arXiv:2603.17703

Independent Trivariate Bicycle Codes

Aygul Azatovna Galimova

quant-phcs.ITSubmitted Mar 18, 2026• Announced Mar 19, 2026• 0 citations

Abstract

We introduce six independent trivariate bicycle (ITB) codes, which extend the bivariate bicycle framework of Bravyi et al.\ to three cyclic dimensions. Using asymmetric polynomial pairs on three-dimensional tori, we construct four codes including a $[[140,6,14]]$ code with $kd^2/n = 8.40$. In the code-capacity setting, the $[[140,6,14]]$ code achieves a pseudothreshold of $8.0\%$ and $kd^2/n = 8.40$, exceeding the best multivariate bicycle code of Voss et al.\ ($7.9\%$, $kd^2/n = 2.67$). With circuit-level depolarizing noise, pseudothresholds reach $0.59\%$ for $[[140,6,14]]$ and $0.53\%$ for $[[84,6,10]]$. On the SI1000 superconducting noise model, the $[[140,6,14]]$ code achieves a per-round per-observable rate of $5.6 \times 10^{-5}$ at $p = 0.20\%$. We additionally present two self-dual codes with weight-8 stabilizers: $[[54,14,5]]$ ($kd^2/n = 6.48$) and $[[128,20,8]]$ ($kd^2/n = 10.0$). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.

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arXiv:2603.17703

Independent Trivariate Bicycle Codes

Aygul Azatovna Galimova

quant-phcs.ITSubmitted Mar 18, 2026• Announced Mar 19, 2026• 0 citations

arXiv HTML PDF

Abstract

[[140,6,14]]

code with

kd^2/n = 8.40

. In the code-capacity setting, the

[[140,6,14]]

code achieves a pseudothreshold of

8.0\%

and

kd^2/n = 8.40

, exceeding the best multivariate bicycle code of Voss et al.\ (

7.9\%

kd^2/n = 2.67

). With circuit-level depolarizing noise, pseudothresholds reach

0.59\%

for

[[140,6,14]]

and

0.53\%

for

[[84,6,10]]

. On the SI1000 superconducting noise model, the

[[140,6,14]]

code achieves a per-round per-observable rate of

5.6 \times 10^{-5}

p = 0.20\%

. We additionally present two self-dual codes with weight-8 stabilizers:

[[54,14,5]]

(

kd^2/n = 6.48

) and

[[128,20,8]]

(

kd^2/n = 10.0

). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.

Related papers

Quantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven SearchQuantum Bicycle LDPC Codes with High $kd^2/n$ from Divisor-Driven Search

Liangdong Lu, Guanmin Guo, Yang Liu +1 more

Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes $[[42,12,4]]_2$ and $[[62,12,4]]_2$ and produces a family of codes with competitive figure of merit $kd^2/n$ , including $[[66,20,7]]_2$ with $kd^2/n=14.85$ , above the bivariate bicycle code $[[144,12,12]]_2$ ( $kd^2/n=12$ ) at less than half the block length, together with $[[46,2,8]]_2$ , $[[66,2,9]]_2$ , $[[66,4,8]]_2$ , $[[66,6,8]]_2$ and, at $n=90$ , $[[90,16,6]]_2$ , $[[90,18,6]]_2$ , $[[90,20,6]]_2$ . An exhaustive census at $n=48$ delineates the boundary of this picture: we exhibit a $[[48,10,6]]_2$ code from a minimal $48$ -element group (the Aydin--Tamo--Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes $[[48,10,5]]_2$ from the weight- $8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes $[[42,12,4]]_2$ and $[[62,12,4]]_2$ and produces a family of codes with competitive figure of merit $kd^2/n$, including $[[66,20,7]]_2$ with $kd^2/n=14.85$, above the bivariate bicycle code $[[144,12,12]]_2$ ($kd^2/n=12$) at less than half the block length, together with $[[46,2,8]]_2$, $[[66,2,9]]_2$, $[[66,4,8]]_2$, $[[66,6,8]]_2$ and, at $n=90$, $[[90,16,6]]_2$, $[[90,18,6]]_2$, $[[90,20,6]]_2$. An exhaustive census at $n=48$ delineates the boundary of this picture: we exhibit a $[[48,10,6]]_2$ code from a minimal $48$-element group (the Aydin--Tamo--Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes $[[48,10,5]]_2$ from the weight-$8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.

cs.ITCross-listedarXiv:2608.09115Aug 11, 20260 citations

Multivariate bicycle codes

Lukas Voss, Sim Jian Xian, Tobias Haug +1 more

Quantum error correction suppresses noise in quantum systems to allow for high-precision computations. In this Letter, we introduce multivariate bicycle quantum low-density parity-check (QLDPC) codes, via an extension of the framework developed by Bravyi [S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, ], and particularly focus on trivariate bicycle (TB) codes. Unlike the weight-6 codes proposed in their study, we offer concrete examples of weight-5 TB-QLDPC codes which promise to be more amenable to near-term experimental setups. We show that TB-QLDPC codes up to weight 6 have a biplanar structure and often possess a two-dimensional toric layout. Under circuit level noise, we find substantially better encoding rates than comparable surface codes while offering similar error suppression capabilities. For example, we can encode four logical qubits with distance 5 into 60 physical qubits using weight-5 check measurements of circuit depth 7, while a surface code with these parameters requires 200 physical qubits. The high encoding rate and compact layout make our codes highly suitable candidates for near-term hardware implementations, paving the way for a realizable quantum error correction protocol.Quantum error correction suppresses noise in quantum systems to allow for high-precision computations. In this Letter, we introduce multivariate bicycle quantum low-density parity-check (QLDPC) codes, via an extension of the framework developed by Bravyi [S. Bravyi, A. W. Cross, J. M. Gambetta, D. Maslov, P. Rall, and T. J. Yoder, ], and particularly focus on trivariate bicycle (TB) codes. Unlike the weight-6 codes proposed in their study, we offer concrete examples of weight-5 TB-QLDPC codes which promise to be more amenable to near-term experimental setups. We show that TB-QLDPC codes up to weight 6 have a biplanar structure and often possess a two-dimensional toric layout. Under circuit level noise, we find substantially better encoding rates than comparable surface codes while offering similar error suppression capabilities. For example, we can encode four logical qubits with distance 5 into 60 physical qubits using weight-5 check measurements of circuit depth 7, while a surface code with these parameters requires 200 physical qubits. The high encoding rate and compact layout make our codes highly suitable candidates for near-term hardware implementations, paving the way for a realizable quantum error correction protocol.

arXiv:2406.19151Physical Review AJun 6, 20258 citations

Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark

Liangdong Lu, Ruipan Yang, Guanmin Guo

Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight- $6$ checks, exemplified by $[[144,12,12]]$ with $kd^2/n=12$ . We develop the algebraic structure theory of BB-type codes with weight- $8$ checks (weight- $4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula $k=2\dim R/(A,B)$ (forcing even $k$ ), a $4\ell m$ -element symmetry group on generator pairs, an $X/Z$ distance equality $d_X=d_Z$ , and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At $n=144$ the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: $[[144,6,d\ge 15]]$ exceeds the benchmark distance $12$ (certified $d\ge 15$ ), $[[144,10,12]]$ reaches it with weight- $8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at $kd^2/n=11.11$ ( $7.4\%$ below benchmark) while decoding no worse. At $n=72$ , $[[72,14,8]]$ attains $kd^2/n=12.44$ ---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight- $8$ codes at $\approx 0.1\%$ pseudo-threshold versus $\approx 0.4\%$ for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-$6$ checks, exemplified by $[[144,12,12]]$ with $kd^2/n=12$. We develop the algebraic structure theory of BB-type codes with weight-$8$ checks (weight-$4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula $k=2\dim R/(A,B)$ (forcing even $k$), a $4\ell m$-element symmetry group on generator pairs, an $X/Z$ distance equality $d_X=d_Z$, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At $n=144$ the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: $[[144,6,d\ge 15]]$ exceeds the benchmark distance $12$ (certified $d\ge 15$), $[[144,10,12]]$ reaches it with weight-$8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at $kd^2/n=11.11$ ($7.4\%$ below benchmark) while decoding no worse. At $n=72$, $[[72,14,8]]$ attains $kd^2/n=12.44$---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-$8$ codes at $\approx 0.1\%$ pseudo-threshold versus $\approx 0.4\%$ for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.

quant-pharXiv:2609.06572Sep 8, 20260 citations

Self-dual Stacked Quantum Low-Density Parity-Check Codes

Ze-Chuan Liu, Chong-Yuan Xu, Yong Xu

Quantum low-density parity-check (qLDPC) codes are promising candidates for fault-tolerant quantum computation due to their high encoding rates and distances. However, implementing logical operations using qLDPC codes presents significant challenges. Previous research has demonstrated that self-dual qLDPC codes facilitate the implementation of transversal Clifford gates. Here we introduce a method for constructing self-dual qLDPC codes by stacking non-self-dual qLDPC codes. Leveraging this methodology, we develop double-chain bicycle codes, double-layer bivariate bicycle (BB) codes, double-layer twisted BB codes, and double-layer reflection codes, many of which exhibit favorable code parameters. Additionally, we conduct numerical calculations to assess the performance of these codes as quantum memory under the circuit-level noise model, revealing that the logical failure rate can be significantly reduced with high pseudo-thresholds.Quantum low-density parity-check (qLDPC) codes are promising candidates for fault-tolerant quantum computation due to their high encoding rates and distances. However, implementing logical operations using qLDPC codes presents significant challenges. Previous research has demonstrated that self-dual qLDPC codes facilitate the implementation of transversal Clifford gates. Here we introduce a method for constructing self-dual qLDPC codes by stacking non-self-dual qLDPC codes. Leveraging this methodology, we develop double-chain bicycle codes, double-layer bivariate bicycle (BB) codes, double-layer twisted BB codes, and double-layer reflection codes, many of which exhibit favorable code parameters. Additionally, we conduct numerical calculations to assess the performance of these codes as quantum memory under the circuit-level noise model, revealing that the logical failure rate can be significantly reduced with high pseudo-thresholds.

quant-pharXiv:2602.15372Feb 18, 20260 citations

Mirror codes: High-threshold quantum LDPC codes beyond the CSS regime

Andrey Boris Khesin, Jonathan Z. Lu

The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group $G$ and two subsets of $G$ whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find $[[ 60, 4, 10 ]], [[ 36, 6, 6 ]], [[ 48, 8, 6 ]]$ , and $[[ 85, 8, 9 ]]$ codes, all of which are not CSS; we also find several weight-7 codes with $kd > n$ . Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of $0.2\%$ , approximately matching that of the $[[ 144, 12, 12 ]]$ BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.The realization of quantum error correction protocols whose logical error rates are suppressed far below physical error rates relies on an intricate combination: the error-correcting code's efficiency, the syndrome extraction circuit's fault tolerance and overhead, the decoder's quality, and the device's constraints, such as physical qubit count and connectivity. This work makes two contributions towards error-corrected quantum devices. First, we introduce mirror codes, a simple yet flexible construction of LDPC stabilizer codes parameterized by a group $G$ and two subsets of $G$ whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find $[[ 60, 4, 10 ]], [[ 36, 6, 6 ]], [[ 48, 8, 6 ]]$, and $[[ 85, 8, 9 ]]$ codes, all of which are not CSS; we also find several weight-7 codes with $kd > n$. Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of $0.2\%$, approximately matching that of the $[[ 144, 12, 12 ]]$ BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.

quant-pharXiv:2603.05496Mar 6, 20260 citations

Multivariate Multicycle Codes for Complete Single-Shot Decoding

Feroz Ahmed Mian, Owen Gwilliam, Stefan Krastanov

We introduce multivariate multicycle (MM) codes, a new family of quantum error-correcting codes (QECCs) that unifies bivariate bicycle, multivariate bicycle, abelian two-block group algebra, generalized bicycle, trivariate tricycle, and toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length- $\textit{t+1}$ chain complexes with $\textit{$t \ge 4$}$. The chief advantage of these codes is that they possess metachecks and high confinement that permit complete single-shot decoding. We offer a framework that facilitates the construction of long-length chain complexes through the use of Koszul complex. In particular, obtaining explicit boundary maps (parity check and metacheck matrices) is particularly straightforward in our approach. This simple but very general parameterization of codes permitted us to efficiently perform a numerical search, where we identify several MM code candidates that demonstrate these capabilities at high rates and high code distances. Examples of new codes with parameters $[[n,k,d]]$ include $[[96, 12, 8]]$ , $[[144, 12, 12]]$ , $[[216, 12, 14]]$ , $[[288, 12, 16]]$ , $[[324, 12, 20]]$ , $[[432, 12, 27]]$ , $[[486, 24, 12]]$ , $[[630, 70, 9]]$ , and $[[648, 18, 23]]$ . Notably, our codes achieve confinement profiles that surpass all known single-shot-decodable quantum CSS codes of practical blocksize. Our codes are also the first explicit instances of collapsed 5D through 9D higher dimensional QECCs, with check weights significantly lower than those of recent small instances of quantum Tanner codes.We introduce multivariate multicycle (MM) codes, a new family of quantum error-correcting codes (QECCs) that unifies bivariate bicycle, multivariate bicycle, abelian two-block group algebra, generalized bicycle, trivariate tricycle, and toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length-$\textit{t+1}$ chain complexes with $\textit{$t \ge 4$}$. The chief advantage of these codes is that they possess metachecks and high confinement that permit complete single-shot decoding. We offer a framework that facilitates the construction of long-length chain complexes through the use of Koszul complex. In particular, obtaining explicit boundary maps (parity check and metacheck matrices) is particularly straightforward in our approach. This simple but very general parameterization of codes permitted us to efficiently perform a numerical search, where we identify several MM code candidates that demonstrate these capabilities at high rates and high code distances. Examples of new codes with parameters $[[n,k,d]]$ include $[[96, 12, 8]]$, $[[144, 12, 12]]$, $[[216, 12, 14]]$, $[[288, 12, 16]]$, $[[324, 12, 20]]$, $[[432, 12, 27]]$, $[[486, 24, 12]]$, $[[630, 70, 9]]$, and $[[648, 18, 23]]$. Notably, our codes achieve confinement profiles that surpass all known single-shot-decodable quantum CSS codes of practical blocksize. Our codes are also the first explicit instances of collapsed 5D through 9D higher dimensional QECCs, with check weights significantly lower than those of recent small instances of quantum Tanner codes.

quant-pharXiv:2601.18879Jan 28, 20260 citations

[[140,6,14]]

code with

kd^2/n = 8.40

. In the code-capacity setting, the

[[140,6,14]]

code achieves a pseudothreshold of

8.0\%

and

kd^2/n = 8.40

, exceeding the best multivariate bicycle code of Voss et al.\ (

7.9\%

kd^2/n = 2.67

). With circuit-level depolarizing noise, pseudothresholds reach

0.59\%

for

[[140,6,14]]

and

0.53\%

for

[[84,6,10]]

. On the SI1000 superconducting noise model, the

[[140,6,14]]

code achieves a per-round per-observable rate of

5.6 \times 10^{-5}

p = 0.20\%

. We additionally present two self-dual codes with weight-8 stabilizers:

[[54,14,5]]

(

kd^2/n = 6.48

) and

[[128,20,8]]

(

kd^2/n = 10.0

). These results expand the design space of algebraic quantum LDPC codes and demonstrate that the third cyclic dimension yields competitive candidates for practical fault-tolerant implementations.

Quantum Bicycle LDPC Codes with High

kd^2/n

from Divisor-Driven Search

[[42,12,4]]_2

and

[[62,12,4]]_2

and produces a family of codes with competitive figure of merit

kd^2/n

, including

[[66,20,7]]_2

with

kd^2/n=14.85

, above the bivariate bicycle code

[[144,12,12]]_2

(

kd^2/n=12

) at less than half the block length, together with

[[46,2,8]]_2

[[66,2,9]]_2

[[66,4,8]]_2

[[66,6,8]]_2

and, at

n=90

[[90,16,6]]_2

[[90,18,6]]_2

[[90,20,6]]_2

. An exhaustive census at

n=48

delineates the boundary of this picture: we exhibit a

[[48,10,6]]_2

code from a minimal

48

-element group (the Aydin--Tamo--Barg realization uses

72

elements), and prove that distance

5

forces a stabilizer-rank loss, which excludes

[[48,10,5]]_2

from the weight-

8

symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.

Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-

6

checks, exemplified by

[[144,12,12]]

with

kd^2/n=12

. We develop the algebraic structure theory of BB-type codes with weight-

8

checks (weight-

4

generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula

k=2\dim R/(A,B)

(forcing even

k

), a

4\ell m

-element symmetry group on generator pairs, an

X/Z

distance equality

d_X=d_Z

, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At

n=144

the pipeline returns a census of

53

codes whose strongest members surpass the BB benchmark:

[[144,6,d\ge 15]]

exceeds the benchmark distance

12

(certified

d\ge 15

[[144,10,12]]

reaches it with weight-

8

checks, and

[[144,16,10]]

encodes a third more logical qubits at

kd^2/n=11.11

(

7.4\%

below benchmark) while decoding no worse. At

n=72

[[72,14,8]]

attains

kd^2/n=12.44

---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-

8

codes at

\approx 0.1\%

pseudo-threshold versus

\approx 0.4\%

for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.

G

and two subsets of

G

whose total size bounds the check weight. These codes contain all abelian two-block group algebra codes, such as bivariate bicycle (BB) codes. At the same time, they are manifestly not CSS in general, thus deviating substantially from most prior constructions. Fixing a check weight of 6, we find

[[ 60, 4, 10 ]], [[ 36, 6, 6 ]], [[ 48, 8, 6 ]]

, and

[[ 85, 8, 9 ]]

codes, all of which are not CSS; we also find several weight-7 codes with

kd > n

. Next, we construct syndrome extraction circuits that trade overhead for provable fault tolerance. These circuits use 1-2, 3, and 6 ancillae per check, and respectively are partially fault-tolerant (FT), provably FT on weight-6 CSS codes, and provably FT on \emph{all} weight-6 stabilizer codes. Using our constructions, we perform end-to-end quantum memory experiments on several representative mirror codes under circuit-level noise. We achieve an error pseudothreshold on the order of

0.2\%

, approximately matching that of the

[[ 144, 12, 12 ]]

BB code under the same model. These findings position mirror codes as a versatile candidate for fault-tolerant quantum memory, especially on smaller-scale devices in the near term.

\textit{t+1}

chain complexes with $\textit{$t \ge 4$}$. The chief advantage of these codes is that they possess metachecks and high confinement that permit complete single-shot decoding. We offer a framework that facilitates the construction of long-length chain complexes through the use of Koszul complex. In particular, obtaining explicit boundary maps (parity check and metacheck matrices) is particularly straightforward in our approach. This simple but very general parameterization of codes permitted us to efficiently perform a numerical search, where we identify several MM code candidates that demonstrate these capabilities at high rates and high code distances. Examples of new codes with parameters

[[n,k,d]]

include

[[96, 12, 8]]

[[144, 12, 12]]

[[216, 12, 14]]

[[288, 12, 16]]

[[324, 12, 20]]

[[432, 12, 27]]

[[486, 24, 12]]

[[630, 70, 9]]

, and

[[648, 18, 23]]

. Notably, our codes achieve confinement profiles that surpass all known single-shot-decodable quantum CSS codes of practical blocksize. Our codes are also the first explicit instances of collapsed 5D through 9D higher dimensional QECCs, with check weights significantly lower than those of recent small instances of quantum Tanner codes.