Fourier’s Quantum Information Processing

Abstract

We demonstrate that quantum information processing (QIP) completely rests on quantum Fourier transform (QFT), and 190 years after his death, the work of Jean-Baptiste Joseph Fourier is more present than ever in one of the most important pillars of physics: QIP. Specifically, this study shows the impact of QFT on the main tool of quantum communication: entanglement, with a particular emphasis on its spectral nature, whose study is completed when entanglement is used in teleportation, and quantum secret sharing. Finally, a better understanding of the spectral nature of quantum entanglement will lead to better protocols for quantum communications, in general, and quantum cryptography, in particular.

On the spectral nature of entanglement

“…It is evident that if we resort to all the equivalences between gates present in the literature [27,30,31], the Fourier expansion in other scopes outside the quantum entanglement and teleportation becomes obvious. This can be seen in detail in a recent work that links QFT with quantum information processing [36].…”

Fourier Behind Entanglement: A Spectral Approach to the Quantum Internet

“…[31] Although ibmq_5_yorktown is a machine of considerable decoherence, the percentage errors of (Equation 20) are practically insignificant, which highlights the validity of the equivalence of Figure 5c, that is, the validity of the spectral facet of the entanglement. [68,69] Currently, IBM Q Experience [31] does not allow us to implement Controlled-S directly so this gate was implemented using the equivalent circuits of Figure 12, where…”

Rough Entanglement: Not Enough Fourier

“…[1] This powerful tool has been consolidated over the years in such critical applications as phase estimation, within the most famous quantum algorithm in the literature, the Shor algorithm. [2] In recent years, it has been shown that QFT is a key piece in the same constitution of quantum entanglement, [3][4][5][6] which is why the latter has two complementary facets, a temporal facet, traditionally known, [1] and another spectral, [3][4][5][6] which will lead to the creation of future algorithms and protocols with higher performance to be used in quantum communications, [7] in general, and quantum cryptography, [8] in particular.…”

“…The spectral manifestation of entanglement has been exploited to the limit in previous works in implementations ranging from quantum teleportation, [3][4][5][6] through quantum secret sharing (QSS), [3][4][5][6] to quantum repeaters. [3][4][5][6] These applications have a marked projection on the quantum key distribution (QKD) [9][10][11][12] protocols, although their greatest contribution is expected to happen in the field of the future quantum Internet.…”

“…The spectral manifestation of entanglement has been exploited to the limit in previous works in implementations ranging from quantum teleportation, [3][4][5][6] through quantum secret sharing (QSS), [3][4][5][6] to quantum repeaters. [3][4][5][6] These applications have a marked projection on the quantum key distribution (QKD) [9][10][11][12] protocols, although their greatest contribution is expected to happen in the field of the future quantum Internet. [13][14][15][16][17][18] Specifically, in the quantum cryptography [8] context, fiber optic cabling for terrestrial implementations of QKD protocols [9][10][11][12] requires quantum repeaters every certain number DOI: 10.1002/qute.202200176 of kilometers, [19,20] which in turn requires a large amount of quantum memory.…”