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arXiv:2610.02392

Singer-Difference-Set Qudit Stabilizer Codes from Non-Degenerate Quadrics in PG(d,q)PG(d,q): Construction, Structural Theorems, and Monte-Carlo Performance

Michel Kulhandjian, Lajos Hanzo

quant-phcs.ITSubmitted Oct 1, 2026• Announced Oct 5, 2026• 0 citations
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Abstract

We propose Q(q,d), a family of q-ary non-CSS qudit stabilizer codes with prime power q and dimension d >= 2, constructed from Singer difference sets and non-degenerate quadrics in PG(d,q). The parity check H = (A | M_Q A) couples the Singer-circulant incidence matrix A with the quadric circulant M_Q, and a Cancellation Lemma uses the identity (k - lambda) = q^{d-1} = 0 mod p to make H symplectic-isotropic over F_q. Five structural theorems carry the novelty: a cross-correlation modular-constancy identity nu = 1 mod q at q odd, d >= 3, with a two-value pattern that proves a 2-torsion distance ceiling; a CRT zero-count identity; an explicit Clifford bridge identifying Q(q,2) as a graph-state encoding of the trivial Singer CSS code, with the 2021 low-density-spreading (LDS) sign-flip pattern realised by a symmetric circulant; a paired column-rigidity equality d_min = R_P + 1; and a lower bound d_min(Q(7,2)) >= 8 from an exhaustive Singer-Frobenius search. A Hamada-Smith specialisation yields the closed-form logical dimension k(p,d) = theta_d(p) - C(p+d-1, d) - 1, and the flagship Q(p,2) = [[p^2 + p + 1, p(p+1)/2, d_min]]_p attains asymptotic rate one-half. We conjecture d_min(Q(p,2)) = p + 1 for odd primes, proven at p = 5 and lower-bounded to >= 8 at p = 7. Instances include Q(3,2) = [[13,6,3]]_3, Q(5,2) = [[31,15,6]]_5, Q(7,2) = [[57,28, d>=8]]_7, and Q(3,4) = [[121,105,3]]_3. Monte-Carlo over 1.5 x 10^6 trials shows Q(5,2) achieves a 70-fold per-logical-qudit improvement over Steane [[7,1,3]]_2 at depolarising rate 10^-3 with zero silent logical errors. A textbook bounded-distance decoder is used; Berlekamp-Massey-style cyclic decoding is left as the natural sequel.

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Accepted for publication in IEEE Open Journal of the Communications Society

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Quantum XYZ Stabilizer Codes

Alessio Baldelli, Davide Orsucci, Francisco Lázaro +1 more

Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define $X$ and $Z$-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with $X$-, $Y$-, and $Z$-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the $Y$-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ$^2$ hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.

cs.ITCross-listedarXiv:2607.14988•Jul 17, 2026•0 citations

Singer-Difference-Set Qudit Stabilizer Codes From Non-Degenerate Quadrics in PG( d , q ): Construction, Structural Theorems, and Monte-Carlo Performance

Michel Kulhandjian, Lajos Hanzo

We propose $\mathcal {Q}(q,d)$ , a new family of $\mathbb {F}_{q}$ -ary non-CSS qudit stabilizer codes parameterised by a prime power $q$ and dimension $d\ge 2$ , constructed from Singer difference sets and non-degenerate quadrics in the projective geometry $\mathop {\mathrm {PG}}\nolimits (d,q)$ . The parity-check matrix $H {\,}\text {=}{\,}(A\mid M_{Q}A)$ couples the Singer-circulant point-hyperplane incidence matrix $A$ with the formal trace-of-square quadric circulant $M_{Q}$ , and a Cancellation Lemma exploits the design-theoretic identity $(k-\lambda){\,}\text {=}{\,}q^{d-1}\equiv 0\pmod p$ to make $H$ symplectic-isotropic over $\mathbb {F}_{q}$ for every $(q,d)$ . The novelty of the construction is concentrated in five original structural theorems: (i) an unconditional cross-correlation modular-constancy theorem $\nu \equiv 1\pmod q$ at $q$ odd, $d\ge 3$ with a sharp two-value pattern $V_{2}-V_{1}{\,}\text {=}{\,}q^{(d-1)/2}$ , proving a 2-torsion distance ceiling; (ii) a Chinese-Remainder-Theorem (CRT) zero-count theorem $\#\{f:\tau _{H,f}{\,}\text {=}{\,}0\}{\,}\text {=}{\,}(|\mathcal {F}|-1)/2$ ; (iii) an explicit Clifford bridge $U_{M}{\,}\text {=}{\,}H^{\otimes n}{\,}\mathrm {CZ}^{(M)}{\,}H^{\otimes n}$ that identifies $\mathcal {Q}(q,2)$ as Clifford-equivalent to a graph-state encoding of the trivial Singer CSS code, with the classical 2021 low-density-spreading (LDS) sign-flip pattern realised by the symmetric circulant $M$ ; (iv) a paired column-rigidity equality $d_{\min }{\,}\text {=}{\,}R_{P}+1$ exactly; and (v) a rigorous lower bound $d_{\min }(\mathcal {Q}(7,2))\ge 8$ from an exhaustive Singer-Frobenius search. Via a Hamada-Smith specialisation we obtain the closed-form logical dimension $k(p,d){\,}\text {=}{\,}\theta _{d}(p)-\binom {p+d-1}{d}-1$ , with the flagship $d{\,}\text {=}{\,}2$ family $\mathcal {Q}(p,2){\,}\text {=}{\,}[[p^{2}+p+1,{\,}p(p+1)/2,{\,}d_{\min }]]_{p}$ attaining asymptotic rate $\tfrac {1}{2}$ . For general odd primes $p\ge 3$ the flagship distance is conjectured to satisfy $d_{\min }(\mathcal {Q}(p,2)){\,}\text {=}{\,}p+1$ ; this conjecture is analytically proven exact at $p{\,}\text {=}{\,}5$ and exhaustively lower-bounded to $d_{\min }\ge 8$ at $p{\,}\text {=}{\,}7$ , and remains open for general $p$ . Concrete instances include $\mathcal {Q}(3,2){\,}\text {=}{\,}[[{13,6,3}]]_{3}$ , $\mathcal {Q}(5,2){\,}\text {=}{\,}[[{31,15,6}]]_{5}$ (double-error correcting), $\mathcal {Q}(7,2){\,}\text {=}{\,}[[57,28,d{\ge }8]]_{7}$ , and $\mathcal {Q}(3,4){\,}\text {=}{\,}[[{121,105,3}]]_{3}$ . Monte-Carlo simulations over $1.5{\times }10^{6}$ trials show that $\mathcal {Q}(5,2)$ achieves a $\sim 70\times $ per-logical-qudit improvement over Steane $[[{7,1,3}]]_{2}$ at depolarising rate $p_{\mathrm {ch}}{\,}\text {=}{\,}10^{-3}$ with zero silent logical errors (the “erase-don’t-mistake” property). We use a textbook bounded-distance decoder throughout, leaving Berlekamp-Massey-style cyclic decoding exploiting the Singer-circulant structure as the natural sequel. A fifteen-row axis-by-axis comparison against the closest prior constructions establishes that the Singer-difference-set + quadric-Gram-matrix combination, together with the closed-form $k$ , the cross-correlation and CRT zero-count theorems, the explicit Clifford bridge to the 2021 LDS construction, and the proven 2-torsion ceiling, is new.

IEEE Open Journal of the Communications Society•Jan 1, 2026•0 citations

Active Stabilizers In Reed–Muller CSS Codes From Cayley–Dickson Zero-Divisors

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Local-dimension-invariant low-density parity-check stabilizer codes

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The protection of quantum information in systems leveraging more than two basis states (qudits) is an ongoing challenge. A tool for transforming the plethora of qubit stabilizer codes into qudit stabilizer codes is the local-dimension-invariant (LDI) extension to the stabilizer formalism. A drawback of the LDI method is that generally in order to ensure the distance of the resultant code is at least preserved the newly selected local-dimension will need to be any prime bigger than a large cutoff value. In this work we consider the repercussions of this large cutoff value, then by focusing on quantum low-density parity-check (qLDPC) codes we are able to bring down this cutoff value. Notably, we are able to reduce it in some cases to the initial local-dimension and for arbitrary CSS qubit-derived codes to singly exponential in $d$ with no dependence on the number of logicals $k$. This reduces the size of the caveat in LDI forms in their distance preservation condition.

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arXiv:2502.20297•Feb 27, 2025•0 citations

Generalized $\mathbb{Z}_p$ toric codes as qudit low-density parity-check codes

Zijian Liang, Yu-An Chen

We study two-dimensional translation-invariant CSS stabilizer codes over prime-dimensional qudits on the square lattice under twisted boundary conditions, generalizing the Kitaev $\mathbb{Z}_p$ toric code by augmenting each stabilizer with two additional qudits. Using the Laurent-polynomial formalism, we adapt the Gröbner basis to compute the logical dimension $k$ efficiently, without explicitly constructing large parity-check matrices. We then perform a systematic search over various stabilizer realizations and lattice geometries for $p\in\{3,5,7,11\}$, identifying qudit low-density parity-check codes with the optimal finite-size performance. Representative examples include $[[242,10,22]]_3$ and $[[120,6,20]]_{11}$, both achieving $k d^{2}/n=20$. Across the searched regime, the best observed $k d^{2}$ at fixed $n$ increases with $p$, with an empirical relation $k d^{2} = 0.0541 \, n^{2}\ln p + 3.84 \, n$, compatible with a Bravyi--Poulin--Terhal-type tradeoff when the interaction range grows with system size.

quant-pharXiv:2602.20158•Feb 24, 2026•0 citations
We propose Q(q,d), a family of q-ary non-CSS qudit stabilizer codes with prime power q and dimension d >= 2, constructed from Singer difference sets and non-degenerate quadrics in PG(d,q). The parity check H = (A | M_Q A) couples the Singer-circulant incidence matrix A with the quadric circulant M_Q, and a Cancellation Lemma uses the identity (k - lambda) = q^{d-1} = 0 mod p to make H symplectic-isotropic over F_q.
Five structural theorems carry the novelty: a cross-correlation modular-constancy identity nu = 1 mod q at q odd, d >= 3, with a two-value pattern that proves a 2-torsion distance ceiling; a CRT zero-count identity; an explicit Clifford bridge identifying Q(q,2) as a graph-state encoding of the trivial Singer CSS code, with the 2021 low-density-spreading (LDS) sign-flip pattern realised by a symmetric circulant; a paired column-rigidity equality d_min = R_P + 1; and a lower bound d_min(Q(7,2)) >= 8 from an exhaustive Singer-Frobenius search.
A Hamada-Smith specialisation yields the closed-form logical dimension k(p,d) = theta_d(p) - C(p+d-1, d) - 1, and the flagship Q(p,2) = [[p^2 + p + 1, p(p+1)/2, d_min]]_p attains asymptotic rate one-half. We conjecture d_min(Q(p,2)) = p + 1 for odd primes, proven at p = 5 and lower-bounded to >= 8 at p = 7. Instances include Q(3,2) = [[13,6,3]]_3, Q(5,2) = [[31,15,6]]_5, Q(7,2) = [[57,28, d>=8]]_7, and Q(3,4) = [[121,105,3]]_3. Monte-Carlo over 1.5 x 10^6 trials shows Q(5,2) achieves a 70-fold per-logical-qudit improvement over Steane [[7,1,3]]_2 at depolarising rate 10^-3 with zero silent logical errors. A textbook bounded-distance decoder is used; Berlekamp-Massey-style cyclic decoding is left as the natural sequel.
Stabilizer codes are often constructed within the Calderbank--Shor--Steane (CSS) framework, where two mutually orthogonal binary classical codes define XXX and ZZZ-type stabilizer generators. While this structure is algebraically convenient, additional non-CSS constraints may help suppress low-weight logical operators and improve decoding performance in the finite-length regime. We thus introduce quantum XYZ stabilizer codes, whose parity-check matrix (PCM) is built from three pairwise orthogonal binary PCMs associated with XXX-, YYY-, and ZZZ-type stabilizer generators. A nontrivial point is that an XYZ code instance is not automatically genuinely non-CSS: the same stabilizer group may admit a CSS generating set. We characterize this collapse, obtaining algebraic and rank conditions for deciding when the YYY-type checks are redundant and when they define genuinely non-CSS stabilizer constraints. We also derive upper and lower bounds on the quantum minimum distance, including bounds for mixed Pauli logical operators. The novel framework includes a known non-CSS topological code, namely the XYZ2^22 hexagonal code, and yields also sparse finite-length quantum low-density parity-check (qLDPC) constructions from intersecting-subset and quasi-dyadic code families. Simulations under depolarizing code-capacity noise and quaternary belief propagation decoding show that the proposed XYZ qLDPC instances can outperform representative CSS qLDPC instances with similar finite-length parameters.
We propose Q(q,d)\mathcal {Q}(q,d)Q(q,d) , a new family of Fq\mathbb {F}_{q}Fq​ -ary non-CSS qudit stabilizer codes parameterised by a prime power qqq and dimension d≥2d\ge 2d≥2 , constructed from Singer difference sets and non-degenerate quadrics in the projective geometry PG(d,q)\mathop {\mathrm {PG}}\nolimits (d,q)PG(d,q) . The parity-check matrix H = (A∣MQA)H {\,}\text {=}{\,}(A\mid M_{Q}A)H=(A∣MQ​A) couples the Singer-circulant point-hyperplane incidence matrix AAA with the formal trace-of-square quadric circulant MQM_{Q}MQ​ , and a Cancellation Lemma exploits the design-theoretic identity (k−λ) = qd−1≡0(modp)(k-\lambda){\,}\text {=}{\,}q^{d-1}\equiv 0\pmod p(k−λ)=qd−1≡0(modp) to make HHH symplectic-isotropic over Fq\mathbb {F}_{q}Fq​ for every (q,d)(q,d)(q,d) . The novelty of the construction is concentrated in five original structural theorems: (i) an unconditional cross-correlation modular-constancy theorem ν≡1(modq)\nu \equiv 1\pmod qν≡1(modq) at qqq odd, d≥3d\ge 3d≥3 with a sharp two-value pattern V2−V1 = q(d−1)/2V_{2}-V_{1}{\,}\text {=}{\,}q^{(d-1)/2}V2​−V1​=q(d−1)/2 , proving a 2-torsion distance ceiling; (ii) a Chinese-Remainder-Theorem (CRT) zero-count theorem #{f:τH,f = 0} = (∣F∣−1)/2\#\{f:\tau _{H,f}{\,}\text {=}{\,}0\}{\,}\text {=}{\,}(|\mathcal {F}|-1)/2#{f:τH,f​=0}=(∣F∣−1)/2 ; (iii) an explicit Clifford bridge UM = H⊗n CZ(M) H⊗nU_{M}{\,}\text {=}{\,}H^{\otimes n}{\,}\mathrm {CZ}^{(M)}{\,}H^{\otimes n}UM​=H⊗nCZ(M)H⊗n that identifies Q(q,2)\mathcal {Q}(q,2)Q(q,2) as Clifford-equivalent to a graph-state encoding of the trivial Singer CSS code, with the classical 2021 low-density-spreading (LDS) sign-flip pattern realised by the symmetric circulant MMM ; (iv) a paired column-rigidity equality dmin⁡ = RP+1d_{\min }{\,}\text {=}{\,}R_{P}+1dmin​=RP​+1 exactly; and (v) a rigorous lower bound dmin⁡(Q(7,2))≥8d_{\min }(\mathcal {Q}(7,2))\ge 8dmin​(Q(7,2))≥8 from an exhaustive Singer-Frobenius search. Via a Hamada-Smith specialisation we obtain the closed-form logical dimension k(p,d) = θd(p)−(p+d−1d)−1k(p,d){\,}\text {=}{\,}\theta _{d}(p)-\binom {p+d-1}{d}-1k(p,d)=θd​(p)−(dp+d−1​)−1 , with the flagship d = 2d{\,}\text {=}{\,}2d=2 family Q(p,2) = [[p2+p+1, p(p+1)/2, dmin⁡]]p\mathcal {Q}(p,2){\,}\text {=}{\,}[[p^{2}+p+1,{\,}p(p+1)/2,{\,}d_{\min }]]_{p}Q(p,2)=[[p2+p+1,p(p+1)/2,dmin​]]p​ attaining asymptotic rate 12\tfrac {1}{2}21​ . For general odd primes p≥3p\ge 3p≥3 the flagship distance is conjectured to satisfy dmin⁡(Q(p,2)) = p+1d_{\min }(\mathcal {Q}(p,2)){\,}\text {=}{\,}p+1dmin​(Q(p,2))=p+1 ; this conjecture is analytically proven exact at p = 5p{\,}\text {=}{\,}5p=5 and exhaustively lower-bounded to dmin⁡≥8d_{\min }\ge 8dmin​≥8 at p = 7p{\,}\text {=}{\,}7p=7 , and remains open for general ppp . Concrete instances include Q(3,2) = [[13,6,3]]3\mathcal {Q}(3,2){\,}\text {=}{\,}[[{13,6,3}]]_{3}Q(3,2)=[[13,6,3]]3​ , Q(5,2) = [[31,15,6]]5\mathcal {Q}(5,2){\,}\text {=}{\,}[[{31,15,6}]]_{5}Q(5,2)=[[31,15,6]]5​ (double-error correcting), Q(7,2) = [[57,28,d≥8]]7\mathcal {Q}(7,2){\,}\text {=}{\,}[[57,28,d{\ge }8]]_{7}Q(7,2)=[[57,28,d≥8]]7​ , and Q(3,4) = [[121,105,3]]3\mathcal {Q}(3,4){\,}\text {=}{\,}[[{121,105,3}]]_{3}Q(3,4)=[[121,105,3]]3​ . Monte-Carlo simulations over 1.5×1061.5{\times }10^{6}1.5×106 trials show that Q(5,2)\mathcal {Q}(5,2)Q(5,2) achieves a $\sim 70\times $ per-logical-qudit improvement over Steane [[7,1,3]]2[[{7,1,3}]]_{2}[[7,1,3]]2​ at depolarising rate pch = 10−3p_{\mathrm {ch}}{\,}\text {=}{\,}10^{-3}pch​=10−3 with zero silent logical errors (the “erase-don’t-mistake” property). We use a textbook bounded-distance decoder throughout, leaving Berlekamp-Massey-style cyclic decoding exploiting the Singer-circulant structure as the natural sequel. A fifteen-row axis-by-axis comparison against the closest prior constructions establishes that the Singer-difference-set + quadric-Gram-matrix combination, together with the closed-form kkk , the cross-correlation and CRT zero-count theorems, the explicit Clifford bridge to the 2021 LDS construction, and the proven 2-torsion ceiling, is new.
We establish three results connecting the zero-divisor structure of the Cayley–Dickson algebras Aₙ to the stabilizer formalism of quantum error-correcting codes. Step 1 (Bijection, n=4). The 7 lo-flats of ZD(4) — weight-4 subsets {a,b,c,d} ⊂ {1,...,7} with a⊕b⊕c⊕d=0 — are in explicit bijection with the 7 non-identity elements of the X-stabilizer group of the Steane [[7,1,3]] code. All 7 lo-flats carry identical sign-parity pattern: π(EE|FF)=−1 (inactive), π(EF₁)=π(EF₂)=+1 (active). Step 2 (Operator translation). For every mixed flat F={i,j,8+k,8+l} in A₄, the algebraic identity π_A·π_B·π_C=−1 translates into the Clifford-group identity (Z_αZ_β)(X_αX_γ)(X_βX_γ) = −Y_αY_β. The global phase −1 is the precise Clifford avatar of π_Aπ_Bπ_C=−1. The operator −Y_αY_β is a detectable error with full-group syndrome weight 2^(n−2), proved for all n≥4 and verified on QPU. Step 3 (Extension to n=5, verified on QPU). The 15 Fano planes of PG(3,F₂) are in explicit bijection with the 15 non-identity elements of the X-stabilizer group S_X ≅ (Z/2Z)⁴ of the [[15,7,3]] code (4 generators, 2⁴−1=15 non-identity elements). The phase identity and syndrome formula extend to all n≥4 (Universal Phase Lemma). Syndrome structure verified on ibm_kingston (156-qubit QPU, job d8bm67ijki0s73aq6idg, 27 May 2026). The Type-A/Type-B criterion for Fano planes in PG(3,F₂) is resolved: a plane W is Type-B if and only if 8∈W or W={1,...,7}, where 8=(1,0,0,0)∈F₂⁴ is the bridge element between A₃ and A₄ in the Cayley–Dickson tower. All algebraic results are cited from the companion papers (doi:10.5281/zenodo.20315751, doi:10.5281/zenodo.20417859).Every claim is labelled [VERIFIED], [THEOREM], or [OPEN] throughout.
The protection of quantum information in systems leveraging more than two basis states (qudits) is an ongoing challenge. A tool for transforming the plethora of qubit stabilizer codes into qudit stabilizer codes is the local-dimension-invariant (LDI) extension to the stabilizer formalism. A drawback of the LDI method is that generally in order to ensure the distance of the resultant code is at least preserved the newly selected local-dimension will need to be any prime bigger than a large cutoff value. In this work we consider the repercussions of this large cutoff value, then by focusing on quantum low-density parity-check (qLDPC) codes we are able to bring down this cutoff value. Notably, we are able to reduce it in some cases to the initial local-dimension and for arbitrary CSS qubit-derived codes to singly exponential in ddd with no dependence on the number of logicals kkk. This reduces the size of the caveat in LDI forms in their distance preservation condition.
We introduce new families of quantum Tanner codes, a class of quantum codes that first appeared in the work of Leverrier and Zémor (FOCS 2022). These codes are built from two classical Tanner codes, for which the underlying graphs are extracted from coverings of 2D geometrical complexes, and the local linear codes are tensor-products of cyclic or double-circulant linear codes. The advantage of code lifting is that, for any lift of odd index ttt of an [[n,k,d]][[n,k,d]][[n,k,d]]-code, we can adapt the study of the transfer homomorphism arising in cellular homology to describe symmetries of its logical operators and to establish that its dimension is lower bounded by kkk, and its distance is upper bounded by t⋅dt\cdot dt⋅d. Moreover, when the dimension of the lifted code is equal to kkk, its distance is lower bounded by ddd. These parameter bounds also apply to the previous methods of code lifting of Guémard (IEEE Trans. Inf. Theory, 2025). Finally, We present several explicit families, and identify instances of moderate length quantum codes which are degenerate, have low check weight, and whose distance surpasses the square root of the code length. Among them, we report the existence of a [[96,2,12]][[96,2,12]][[96,2,12]]-code whose distance growth saturates our bound, and for which half of the checks are of weight 8 and the other half of weight 4.
Generalized Zp\mathbb{Z}_pZp​ toric codes as qudit low-density parity-check codes
We study two-dimensional translation-invariant CSS stabilizer codes over prime-dimensional qudits on the square lattice under twisted boundary conditions, generalizing the Kitaev Zp\mathbb{Z}_pZp​ toric code by augmenting each stabilizer with two additional qudits. Using the Laurent-polynomial formalism, we adapt the Gröbner basis to compute the logical dimension kkk efficiently, without explicitly constructing large parity-check matrices. We then perform a systematic search over various stabilizer realizations and lattice geometries for p∈{3,5,7,11}p\in\{3,5,7,11\}p∈{3,5,7,11}, identifying qudit low-density parity-check codes with the optimal finite-size performance. Representative examples include [[242,10,22]]3[[242,10,22]]_3[[242,10,22]]3​ and [[120,6,20]]11[[120,6,20]]_{11}[[120,6,20]]11​, both achieving kd2/n=20k d^{2}/n=20kd2/n=20. Across the searched regime, the best observed kd2k d^{2}kd2 at fixed nnn increases with ppp, with an empirical relation kd2=0.0541 n2ln⁡p+3.84 nk d^{2} = 0.0541 \, n^{2}\ln p + 3.84 \, nkd2=0.0541n2lnp+3.84n, compatible with a Bravyi--Poulin--Terhal-type tradeoff when the interaction range grows with system size.