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

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

doi:10.1103/physreva.96.012302

Cited by 40 publications

(73 citation statements)

References 30 publications

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“…A scheme for MBQC with 1D-SPT phases that leverages the symmetry to do universal MBQC at arbitrary points in the phase was proposed in Refs. [30][31][32]. Note, however, that this approach cannot be immediately applied to 2D SPTO because strict single-site locality of measurements is required for MBQC.…”

Section: Phases Of Symmetry-protected Topological Ordermentioning

confidence: 99%

“…[25] that all ground states of a certain 1D SPTO phase, known as the Haldane phase [26,27], have an equivalent computational capacity provided that the symmetries remain unbroken. Furthermore, any ground state residing in a 1D SPTO phase protected by a finite Abelian symmetry group has been shown to act as a 1D MBQC resource [28][29][30][31][32]. Recently, these results have been extended to 2D resource states lying in a quasi-1D SPTO phase, protected by so-called subsystem symmetries, giving rise to quantum phases of matter in which every state is universal resource for * AustinDaniel@unm.edu MBQC.…”

Section: Introductionmentioning

confidence: 99%

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Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

Daniel

Alexander

Miyake

2020

Quantum

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What kinds of symmetry-protected topologically ordered (SPTO) ground states can be used for universal measurement-based quantum computation in a similar fashion to the 2D cluster state? 2D SPTO states are classified not only by global on-site symmetries but also by subsystem symmetries, which are fine-grained symmetries dependent on the lattice geometry. Recently, all states within so-called SPTO cluster phases on the square and hexagonal lattices have been shown to be universal, based on the presence of subsystem symmetries and associated structures of quantum cellular automata. Motivated by this observation, we analyze the computational capability of SPTO cluster phases on all vertex-translative 2D Archimedean lattices. There are four subsystem symmetries here called ribbon, cone, fractal, and 1-form symmetries, and the former three are fundamentally in one-to-one correspondence with three classes of Clifford quantum cellular automata. We conclude that nine out of the eleven Archimedean lattices support universal cluster phases protected by one of the former three symmetries, while the remaining lattices possess 1-form symmetries and have a different capability related to error correction.

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Section: Phases Of Symmetry-protected Topological Ordermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

Daniel

Alexander

Miyake

2020

Quantum

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“…A complementing approach to topological quantum computation uses Majorana fermions located at the edges of one-dimensional (1D) chains with symmetryprotected topological (SPT) order [4][5][6]. On the other hand, quantum states exhibiting SPT order can be used as resources for instances of measurement-based quantum computation [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] -an insight most relevant to the present work. Various further examples can be found [23][24][25]; indeed, every time a new type of quantum order is discovered, it is not long before its uses in notions of quantum computation are being investigated.…”

mentioning

confidence: 99%

“…The same symmetries also aid in un-derstanding a quantum state's usefulness for quantum computation. This is especially true when considering SPT order and MBQC, wherein the group of tensor symmetries that define 1D SPT order is essentially the only ingredient needed to construct a scheme of MBQC, and hence the possible MBQC schemes inherit the same algebraic structure that classifies 1D SPT order [14,15]. In Ref.…”

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confidence: 99%

Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter

Stephen

Nautrup

Bermejo-Vega

et al. 2019

Quantum

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104

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Quantum phases of matter are resources for notions of quantum computation. In this work, we establish a new link between concepts of quantum information theory and condensed matter physics by presenting a unified understanding of symmetry-protected topological (SPT) order protected by subsystem symmetries and its relation to measurement-based quantum computation (MBQC). The key unifying ingredient is the concept of quantum cellular automata (QCA) which we use to define subsystem symmetries acting on rigid lowerdimensional lines or fractals on a 2D lattice. Notably, both types of symmetries are treated equivalently in our framework. We show that states within a non-trivial SPT phase protected by these symmetries are indicated by the presence of the same QCA in a tensor network representation of the state, thereby characterizing the structure of entanglement that is uniformly present throughout these phases. By also formulating schemes of MBQC based on these QCA, we are able to prove that most of the phases we construct are computationally universal phases of matter, in which every state is a resource for universal MBQC. Interestingly, our approach allows us to construct computational phases which have practical advantages over previous examples, including a computational speedup. The significance of the approach stems from constructing novel computationally universal phases of matter and showcasing the power of tensor networks and quantum information theory in classifying subsystem SPT order.The fields of study of quantum phases of matter and of quantum computation have been evolving alongside each other for over a decade, such that they are now deeply intertwined. This is on the one hand because many instances of non-trivial quantum order are key to storing or processing quantum informa-Figure 1: Interrelation between symmetry-protected topological (SPT) order, measurement-based quantum computation (MBQC), and quantum cellular automata (QCA). By framing both subsystem SPT order and MBQC in terms of QCA, we develop a new framework for characterizing subsystem SPT order and constructing computationally universal phases of matter.tion, an idea that has stimulated a plethora of theoretical and experimental research in both fields. Perhaps the most familiar example is the idea of topological quantum computation which leverages the anyonic excitations of topologically ordered systems to perform error-resilient quantum computation [1][2][3]. A complementing approach to topological quantum computation uses Majorana fermions located at the edges of one-dimensional (1D) chains with symmetryprotected topological (SPT) order [4][5][6]. On the other hand, quantum states exhibiting SPT order can be used as resources for instances of measurement-based quantum computation [7-22] -an insight most relevant to the present work. Various further examples can be found [23][24][25]; indeed, every time a new type of quantum order is discovered, it is not long before its uses in notions of quantum computation are being investigated.The m...

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“…Examples include the 1D cluster state and the 1D spin-1 AKLT state, which can be used for arbitrary single-qubit gates [33,34]. Such utility in quantum gates has been further established [35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Universal measurement-based quantum computation in two-dimensional symmetry-protected topological phases

Wei

Huang

2017

Phys. Rev. A

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Recent progress in characterization for gapped quantum phases has also triggered the search of universal resource for quantum computation in symmetric gapped phases. Prior works in one dimension suggest that it is a feature more common than previously thought that nontrivial 1D symmetry-protected topological (SPT) phases provide quantum computational power characterized by the algebraic structure defining these phases. Progress in two and higher dimensions so far has been limited to special fixed points in SPT phases. Here we provide two families of 2D Z2 symmetric wave functions such that there exists a finite region of the parameter in the SPT phases that supports universal quantum computation. The quantum computational power loses its universality at the boundary between the SPT and symmetry-breaking phases.

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

Cited by 40 publications

References 30 publications

Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

Computational universality of symmetry-protected topologically ordered cluster phases on 2D Archimedean lattices

Subsystem symmetries, quantum cellular automata, and computational phases of quantum matter

Universal measurement-based quantum computation in two-dimensional symmetry-protected topological phases

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