Quantum Complexity as Hydrodynamics

As a new step towards defining complexity for quantum field theories, we map Nielsen operator complexity for \(SU(N)\) gates to two-dimensional hydrodynamics. We develop a tractable large \(N\) limit that leads to regular geometries on the manifold of unitaries as \(N\) is taken to infinity. To achi...

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Bibliographic Details
Published in:arXiv.org 2022-02
Main Authors: Basteiro, Pablo, Erdmenger, Johanna, Fries, Pascal, Goth, Florian, Matthaiakakis, Ioannis, Meyer, René
Format: Article
Language:English
Subjects:
Complexity
Computational fluid dynamics
Conjugate points
Cost function
Fluid flow
Fluid mechanics
Holography
Hydrodynamics
Mathematical analysis
Operators (mathematics)
Polynomials
Quantum theory
Toruses
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Summary:As a new step towards defining complexity for quantum field theories, we map Nielsen operator complexity for \(SU(N)\) gates to two-dimensional hydrodynamics. We develop a tractable large \(N\) limit that leads to regular geometries on the manifold of unitaries as \(N\) is taken to infinity. To achieve this, we introduce a basis of non-commutative plane waves for the \(\mathfrak{su}(N)\) algebra and define a metric with polynomial penalty factors. Through the Euler-Arnold approach we identify incompressible inviscid hydrodynamics on the two-torus as a novel effective theory of large-qudit operator complexity. For large \(N\), our cost function captures two essential properties of holographic complexity measures: ergodicity and conjugate points.
ISSN:2331-8422
DOI:10.48550/arxiv.2109.01152