We determine conditions on classical group algebra codes so that they have pre-orientation for cup products and copy-cup gates. This defines quantum codes that have constant-depth $\operatorname{CZ}$ and $\operatorname{CCZ}$ gates constructed via tensor products of classical group algebra codes, including hypergraph and balanced products. We show that determining the conditions relies on solving the perfect matching problem in graph theory. Conditions are fully determined for the 2- and 3-copy-cup gates, for group algebra codes up to weight 4, including for codes with odd check weight. These include the bivariate bicycle codes, which we show do not have the pre-orientation for either type of copy-cup gate. We show that abelian weight 4 group algebra codes satisfying the non-associative 3-copy-cup gate necessarily have a code distance of 2, whereas codes that satisfy conditions for the symmetric 3-copy-cup gate can have higher distances, and in fact also satisfy conditions for the 2-copy-cup gate. Finally we find examples of quantum codes from the product of abelian group algebra codes that have inter-code constant-depth $\operatorname{CZ}$ and $\operatorname{CCZ}$ gates.We determine conditions on classical group algebra codes so that they have pre-orientation for cup products and copy-cup gates. This defines quantum codes that have constant-depth $\operatorname{CZ}$ and $\operatorname{CCZ}$ gates constructed via tensor products of classical group algebra codes, including hypergraph and balanced products. We show that determining the conditions relies on solving the perfect matching problem in graph theory. Conditions are fully determined for the 2- and 3-copy-cup gates, for group algebra codes up to weight 4, including for codes with odd check weight. These include the bivariate bicycle codes, which we show do not have the pre-orientation for either type of copy-cup gate. We show that abelian weight 4 group algebra codes satisfying the non-associative 3-copy-cup gate necessarily have a code distance of 2, whereas codes that satisfy conditions for the symmetric 3-copy-cup gate can have higher distances, and in fact also satisfy conditions for the 2-copy-cup gate. Finally we find examples of quantum codes from the product of abelian group algebra codes that have inter-code constant-depth $\operatorname{CZ}$ and $\operatorname{CCZ}$ gates.

quant-pharXiv:2602.23307Physical Review AFeb 27, 20260 citations

Abelian groups and the Weyl approach to kinematics nonlocal function-algebras

Jan Jerzy Sławianowski

Reports on Mathematical PhysicsSep 1, 19742 citations

Lifted surgery for non-Abelian two-block group-algebra codes

Tushar Pandey

Code surgery measures a logical operator of a quantum LDPC code with $O(d)$ rounds of syndrome extraction. Lifted surgery amortises this cost on Abelian group-algebra codes by measuring an orbit of logical operators under a translation symmetry in one merged code. We extend it to two-block group-algebra codes over non-Abelian groups and ask whether non-commutativity lets one merged code measure more operators than any Abelian group of its symmetries. We classify the block-affine automorphisms of these codes and find that, measured against this full group, most apparent non-Abelian gains disappear. We prove that the gain is at most the index of a largest Abelian subgroup. We find codes with exact distance up to $13$ where the gain is two, including codes in which one merged code reads out every logical qubit, and codes over products of $A_4$ , $S_4$ and SL(2,3) with gain three for a best logical, up to $[[336,26,12]]$ . A merged-distance lemma gives a simple condition for the gadget to preserve the code distance. In circuit-level simulations with Relay-BP decoding, at the error rates we can resolve, the non-Abelian gadget is as reliable as, or more reliable than, the Abelian gadgets it replaces within statistical error, while using two to three times fewer rounds.Code surgery measures a logical operator of a quantum LDPC code with $O(d)$ rounds of syndrome extraction. Lifted surgery amortises this cost on Abelian group-algebra codes by measuring an orbit of logical operators under a translation symmetry in one merged code. We extend it to two-block group-algebra codes over non-Abelian groups and ask whether non-commutativity lets one merged code measure more operators than any Abelian group of its symmetries. We classify the block-affine automorphisms of these codes and find that, measured against this full group, most apparent non-Abelian gains disappear. We prove that the gain is at most the index of a largest Abelian subgroup. We find codes with exact distance up to $13$ where the gain is two, including codes in which one merged code reads out every logical qubit, and codes over products of $A_4$, $S_4$ and SL(2,3) with gain three for a best logical, up to $[[336,26,12]]$. A merged-distance lemma gives a simple condition for the gadget to preserve the code distance. In circuit-level simulations with Relay-BP decoding, at the error rates we can resolve, the non-Abelian gadget is as reliable as, or more reliable than, the Abelian gadgets it replaces within statistical error, while using two to three times fewer rounds.

quant-pharXiv:2610.05307Oct 6, 20260 citations

Lower Bounds of Functions on Finite Abelian Groups

Jianting Yang, Ke Ye, Lihong Zhi

arXiv:2302.02188Lecture notes in computer scienceDec 8, 20233 citations

Quadratic tensors as a unification of Clifford, Gaussian, and free-fermion physics

Andreas Bauer, Seth Lloyd

Certain families of quantum mechanical models can be described and solved efficiently on a classical computer, including qubit or qudit Clifford circuits and stabilizer codes, free-boson or free-fermion models, and certain rotor and GKP codes. We show that all of these families can be described as instances of the same algebraic structure, namely quadratic functions over abelian groups, or more generally over (super) Hopf algebras. Different kinds of degrees of freedom correspond to different "elementary" abelian groups or Hopf algebras: $\mathbb{Z}_2$ for qubits, $\mathbb{Z}_d$ for qudits, $\mathbb{R}$ for continuous variables, both $\mathbb{Z}$ and $\mathbb{R}/\mathbb{Z}$ for rotors, and a super Hopf algebra $\mathcal F$ for fermionic modes. Objects such as states, operators, superoperators, or projection-operator valued measures, etc, are tensors. For the solvable models above, these tensors are quadratic tensors based on quadratic functions. Quadratic tensors with $n$ degrees of freedom are fully specified by only $O(n^2)$ coefficients. Tensor networks of quadratic tensors can be contracted efficiently on the level of these coefficients, using an operation reminiscent of the Schur complement. Our formalism naturally includes models with mixed degrees of freedom, such as qudits of different dimensions. We also use quadratic functions to define generalized stabilizer codes and Clifford gates for arbitrary abelian groups. Finally, we give a generalization from quadratic (or 2nd order) to $i$ th order tensors, which are specified by $O(n^i)$ coefficients but cannot be contracted efficiently in general.Certain families of quantum mechanical models can be described and solved efficiently on a classical computer, including qubit or qudit Clifford circuits and stabilizer codes, free-boson or free-fermion models, and certain rotor and GKP codes. We show that all of these families can be described as instances of the same algebraic structure, namely quadratic functions over abelian groups, or more generally over (super) Hopf algebras. Different kinds of degrees of freedom correspond to different "elementary" abelian groups or Hopf algebras: $\mathbb{Z}_2$ for qubits, $\mathbb{Z}_d$ for qudits, $\mathbb{R}$ for continuous variables, both $\mathbb{Z}$ and $\mathbb{R}/\mathbb{Z}$ for rotors, and a super Hopf algebra $\mathcal F$ for fermionic modes. Objects such as states, operators, superoperators, or projection-operator valued measures, etc, are tensors. For the solvable models above, these tensors are quadratic tensors based on quadratic functions. Quadratic tensors with $n$ degrees of freedom are fully specified by only $O(n^2)$ coefficients. Tensor networks of quadratic tensors can be contracted efficiently on the level of these coefficients, using an operation reminiscent of the Schur complement. Our formalism naturally includes models with mixed degrees of freedom, such as qudits of different dimensions. We also use quadratic functions to define generalized stabilizer codes and Clifford gates for arbitrary abelian groups. Finally, we give a generalization from quadratic (or 2nd order) to $i$th order tensors, which are specified by $O(n^i)$ coefficients but cannot be contracted efficiently in general.

quant-pharXiv:2601.15396Jan 23, 20260 citations

We determine conditions on classical group algebra codes so that they have pre-orientation for cup products and copy-cup gates. This defines quantum codes that have constant-depth

\operatorname{CZ}

and

\operatorname{CCZ}

gates constructed via tensor products of classical group algebra codes, including hypergraph and balanced products. We show that determining the conditions relies on solving the perfect matching problem in graph theory. Conditions are fully determined for the 2- and 3-copy-cup gates, for group algebra codes up to weight 4, including for codes with odd check weight. These include the bivariate bicycle codes, which we show do not have the pre-orientation for either type of copy-cup gate. We show that abelian weight 4 group algebra codes satisfying the non-associative 3-copy-cup gate necessarily have a code distance of 2, whereas codes that satisfy conditions for the symmetric 3-copy-cup gate can have higher distances, and in fact also satisfy conditions for the 2-copy-cup gate. Finally we find examples of quantum codes from the product of abelian group algebra codes that have inter-code constant-depth

\operatorname{CZ}

and

\operatorname{CCZ}

gates.

Code surgery measures a logical operator of a quantum LDPC code with

O(d)

rounds of syndrome extraction. Lifted surgery amortises this cost on Abelian group-algebra codes by measuring an orbit of logical operators under a translation symmetry in one merged code. We extend it to two-block group-algebra codes over non-Abelian groups and ask whether non-commutativity lets one merged code measure more operators than any Abelian group of its symmetries. We classify the block-affine automorphisms of these codes and find that, measured against this full group, most apparent non-Abelian gains disappear. We prove that the gain is at most the index of a largest Abelian subgroup. We find codes with exact distance up to

13

where the gain is two, including codes in which one merged code reads out every logical qubit, and codes over products of

A_4

S_4

and SL(2,3) with gain three for a best logical, up to

[[336,26,12]]

. A merged-distance lemma gives a simple condition for the gadget to preserve the code distance. In circuit-level simulations with Relay-BP decoding, at the error rates we can resolve, the non-Abelian gadget is as reliable as, or more reliable than, the Abelian gadgets it replaces within statistical error, while using two to three times fewer rounds.

\mathbb{Z}_2

for qubits,

\mathbb{Z}_d

for qudits,

\mathbb{R}

for continuous variables, both

\mathbb{Z}

and

\mathbb{R}/\mathbb{Z}

for rotors, and a super Hopf algebra

\mathcal F

for fermionic modes. Objects such as states, operators, superoperators, or projection-operator valued measures, etc, are tensors. For the solvable models above, these tensors are quadratic tensors based on quadratic functions. Quadratic tensors with

n

degrees of freedom are fully specified by only

O(n^2)

coefficients. Tensor networks of quadratic tensors can be contracted efficiently on the level of these coefficients, using an operation reminiscent of the Schur complement. Our formalism naturally includes models with mixed degrees of freedom, such as qudits of different dimensions. We also use quadratic functions to define generalized stabilizer codes and Clifford gates for arbitrary abelian groups. Finally, we give a generalization from quadratic (or 2nd order) to

i

th order tensors, which are specified by

O(n^i)

coefficients but cannot be contracted efficiently in general.