Quantum complexity of parametric integration

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al.\ that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a \#P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

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Reduction of Feynman integrals in the parametric representation II: reduction of tensor integrals

The European Physical Journal C ◽

10.1140/epjc/s10052-021-09036-5 ◽

2021 ◽

Vol 81 (3) ◽

Author(s):

Wen Chen

Keyword(s):

Gröbner Basis ◽

Parametric Representation ◽

Polynomial Equations ◽

Grobner Basis ◽

Spacetime Dimension ◽

Integration By Parts ◽

Feynman Integrals ◽

Parametric Integrals ◽

Parametric Integration

AbstractIn a recent paper by the author (Chen in JHEP 02:115, 2020), the reduction of Feynman integrals in the parametric representation was considered. Tensor integrals were directly parametrized by using a generator method. The resulting parametric integrals were reduced by constructing and solving parametric integration-by-parts (IBP) identities. In this paper, we furthermore show that polynomial equations for the operators that generate tensor integrals can be derived. Based on these equations, two methods to reduce tensor integrals are developed. In the first method, by introducing some auxiliary parameters, tensor integrals are parametrized without shifting the spacetime dimension. The resulting parametric integrals can be reduced by using the standard IBP method. In the second method, tensor integrals are (partially) reduced by using the technique of Gröbner basis combined with the application of symbolic rules. The unreduced integrals can further be reduced by solving parametric IBP identities.

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Quantum complexity associated with tunneling

Quantum Information and Computation ◽

10.26421/qic19.3-4-3 ◽

2019 ◽

Vol 19 (3&4) ◽

pp. 222-236

Author(s):

Ofir Flom ◽

Asher Yahalom ◽

Haggai Zilberberg ◽

L.P. Horwitz ◽

Jacob Levitan

Keyword(s):

Quantum Effect ◽

Entropy Function ◽

Potential Well ◽

Rapid Rise ◽

Dimensional Model ◽

One Dimensional ◽

Spatial Entropy ◽

Quantum Complexity ◽

Entropy Growth ◽

Infinite Potential Well

We use a one dimensional model of a square barrier embedded in an infinite potential well to demonstrate that tunneling leads to a complex behavior of the wave function and that the degree of complexity may be quantified by use of a locally defined spatial entropy function defined by S=-\int |\Psi(x,t)|^2 \ln |\Psi(x,t)|^2 dx . We show that changing the square barrier by increasing the height or width of the barrier not only decreases the tunneling but also slows down the rapid rise of the entropy function, indicating that the locally defined entropy growth is an essentially quantum effect.

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Quantum networks: topology and spectral characterization

EPJ Web of Conferences ◽

10.1051/epjconf/201818202014 ◽

2018 ◽

Vol 182 ◽

pp. 02014

Author(s):

Vesna Berec

Keyword(s):

Quantum Information Processing ◽

Theoretic Approach ◽

Quantum State Transfer ◽

Complexity Measures ◽

Quantum Networks ◽

Graph Theoretic ◽

Topological Features ◽

State Transfer ◽

Quantum Complexity ◽

And Control

To utilize a scalable quantum network and perform a quantum state transfer within distant arbitrary nodes, coherence and control of the dynamics of couplings between the information units must be achieved as a prerequisite ingredient for quantum information processing within a hierarchical structure. Graph theoretic approach provides a powerful tool for the characterization of quantum networks with non-trivial clustering properties. By encoding the topological features of the underlying quantum graphs, relations between the quantum complexity measures are presented revealing the intricate links between a quantum and a classical networks dynamics.

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