General phase spaces: from discrete variables to rotor and continuum limits

Albert, Victor V.; Pascazio, Saverio; Devoret, Michel

doi:10.1088/1751-8121/aa9314

Cited by 27 publications

(18 citation statements)

References 111 publications

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“…For our purposes, a quantum rotor [also, an Oð2Þ or planar quantum rotor] is simply a system with a basis fjxig that is labeled by an integer x ∈ Z indexing representations of Uð1Þ [46]. Consider the three-rotor code given in Ref.…”

Section: Rotor Version Of the Qutrit Secret-sharing Codementioning

confidence: 99%

Continuous Symmetries and Approximate Quantum Error Correction

et al. 2020

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Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here, we study the compatibility of these two important principles. If a logical quantum system is encoded into n physical subsystems, we say that the code is covariant with respect to a symmetry group G if a G transformation on the logical system can be realized by performing transformations on the individual subsystems. For a G-covariant code with G a continuous group, we derive a lower bound on the errorcorrection infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems n or the dimension d of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with n or d as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem for quantum computation: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large d (using random codes) or n (using codes based on W states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.

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Section: Rotor Version Of the Qutrit Secret-sharing Codementioning

confidence: 99%

Continuous Symmetries and Approximate Quantum Error Correction

et al. 2020

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“…We may measure the quantum dimension d a µ n associated with each µ topological charge a µ n through Eq. (81). Further, this leads to a notion of quantum dimension of any particle type a = (q, a µ n ) = (q, g µ n , µ n ), defined by…”

Section: Braiding Of 2d Mobile Quasiparticlesmentioning

confidence: 99%

Twisted fracton models in three dimensions

et al. 2019

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We study novel three-dimensional gapped quantum phases of matter which support quasiparticles with restricted mobility, including immobile "fracton" excitations. So far, most existing fracton models may be instructively viewed as generalized Abelian lattice gauge theories. Here, by analogy with Dijkgraaf-Witten topological gauge theories, we discover a natural generalization of fracton models, obtained by twisting the gauge symmetries. Introducing generalized gauge transformation operators carrying an extra phase factor depending on local configurations, we construct a plethora of exactly solvable three-dimensional models, which we dub "twisted fracton models." A key result of our approach is to demonstrate the existence of rich non-Abelian fracton phases of distinct varieties in a three-dimensional system with finite-range interactions. For an accurate characterization of these novel phases, the notion of being inextricably non-Abelian is introduced for fractons and quasiparticles with one-dimensional mobility, referring to their new behavior of displaying braiding statistics that is, and remains, non-Abelian regardless of which quasiparticles with higher mobility are added to or removed from them. We also analyze these models by embedding them on a three-torus and computing their ground state degeneracies, which exhibit a surprising and novel dependence on the system size in the non-Abelian fracton phases. Moreover, as an important advance in the study of fracton order, we develop a general mathematical framework which systematically captures the fusion and braiding properties of fractons and other quasiparticles with restricted mobility. CONTENTS

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“…In this sense, a fracton is only a fraction of a conventional mobile excitation. The condensed matter literature has seen a flurry of recent activity fleshing out the properties of these strange new particles [27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45]. 1 The smallest mobile bound state of fractons is usually also a non-trivial excitation of the system, which cannot decay directly into the vacuum, and therefore exists as a stable particle.…”

Section: Introductionmentioning

confidence: 99%

Emergent phases of fractonic matter

2018

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Fractons are emergent particles which are immobile in isolation, but which can move together in dipolar pairs or other small clusters. These exotic excitations naturally occur in certain quantum phases of matter described by tensor gauge theories. Previous research has focused on the properties of small numbers of fractons and their interactions, effectively mapping out the "Standard Model" of fractons. In the present work, however, we consider systems with a finite density of either fractons or their dipolar bound states, with a focus on the U (1) fracton models. We study some of the phases in which emergent fractonic matter can exist, thereby initiating the study of the "condensed matter" of fractons. We begin by considering a system with a finite density of fractons, which we show can exhibit microemulsion physics, in which fractons form small-scale clusters emulsed in a phase dominated by long-range repulsion. We then move on to study systems with a finite density of mobile dipoles, which have phases analogous to many conventional condensed matter phases. We focus on two major examples: Fermi liquids and quantum Hall phases. A finite density of fermionic dipoles will form a Fermi surface and enter a Fermi liquid phase. Interestingly, this dipolar Fermi liquid exhibits a finite-temperature phase transition, corresponding to an unbinding transition of fractons. Finally, we study chiral two-dimensional phases corresponding to dipoles in "quantum Hall" states of their emergent magnetic field. We study numerous aspects of these generalized quantum Hall systems, such as their edge theories and ground state degeneracies. arXiv:1709.09673v2 [cond-mat.str-el]

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General phase spaces: from discrete variables to rotor and continuum limits

Cited by 27 publications

References 111 publications

Continuous Symmetries and Approximate Quantum Error Correction

Continuous Symmetries and Approximate Quantum Error Correction

Twisted fracton models in three dimensions

Emergent phases of fractonic matter

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