Pair-Partition Constructions for CPM- and GPM-Based Quantum LDPC Codes

Koki Okada and Kenta Kasai. Updated August 17, 2026.

This page lists 44 exact-distance PP-based CSS quantum LDPC codes of girth at least 6. It provides their construction and distance-verification data.

arXiv:2607.14091 Software on GitHub SHA-256 manifest Girth-8 seed randomization Rank-bound tip Girth-8 tip Citation

This webpage is the authoritative catalogue for the reported code parameters, fixed instances, and verification records. The GitHub repository is maintained separately for construction, search, and verification software; any instance files kept there serve only as program inputs and regression fixtures.

Recent Certified PP Additions (August 2026)

This release adds completed PP artifacts produced after the previous APM-PP update. Exact-distance codes enter the catalogue; alternative AOD and computation-oriented presentations are linked without being counted as new binary equivalence classes. The already published [[576,294,12]] package is not duplicated, and unfinished searches are excluded.

[[560,286,12]]

(J,L,P)=(4,16,35), CPM-PP/APM-PP, exact dX=dZ=12.

This is the updated (4,16) length-distance frontier: it replaces [[576,294,12]] at exact distance 12, using 16 fewer qubits.

APMCPMexact/frontier certificate

[[512,132,14]]

(J,L,P)=(3,8,64), APM-PP, girth 8, exact dX=dZ=14.

The absolute AOD schedule reduces distinct cyclic shifts from (24,27) to (17,17). A separate relative-schedule realization is provided under its own objective.

absolute-AOD NPZabsolute certificaterelative-AOD NPZrelative certificate

[[2048,516,24]]

(J,L,P)=(3,8,256), genuine APM-PP, girth 8, exact d=24.

The construction audit excludes CPM gauge reduction. A weight-24 X logical and independent lower certificates establish the quantum distance.

NPZexact-distance auditindependent lower audit

[[3840,1540,18]]

(J,L,P)=(3,10,384), genuine APM-PP, girth 6, exact dX=dZ=18.

Construction and distance audits, complete lower searches through weight 16, and explicit weight-18 logicals are included.

NPZindependent auditdistance summary

[[472,122,16]] strict APM

(J,L,P)=(3,8,59), strict all-nonzero APM-PP, girth 8.

An explicit qubit permutation transfers the existing exact-distance certificate. This is a computation-oriented presentation, not a new binary equivalence class.

strict-APM NPZcertificateFTQC architecture data

[[1122,148,d]] compute presentation

Sparse APM-PP representation with computation artifacts; the current source record supplies only d<=20.

It is therefore published as a distance-pending representation and is deliberately omitted from the exact-distance catalogue.

certificateblock maps

machine-readable indexSHA-256figure (PNG)

Promising Lower-Bound and Distance-Pending Candidates

These fixed PP instances include certified lower-bound improvements for the \((J,L)=(4,12)\) family and two minimal-twist \((3,8)\) APM-PP codes. Their exact distances are still being determined, so every graph places each such point at its proved lower bound. The \((4,16)\) GPM-PP entry below is different: its distance is still pending, and its open plot marker at \(d_{\rm target}=18\) is a search target, not a certified lower bound.

[[1536,388,d≥26]]

(J,L,P)=(3,8,192), minimal-twist Twisted APM-PP, girth 8, rate 0.253.

Complete exclusion through weight 24 on both CSS sides, together with the all-one row-space parity certificate, gives d≥26. The 48 affine blocks use only slopes 1 and 97; 32 blocks are twisted and the remaining 16 are ordinary CPM shifts. This Okada construction is a Z₂-twisted double cover of a valid [[768,196]] CPM-PP code and improves the certified \((3,8)\) length-distance frontier. No matching upper-bound logical is included.

READMEAPM-PP inputX checksZ checksverifierX lower certificateZ lower certificatesource bundleSHA-256

[[2304,580,d≥26]]

(J,L,P)=(3,8,288), minimal-twist APM-PP, girth 8, rate 0.252.

Complete exclusion through weight 24 on both CSS sides, together with the all-one row-space parity certificate, gives d≥26. The 48 affine blocks use only slopes 1 and 145; 20 blocks are twisted and the remaining 28 are ordinary CPM shifts. No matching upper-bound logical is included.

READMEAPM-PP inputX checksZ checksverifierX lower certificateZ lower certificatesource bundleSHA-256

[[876,298,d≥20]]

(J,L,P)=(4,12,73), girth 6, rate 0.340.

Complete exclusion through weight 18; the weight-20 search is continuing.

CPMverification indexlower-bound audit

[[948,322,d≥22]]

(J,L,P)=(4,12,79), girth 6, rate 0.340.

Complete exclusion through weight 20; the weight-22 search is continuing.

CPMverification indexlower-bound audit

[[1068,362,d≥26]]

(J,L,P)=(4,12,89), girth 6, rate 0.339.

Complete exclusion through weight 24 on both CSS sides; the even-weight certificate gives d≥26, while verified weight-84 logicals give d≤84.

CPMverification indexlower-bound auditcertificate bundleSHA-256

[[784,398,d pending]]

(J,L,P)=(4,16,49), \(C_7\times C_7\) GPM-PP, girth 6, rate 0.508.

Exact exclusion through weight 17 is complete for 9 of 16 symmetry-representative roots on each CSS side. No weight-at-most-17 witness was found in the completed shards, but the unfinished roots prevent a global lower-bound claim. A verified X-type logical gives \(d\le78\).

GPM-PP NPZconstruction auditpartial distance statusweight-78 logicalSHA-256

Distance versus Block Length by \((J,L)\)

The plot contains only the \((n,d)\)-nondominated points within each \((J,L)\) family. It derives those points from the complete catalogue and the girth-4 graph-only set; non-frontier codes remain available in the complete table below. Choose a \((J,L)\) family, then point to or select a code to display its construction parameters and data files. Pointing to a code highlights only its \((J,L)\) family and the corresponding \(d=\delta_{J,L}n\) line. Exact-distance points have both a complete lower-bound search and a matching logical vector. Promising lower-bound candidates are plotted at their certified lower bounds, not at their known upper bounds. The open \((784,18)\) marker is explicitly provisional: 18 is the current search target and is not a distance claim. Starting at the origin \((0,0)\), the dashed line for each family is \(d=\delta_{J,L}n\), where \(\delta_{J,L}\) is the typical relative-distance baseline of the corresponding regular one-edge LDPC configuration ensemble.

Catalogue of Verified Codes and Promising Candidates

The yellow rows are the six new GPM-PP search additions; two are cyclic-equivalent composite lifts and four use genuinely noncyclic groups. The purple rows are exact-distance APM-PP additions. Only their nondominated members appear in the automatically generated figures; every certified point remains in this table. The green row gives a girth-6 small-lift frontier point found by searching prime lift sizes in increasing order. All girth-4 instances are omitted For every displayed point, both CSS matrices and the positive encoded dimension were independently checked. Exact-distance rows additionally have a matching verified logical vector; rows marked with \(d\geq d_0\) state only the independently certified lower bound \(d_0\). The girth is shown separately for each row. The displayed \(P\) for \(\lbrack\!\lbrack 1414,812,12\rbrack\!\rbrack\) is the smallest found in the sampled search, not a proof of global optimality over every PP array. The remaining rows agree with Table I of the current paper. Among those paper codes, the highlighted \(\lbrack\!\lbrack 472,122,16\rbrack\!\rbrack\), \(\lbrack\!\lbrack 584,150,18\rbrack\!\rbrack\), \(\lbrack\!\lbrack 1112,282,20\rbrack\!\rbrack\), \(\lbrack\!\lbrack 1336,338,22\rbrack\!\rbrack\), \(\lbrack\!\lbrack 1630,656,20\rbrack\!\rbrack\), \(\lbrack\!\lbrack 1784,450,24\rbrack\!\rbrack\), and \(\lbrack\!\lbrack 2230,896,24\rbrack\!\rbrack\) codes have girth 8 on both sides. The first three codes were provided by Nishad Maskara and independently verified for this work.

Every catalogue row now links a fixed-input certificate bundle, a verification index, and SHA-256 checksums. The exact-distance label is used only when the lower-bound payload and an explicit logical vector have both passed independent verification.

complete row-by-row audit (JSON) audit table (Markdown) master manifest

Worked Construction Example

The following steps trace the construction of the published \(\lbrack\!\lbrack 472,122,14\rbrack\!\rbrack\) instance. They show how a pair-partition design becomes two binary CSS check matrices and how the stated code parameters are then verified.

1. Choose the block parameters. Set \(J=3\), \(L=8\), and \(P=59\). Thus each CSS check matrix will have \(JP=177\) rows and \(LP=472\) columns, with column weight 3 and row weight 8.
2. Generate and screen the pair-partition array M. Each of the nine cells \(M_{ij}\) is a perfect matching of \(\{0,\ldots,7\}\). For example,
   \[ M_{00}=\bigl\{\{0,5\},\{1,7\},\{2,4\},\{3,6\}\bigr\}. \]
   The matchings are arranged so that no unordered pair is reused within an array row or column. The associated pairing graphs are checked before any exponent search; a design that already forces a forbidden short cycle is discarded here.
3. Form the linear system from M. Write \(E=(e_{i,\ell})\) and \(D=(d_{j,\ell})\). Every pair \(\{u,v\}\) in \(M_{ij}\) gives
   \[ d_{j,u}-e_{i,u}\equiv d_{j,v}-e_{i,v}\pmod P. \]
   Here there are 48 exponent variables and 36 raw equations. The pair \(\{0,5\}\) in \(M_{00}\), for example, gives
   \[ \begin{aligned} d_{0,0}-e_{0,0} &\equiv d_{0,5}-e_{0,5}\pmod{59},\\ 11-53 &\equiv 2-44 \equiv 17\pmod{59}. \end{aligned} \]
   Thus the published exponents satisfy this row of \(A_Mx=0\).
4. Solve over \(\mathbb F_{59}\) and screen the exponents. One surviving solution is
   \[ \begin{aligned} E&=\begin{bmatrix} 53&43&36&43&56&44&49&9\\ 56&8&52&24&2&45&29&47\\ 5&14&44&22&23&39&33&32 \end{bmatrix},\\[4pt] D&=\begin{bmatrix} 11&36&9&51&29&2&57&2\\ 9&56&3&11&12&57&16&36\\ 51&48&18&43&56&26&54&7 \end{bmatrix}. \end{aligned} \]
   Candidates that fail the same-side CPM cycle tests or reproduce a stored low-weight logical pattern are rejected. The stored patterns accelerate construction; they are not used as a distance proof.
5. Lift the exponents to CPMs and verify the CSS code. Replace every entry of \(E\) and \(D\) by the corresponding 59-by-59 circulant permutation matrix. Direct checks give \(H_XH_Z^{\mathsf T}=0\), girth 6 on both sides, and \(\operatorname{rank}(H_X)=\operatorname{rank}(H_Z)=175\). Hence
   \[ n=8\cdot59=472,\qquad k=472-175-175=122. \]
6. Certify the distance. Complete searches find no non-stabilizer zero-syndrome vector through weight 12 on either CSS side. All such vectors have even weight, and explicit weight-14 representatives are found on both sides. Therefore the exact distance is \(d=14\).

M, E, and D construction audit X lower-bound record Z lower-bound record

Data Format

CPM_CSS_V1
qc_590_240_12 59 10 3 3
e_00 e_01 ... e_09
e_10 e_11 ... e_19
e_20 e_21 ... e_29
d_00 d_01 ... d_09
d_10 d_11 ... d_19
d_20 d_21 ... d_29

Distance Certification Summary

CSS-Side Isometries and Fold-Transversal Clifford Gates

The public verification records for [[574,252,18]], [[944,478,20]], [[516,178,20]], [[1784,450,24]], and [[2110,848,22]] include Hamming-isometric coordinate permutations \(\pi\) that exchange the two stabilizer row spaces:

Besides transferring distance information between the two CSS sides, this implies that the physical operation \(P_\pi H^{\otimes n}\) normalizes the stabilizer group. At the code level it therefore implements a logical Clifford using only single-qubit Hadamards and a qubit permutation: a Hadamard-type fold-transversal gate.

• For [[574,252,18]], the verified permutation is a fixed-point-free involution consisting of 287 disjoint transpositions (verification record).
• For [[944,478,20]], it is likewise a fixed-point-free involution, consisting of 472 disjoint transpositions (verification record).
• For [[516,178,20]], a verified CPM-affine coordinate permutation exchanges the two CSS sides and transfers the complete weight-18 exclusion (verification record).
• For [[1784,450,24]], a verified involution exchanges the two row spaces and transfers the complete weight-22 exclusion from the X side to the Z side (verification record).
• For [[2110,848,22]], a verified involutive CPM coordinate permutation exchanges all 633 rows on the two CSS sides and transfers the complete weight-20 exclusion (verification record).

This stabilizer-level result does not by itself identify the logical gate with independent Hadamards on all encoded qubits. The induced logical symplectic action may permute or mix logical degrees of freedom and must be computed separately. The physical cost and fault-tolerance implications of realizing the qubit permutation depend on the hardware connectivity and implementation.

Code Data Files

Memory Efficiency Relative to the Surface Code

Following the comparison used in Figure 1 (left) of arXiv:2607.27644, the vertical axis is \(\eta=kd^2/n\) and the horizontal axis is the block length \(n\). The rotated surface-code baseline is \(\eta=1\). Thus, a value \(\eta\) means that the PP code uses \(1/\eta\) as many data qubits as \(k\) independent distance-\(d\) surface-code patches under this comparison. Both axes are logarithmic. This data-qubit metric does not include syndrome ancillas and does not by itself predict circuit-level logical error rates. As in the distance plot, only the per-family Pareto frontier for the plotted axes is shown. The semi-transparent girth-4 frontier points appear only in the graph and are not included in the catalogue table.

Citation

@unpublished{okadaKasaiPairPartitionCPM2026,
  author = {Koki Okada and Kenta Kasai},
  title  = {Pair-Partition Constructions for CPM-Based Quantum LDPC Codes},
  note   = {arXiv:2607.14091},
  year   = {2026}
}