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

Physical Review Research

Symmetry-protected topological order as a requirement for measurement-based quantum gate teleportation

Zhuohao Liu, Emma C. Johnson, David L. Feder

Published Feb 1, 2024• 1 citation
Publisher

Abstract

All known resource states for measurement-based quantum teleportation in correlation space possess symmetry-protected topological order, but is this a sufficient or even necessary condition? This work considers two families of one-dimensional qubit states to answer this question in the negative. The first is a family of matrix-product states with bond dimension two that includes the cluster state as a special case, protected by a global non-on-site symmetry, which is characterized by a finite correlation length and a degenerate entanglement spectrum in the thermodynamic limit but which is unable to deterministically teleport a universal set of single-qubit gates. The second are states with bond dimension four that are a resource for deterministic universal teleportation of finite single-qubit gates, but which possess no symmetry. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.013134

Related papers

Symmetry protected topological order as a requirement for measurement-based quantum gate teleportation

Zhuohao Liu, Emma C. Johnson, David L. Feder

All known resource states for measurement-based quantum teleportation in correlation space possess symmetry protected topological order, but is this a sufficient or even necessary condition? This work considers two families of one-dimensional qubit states to answer this question in the negative. The first is a family of matrix-product states with bond dimension two that includes the cluster state as a special case, protected by a global non-onsite symmetry, which is characterized by a finite correlation length and a degenerate entanglement spectrum in the thermodynamic limit but which is unable to deterministically teleport a universal set of single-qubit gates. The second are states with bond dimension four that are a resource for deterministic universal teleportation of finite single-qubit gates, but which possess no symmetry.

arXiv:2310.10561•Oct 16, 2023•0 citations

Quantum teleportation implies symmetry-protected topological order

Yifan Hong, David T. Stephen, Aaron J. Friedman

We constrain a broad class of teleportation protocols using insights from locality. In the ``standard'' teleportation protocols we consider, all outcome-dependent unitaries are Pauli operators conditioned on linear functions of the measurement outcomes. We find that all such protocols involve preparing a ``resource state'' exhibiting symmetry-protected topological (SPT) order with Abelian protecting symmetry Gk=(Z2×Z2)k . The k logical states are teleported between the edges of the chain by measuring the corresponding 2k string order parameters in the bulk and applying outcome-dependent Paulis. Hence, this single class of nontrivial SPT states is both necessary and sufficient for the standard teleportation of k qubits. We illustrate this result with several examples, including the cluster state, variants thereof, and a nonstabilizer hypergraph state.

arXiv:2310.12227•Quantum•Oct 10, 2024•4 citations

Qudit quantum computation on matrix product states with global symmetry

Dongsheng Wang, David T. Stephen, Robert Raussendorf

Resource states that contain nontrivial symmetry-protected topological order are identified for universal single-qudit measurement-based quantum computation. Our resource states fall into two classes: one as the qudit generalizations of the one-dimensional qubit cluster state, and the other as the higher-symmetry generalizations of the spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) state, namely, with unitary, orthogonal, or symplectic symmetry. The symmetry in cluster states protects information propagation (identity gate), while the higher symmetry in AKLT-type states enables nontrivial gate computation. This work demonstrates a close connection between measurement-based quantum computation and symmetry-protected topological order.

arXiv:1609.07174•Physical Review A•Mar 9, 2017•33 citations

Redundant string symmetry-based error correction: Demonstrations on quantum devices

Zhangjie Qin, Daniel Azses, Eran Sela +2 more

Computational power in measurement-based quantum computing stems from the symmetry-protected topological (SPT) order of entangled resource states. However, resource states are prone to preparation errors. We introduce a quantum error correction approach using redundant nonlocal symmetry of the resource state. We demonstrate it within a teleportation protocol based on extending the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry of one-dimensional cluster states to other graph states. Qubit ZZ-crosstalk errors, which are prominent in quantum devices, degrade the teleportation fidelity of the usual cluster state. However, as we demonstrate on quantum hardware, once we grow graph states with redundant symmetry, perfect teleportation fidelity is restored. We identify the underlying redundant-SPT order as error-protected degeneracies in the entanglement spectrum.

arXiv:2310.12854•Physical Review A•Jul 31, 2024•2 citations

Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases

Nathanan Tantivasadakarn, Ryan Thorngren, Ashvin Vishwanath +1 more

A fundamental distinction between many-body quantum states are those with short- and long-range entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schrödinger cat states, topological order, and quantum criticality. Remarkably, examples are known where LRE is obtained by performing single-site measurements on SRE, such as the toric code from measuring a sublattice of a 2D cluster state. However, a systematic understanding of when and how measurements of SRE give rise to LRE is still lacking. Here, we establish that LRE appears upon performing measurements on symmetry-protected topological (SPT) phases—of which the cluster state is one example. For instance, we show how to implement the Kramers-Wannier transformation by adding a cluster SPT to an input state followed by measurement. This transformation naturally relates states with SRE and LRE. An application is the realization of double-semion order when the input state is the Z 2 Levin-Gu SPT. Similarly, the addition of fermionic SPTs and measurement leads to an implementation of the Jordan-Wigner transformation of a general state. More generally, we argue that a large class of SPT phases protected by G × H symmetry gives rise to anomalous LRE upon measuring G -charges, and we prove that this persists for generic points in the SPT phase under certain conditions. Our work introduces a new practical tool for using SPT phases as resources for creating LRE, and we uncover the classification result that all states related by sequentially gauging Abelian groups or by Jordan-Wigner transformation are in the same equivalence class, once we augment finite-depth circuits with single-site measurements. In particular, any topological or fracton order with a solvable finite gauge group can be obtained from a product state in this way. Published by the American Physical Society 2024

arXiv:2112.01519•Physical Review X•Jun 7, 2024•126 citations

Symmetry-protected topologically ordered states for universal quantum computation

Hendrik Poulsen Nautrup, Tzu-Chieh Wei

Measurement-based quantum computation is a model for quantum information processing utilizing local measurements on suitably entangled resource states for the implementation of quantum gates. A complete characterization for universal resource states is still missing. It has been shown that symmetry-protected topological order in one dimension can be exploited for the protection of certain quantum gates in measurement-based quantum computation. In this paper we show that the two-dimensional (2D) plaquette states on arbitrary lattices exhibit nontrivial symmetry-protected topological order in terms of symmetry fractionalization and that they are universal resource states for quantum computation. Our results of the nontrivial symmetry-protected topological order on arbitrary 2D lattices are based on an extension of the recent construction by Chen et al. [Phys. Rev. B 87, 155114 (2013)] on the square lattice.

arXiv:1509.02947•Physical Review A•Nov 9, 2015•34 citations
All known resource states for measurement-based quantum teleportation in correlation space possess symmetry-protected topological order, but is this a sufficient or even necessary condition? This work considers two families of one-dimensional qubit states to answer this question in the negative. The first is a family of matrix-product states with bond dimension two that includes the cluster state as a special case, protected by a global non-on-site symmetry, which is characterized by a finite correlation length and a degenerate entanglement spectrum in the thermodynamic limit but which is unable to deterministically teleport a universal set of single-qubit gates. The second are states with bond dimension four that are a resource for deterministic universal teleportation of finite single-qubit gates, but which possess no symmetry. Published by the American Physical Society 2024
All known resource states for measurement-based quantum teleportation in correlation space possess symmetry protected topological order, but is this a sufficient or even necessary condition? This work considers two families of one-dimensional qubit states to answer this question in the negative. The first is a family of matrix-product states with bond dimension two that includes the cluster state as a special case, protected by a global non-onsite symmetry, which is characterized by a finite correlation length and a degenerate entanglement spectrum in the thermodynamic limit but which is unable to deterministically teleport a universal set of single-qubit gates. The second are states with bond dimension four that are a resource for deterministic universal teleportation of finite single-qubit gates, but which possess no symmetry.
We constrain a broad class of teleportation protocols using insights from locality. In the ``standard'' teleportation protocols we consider, all outcome-dependent unitaries are Pauli operators conditioned on linear functions of the measurement outcomes. We find that all such protocols involve preparing a ``resource state'' exhibiting symmetry-protected topological (SPT) order with Abelian protecting symmetry Gk=(Z2×Z2)k . The k logical states are teleported between the edges of the chain by measuring the corresponding 2k string order parameters in the bulk and applying outcome-dependent Paulis. Hence, this single class of nontrivial SPT states is both necessary and sufficient for the standard teleportation of k qubits. We illustrate this result with several examples, including the cluster state, variants thereof, and a nonstabilizer hypergraph state.
Resource states that contain nontrivial symmetry-protected topological order are identified for universal single-qudit measurement-based quantum computation. Our resource states fall into two classes: one as the qudit generalizations of the one-dimensional qubit cluster state, and the other as the higher-symmetry generalizations of the spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) state, namely, with unitary, orthogonal, or symplectic symmetry. The symmetry in cluster states protects information propagation (identity gate), while the higher symmetry in AKLT-type states enables nontrivial gate computation. This work demonstrates a close connection between measurement-based quantum computation and symmetry-protected topological order.
Computational power in measurement-based quantum computing stems from the symmetry-protected topological (SPT) order of entangled resource states. However, resource states are prone to preparation errors. We introduce a quantum error correction approach using redundant nonlocal symmetry of the resource state. We demonstrate it within a teleportation protocol based on extending the ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry of one-dimensional cluster states to other graph states. Qubit ZZ-crosstalk errors, which are prominent in quantum devices, degrade the teleportation fidelity of the usual cluster state. However, as we demonstrate on quantum hardware, once we grow graph states with redundant symmetry, perfect teleportation fidelity is restored. We identify the underlying redundant-SPT order as error-protected degeneracies in the entanglement spectrum.
A fundamental distinction between many-body quantum states are those with short- and long-range entanglement (SRE and LRE). The latter cannot be created by finite-depth circuits, underscoring the nonlocal nature of Schrödinger cat states, topological order, and quantum criticality. Remarkably, examples are known where LRE is obtained by performing single-site measurements on SRE, such as the toric code from measuring a sublattice of a 2D cluster state. However, a systematic understanding of when and how measurements of SRE give rise to LRE is still lacking. Here, we establish that LRE appears upon performing measurements on symmetry-protected topological (SPT) phases—of which the cluster state is one example. For instance, we show how to implement the Kramers-Wannier transformation by adding a cluster SPT to an input state followed by measurement. This transformation naturally relates states with SRE and LRE. An application is the realization of double-semion order when the input state is the Z 2 Levin-Gu SPT. Similarly, the addition of fermionic SPTs and measurement leads to an implementation of the Jordan-Wigner transformation of a general state. More generally, we argue that a large class of SPT phases protected by G × H symmetry gives rise to anomalous LRE upon measuring G -charges, and we prove that this persists for generic points in the SPT phase under certain conditions. Our work introduces a new practical tool for using SPT phases as resources for creating LRE, and we uncover the classification result that all states related by sequentially gauging Abelian groups or by Jordan-Wigner transformation are in the same equivalence class, once we augment finite-depth circuits with single-site measurements. In particular, any topological or fracton order with a solvable finite gauge group can be obtained from a product state in this way. Published by the American Physical Society 2024
Measurement-based quantum computation is a model for quantum information processing utilizing local measurements on suitably entangled resource states for the implementation of quantum gates. A complete characterization for universal resource states is still missing. It has been shown that symmetry-protected topological order in one dimension can be exploited for the protection of certain quantum gates in measurement-based quantum computation. In this paper we show that the two-dimensional (2D) plaquette states on arbitrary lattices exhibit nontrivial symmetry-protected topological order in terms of symmetry fractionalization and that they are universal resource states for quantum computation. Our results of the nontrivial symmetry-protected topological order on arbitrary 2D lattices are based on an extension of the recent construction by Chen et al. [Phys. Rev. B 87, 155114 (2013)] on the square lattice.