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arXiv:2404.15883 · Quantum

Non-onsite symmetries and quantum teleportation in split-index matrix product states

David T. Stephen

Published May 12, 2025• 2 citations
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Abstract

We describe a class of spin chains with new physical and computational properties. On the physical side, the spin chains give examples of symmetry-protected topological phases that are defined by non-onsite symmetries, i.e., symmetries that are not a tensor product of single-site operators. These phases can be detected by string-order parameters, but notably do not exhibit entanglement spectrum degeneracy. On the computational side, the spin chains represent a new class of states that can be used to deterministically teleport information across long distances, with the novel property that the necessary classical side processing is a non-linear function of the measurement outcomes. We also give examples of states that can serve as universal resources for measurement-based quantum computation, providing the first examples of such resources without entanglement spectrum degeneracy. The key tool in our analysis is a new kind of tensor network representation which we call split-index matrix product states (SIMPS). We develop the basic formalism of SIMPS, compare them to matrix product states, show how they are better equipped to describe certain kinds of non-onsite symmetries including anomalous symmetries, and discuss how they are also well-suited to describing quantum teleportation and constrained spin chains.

DOI
10.22331/q-2025-05-12-1738

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We describe a class of spin chains with new physical and computational properties. On the physical side, the spin chains give examples of symmetry-protected topological phases that are defined by non-onsite symmetries, i.e., symmetries that are not a tensor product of single-site operators. These phases can be detected by string-order parameters, but notably do not exhibit entanglement spectrum degeneracy. On the computational side, the spin chains represent a new class of states that can be used to deterministically teleport information across long distances, with the novel property that the necessary classical side processing is a non-linear function of the measurement outcomes. We also give examples of states that can serve as universal resources for measurement-based quantum computation, providing the first examples of such resources without entanglement spectrum degeneracy. The key tool in our analysis is a new kind of tensor network representation which we call split-index matrix product states (SIMPS). We develop the basic formalism of SIMPS, compare them to matrix product states, show how they are better equipped to describe certain kinds of non-onsite symmetries including anomalous symmetries, and discuss how they are also well-suited to describing quantum teleportation and constrained spin chains.
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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
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.
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.
Quantum measurements often destroy the “quantumness” of a state. Combining measurements and entanglement, however, unlocks new opportunities in quantum computing, as it transcends the boundaries between quantum phases. Measurement-based quantum computation is a perfect example: from an array of locally entangled qubits, individual measurements teleport and modify some logical information. Since quantum operations are prone to errors, this information is encoded in the topological Toric code, thus enabling fault-tolerant computation. Since the Toric code has a 2-dimensional structure, the resource state needed to perform measurement-based quantum computation is necessarily 3-dimensional and is called the 3D cluster state. In this thesis, we provide new tools to enlarge the capabilities of topological measurement-based quantum computation and reduce its operational cost. First, we introduce a lattice defect in the underlying cluster state that improves the expressivity of the computation. Second, we develop general ideas towards circuit compaction and quantify the resources used by the new methods. Furthermore, here, the Toric code is created thanks to local measurements, while it cannot be created with local unitary operations only. We investigate the mathematical roots of this intriguing property, and we show that long-range entanglement arises from locally measuring some general short-range entangled state. Specifically, we assume that the initial short-range entangled state is in a non-trivial symmetry protected topological phase for an abelian group on an infinite spin chain. That is, there is a global operation generated by local unitary operators labeled by the element of the group that leaves the state invariant. On infinite quantum spin chains, these symmetry protected topological phases have been completely characterized. We then perform on-site measurements on increasing intervals of the chain. This creates a sequence of measured states that are eigenstates of the local unitaries associated with elements of a subgroup of the symmetry group. Using the algebraic properties of the initial state, we show that this sequence of measured states has increasing long-range correlations. The entanglement bounds necessarily deteriorate with the size of the measured intervals; thus, the measured states cannot be uniformly short-range entangled.
We investigate the phase diagram of a quantum spin-1 chain whose Hamiltonian is invariant under a global on-site A4{A}_{4}A4​, translation, and lattice inversion symmetries. We detect different gapped phases characterized by a symmetry protected topological (SPT) order and symmetry breaking using matrix product state order parameters. We observe a rich variety of phases of matter characterized by a combination of symmetry breaking and symmetry fractionalization and also the interplay between the on-site and spatial symmetries. Examples of continuous phase transitions directly between topologically nontrivial SPT phases are also observed.