Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2609.27318

Minimal representations of topology-preserving quantum-like states

William Horvat, Debadrita Saha, Nicolás Gigena, Prineha Narang, Gregory D. Scholes

quant-phmath.COSubmitted Sep 23, 2026• Announced Sep 24, 2026• 0 citations• Revised Oct 6, 2026 (v2)
v1 · Sep 23, 2026v2 · Oct 6, 2026
arXivHTMLPDF

Abstract

We provide an equitable partition that gives an exact, minimal representation for the graph Cartesian product formed from quantum-like bits that preserves the relevant spectral and topological properties. We show that this result follows from the fact that the operations of taking the Cartesian product of graphs and constructing equitable partition of the graphs commute. Numerical simulations illustrate the preserved emergent eigenstates in the reduced structures. Additionally, we provide a construction of the minimal structure without passing through the Cartesian product. Finally, we frame quantum-like structures in the language of topology and fibrations.

Comments
Accepted for publication in Physical Review E

Related papers

PERFECT STATE TRANSFER, GRAPH PRODUCTS AND EQUITABLE PARTITIONS

Ge Yang, Benjamin Greenberg, Oscar Perez +1 more

We describe new constructions of graphs which exhibit perfect state transfer on continuous-time quantum walks. Our constructions are based on generalizations of the double cones and variants of the Cartesian graph products (which include the hypercube). We also describe a generalization of the path collapsing argument (which reduces questions about perfect state transfer to simpler weighted multigraphs) for graphs with equitable distance partitions.

arXiv:1009.1340•International Journal of Quantum Information•Apr 1, 2011•31 citations

Quantumlike Product States Constructed from Classical Networks

Gregory D. Scholes, Graziano Amati

Can complex classical systems be designed to exhibit superpositions of tensor products of basis states, thereby mimicking quantum states? We exhibit a one-to-one map between the product basis of quantum states comprising an arbitrary number of qubits and the eigenstates of a construction comprising classical oscillator networks. Specifically, we prove the existence of this map based on Cartesian products of graphs, where the graphs depict the layout of oscillator networks. We show how quantumlike gates can act on the classical networks to allow quantumlike operations in the state space.

arXiv:2406.19221•Physical Review Letters•Feb 13, 2025•13 citations

Encoding matroids into quantum states

Nathan Ferreira, Alison A. Silva, Giuliano G. La Guardia +1 more

Efficient representations of multipartite quantum states play a fundamental role in quantum information theory, providing both conceptual insight and practical tools for characterizing entanglement. Motivated by the axiomatic framework for graph states [Phys. Rev. A 85, 062313 (2012)] and its subsequent extension to hypergraph states [Phys. Rev. A 87, 022311 (2013)], we introduce an axiomatic construction of \emph{matroid states}, a new family of multipartite quantum states associated with matroids. Our constructions are based on a set of axioms analogous to those that define graph and hypergraph states, yielding a consistent quantum representation of arbitrary matroids. Two ways of constructing matroid states are proposed: the first is defined in terms of circuits, and the second in terms of independent sets. In both approaches, we establish the existence of universal global operators that satisfy desirable properties such as locality, symmetry, commutativity, and are associated with the combinatorial structure of matroids. Furthermore, we establish a hierarchy connecting graph, matroid, and hypergraph states within a unified framework. Additionally, we show how to obtain an arbitrary graph state by applying suitable families of matroid states, whose corresponding operators are the generators of the stabilizer subgroup of the graph state. These results identify matroid theory as a natural combinatorial language for the efficient description of multipartite quantum states and open new perspectives for the investigation of quantum entanglement and related combinatorial structures.

quant-pharXiv:2607.02736•Jul 6, 2026•0 citations

Vector representations of graphs and distinguishing quantum product states with one-way LOCC

David W. Kribs, Comfort Mintah, Michael Nathanson +1 more

Distinguishing sets of quantum states shared by two parties using only local operations and classical communication measurements is a fundamental topic in quantum communication and quantum information theory. We introduce a graph-theoretic approach, based on the theory of vector representations of graphs, to the core problem of distinguishing product states with one-way LOCC. We establish a number of results that show how distinguishing such states can be framed in terms of properties of the underlying graphs associated with a set of vector product states. We also present a number of illustrative examples.

arXiv:1912.10483•Linear Algebra and its Applications•May 19, 2020•4 citations

Symmetry-Protected Topological Structures in Quantum Many-Body Scar Subspaces

Chihiro Matsui, Thomas Quella, Naoto Tsuji

We investigate how symmetry-associated topological properties, originally defined for ground states, can be consistently extended to dynamically isolated subspaces of quantum many-body systems. Focusing on quantum many-body scars (QMBS), we show that a scar subspace is invariant under the relevant symmetries, and it can support well-defined topological properties inherited from an underlying symmetry-protected topological (SPT) ground state, even though individual scar states need not themselves be symmetric. We refer to such structures as SPT-like scar subspaces, emphasizing that the topological properties are defined at the level of the symmetry-invariant subspace rather than on a state-by-state basis. As a concrete example, we analyze the spin-$1$ Affleck--Kennedy--Lieb--Tasaki (AKLT) model, where a restricted spectrum-generating algebra (rSGA) generates a tower of QMBS on top of a nontrivial SPT ground state admitting a matrix product state (MPS) representation. We demonstrate that the resulting symmetry-invariant scar subspace inherits well-defined topological characteristics, encoded in bond-space symmetry representations, expectation values of symmetry-twist operators, and long-range string order in the dilute regime. We further show that these features persist in a broader class of models, including deformations and higher-spin generalizations that preserve the relevant symmetries and admit nontrivial SPT ground states with MPS representations. Our results provide a general framework in which symmetry-associated topological properties, while inherited from the ground state, are consistently realized at the level of symmetry-invariant nonthermal subspaces. This perspective clarifies how topological structures can be meaningfully defined beyond individual eigenstates.

cond-mat.str-elCross-listedarXiv:2512.11216•Dec 15, 2025•0 citations

An efficient algebraic representation for graph states for measurement-based quantum computing

Sebastiano Corli, Enrico Prati

Graph states are the main computational building blocks of measurement-based computation and a useful tool for error correction in the gate model architecture. The graph states form a class of quantum states which are eigenvectors for the abelian group of stabilizer operators. They own topological properties, arising from their graph structure, including the presence of highly connected nodes, called hubs. Starting from hub nodes, we demonstrate how to efficiently express a graph state through the generators of the stabilizer group. We provide examples by expressing the ring and the star topology, for which the number of stabilizers reduces from$n$to$\lceil\frac{n}{2}\rceil$, and from$n$to 1, respectively. We demonstrate that the graph states can be generated by a subgroup of the stabilizer group. Therefore, we provide an algebraic framework to manipulate the graph states with a reduced number of stabilizers.

arXiv:2212.12102•IEEE International Conference on Rebooting Computing (ICRC)•Dec 1, 2022•4 citations
We provide an equitable partition that gives an exact, minimal representation for the graph Cartesian product formed from quantum-like bits that preserves the relevant spectral and topological properties. We show that this result follows from the fact that the operations of taking the Cartesian product of graphs and constructing equitable partition of the graphs commute. Numerical simulations illustrate the preserved emergent eigenstates in the reduced structures. Additionally, we provide a construction of the minimal structure without passing through the Cartesian product. Finally, we frame quantum-like structures in the language of topology and fibrations.
Can complex classical systems be designed to exhibit superpositions of tensor products of basis states, thereby mimicking quantum states? We exhibit a one-to-one map between the product basis of quantum states comprising an arbitrary number of qubits and the eigenstates of a construction comprising classical oscillator networks. Specifically, we prove the existence of this map based on Cartesian products of graphs, where the graphs depict the layout of oscillator networks. We show how quantumlike gates can act on the classical networks to allow quantumlike operations in the state space.
Efficient representations of multipartite quantum states play a fundamental role in quantum information theory, providing both conceptual insight and practical tools for characterizing entanglement. Motivated by the axiomatic framework for graph states [Phys. Rev. A 85, 062313 (2012)] and its subsequent extension to hypergraph states [Phys. Rev. A 87, 022311 (2013)], we introduce an axiomatic construction of \emph{matroid states}, a new family of multipartite quantum states associated with matroids. Our constructions are based on a set of axioms analogous to those that define graph and hypergraph states, yielding a consistent quantum representation of arbitrary matroids. Two ways of constructing matroid states are proposed: the first is defined in terms of circuits, and the second in terms of independent sets. In both approaches, we establish the existence of universal global operators that satisfy desirable properties such as locality, symmetry, commutativity, and are associated with the combinatorial structure of matroids. Furthermore, we establish a hierarchy connecting graph, matroid, and hypergraph states within a unified framework. Additionally, we show how to obtain an arbitrary graph state by applying suitable families of matroid states, whose corresponding operators are the generators of the stabilizer subgroup of the graph state. These results identify matroid theory as a natural combinatorial language for the efficient description of multipartite quantum states and open new perspectives for the investigation of quantum entanglement and related combinatorial structures.
Distinguishing sets of quantum states shared by two parties using only local operations and classical communication measurements is a fundamental topic in quantum communication and quantum information theory. We introduce a graph-theoretic approach, based on the theory of vector representations of graphs, to the core problem of distinguishing product states with one-way LOCC. We establish a number of results that show how distinguishing such states can be framed in terms of properties of the underlying graphs associated with a set of vector product states. We also present a number of illustrative examples.
We investigate how symmetry-associated topological properties, originally defined for ground states, can be consistently extended to dynamically isolated subspaces of quantum many-body systems. Focusing on quantum many-body scars (QMBS), we show that a scar subspace is invariant under the relevant symmetries, and it can support well-defined topological properties inherited from an underlying symmetry-protected topological (SPT) ground state, even though individual scar states need not themselves be symmetric. We refer to such structures as SPT-like scar subspaces, emphasizing that the topological properties are defined at the level of the symmetry-invariant subspace rather than on a state-by-state basis. As a concrete example, we analyze the spin-111 Affleck--Kennedy--Lieb--Tasaki (AKLT) model, where a restricted spectrum-generating algebra (rSGA) generates a tower of QMBS on top of a nontrivial SPT ground state admitting a matrix product state (MPS) representation. We demonstrate that the resulting symmetry-invariant scar subspace inherits well-defined topological characteristics, encoded in bond-space symmetry representations, expectation values of symmetry-twist operators, and long-range string order in the dilute regime. We further show that these features persist in a broader class of models, including deformations and higher-spin generalizations that preserve the relevant symmetries and admit nontrivial SPT ground states with MPS representations. Our results provide a general framework in which symmetry-associated topological properties, while inherited from the ground state, are consistently realized at the level of symmetry-invariant nonthermal subspaces. This perspective clarifies how topological structures can be meaningfully defined beyond individual eigenstates.
Graph states are the main computational building blocks of measurement-based computation and a useful tool for error correction in the gate model architecture. The graph states form a class of quantum states which are eigenvectors for the abelian group of stabilizer operators. They own topological properties, arising from their graph structure, including the presence of highly connected nodes, called hubs. Starting from hub nodes, we demonstrate how to efficiently express a graph state through the generators of the stabilizer group. We provide examples by expressing the ring and the star topology, for which the number of stabilizers reduces fromnnnto⌈n2⌉\lceil\frac{n}{2}\rceil⌈2n​⌉, and fromnnnto 1, respectively. We demonstrate that the graph states can be generated by a subgroup of the stabilizer group. Therefore, we provide an algebraic framework to manipulate the graph states with a reduced number of stabilizers.