A converse to Lieb-Robinson bounds in one dimension using index theory

Ranard, Daniel; Walter, Michael; Witteveen, Freek

doi:10.48550/arxiv.2012.00741

Cited by 10 publications

(22 citation statements)

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“…In 1D, QCA are equivalent to matrix-product unitaries 212 and are fully classified by a topological invariant called (chiral) index, 213 which is quantized as log Q + and is additive upon tensoring and composition. While there are several ways to define the index for a 1D QCA Û , one elegant definition is given by the following entropy formula: 214,215,216…”

Section: Specific Lower Boundsmentioning

confidence: 99%

Bounds in Nonequilibrium Quantum Dynamics

Gong¹,

Hamazaki²

2022

Preprint

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We review various bounds concerning out-of-equilibrium dynamics in few-level and manybody quantum systems. We primarily focus on closed quantum systems but will also mention some related results for open quantum systems and classical stochastic systems. We start from the speed limits, the universal bounds on the speeds of (either quantum or classical) dynamical evolutions. We then turn to review the bounds that address how good and how long would a quantum system equilibrate or thermalize. Afterward, we focus on the stringent constraint set by locality in many-body systems, rigorously formalized as the Lieb-Robinson bound. We also review the bounds related to the dynamics of entanglement, a genuine quantum property. Apart from some other miscellaneous topics, several notable error bounds for approximated quantum dynamics are discussed. While far from comprehensive, this topical review covers a considerable amount of recent progress and thus could hopefully serve as a convenient starting point and up-to-date guidance for interested readers.

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Section: Specific Lower Boundsmentioning

confidence: 99%

Bounds in Nonequilibrium Quantum Dynamics

Gong¹,

Hamazaki²

2022

Preprint

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“…[10], is that QCA are fully classified by a genuinely dynamical topological index. This result was recently generalized to include the more realistic case where the causal cone is only approximate [17]. Importantly, this topological index is zero if and only if the evolution is generated by a (quasi-)local Hamiltonian.…”

mentioning

confidence: 94%

“…Although the index was initially defined in terms of abstract operator algebras [10], an equivalent definition, which reflects an intuitive picture of quantum-information flow [7,16,17], was recently put forward. In turn, this made it possible to establish a lower bound on quantum scrambling in terms of the index, building a bridge between genuinely dynamical topological invariants and quantum chaos [16].…”

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confidence: 99%

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Coarse-grained Entanglement and Operator Growth in Anomalous Dynamics

Gong,

Nahum,

Piroli

2021

Preprint

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In two-dimensional Floquet systems, many-body localized dynamics in the bulk may give rise to a chaotic evolution at the one-dimensional edges that is characterized by a nonzero chiral topological index. Such anomalous dynamics is qualitatively different from local-Hamiltonian evolution. Here we show how the presence of a nonzero index affects entanglement generation and the spreading of local operators, focusing on the coarsegrained description of generic systems. We tackle this problem by analyzing exactly solvable models of random quantum cellular automata (QCA) which generalize random circuits. We find that a nonzero index leads to asymmetric butterfly velocities with different diffusive broadening of the light cones, and to a modification of the order relations between the butterfly and entanglement velocities. We propose that these results can be understood via a generalization of the recently-introduced entanglement membrane theory, by allowing for a spacetime entropy current, which in the case of a generic QCA is fixed by the index. We work out the implications of this current on the entanglement "membrane tension" and show that the results for random QCA are recovered by identifying the topological index with a background velocity for the coarse-grained entanglement dynamics.

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“…1 They originally arose in the context of quantum simulation and as a model for quantum computation [1][2][3][4][5]. However, in recent years, QCAs have seen wide-ranging applications from discretized quantum field theories [6,7] to the classification of Floquet phases [8][9][10][11][12][13] and tensor network unitary operators [14][15][16][17][18], entanglement growth in quantum dynamics [19][20][21][22], and the construction of symmetry-protected topological states and their anomalous boundaries [23,24]. Beyond such applications, QCAs represent a fundamental class of mathematical objects in the quantum many-body setting, meshing the notions of unitarity and locality.…”

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confidence: 99%

Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond

Shirley¹,

Chen²,

Dua³

et al. 2022

Preprint

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We construct a novel three-dimensional quantum cellular automaton (QCA) based on a system with short-range entangled bulk and chiral semion boundary topological order. We argue that either the QCA is nontrivial, i.e., not a finite-depth circuit of local quantum gates, or there exists a two-dimensional commuting projector Hamiltonian realizing the chiral semion topological order (characterized by U (1)2 Chern-Simons theory). Our QCA is obtained by first constructing the Walker-Wang Hamiltonian of a certain premodular tensor category of order four, then condensing the deconfined bulk boson at the level of lattice operators. We show that the resulting Hamiltonian hosts chiral semion surface topological order in the presence of a boundary and can be realized as a non-Pauli stabilizer code on qubits, from which the QCA is defined. The construction is then generalized to a class of QCAs defined by non-Pauli stabilizer codes on 2 n -dimensional qudits that feature surface anyons described by U (1)2n Chern-Simons theory. Our results support the conjecture that the group of nontrivial three-dimensional QCAs is isomorphic to the Witt group of non-degenerate braided fusion categories.

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A converse to Lieb-Robinson bounds in one dimension using index theory

Cited by 10 publications

References 0 publications

Bounds in Nonequilibrium Quantum Dynamics

Bounds in Nonequilibrium Quantum Dynamics

Coarse-grained Entanglement and Operator Growth in Anomalous Dynamics

Three-dimensional quantum cellular automata from chiral semion surface topological order and beyond

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