We address the question of whether symmetry-protected topological (SPT) order can persist at nonzero temperature, with a focus on understanding the thermal stability of several models studied in the theory of quantum computation. We present three results in this direction. First, we prove that nontrivial SPT order protected by a global onsite symmetry cannot persist at nonzero temperature, demonstrating that several quantum computational structures protected by such onsite symmetries are not thermally stable. Second, we prove that the three-dimensional (3D) cluster-state model used in the formulation of topological measurement-based quantum computation possesses a nontrivial SPT-ordered thermal phase when protected by a generalized (1-form) symmetry. The SPT order in this model is detected by long-range localizable entanglement in the thermal state, which compares with related results characterizing SPT order at zero temperature in spin chains using localizable entanglement as an order parameter. Our third result is to demonstrate that the high-error tolerance of this 3D cluster-state model for quantum computation, even without a protecting symmetry, can be understood as an application of quantum error correction to effectively enforce a 1-form symmetry.
We address the question of whether symmetry-protected topological (SPT) order can persist at nonzero temperature, with a focus on understanding the thermal stability of several models studied in the theory of quantum computation. We present three results in this direction. First, we prove that nontrivial SPT order protected by a global onsite symmetry cannot persist at nonzero temperature, demonstrating that several quantum computational structures protected by such onsite symmetries are not thermally stable. Second, we prove that the three-dimensional (3D) cluster-state model used in the formulation of topological measurement-based quantum computation possesses a nontrivial SPT-ordered thermal phase when protected by a generalized (1-form) symmetry. The SPT order in this model is detected by long-range localizable entanglement in the thermal state, which compares with related results characterizing SPT order at zero temperature in spin chains using localizable entanglement as an order parameter. Our third result is to demonstrate that the high-error tolerance of this 3D cluster-state model for quantum computation, even without a protecting symmetry, can be understood as an application of quantum error correction to effectively enforce a 1-form symmetry.
In recent years, topological phases of matter have presented exciting new avenues to achieve scalable quantum computation. In this thesis, we investigate a class of quantum many-body spin models known as symmetry-protected topological (SPT) phases for use in quantum information processing and storage. We explore the fault-tolerant properties of SPT phases, and how they can be utilized in the design of a quantum computer. Of central importance in this thesis is the concept of quantum error-correction, which in addition to its importance in fault-tolerant quantum computation, is used to characterise the stability of topological phases at finite temperature. We begin with an introduction to quantum computation, quantum error correction, and topological phases of matter. We then focus on the fundamental question of whether symmetry-protected topological phases of matter can exist in thermal equilibrium; we prove that systems protected by global onsite symmetries cannot be ordered at nonzero temperature. Subsequently, we show that certain three-dimensional models with generalised higher-form symmetries can be thermally SPT ordered, and we relate this order to the ability to perform fault-tolerant measurement-based quantum computation. Following this, we assess feasibility of these phases as quantum memories, motivated by the fact that SPT phases in three dimensions can possess protected topological degrees of freedom on their boundary. We find that certain SPT ordered systems can be self-correcting, allowing quantum information to be stored for arbitrarily long times without requiring active error correction. Finally, we develop a framework to construct new schemes of fault-tolerant measurement-based quantum computation. As a notable example, we develop a cluster-state scheme that simulates the braiding and fusion of surface-code defects, offering novel alternative methods to achieve fault-tolerant universal quantum computation.
We propose an order parameter for the symmetry-protected topological (SPT) phases which are protected by Abelian on-site symmetries. This order parameter, called the SPT entanglement, is defined as the entanglement between A and B, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region C is measured and known, where C is a connected region surrounded by A,B, and the boundaries of the system. In the case of one-dimensional systems we prove that in the limit where A and B are large and far from each other compared to the correlation length, the SPT entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the nontrivial phases. Moreover, we show that the SPT entanglement is invariant under the low-depth quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher dimensions as well. Also, we show that there is an intriguing connection between SPT entanglement and the Fourier transform of the string order parameters, which are the traditional tool for detecting SPT phases. This leads to an algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters. Finally, we discuss implications of our results in the context of measurement-based quantum computation.
Topologically-ordered phases of matter at nonzero temperature are conjectured to exhibit universal patterns of long-range entanglement, which can be detected using the entanglement negativity, a mixed-state entanglement measure. In this paper, we show that the entanglement negativity in certain topological orders can be understood through the properties of an emergent symmetry-protected topological (SPT) order that is localized on the entanglement bipartition. This connection leads to an understanding of (i) universal contributions to the entanglement negativity, which diagnose finite-temperature topological order and (ii) the behavior of the entanglement negativity across certain phase transitions in which thermal fluctuations eventually destroy long-range entanglement across the bipartition surface. Within this correspondence, the universal patterns of entanglement in the finite-temperature topological order are related to the stability of an emergent SPT order against a symmetry-breaking field. SPT orders protected by higher-form symmetries---which arise, for example, in the description of the entanglement negativity for Z2 topological order in d=4 spatial dimensions---remain robust even in the presence of a weak symmetry-breaking perturbation, leading to long-range entanglement at nonzero temperature for certain topological orders.
Whether self-correcting quantum memories can exist at nonzero temperature in a physically reasonable setting remains a great open problem. It has recently been argued [S. Roberts and S. D. Bartlett, Phys. Rev. X 10, 031041 (2020)] that symmetry-protected topological (SPT) systems in three space dimensions subject to a strong constraint---that the quantum dynamics respect a 1-form symmetry---realize such a quantum memory. We illustrate how this works in Walker-Wang codes, which provide a specific realization of these desiderata. In this setting we show that it is sufficient for the 1-form symmetry to be enforced on a subvolume of the system. This strongly suggests that the SPT character of the state is not essential. We confirm this by constructing an explicit example with a trivial (paramagnetic) bulk that realizes a self-correcting quantum memory. We therefore show that the enforcement of a 1-form symmetry on a measure-zero subvolume of a three-dimensional system can be sufficient to stabilize a self-correcting quantum memory at nonzero temperature.
Bell theorems of many-body nonlocality and contextuality serve as a benchmark for proving quantum advantage in that a quantum computer outperforms a classical computer for a certain problem. In practice, however, near-term quantum devices do not prepare perfectly pure states but rather mixed states produced from noisy channels. We investigate noisy quantum advantage by considering thermal mixed states of one-dimensional many-body systems with a symmetry-protected topological (SPT) order. In the pure-state (or zero-temperature) case, these states are known to be useful for measurement-based quantum computation, and to outperform classical computers in a many-body contextuality game, provided string order parameters (SOPs) of SPT are sufficiently large. Here, we show that quantum advantage in mixed states is measured by a combination of twisted SOP and symmetry representation expectation values. Using the minimally entangled typical thermal states algorithm, it is demonstrated that quantum advantage persists to a nonzero critical temperature for finite-sized instances of the many-body contextuality game. While this critical temperature goes to zero in the thermodynamic limit, it is relatively robust to system size, suggesting that these states remain useful for demonstrating genuine "quantumness" of noisy hardware in a scalable fashion. Finally, we show that the quantum winning probability is lower bounded by the global fidelity with the 1D cluster state, so that our contextuality game can provide an operational meaning to benchmark the capacity to create long-range order like SPT states in near-term experimental devices.
We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature-it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic pointlike excitations. The fermionic toric code, importantly, possesses an anomalous two-form symmetry, associated with the spacelike Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to a phase of matter that, in the context of equilibrium phases of matter, only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous two-form symmetries. Our Letter therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions.