Symmetry-protected self-correcting quantum memories

A self-correcting quantum memory can store and protect quantum information for a time that increases without bound with the system size and without the need for active error correction. We demonstrate that symmetry can lead to self-correction in 3D spin-lattice models. In particular, we investigate codes given by 2D symmetry-enriched topological (SET) phases that appear naturally on the boundary of 3D symmetry-protected topological (SPT) phases. We find that while conventional on-site symmetries are not sufficient to allow for self-correction in commuting Hamiltonian models of this form, a generalized type of symmetry known as a 1-form symmetry is enough to guarantee self-correction. We illustrate this fact with the 3D "cluster-state" model from the theory of quantum computing. This model is a self-correcting memory, where information is encoded in a 2D SET-ordered phase on the boundary that is protected by the thermally stable SPT ordering of the bulk. We also investigate the gauge color code in this context. Finally, noting that a 1-form symmetry is a very strong constraint, we argue that topologically ordered systems can possess emergent 1-form symmetries, i.e., models where the symmetry appears naturally, without needing to be enforced externally.Comment: 39 pages, 16 figures, comments welcome; v2 includes much more explicit detail on the main example model, including boundary conditions and implementations of logical operators through local moves; v3 published versio

Symmetry-Protected Topological Phases for Robust Quantum Computation

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

Symmetry protected topological order at nonzero temperature

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 on-site symmetry cannot persist at nonzero temperature, demonstrating that several quantum computational structures protected by such on-site symmetries are not thermally stable. Second, we prove that the 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 global 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.Comment: 42 pages, 10 figures, comments welcome; v2 published versio

Reflection and potentialism

It was widely thought that the paradoxes of Russell, Cantor, and Burali-Forti had been solved by the iterative conception of set. According to this conception, the sets occur in a well-ordered transfinite series of stages. On standard articulations – for example, those in Boolos (1971, 1989) – the sets are implicitly taken to constitute a plurality. Although sets may fail to exist at certain stages, they all exist simpliciter. But if they do constitute a plurality, what could stop them from forming a set? Without a satisfactory answer to this question, the paradoxes threaten to reemerge. In response, it has been argued that we should think of the sets as an inherently potential totality: whatever things there are, there could have been a set of them. In other words, any plurality could have formed a set. Call this potentialism. Actualism, in contrast, is the view that there could not have been more sets than there are: whatever sets there could have been, there are. This thesis explores a particular consideration in favour of actualism; namely, that certain desirable second-order resources are available to the acutalist but not the potentialist. In the first part of chapter 1 I introduce the debate between potentialism and actualism and argue that some prominent considerations in favour of potentialism are inconclusive. In the second part I argue that potentialism is incompatible with the potentialist version of the second-order comprehension schema and point out that this schema appears to be required by strong set-theoretic reflection principles. In chapters 2 and 3 I explore the possibilities for reflection principles which are compatible with potentialism. In particular, in chapter 2 I consider a recent suggestion by Geoffrey Hellman for a modal structural reflection principle, and in chapter 3 I consider some influential proposals by William Reinhardt for modal reflection principles

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On the Ramsey numbers R(3, 8) and R(3, 9)

AbstractUsing methods developed by Graver and Yackel, and various computer algorithms, we show that 28 ≤ R(3, 8) ≤ 29, and R(3, 9) = 36, where R(k, l) is the classical Ramsey number for 2-coloring the edges of a complete graph