# Quantum geometry and quantum algorithms

@article{Garnerone2007QuantumGA, title={Quantum geometry and quantum algorithms}, author={Silvano Garnerone and Annalisa Marzuoli and Mario Rasetti}, journal={Journal of Physics A: Mathematical and Theoretical}, year={2007}, volume={40}, pages={3047 - 3066} }

Motivated by algorithmic problems arising in quantum field theories whose dynamical variables are geometric in nature, we provide a quantum algorithm that efficiently approximates the coloured Jones polynomial. The construction is based on the complete solution of the Chern–Simons topological quantum field theory and its connection to Wess–Zumino–Witten conformal field theory. The coloured Jones polynomial is expressed as the expectation value of the evolution of the q-deformed spin-network…

## 17 Citations

Combinatorial Framework for Topological Quantum Computing

- Computer Science
- 2012

In this chapter we describe a combinatorial framework for topological quantum computation, and illustrate a number of algorithmic questions in knot theory and in the theory of finitely presented…

Quantum and semiclassical spin networks: from atomic and molecular physics to quantum computing and gravity

- Physics
- 2008

The mathematical apparatus of quantum-mechanical angular momentum (re)coupling, developed originally to describe spectroscopic phenomena in atomic, molecular, optical and nuclear physics, is embedded…

Quantum automata, braid group and link polynomials

- Computer ScienceQuantum Inf. Comput.
- 2007

In this combinatorial framework, families of finite-states and discrete-time quantum automata capable of accepting the language generated by the braid group are implemented, whose transition amplitudes are colored Jones polynomials.

TOWARDS A QUANTUM ALGORITHM FOR THE PERMANENT

- Computer Science, Physics
- 2007

We resort to considerations based on topological quantum field theory to outline the development of a possible quantum algorithm for the evaluation of the permanent of a 0 - 1 matrix. Such an…

Topology, Formal Languages and Quantum Information

- Computer Science
- 2010

The paper provides a critical overview of the basic conceptual tools and algorithms of quantum information theory underlying the construction of quantum invariants of links and 3-manifolds as well as…

Quantum Physics, Topology, Formal Languages, Computation: A Categorical View as Homage to David Hilbert

- Computer SciencePerspectives on Science
- 2014

The deep structural properties of a quantum information theoretic approach to formal languages and universal computation, as well as those of the topology problem of defining the presentation of the Mapping Class Group of a smooth, compact manifold are shown to be grounded in the common categorical features of the two problems.

Spin Network Quantum Circuits

- MathematicsInt. J. Circuit Theory Appl.
- 2017

In this manuscript, a recent approach to quantum computation promoted by the authors, based on the theory of recoupling of quantum angular momenta instead of the conventional notion of q‐bit (that…

Combinatorics of angular momentum recoupling theory: spin networks, their asymptotics and applications

- Physics
- 2009

The quantum theory of angular momentum and the associated Racah–Wigner algebra of the Lie group SU(2) have been widely used in many branches of theoretical and applied physics, chemical physics, and…

Quantum Fourier Transforms and the Complexity of Link Invariants for Quantum Doubles of Finite Groups

- MathematicsArXiv
- 2012

It is proved that, for groups which satisfy certain properties, the probability of success of any randomized computation can be approximated to within any $${\varepsilon}$$ε by the plat closure, and the question of simulating anyonic computation in groups uniformly as a function of the group size is made partial progress.

## References

SHOWING 1-10 OF 88 REFERENCES

Quantum automata, braid group and link polynomials

- Computer ScienceQuantum Inf. Comput.
- 2007

In this combinatorial framework, families of finite-states and discrete-time quantum automata capable of accepting the language generated by the braid group are implemented, whose transition amplitudes are colored Jones polynomials.

Quantum field theory and the Jones polynomial

- Mathematics
- 1989

It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones…

Approximate Counting and Quantum Computation

- Computer Science, MathematicsCombinatorics, Probability and Computing
- 2005

A form of additive approximation which can be used to simulate a function in BQP is introduced and it is shown that all functions in the classes #P and GapP have such an approximation scheme under certain natural normalizations.

Efficient discrete approximations of quantum gates

- Computer Science
- 2002

Here it is proved that using certain sets of base gates quantum compiling requires a string length that is linear in log 1/e, a result which matches the lower bound from counting volume up to constant factor.

The Quantum Theory of Fields, Vol. 1: Foundations

- Physics
- 1995

Cambridge University Press) Available for the first time in paperback, The Quantum Theory of Fields is a self-contained, comprehensive, and up-to-date introduction to quantum field theory from Nobel…

A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

- Computer ScienceSTOC '06
- 2006

An explicit and simplePolynomial quantum algorithm to approximate the Jones polynomial of an n strands braid with m crossings at any primitive root of unity e2πi/k, where the running time of the algorithm is polynometric in m, n and k.

Introduction to the Yang-Mills quantum theory

- Physics
- 1980

A pedagogical discussion of the Yang-Mills quantum theory is presented. A somewhat unconventional description makes use of physical, quantum-mechanical ideas, rather than of formal, mathematical…

Topological quantum field theories

- Physics
- 2000

Following my plenary lecture on ICMP2000 I review my results concerning two closely related topics: topological quantum field theories and the problem of quantization of gauge theories. I start with…

Quantum Gravity in 2+1 Dimensions

- Physics
- 1998

1. Why (2+1)-dimensional gravity? 2. Classical general relativity in 2+1 dimensions 3. A field guide to the (2+1)-dimensional spacetimes 4. Geometric structures and Chern-Simons theory 5. Canonical…