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Simulation of topological field theories by quantum computers
Quantum Physics · Submitted 20 Jan 2000
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim TQFTs cannot be used to define a model of computation stronger than the usual quantum model BQP. 2.
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The abstract, sentence by sentence
Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned topological models having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H = 0. These are called topological quantum filed theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model BQP. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. | background 0.99 | 0.07 | 0.02 | asserted 0.75 |
| s1 | In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned topological models having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H = 0. | background 0.99 | 0.06 | 0.03 | asserted 0.99 |
| s2 | These are called topological quantum filed theories (TQFTs). | background 1.00 | 0.07 | 0.03 | asserted 0.97 |
| s3 | These exotic physical systems are proved to be efficiently simulated on a quantum computer. | result 0.86 | 0.64 | 0.14 | asserted 1.00 |
| s4 | The conclusion is two-fold: 1. | result 0.71 | 0.45 | 0.16 | unclear 0.87 |
| s5 | TQFTs cannot be used to define a model of computation stronger than the usual quantum model BQP. 2. | result 0.83 | 0.69 | 0.46 | asserted 1.00 |
| s6 | TQFTs provide a radically different way of looking at quantum computation. | interpretation 0.94 | 0.39 | 0.27 | asserted 1.00 |
| s7 | The rich mathematical structure of TQFTs might suggest a new quantum algorithm. | interpretation 0.76 | 0.12 | 0.06 | speculative 0.89 |
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The paper
Original abstract
Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a time t. In contrast to this quantum engineering, the most abstract reaches of theoretical physics has spawned topological models having a finite dimensional internal state space with no natural tensor product structure and in which the evolution of the state is discrete, H = 0. These are called topological quantum filed theories (TQFTs). These exotic physical systems are proved to be efficiently simulated on a quantum computer. The conclusion is two-fold: 1. TQFTs cannot be used to define a model of computation stronger than the usual quantum model BQP. 2. TQFTs provide a radically different way of looking at quantum computation. The rich mathematical structure of TQFTs might suggest a new quantum algorithm.
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How the literature uses it
16 arXiv papers cite this in 19 sentences
Papers citing it · 12 sentences without a confident label
Compares with it 1 paper
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Similarly, estimating the Jones polynomial of the trace closure of a braid is DQC1-complete [ref] , while estimating the Jones polynomial of the plat closure of a braid is BQP-complete [ref] .
Approximating the Turaev-Viro Invariant of Mapping Tori is Complete for One Clean Qubit · Analogy with Jones polynomials · 0.98
Cites it as background 6 papers
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Freedman, Kitaev, Larsen and Wang established this quantum computing paradigm by showing that there exist unitary topological quantum field theories (TQFTs) based on evaluations of the Jones polynomial at roots of unity for which certain approximations of knot and link invariants are equivalent in power to the usual quantum circuit model of $$ [ref] .
Topological quantum computation is hyperbolic · Introduction · 0.92
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The field of topological quantum computation is based on the recognition that one can exploit the robustness to external perturbations of topological degrees of freedom in $2+1$ dimensional systems, such as the quantum hall fluid, in order to perform fault-tolerant quantum computation [ref] .
Elementary Particles as Gates for Universal Quantum Computation · Quantum Computation with Braids · 0.99
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In particular, approximating the Jones polynomial, a famous invariant of links, is a BQP-complete problem [ref] .
Approximating the Turaev-Viro Invariant of Mapping Tori is Complete for One Clean Qubit · Introduction · 0.86
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