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arXiv:1611.08053 · Physical Review Letters

Computational Power of Symmetry-Protected Topological Phases

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

Published Jul 5, 2017• 86 citations
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Abstract

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.

DOI
10.1103/physrevlett.119.010504

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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 identify certain classes of 1D symmetry-protected topological (SPT) phases in which almost every state is universal for measurement based quantum computation (MBQC) in 1D. By developing a general scheme to extract the useful entanglement from SPT ordered states, we can associate to each of these phases a Lie group of executable gates. We then show how to determine this Lie group from the cohomological description of SPT phases, thereby affirming the compelling link between SPT order and the computational power of 1D resource states.
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.
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.
It is a challenging problem to correctly characterize the symmetry-protected topological (SPT) phases in open quantum systems. As the measurement-based quantum computation (MBQC) utilizes non-trivial edge states of the SPT phases as the logical qubit, its computational power is closely tied to the non-trivial topological nature of the phases. In this paper, we propose to use the gate fidelity which is a measure of the computational power of the MBQC to identify the SPT phases in mixed-state settings. Specifically, we investigate the robustness of the Haldane phase by considering the MBQC on the Affleck-Kennedy-Lieb-Tasaki state subject to different types of noises. To illustrate how our criterion works, we analytically and numerically calculated the gate fidelity to find that its behavior depends crucially on whether the noises satisfy a certain symmetry condition with respect to the on-site Z2×Z2\mathbb{Z}_2 \times \mathbb{Z}_2Z2​×Z2​ symmetry. In particular, the fidelity for the identity gate, which is given by the sum of the non-local string order parameters, plays an important role. Furthermore, we demonstrate that a stronger symmetry conditions are required to be able to perform other (e.g., the ZZZ-rotation gate) gates with high fidelity. By examining which unitary gates can be implemented with the MBQC on the decohered states, we can gain a useful insight into the richer structure of noisy SPT states that cannot be captured solely by the string order parameters.