An old question in sphere-packing turns out to yield an unexpected construction in quantum error correction. The face-centered cubic lattice densest sphere packing in three dimensions, proved by Hales generates a CSS code with encoding rate 2/3. We show that this construction extends to every Dn root lattice (n >= 3), yielding codes with parameters [n(n-1)L^n/2, (n-2)(n+1)L^n/2 + 2, 3] and encoding rate converging to 1 - 4/K as L -> infinity where K = 2n(n-1) is the coordination number. CSS validity, the rank formula, and the rate are proved analytically for all n >= 3. The first two members are computationally verified: the D3 (FCC) code and the D4 code [1536, 1282, 3] at L=4 confirmed at L=6 as [7776, 6482, 3]. For D4 at L=4, all 4096 triangles are logical operators, establishing d=3 exactly. As K grows along the densest-packing hierarchy the protected fraction approaches unity, making this, to our knowledge, the first proven infinite family of topological CSS codes whose rate converges to 1 at fixed distance.
An old question in sphere-packing turns out to yield an unexpected construction in quantum error correction. The face-centered cubic lattice densest sphere packing in three dimensions, proved by Hales generates a CSS code with encoding rate 2/3. We show that this construction extends to every Dn root lattice (n >= 3), yielding codes with parameters [n(n-1)L^n/2, (n-2)(n+1)L^n/2 + 2, 3] and encoding rate converging to 1 - 4/K as L -> infinity where K = 2n(n-1) is the coordination number. CSS validity, the rank formula, and the rate are proved analytically for all n >= 3. The first two members are computationally verified: the D3 (FCC) code and the D4 code [1536, 1282, 3] at L=4 confirmed at L=6 as [7776, 6482, 3]. For D4 at L=4, all 4096 triangles are logical operators, establishing d=3 exactly. As K grows along the densest-packing hierarchy the protected fraction approaches unity, making this, to our knowledge, the first proven infinite family of topological CSS codes whose rate converges to 1 at fixed distance.
The [[3L3,2L3+2,3]] CSS code on the Face Centered Cubic (FCC) lattice reaches a 67.7% encoding rate with weight-12 stabilizers. Its distance is 3. We classify every weight-3 logical operator of the code. The X-type ones are the 4L3 triangular faces of the lattice. The Z-type ones are the 4L3 claws, each formed by the three edges of a tetrahedron meeting at one of its vertices. The two families are exact duals under the exchange of vertices and octahedral voids, and each spans 2L3−1 independent logical classes. We prove this count analytically from the homology of the 3-torus, not only numerically: the cycle space has dimension 25L3+1 , the boundary space has dimension 25L3−2 , and subtracting the stabilizer rank leaves 2L3−1 for every even L. So k=(2L3−1)+3 , and the excess rate is carried entirely by weight-3 operators. We then show that this sector cannot be protected. The three-body parity ZeaZebZec at a face anticommutes with four octahedral stabilizers, so it is not measurable, and no measurement of any kind can distinguish ∣ψ⟩ from L^T∣ψ⟩ for arbitrary encoded ∣ψ⟩ . Removing the sector requires adjoining all 2L3−1 independent directions, which returns k=3 . Verified exactly over GF(2) at L=4,6,8,10 .
We construct a three-dimensional Calderbank-Shor-Steane (CSS) stabilizer code on the Face-Centered Cubic (FCC) lattice. Physical qubits reside on the edges of the lattice (coordination K=12); X-stabilizers act on octahedral voids and Z-stabilizers on vertices, both with uniform weight 12. Computational verification confirms CSS validity (HXHZT=0 over GF(2)) and reveals k=2L3+2 logical qubits: k=130 at L=4 and k=434 at L=6, yielding encoding rates of 67.7% and 67.0% respectively. The minimum distance d=3 is proven exactly by exhaustive elimination of all weight-≤2 candidates combined with constructive weight-3 non-stabilizer codewords. The code parameters are [[192, 130, 3]] at L=4 and [[648, 434, 3]] at L=6. This rate is 24x higher than the cubic 3D toric code (2.8% at d=4), though at a lower distance (d=3 vs. d=4); the comparison is across different distances. The high rate originates in a structural surplus: the FCC lattice has 3L3 edges but only L3−2 independent stabilizer constraints, leaving k=2L3+2 logical degrees of freedom. We provide a minimum-weight perfect matching (MWPM) decoder adapted to the FCC geometry, demonstrate a 10x coding gain at p=0.001 (and 63x at p=0.0005), and discuss implications for fault-tolerant quantum computing on neutral-atom and photonic platforms.
The [[3L3, 2L3+2, 3]] CSS code on the Face-Centered Cubic (FCC) lattice achieves 67.7% encoding rate with weight-12 stabilizers, but its distance is capped at d = 3 by weight-3 logical operators sitting at tetrahedral interstitial voids. These logicals carry a geometric ngerprintone error per triad sheet, all three converging at a known voidthat makes them detectable by a ag ancilla measuring the three-body parity Zea Zeb Zec . The weight-3 logical ips this parity from +1 to −1; the stabilizer syndrome stays silent, but the ag sees it. Running R independent ag rounds per syndrome cycle pushes the logical failure rate to p3+R, so the eective distance is simply deff = 3 + Radjustable at runtime, with no changes to the code or its logical count. At R = 4 the FCC+ags code suppresses errors as well as a distance-7 surface code, while encoding 130 logical qubits at 29.0% rate versus 1 logical qubit at 1.0%. An adaptive scheduling protocol cuts the ag overhead to under 0.1% of measurement cycles at realistic noise levels.
We construct a three-dimensional Calderbank-Shor-Steane (CSS) stabilizer code on the Face-Centered Cubic (FCC) lattice. Physical qubits reside on the edges of the lattice (coordination K = 12); X-stabilizers act on octahedral voids and Z-stabilizers on vertices, both with uniform weight 12. Computational verication conrms CSS validity (HXHT Z = 0 over GF(2)) and reveals k = 2L3 + 2 logical qubits: k = 130 at L = 4 and k = 434 at L = 6, yielding encoding rates of 67.7% and 67.0% respectively. Exhaustive search over all kernel cosets establishes the minimum distance as d = 3, independent of L. The veried code parameters are [[192, 130, 3]] at L = 4 and [[648, 434, 3]] at L = 6. This rate is 24×higher than the cubic 3D toric code (2.8% at the same distance) and, to our knowledge, the highest encoding rate reported for any three-dimensional topological stabilizer code. The high rate originates in a structural decit: the FCC lattice has 6L3 edges but only 2 ×(L3/2−1) independent stabilizer constraints, leaving ∼2L3/3 logical degrees of freedom. The minimum distance is d = 3, independent of L, established by exhaustive coset search. We provide the complete reproducible simulation code and discuss implications for fault-tolerant quantum computing on neutral-atom and photonic platforms.
I introduce a method to generate families of CSS codes with interesting code parameters. The object of study is Coxeter groups, both finite and infinite (reducible or not), and a geometrically motivated partial order of Coxeter group elements named after Bruhat. The Bruhat order is known to provide a link to algebraic topology -- it doubles as a face poset capturing the inclusion relations of the p-dimensional cells of a regular CW~complex and that is what makes it interesting for QEC code design. Assisted by the Bruhat face poset interval structure unique to Coxeter groups I show that the corresponding chain complexes can be turned into multitudes of CSS codes. Depending on the approach, I obtain CSS codes (and their families) with controlled stabilizer weights, for example [6006,924,{≤14,≤7}] (stabilizer weights~14 and 9) and [22880,3432,{≤8,≤16}] (weights 16 and 10), and CSS codes with highly irregular stabilizer weight distributions such as [571,199,{5,5}]. For the latter, I develop a weight-reduction method to deal with rare heavy stabilizers. Finally, I show how to extract four-term (length three) chain complexes that can be interpreted as CSS codes with a metacheck.
Abstract The [[3𝐿 3 , 2𝐿 3 +2, 3]] CSS code on the Face-Centered Cubic (FCC) lattice achieves 67.7% encoding rate with weight-12 stabilizers, but its distance is capped at 𝑑 = 3 by weight-3 logical operators sitting at tetrahedral interstitial voids. These logicals carry a geometric fingerprint—one error per triad sheet, all three converging at a known void—that makes them detectable by a flag ancilla measuring the three-body parity 𝑍 𝑒𝑎 𝑍 𝑒𝑏 𝑍 𝑒𝑐 . The weight-3 logical flips this parity from +1 to −1; the stabilizer syndrome stays silent, but the flag sees it. Running 𝑅 independent flag rounds per syndrome cycle pushes the logical failure rate to 𝑝 3+𝑅 , so the effective distance is simply 𝑑eff = 3+𝑅—adjustable at runtime, with no changes to the code or its logical count. At 𝑅 = 4 the FCC+flags code achieves the same 𝑝 7 distance scaling as a distance-7 surface code, with a prefactor 𝑁void = 256 versus ∼ 10 for the surface code—the cost of encoding 130 logical qubits at 29.0% rate versus 1 logical qubit at 1.0%. An adaptive scheduling protocol cuts the flag overhead to under 0.1% of measurement cycles at realistic noise levels.