arXiv:2610.00561

Optimal two-mode bosonic loss codes from finite group symmetry

Argyris Giannisis Manes, Mahadevan Subramanian, Liang Jiang

quant-phSubmitted Sep 30, 2026• Announced Oct 2, 2026• 0 citations

Abstract

Photon loss is a dominant noise process in bosonic quantum hardware, including superconducting cavities. Fixed-total-photon-number qubit encodings in two bosonic modes retain the loss-detection advantage of dual-rail qubits while supporting photon loss correction. Optimizing entanglement fidelity in this setting for $4\leq n\leq25$ over arbitrary encoders and decoders reveals finite-group structure in every best-found code: 11 correspond to two-dimensional irreducible representations and 11 to reducible ones. Motivated by this emergence, we derive the necessary-and-sufficient Knill-Laflamme conditions for arbitrary finite-group-invariant codes, reducing their construction to equations on representation multiplicity spaces. For every finite subgroup of $SU(2)$ and every tensor rank, we explicitly construct the minimum number of operators required to impose all corresponding loss-correction constraints. These results produce analytic counterparts to most numerical codes and predict constructions missed by the initial search. Exact MacWilliams-Farkas certificates show that our codes achieve the maximum possible loss distance in 21 of the 22 sectors, and we prove that the resulting distance bound is monotone in total photon number. Beyond the scan, we identify a binary-polyhedral sequence with photon number $n_d=\lceil(3d^2+1)/4\rceil$ and construct each corresponding code through distance $d=10$. To our knowledge, the constructed $(n,d)=(28,6),(49,8),(76,10)$ codes give the smallest reported $n$ for their respective distances; all members through $d=9$ attain the fixed-$n$ LP distance bound. Together, these constructions, certificates, and a symmetry-reduced constraint count provide evidence for an infinite code family conjectured to attain every distance at the minimum photon number.

Comments: 7-page Letter and 65-page Supplemental Material; 3 main figures and 28 supplementary figures

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arXiv:2610.00561

Optimal two-mode bosonic loss codes from finite group symmetry

Argyris Giannisis Manes, Mahadevan Subramanian, Liang Jiang

quant-phSubmitted Sep 30, 2026• Announced Oct 2, 2026• 0 citations

arXiv HTML PDF

Abstract

4\leq n\leq25

over arbitrary encoders and decoders reveals finite-group structure in every best-found code: 11 correspond to two-dimensional irreducible representations and 11 to reducible ones. Motivated by this emergence, we derive the necessary-and-sufficient Knill-Laflamme conditions for arbitrary finite-group-invariant codes, reducing their construction to equations on representation multiplicity spaces. For every finite subgroup of

SU(2)

and every tensor rank, we explicitly construct the minimum number of operators required to impose all corresponding loss-correction constraints. These results produce analytic counterparts to most numerical codes and predict constructions missed by the initial search. Exact MacWilliams-Farkas certificates show that our codes achieve the maximum possible loss distance in 21 of the 22 sectors, and we prove that the resulting distance bound is monotone in total photon number. Beyond the scan, we identify a binary-polyhedral sequence with photon number

n_d=\lceil(3d^2+1)/4\rceil

and construct each corresponding code through distance

d=10

. To our knowledge, the constructed

(n,d)=(28,6),(49,8),(76,10)

codes give the smallest reported

n

for their respective distances; all members through

d=9

attain the fixed-

n

LP distance bound. Together, these constructions, certificates, and a symmetry-reduced constraint count provide evidence for an infinite code family conjectured to attain every distance at the minimum photon number.

Comments: 7-page Letter and 65-page Supplemental Material; 3 main figures and 28 supplementary figures

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Error Correction Properties of Covariant Bosonic Encodings

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Bosonic codes offer a promising approach towards hardware-efficient fault tolerance, with many leading examples such as the cat-code and the GKP code organized by an underlying symmetry. We build on a representation-theoretic framework of quantum error correction to formalize the construction and analysis of multimode symmetric bosonic codes, focusing on finite groups. Here, a physical, logical representation and an initial seed state together define the codewords via a covariant encoding. A central observation is that QEC matrices can be viewed as morphisms of group representations, so that Schur's lemma forces entire irreducible representations blocks to vanish. We recover several known single- and two-mode bosonic codes and build and analyze the error protection capabilities of three codes based on finite subgroups of SU(2). We analyze the impact of reducibility of the logical representation and optimize the seed state using the near-optimal fidelity.Bosonic codes offer a promising approach towards hardware-efficient fault tolerance, with many leading examples such as the cat-code and the GKP code organized by an underlying symmetry. We build on a representation-theoretic framework of quantum error correction to formalize the construction and analysis of multimode symmetric bosonic codes, focusing on finite groups. Here, a physical, logical representation and an initial seed state together define the codewords via a covariant encoding. A central observation is that QEC matrices can be viewed as morphisms of group representations, so that Schur's lemma forces entire irreducible representations blocks to vanish. We recover several known single- and two-mode bosonic codes and build and analyze the error protection capabilities of three codes based on finite subgroups of SU(2). We analyze the impact of reducibility of the logical representation and optimize the seed state using the near-optimal fidelity.

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quant-pharXiv:2511.14840Nov 20, 20250 citations

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Rowan Brad Quni-Gudzinas

Bosonic algebraically-restricted constellation (BARC) codes encode quantum information in finite superpositions of coherent states whose constellation points are solution sets of multivariate complex polynomial systems, with the polynomial constraints chosen to reflect photon-gain and photon-loss error symmetries [1],[2]. This paper develops a self-contained geometric analysis of the single-mode BARC construction. We restate the polynomial-to-symmetry pipeline, derive explicitly the upper bound on minimum pairwise separation for N equally weighted points on a circle of fixed mean photon number, and evaluate it for the hexagonal BARC family: for N = 6 points at mean photon number n̄, the maximal minimum chord is d = √n̄, giving worst-case coherent-state overlap |⟨α|β⟩|² = exp(−n̄/2). We then compute, with full arithmetic, the exact pairwise overlaps and the loss-error Gram matrix entries for a concrete hexagonal instance at n̄ = 8, obtaining |⟨αᵢ|αⱼ⟩|² = e⁻⁴ ≈ 0.0183 and post-loss overlaps e^(−3.8) ≈ 0.0224 at transmissivity η = 0.95, and we verify the approximate Knill–Laflamme diagonal condition, finding that the off-diagonal-to-diagonal deviation for the loss error is ~13.5% (ratio 0.1353353, O(10⁻¹)), while the next-nearest and opposite-pair contributions are individually bounded at O(10⁻²) or below. We compare these numbers against the qualitative expectations for four-point and cubature-style constellations, and situate BARC codes within the broader resource-commensurable landscape of bosonic encodings [10]. We argue that the polynomial-symmetry viewpoint is valuable primarily as a *generator* of candidate constellations, and that its ultimate worth must be decided by entanglement-fidelity benchmarks rather than by algebraic elegance. ## 1. Introduction Continuous-variable quantum information requires encodings that live in infinite-dimensional Hilbert space yet remain correctable under physically dominant errors. For optical and microwave cavities, those err Full text and updates: papers.qnfo.org/papers/barc-codes-as-algebraically-restricted-coherent-state-constellations-geometry-se/Bosonic algebraically-restricted constellation (BARC) codes encode quantum information in finite superpositions of coherent states whose constellation points are solution sets of multivariate complex polynomial systems, with the polynomial constraints chosen to reflect photon-gain and photon-loss error symmetries [1],[2]. This paper develops a self-contained geometric analysis of the single-mode BARC construction. We restate the polynomial-to-symmetry pipeline, derive explicitly the upper bound on minimum pairwise separation for N equally weighted points on a circle of fixed mean photon number, and evaluate it for the hexagonal BARC family: for N = 6 points at mean photon number n̄, the maximal minimum chord is d = √n̄, giving worst-case coherent-state overlap |⟨α|β⟩|² = exp(−n̄/2). We then compute, with full arithmetic, the exact pairwise overlaps and the loss-error Gram matrix entries for a concrete hexagonal instance at n̄ = 8, obtaining |⟨αᵢ|αⱼ⟩|² = e⁻⁴ ≈ 0.0183 and post-loss overlaps e^(−3.8) ≈ 0.0224 at transmissivity η = 0.95, and we verify the approximate Knill–Laflamme diagonal condition, finding that the off-diagonal-to-diagonal deviation for the loss error is ~13.5% (ratio 0.1353353, O(10⁻¹)), while the next-nearest and opposite-pair contributions are individually bounded at O(10⁻²) or below. We compare these numbers against the qualitative expectations for four-point and cubature-style constellations, and situate BARC codes within the broader resource-commensurable landscape of bosonic encodings [10]. We argue that the polynomial-symmetry viewpoint is valuable primarily as a *generator* of candidate constellations, and that its ultimate worth must be decided by entanglement-fidelity benchmarks rather than by algebraic elegance. ## 1. Introduction Continuous-variable quantum information requires encodings that live in infinite-dimensional Hilbert space yet remain correctable under physically dominant errors. For optical and microwave cavities, those err Full text and updates: papers.qnfo.org/papers/barc-codes-as-algebraically-restricted-coherent-state-constellations-geometry-se/

Oct 5, 20260 citations

4\leq n\leq25

SU(2)

n_d=\lceil(3d^2+1)/4\rceil

and construct each corresponding code through distance

d=10

. To our knowledge, the constructed

(n,d)=(28,6),(49,8),(76,10)

codes give the smallest reported

n

for their respective distances; all members through

d=9

attain the fixed-

n

\mathrm{SU}(2)

as its transversal operations. We introduce a systematic framework for searching codes with specified transversal groups by parametrizing their logical subspaces on the Stiefel manifold and minimizing a composite loss that enforces both the Knill-Laflamme conditions and a target transversal-group structure. Applying this method, we uncover a new

((6,2,3))

code admitting a transversal

Z\bigl(\tfrac{2π}{5}\bigr)

gate (transversal group

\mathrm{C}_{10}

), the smallest known distance

3

code supporting non-Clifford transversal gates, as well as several new

((7,2,3))

codes realizing the binary icosahedral group

2I

. We further propose the \emph{Subset-Sum-Linear-Programming} (SS-LP) construction for codes with transversal \emph{diagonal} gates, which dramatically shrinks the search space by reducing to integer partitions subject to linear constraints. In a more constrained form, the method also applies directly to the binary-dihedral groups

\mathrm{BD}_{2m}

. Specializing to

n=7

, the SS-LP method yields codes for all

\mathrm{BD}_{2m}

with

2m\le 36

, including the first

((7,2,3))

examples supporting transversal

T

gate (

\mathrm{BD}_{16}

) and

\sqrt{T}

gate (

\mathrm{BD}_{32}

), improving on the previous smallest examples

((11,2,3))

and

((19,2,3))

. Extending the SS-LP approach to

((8,2,3))

, we construct new codes for

2m>36

, including one supporting a transversal

T^{1/4}

gate (

\mathrm{BD}_{64}

). These results reveal a far richer landscape of nonadditive codes than previously recognized and underscore a deeper connection between quantum error correction and the algebraic constraints on transversal gate groups.

This paper develops a MacWilliams-type enumerator theory and semidefinite programming bounds for intrinsic quantum codes. An intrinsic code is a subspace

\mathcal{C}\subset V

of a finite-dimensional unitary representation

V

of a group

G

, with errors organized into the isotypic components of

\mathscr{L}(V)

under conjugation. The Knill-Laflamme conditions persist under every

G

-equivariant realization of

V

, aallowing intrinsic bounds to apply across diverse physical encodings. Without a group action, we first construct projector and twirl quadratic enumerators for an arbitrary orthogonal decomposition of

\mathscr{L}(V)

, analogous respectively to the

A

and

B

enumerators of Shor-Laflamme theory. These enumerators satisfy positivity, normalization, and Knill-Laflamme inequalities, but the two superoperator families need not span the same space and therefore admit no canonical duality. A group action places their generalized versions in the same equivariant intertwiner algebra, and the change of basis between them defines the intrinsic MacWilliams transform. When the conjugation representation has multiplicities, this algebra is noncommutative and the enumerators are matrix-valued; positivity and detection are encoded by Loewner-order constraints, making the natural programming bound a semidefinite program. The multiplicity-free specialization recovers the known scalar MacWilliams transform and linear-programming framework. For symmetric powers, intrinsic depth equals ordinary qudit distance, yielding extremality certificates for permutation-invariant qudit codes. Finally, for the

27

-dimensional

\mathrm{SU}(3)

representation

V=(2,2)

, an exact dual certificate proves the sharp bound

K\leq5

for codes detecting the full adjoint isotypic component. An explicit

5

-dimensional code attains the bound and, under the canonical embedding, gives a four-qutrit distance-

2

code.

\mathrm{SU}(2)

construction that unifies permutation-invariant qubit codes and bosonic codes, and higher-dimensional constructions exhibiting transversal Clifford symmetries and realizations beyond qubit systems.