arXiv subjects

Eric Kubischta

Publications and source records attributed to Eric Kubischta.

16 recordsLinked to original sources

arXiv2026

Time-Reversal Selection Rules for Quantum Error Correction

We apply time-reversal symmetry to quantum codes and show that it imposes parity selection rules on the physical error algebra. A time-reversal-invariant logical qubit on an odd number of spins is a Kramers doublet, forcing every even-weight Pauli to act as a scalar. Consequently, all even-weight Knill--Laflamme conditions hold automatically, so single-qubit error detection implies correction. We then reinterpret the Rains shadow enumerator through time reversal: each coefficient is a sum of error-resolved overlaps between a code and its time-reversed image.

quant-ph↗

arXiv2026

Nuclear-Spin Statistical Weights from Young Diagrams

Nuclear-spin statistical weights and selection rules in molecular spectroscopy are governed by the permutation symmetry of identical nuclei, although rigid and semi-rigid molecules typically realize only a proper subgroup of the full symmetric group. Building on the Schur-Weyl framework of Schmiedt, Jensen, and Schlemmer, we extend this approach to molecular geometries whose rovibrational motion realizes cyclic and dihedral permutation subgroups. By combining the Schur-Weyl decomposition of the $n$-spin Hilbert space with the Kra\'skiewicz-Weyman-Adin-Roichman major-index branching rule for the restriction $S_N \downarrow C_m$, we obtain a fully combinatorial method for determining nuclear-spin symmetry species and statistical weights. Each standard Young tableau corresponds to a definite cyclic character determined by its major index, with reflections yielding the associated dihedral symmetry, so that nuclear-spin species follow directly from tableau data without projection operators or case-specific constructions. Applications to $XeOF_4$, $SF_4$, benzene, deuterated benzene, and the tropylium cation demonstrate that the method applies uniformly to realistic molecular point groups and reproduces established statistical weights while providing a transparent spin-resolved structure. The approach supplies the symmetry information required for rovibrational line-intensity modeling and provides a practical, automatable framework for nuclear-spin symmetry analysis in polyatomic molecules.

physics.chem-ph↗

arXiv2026

Classical and Quantum MacWilliams Transforms as Spin Kinematics

Spin is the hidden engine behind the zoo of MacWilliams transforms in weight enumerator theories - not only for qubits and qudits, but even for classical codes. From nothing more than a split into trivial and nontrivial errors, we kinematically derive the MacWilliams transform as a Wigner-$D$ rotation between two canonical bases. Within each classical and quantum theory, changing the length $n$ leaves the rotation untouched: the same element simply reappears at spin $n/2$. And at fixed $n$, changing the rotation axis simply moves between the various classical and quantum theories.

quant-ph↗

arXiv2026

Minimal Permutation-Invariant Qudit Codes from Edge-Colorings of Complete Graphs

We study permutation-invariant quantum codes in the symmetric subspace $\mathrm{Sym}^n(\mathbb{C}^q) $ of $n$ qudits of local dimension $q$. For every integer $q\geq 2$, we construct a permutation-invariant code with parameters $((4,q,2))_q$. Thus four physical qudits suffice to encode one logical qudit with distance two in the symmetric sector for every local dimension. We also show, using linear-programming constraints for permutation-invariant quantum codes, that no permutation-invariant code of dimension $q$ and distance at least $2$ exists in $\mathrm{Sym}^n(\mathbb{C}^q)$ for $n\leq 3$. Hence four qudits are necessary and sufficient. The construction has a simple representation-theoretic and combinatorial description. In the irreducible $\mathrm{SU}(q)$-module $\mathrm{Sym}^4(\mathbb{C}^q)$, the distance-two Knill-Laflamme conditions split into root and Cartan parts. By restricting supports to the even-entry occupation layer, all root-error conditions vanish automatically. The remaining Cartan conditions reduce to linear balancing constraints on packets of occupation vectors. These packets admit a natural graph-theoretic interpretation in terms of the vertices and edges of the complete graph $K_q$: for odd $q$, they are organized by the midpoint rule, while for even $q$, they are organized by a decomposition of $K_q$ into perfect matchings. In this way, the existence of minimal $((4,q,2))_q$ permutation-invariant codes is reduced to a parity-dependent edge-coloring problem on $K_q$.

quant-ph↗

arXiv2026

MacWilliams Identities for Intrinsic Quantum Codes

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.

quant-ph↗

arXiv2025

Intrinsic Quantum Codes

We introduce an intrinsic formulation of quantum error correction based on representation theory, in which error-protection structure is encoded directly in a unitary group representation, rather than being tied to a particular embedding into a larger Hilbert space. In this framework, error models are classified according to the isotypic decomposition of the conjugation action on the operator algebra. Our main result, the \emph{Schur bootstrap}, shows that if an intrinsic code satisfies the Knill--Laflamme conditions on a given symmetry sector, then the same error-protection relations hold for every extrinsic realization obtained from a group-equivariant isometric embedding into a larger Hilbert space. Thus a single intrinsic verification certifies the corresponding symmetry-resolved error-correction conditions across an entire family of physical realizations. We further introduce an intrinsic notion of distance, called depth, defined via adjoint order. For standard multi-qudit systems this coincides with conventional code distance, while for more general representations it refines the usual weight-based notion. We also prove an intrinsic Eastin--Knill theorem: any intrinsic code of depth at least two has a discrete logical symmetry group, with the obstruction to continuous covariant gates arising from the representation-theoretic structure of the adjoint action. We illustrate the framework with several examples, including a minimal $\mathrm{SU}(2)$ construction that unifies permutation-invariant qubit codes and bosonic codes, and higher-dimensional constructions exhibiting transversal Clifford symmetries and realizations beyond qubit systems.

quant-ph↗

arXiv2024

Classification of the Subgroups of the Two-Qubit Clifford Group

We perform a complete classification of all 56 subgroups of the two-qubit Clifford group containing the two-qubit Pauli group. We provide generators for these groups using gates familiar to the quantum information community and we reference these groups against the group libraries provided in GAP. We also list several families of groups in higher levels of the two-qubit Clifford hierarchy.

quant-ph↗

arXiv2024

Flexible Fault Tolerant Gate Gadgets

We design flexible fault tolerant gate gadgets that allow the data and the ancilla to be encoded using different codes. By picking a stabilizer code for the ancilla we are able to perform both Clifford and non-Clifford gates fault tolerantly on generic quantum codes, including both stabilizer and non-additive codes. This allows us to demonstrate the first universal fault tolerant gate set for non-additive codes. We consider fault tolerance both with respect to a dephasing channel and a depolarizing channel.

quant-ph↗

arXiv2024

Quantum Codes from Twisted Unitary $t$-groups

We introduce twisted unitary $t$-groups, a generalization of unitary $t$-groups under a twisting by an irreducible representation. We then apply representation theoretic methods to the Knill-Laflamme error correction conditions to show that twisted unitary $t$-groups automatically correspond to quantum codes with distance $d=t+1$. By construction these codes have many transversal gates, which naturally do not spread errors and thus are useful for fault tolerance.

quant-ph↗

arXiv2024

Quantum Codes and Irreducible Products of Characters

In a recent paper, we defined twisted unitary $1$-groups and showed that they automatically induced error-detecting quantum codes. We also showed that twisted unitary $1$-groups correspond to irreducible products of characters thereby reducing the problem of code-finding to a computation in the character theory of finite groups. Using a combination of GAP computations and results from the mathematics literature on irreducible products of characters, we identify many new non-trivial quantum codes with unusual transversal gates.

quant-ph↗

arXiv2024

Quantum theory of molecular orientations

We formulate a quantum phase space for rotational and nuclear-spin states of rigid molecules. For each nuclear spin isomer, we re-derive the isomer's admissible angular momentum states from molecular geometry and nuclear-spin data, introduce its angular position states using quantization theory, and develop a generalized Fourier transform converting between the two. We classify molecules into three types -- asymmetric, rotationally symmetric, and perrotationally symmetric -- with the last type having no macroscopic analogue due to nuclear-spin statistics constraints. We discuss two general features in perrotationally symmetric state spaces that are Hamiltonian-independent and induced solely by symmetry and spin statistics. First, we quantify when and how an isomer's state space is completely rotation-spin entangled, meaning that it does not admit any separable states. Second, we identify isomers whose position states house an internal pseudo-spin or "fiber" degree of freedom, and the fiber's Berry phase or matrix after adiabatic changes in position yields naturally robust operations, akin to braiding anyonic quasiparticles or realizing fault-tolerant quantum gates. We outline how the fiber can be used as a quantum error-correcting code and discuss scenarios where these features can be experimentally probed

quant-ph↗

arXiv2024

Quantum Weight Enumerators for Real Codes with $X$ and $Z$ Exactly Transversal

In this note we show that the weight enumerators of a real quantum error correcting code with $ X $ and $ Z $ exactly transversal must satisfy certain identities. One consequence of these identities is that if the code is error detecting then it is automatically error correcting for free; implying a relationship between transversality and code distance.

quant-ph↗

arXiv2023

A Family of Quantum Codes with Exotic Transversal Gates

Recently an algorithm has been constructed that shows the binary icosahedral group $\2I$ together with a $T$-like gate forms the most efficient single-qubit universal gate set. To carry out the algorithm fault tolerantly requires a code that implements $\ico$ transversally. However, no such code has ever been demonstrated in the literature. We fill this void by constructing a family of distance $d = 3$ codes that all implement $2I$ transversally. A surprising feature of this family is that the codes can be deduced entirely from symmetry considerations that only $2I$ affords.

quant-ph↗

arXiv2023

Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates

With respect to the transversal gate group (an invariant of quantum codes), we demonstrate that non-additive codes can outperform stabilizer codes. We do this by constructing spin codes that correspond to permutation-invariant multiqubit codes that can implement generalized phase gates transversally. Of particular note, we construct permutation-invariant quantum codes that implement a transversal $T$ gate using fewer qubits and with a better minimum distance than is possible with the best known stabilizer codes.

quant-ph↗

arXiv2023

Quantum Error Correction in the Lowest Landau Level

We develop finite-dimensional versions of the quantum error-correcting codes proposed by Albert, Covey, and Preskill (ACP) for continuous-variable quantum computation on configuration spaces with nonabelian symmetry groups. Our codes can be realized by a charged particle in a Landau level on a spherical geometry -- in contrast to the planar Landau level realization of the qudit codes of Gottesman, Kitaev, and Preskill (GKP) -- or more generally by spin coherent states. Our quantum error-correction scheme is inherently approximate, and the encoded states may be easier to prepare than those of GKP or ACP.

quant-ph↗

arXiv2019

Semantic Security on Wiretap Channels using Universal Hashing with Fading Applications

We furnish a procedure based on universal hash families (UHFs) that can convert an error correcting coding scheme (ECC) of rate $R$ into a semantically secure wiretap coding scheme of rate $R - ξ$ where $ξ$ is a parameter derived from the eavesdropper's point-to-point channel. This conversion is shown to be polynomial time efficient with block length and is applicable to any channel. When an ECC is chosen, our procedure induces a wiretap coding scheme that is concrete and efficient as long as the ECC is also such. To prove this induced wiretap coding scheme is semantically secure, we have constructed bounds on the information leaked to the eavesdropper. Our construction is an upgrade of bounds from recent literature: the novelty here being that our leakage bounds hold for any message distribution. Indeed, our wiretap procedure using UHFs and our characterization of its semantic leakage is the first main contribution of this work. The other main contribution of this work is as follows. We apply the aforementioned procedure to a variety of wiretap channels in order to show the procedure's efficacy, and as a result of such applications, our results establish new achievable semantically secure rates.

cs.IT↗