right]\!\right]$	$3$	Exact	–	$1.50$	$1+x^{3}y^{3}$	$1+x^{4}+x^{5}y^{3}$	$0$	$x^{5}y^{3}$	Exh.	3	G	$6\times\left[\!\left[12,2\right]\!\right]$

Appendix

Table S7 : PBB-small compact catalogue (151 displayed of 151 retained proposals; 39 presentation classes). The table includes every proposal in an exact class and one deterministic representative of every nonexact class. Own evidence and Evid. describe the proposal before class merging; Class effect reports when merging makes the displayed

d

and FOM exact, tighter, or rigorously supported. Column abbreviations, filtering, and evidence conventions are defined above.

ID/class	$(\ell,m)$	$\left[\!\left[n,k\right]\!\right]$	$d$	Own evidence	Class effect	FOM	$A$	$B$	$C$	$D$	Evid.	Obs.	Attr.	Decomp.
PL-ba29a9df/P040	$(12,9)$	$\left[\!\left[216,2\right]\!\right]$	$8$	Exact	–	$0.59$	$1+y$	$1+x+x^{2}$	$1+y$	$x$	M	126	G/O/N/F	conn.
PL-39082ddc/P040	$(12,9)$	$\left[\!\left[216,2\right]\!\right]$	$8$	Exact	–	$0.59$	$1+x+x^{2}$	$1+y$	$x$	$1+y$	M	126	G/O/N/F	conn.
PL-9b97a02b/P053	$(12,12)$	$\left[\!\left[288,48\right]\!\right]$	$3$	Exact	–	$1.50$	$1+x^{6}y^{6}$	$1+x^{2}y^{6}+x^{4}$	$1+x^{6}y^{6}$	$1+x^{4}$	Exh.	20	G/O	$24\times\left[\!\left[12,2\right]\!\right]$
PL-78c3e7bf/P052	$(12,12)$	$\left[\!\left[288,24\right]\!\right]$	$4$	Exact	–	$1.33$	$1+x^{3}y^{6}$	$1+x^{2}y^{6}+x^{4}$	$0$	$1+x^{4}$	Exh.	3	G/O	$6\times\left[\!\left[48,4\right]\!\right]$
PL-ad1cc828/P049	$(12,12)$	$\left[\!\left[288,16\right]\!\right]$	$4$	Exact	–	$0.89$	$1+xy$	$1+x^{2}y^{6}+x^{4}$	$0$	$1+x^{4}$	Exh.	28	G/O	$4\times\left[\!\left[72,4\right]\!\right]$
PL-0bbae5c3/P048	$(12,12)$	$\left[\!\left[288,16\right]\!\right]$	$4$	Exact	–	$0.89$	$1+xy^{5}$	$1+x^{2}y^{6}+x^{4}$	$1+xy^{5}$	$x^{2}y^{6}$	Exh.	13	G/O	$4\times\left[\!\left[72,4\right]\!\right]$

Appendix

Table S8 : PBB-large compact catalogue (73 displayed of 79 retained proposals; 56 presentation classes). The table includes every proposal in an exact class and one deterministic representative of every nonexact class. Own evidence and Evid. describe the proposal before class merging; Class effect reports when merging makes the displayed

d

and FOM exact, tighter, or rigorously supported. Column abbreviations, filtering, and evidence conventions are defined above.

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Jun 1, 2026, quant-phJ/K move · Enter open · S save

Evolutionary Discovery of Bivariate Bicycle Codes with LLM-Guided Search

Juan Cruz-Benito, Andrew W. Cross, David Kremer, Ismael Faro

IBM Research, IBM T. J. Watson Research Center, NY, USA

Quantum Error Correction Quantum Annealing Codebooks Alphaevolve Evolutionary Optimization Methods LLM Inference Optimization+1

Open paper AskPDF arXiv

Quantum LDPC code discovery requires searching large algebraic design spaces while reliably certifying the parameters and equivalence classes of any candidates found. We introduce an LLM-guided evolutionary workflow in which language models mutate Python programs that generate bivariate-bicycle and perturbed bivariate-bicycle code ansätze. Across five campaigns, the system performed approximately 1{,}650 evolutionary iterations, screened about

2 \times 10^5

candidate codes, and required

{\sim}140

hours of computation and

{\sim}

US$400 in LLM inference cost. Candidate codes are evaluated through a staged validation pipeline combining

\mathrm{GF}(2)

rank computation, distance estimation and certification, mixed-integer linear programming, BLISS Tanner-graph deduplication, decomposability analysis, and local-Clifford equivalence checks. At block length

n \leq 360

, the workflow identifies 465 distinct candidate codes: 97 CSS bivariate-bicycle codes and 368 non-CSS perturbed variants. The CSS search recovers known high-performing codes and finds new finite-length representatives, including an indecomposable [[288,16,12]] code and higher-weight codes with up to

k = 50

at distance

d = 8

. The non-CSS search produces perturbed codes matching the gross-code figure of merit at [[144,12,12]], along with additional high-distance candidates reported as certified values or upper bounds according to MILP status. Overall, these results show that LLM-guided program evolution can serve as a practical tool for structured quantum-code discovery when paired with independent evaluation.

Abstract

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