# Deploying Harmonic Analysis in Quantum and Nano Engineering

Quantum metamaterials (QMM) are a relatively new concept. Their purpose is to utilize quantum resources, such as quantum entanglement, in order to transcend some limitations of present day technologies. A typical QMM is a structure comprising numerous qubits in a circuit, engineered to serve a specific purpose, e.g. as an electronic element, an antenna, etc. The functional properties of a QMM depend on its architecture but may also be controlled dynamically via its transient quantum state. The structure of a QMM, which involves a plethora of qubits, poses a fundamental challenge: how can we model and design such structures, given that their complexity grows exponentially with the number of qubits? Can any insights be gleaned without resorting to modelling via quantum computers?

In this talk I will discuss some mathematical structures that arise in the modelling of QMMs and other quantum and nano systems. I will initially focus on a specific conceptual model of a QMM and its dynamics. The central feature of this model is a Hilbert space operator displaying fractal self-similarity. Quite surprisingly, it lends itself to multiresolution analysis based on the Haar wavelet. This is but one example of the relevance of Harmonic Analysis (HA) to this area: its methods, from the ubiquitous Fourier transform to the more specialized Wigner-Weyl transform, are indispensable to quantum and nano system analysis. In the second part of my talk I will highlight some recently discovered alternatives to the Fourier transform and its quantum relatives. These new transforms have already enabled an observation and formulation of an intriguing noise-cancelation phenomenon dubbed the Broadband Redundancy (BR). In essence, there is a sequence of periodic “noisy” waveforms (resp. transforms) whose successive equal-weight superpositions converge to the pure sine/cosine wave (resp. Fourier transform). These specific waveforms are obtained via the periodized Riemann zeta function, while the BR as such results from the uniform distribution (modulo one) of the ordinates of the Riemann zeros.

I invite suggestions with regards to experimental implementations of the BR, e.g. in the realm of quantum optical systems.

Main inspirations and sources:

1. A.M. Zagoskin, Quantum Engineering, Theory and Design of Quantum Coherent Structures, Cambridge University Press: Cambridge 2011

2. A.L. Rakhmanov, A.M. Zagoskin, S. Savel'ev, and F. Nori, Quantum metamaterials: Electromagnetic waves in a Josephson qubit line, Phys. Rev. B 77 (2007), 144507

3. A. Sowa, Riemann's zeta function and the broadband structure of pure harmonics, the IMA Journal of Applied Mathematics 82 (2017), 1238-1252

4. A. Sowa, The Dirichlet ring and unconditional bases in L2[0, 2π], Functional Analysis and Its Applications, 47 (2013), 227-232

5. A. Sowa, Factorizing matrices by Dirichlet multiplication, Linear Algebra and its Applications 438 (2013), 2385-2393

6. A. Sowa, Encoding spatial data into quantum observables, arXiv: 1609.01712 [quant-ph]

7. A. Melli, A. Sowa, K. Wahid, Image denoising via redundant quantum channels, in preparation

8. A. Odlyzko, zeta tables, personal website (2015)