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:1609.07549 · Physical Review A

Symmetry-protected topological phases with uniform computational power in one dimension

Robert Raussendorf, Dongsheng Wang, Abhishodh Prakash, Tzu-Chieh Wei, David T. Stephen

Published Jul 5, 2017• 53 citations
arXivHTMLPDF

Abstract

We investigate the usefulness of ground states of quantum spin chains with symmetry-protected topological order (SPTO) for measurement-based quantum computation. We show that, in spatial dimension 1, if an SPTO phase protects the identity gate, then, subject to an additional symmetry condition that is satisfied in all cases so far investigated, it can also be used for quantum computation.

DOI
10.1103/physreva.96.012302

Related papers

Computational Power of Symmetry-Protected Topological Phases

David T. Stephen, Dongsheng Wang, Abhishodh Prakash +2 more

We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order as resources for measurement-based quantum computation (MBQC). We show that, for a wide range of SPT phases, the computational power of ground states is uniform throughout each phase. This computational power, defined as the Lie group of executable gates in MBQC, is determined by the same algebraic information that labels the SPT phase itself. We prove that these Lie groups always contain a full set of single-qubit gates, thereby affirming the long-standing conjecture that general SPT phases can serve as computationally useful phases of matter.

arXiv:1611.08053•Physical Review Letters•Jul 5, 2017•86 citations

Computational power of one-dimensional symmetry-protected topological phases

David T. Stephen

We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order as resources for measurement-based quantum computation (MBQC). Using tensor network methods, we show that SPT phases protected by a finite abelian on-site symmetry group exhibit uniform computational power. That is, any state from a given phase leads to the same Lie group of executable gates when used as a resource for MBQC. This Lie group is determined by the same algebraic information that labels the SPT phase itself, and we give a necessary condition on the phase that guarantees a full set of single-qubit gates. To obtain our results, we construct several new techniques in MBQC and refine the structure of quantum states with abelian SPT order. Our results are analogous to similar results relating topological order and topological quantum computation, and we comments on their implications on the general connection between quantum phases of matter and quantum computation.

•Jan 1, 2017•3 citations

Symmetry protected measurement-based quantum computation in finite spin chains

Arnab Adhikary

We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order and their usefulness as resources for measurement-based quantum computation (MBQC). It is known that SPT phases (applicable to infinite spin chains) protected by a finite abelian symmetry group exhibit uniform computational power. In this work, we extend these ideas to finite spin chains which are more realistic resources for physical computation. Herein, we relate the usefulness of MBQC resource states to a non-vanishing string-order parameter. Furthermore, using the techniques developed, we show that hard-to-think-about regimes of computation are actually more efficient than the textbook prescription. Our results strengthen the connection between condensed matter and quantum computation and also provide the necessary tools to explore such connections in the state-of-the-art noisy small quantum devices. As an outlook, we discuss how this newly developed formalism can potentially be used to identify non-traditional resource states for MBQC and be extended to higher dimensions, leading to universal quantum computation.

•Jan 1, 2021•0 citations

Symmetry-Protected Phases for Measurement-Based Quantum Computation

Dominic V. Else, Ilai Schwarz, Stephen D. Bartlett +1 more

Ground states of spin lattices can serve as a resource for measurement-based quantum computation. Ideally, the ability to perform quantum gates via measurements on such states would be insensitive to small variations in the Hamiltonian. Here, we describe a class of symmetry-protected topological orders in one-dimensional systems, any one of which ensures the perfect operation of the identity gate. As a result, measurement-based quantum gates can be a robust property of an entire phase in a quantum spin lattice, when protected by an appropriate symmetry.

arXiv:1201.4877•Physical Review Letters•Jun 15, 2012•171 citations

Ground states of one-dimensional symmetry-protected topological phases and their utility as resource states for quantum computation

Abhishodh Prakash, Tzu-Chieh Wei

The program of classifying symmetry-protected topological (SPT) phases in one dimension has been recently completed and has opened the doors to studying closely the properties of systems belonging to these phases. It was recently found that being able to constrain the form of ground states of SPT order based on symmetry properties also makes it possible to explore novel resource states for processing of quantum information. In this paper, we generalize the consideration of Else et al. [Phys. Rev. Lett. 108, 240505 (2012)], where it was shown that the ground-state form of spin-1 chains protected by ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry supports perfect operation of the identity gate, important also for long-distance transmission of quantum information. We develop a formalism to constrain the ground-state form of SPT phases protected by any arbitrary finite symmetry group and use it to examine examples of ground states of SPT phases protected by various finite groups for similar gate protections. We construct a particular Hamiltonian invariant under ${A}_{4}$ symmetry transformation, which is one of the groups that allows protected identity operation, and examine its ground states. We find that there is an extended region where the ground state is the AKLT state, which not only supports the identity gate but also arbitrary single-qubit gates.

arXiv:1410.0974•Physical Review A•Aug 6, 2015•26 citations

Measurement-Based Quantum Computation and Symmetry-Protected Topological Order

Jacob Miller

While quantum computers can achieve dramatic speedups over the classical computers familiar to us, identifying the origin of this quantum advantage in physical systems remains a major goal of quantum information science. A useful tool here is measurement-based quantum computation (MQC), a computational framework utilizing the quantum entanglement found in many-body resource states. Not all resource states are useful for quantum computation however, so an important question is what properties of many-body entanglement characterize universal resource states, which can implement any quantum computation. Many-body states are also studied in condensed matter physics, where the collective behavior of quantum many-body systems sometimes define topological phases of matter. These phases are defined by nonlocal many-body entanglement, making topologically-ordered states natural candidates for MQC. We might wonder if these topological phases could be organized as phases of quantum computation, so that every state within the phase is universal for MQC. While phases of symmetry-protected topological order (SPTO) have arisen as natural candidates, previous attempts to demonstrate an MQC-SPTO correspondence were mostly limited to nonuniversal 1D spin chains, leaving the important 2D setting wide open. In this dissertation, we explore the wide and varied connections between MQC and SPTO, and obtain new results for 1D and 2D systems. After identifying a new MQC-SPTO correspondence within 1D spin chains, we move up and explore the operational use of 2D states with two complementary forms of SPTO. We create a new Union Jack resource state, whose different form of SPTO than previous 2D resource states permits a hierarchical notion of MQC universality. This state leads us to consider an idealized model of 2D SPTO, where we show that an additional symmetry condition makes these model states form universal resources for MQC only when they have nontrivial SPTO. We finally study the intrinsic complexity of SPTO-inspired states for classically intractable sampling, and identify inherent advantages of MQC for this purpose. Our work highlights the rich complexity available in states of entangled quantum matter, providing new evidence which sharpens our understanding of the diverse connections between MQC and SPTO.

•Jan 1, 2017•0 citations
We investigate the usefulness of ground states of quantum spin chains with symmetry-protected topological order (SPTO) for measurement-based quantum computation. We show that, in spatial dimension 1, if an SPTO phase protects the identity gate, then, subject to an additional symmetry condition that is satisfied in all cases so far investigated, it can also be used for quantum computation.
We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order as resources for measurement-based quantum computation (MBQC). We show that, for a wide range of SPT phases, the computational power of ground states is uniform throughout each phase. This computational power, defined as the Lie group of executable gates in MBQC, is determined by the same algebraic information that labels the SPT phase itself. We prove that these Lie groups always contain a full set of single-qubit gates, thereby affirming the long-standing conjecture that general SPT phases can serve as computationally useful phases of matter.
We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order as resources for measurement-based quantum computation (MBQC). Using tensor network methods, we show that SPT phases protected by a finite abelian on-site symmetry group exhibit uniform computational power. That is, any state from a given phase leads to the same Lie group of executable gates when used as a resource for MBQC. This Lie group is determined by the same algebraic information that labels the SPT phase itself, and we give a necessary condition on the phase that guarantees a full set of single-qubit gates. To obtain our results, we construct several new techniques in MBQC and refine the structure of quantum states with abelian SPT order. Our results are analogous to similar results relating topological order and topological quantum computation, and we comments on their implications on the general connection between quantum phases of matter and quantum computation.
We consider ground states of quantum spin chains with symmetry-protected topological (SPT) order and their usefulness as resources for measurement-based quantum computation (MBQC). It is known that SPT phases (applicable to infinite spin chains) protected by a finite abelian symmetry group exhibit uniform computational power. In this work, we extend these ideas to finite spin chains which are more realistic resources for physical computation. Herein, we relate the usefulness of MBQC resource states to a non-vanishing string-order parameter. Furthermore, using the techniques developed, we show that hard-to-think-about regimes of computation are actually more efficient than the textbook prescription. Our results strengthen the connection between condensed matter and quantum computation and also provide the necessary tools to explore such connections in the state-of-the-art noisy small quantum devices. As an outlook, we discuss how this newly developed formalism can potentially be used to identify non-traditional resource states for MBQC and be extended to higher dimensions, leading to universal quantum computation.
Ground states of spin lattices can serve as a resource for measurement-based quantum computation. Ideally, the ability to perform quantum gates via measurements on such states would be insensitive to small variations in the Hamiltonian. Here, we describe a class of symmetry-protected topological orders in one-dimensional systems, any one of which ensures the perfect operation of the identity gate. As a result, measurement-based quantum gates can be a robust property of an entire phase in a quantum spin lattice, when protected by an appropriate symmetry.
The program of classifying symmetry-protected topological (SPT) phases in one dimension has been recently completed and has opened the doors to studying closely the properties of systems belonging to these phases. It was recently found that being able to constrain the form of ground states of SPT order based on symmetry properties also makes it possible to explore novel resource states for processing of quantum information. In this paper, we generalize the consideration of Else et al. [Phys. Rev. Lett. 108, 240505 (2012)], where it was shown that the ground-state form of spin-1 chains protected by ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ symmetry supports perfect operation of the identity gate, important also for long-distance transmission of quantum information. We develop a formalism to constrain the ground-state form of SPT phases protected by any arbitrary finite symmetry group and use it to examine examples of ground states of SPT phases protected by various finite groups for similar gate protections. We construct a particular Hamiltonian invariant under A4{A}_{4}A4​ symmetry transformation, which is one of the groups that allows protected identity operation, and examine its ground states. We find that there is an extended region where the ground state is the AKLT state, which not only supports the identity gate but also arbitrary single-qubit gates.
While quantum computers can achieve dramatic speedups over the classical computers familiar to us, identifying the origin of this quantum advantage in physical systems remains a major goal of quantum information science. A useful tool here is measurement-based quantum computation (MQC), a computational framework utilizing the quantum entanglement found in many-body resource states. Not all resource states are useful for quantum computation however, so an important question is what properties of many-body entanglement characterize universal resource states, which can implement any quantum computation. Many-body states are also studied in condensed matter physics, where the collective behavior of quantum many-body systems sometimes define topological phases of matter. These phases are defined by nonlocal many-body entanglement, making topologically-ordered states natural candidates for MQC. We might wonder if these topological phases could be organized as phases of quantum computation, so that every state within the phase is universal for MQC. While phases of symmetry-protected topological order (SPTO) have arisen as natural candidates, previous attempts to demonstrate an MQC-SPTO correspondence were mostly limited to nonuniversal 1D spin chains, leaving the important 2D setting wide open. In this dissertation, we explore the wide and varied connections between MQC and SPTO, and obtain new results for 1D and 2D systems. After identifying a new MQC-SPTO correspondence within 1D spin chains, we move up and explore the operational use of 2D states with two complementary forms of SPTO. We create a new Union Jack resource state, whose different form of SPTO than previous 2D resource states permits a hierarchical notion of MQC universality. This state leads us to consider an idealized model of 2D SPTO, where we show that an additional symmetry condition makes these model states form universal resources for MQC only when they have nontrivial SPTO. We finally study the intrinsic complexity of SPTO-inspired states for classically intractable sampling, and identify inherent advantages of MQC for this purpose. Our work highlights the rich complexity available in states of entangled quantum matter, providing new evidence which sharpens our understanding of the diverse connections between MQC and SPTO.