# Structural stability of the quantum Fourier transform

Structural stability of the quantum Fourier transform While it is important to investigate the negative effects of decoherence on the performance of quantum information processors, Landauer was one of the first to point out that an equally basic problem, i.e., the effects of unavoidable hardware flaws in the real-world implementations of quantum gates, needs to be investigated as well. Following Landauer’s suggestion, we investigated the structural stability of the quantum Fourier transform (QFT) via significantly changing the analytical form of its controlled rotation gates, thus modeling structural flaws in the Hamiltonian of the QFT. Three types of modified rotation gates were investigated, numerically and analytically, changing the exact QFT rotation angles $$\pi /2^j$$ π / 2 j to (1) $$\pi /\alpha ^j$$ π / α j , (2) $$\pi /2 j^{\beta }$$ π / 2 j β , and (3) $$\pi /\log _{\gamma }(j+1)$$ π / log γ ( j + 1 ) , where $$\alpha$$ α , $$\beta$$ β , and $$\gamma$$ γ are constants and $$j$$ j is the integer distance between QFT qubits. Surprisingly good performance is observed in all the three cases for a wide range of $$\alpha$$ α , $$\beta$$ β , and $$\gamma$$ γ . This demonstrates the structural stability of the QFT Hamiltonian. Our results also demonstrate that the precise implementation of QFT rotation angles is not critical as long as the angles (roughly) observe a monotonic decrease in $$j$$ j (hierarchy). This result is important since it indicates that stringent tolerances do not need to be imposed in the actual manufacturing process of quantum information hardware components. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Quantum Information Processing Springer Journals

# Structural stability of the quantum Fourier transform

, Volume 14 (4) – Jan 28, 2015
14 pages

/lp/springer_journal/structural-stability-of-the-quantum-fourier-transform-Pau20dal3h
Publisher
Springer US
Subject
Physics; Quantum Information Technology, Spintronics; Quantum Computing; Data Structures, Cryptology and Information Theory; Quantum Physics; Mathematical Physics
ISSN
1570-0755
eISSN
1573-1332
D.O.I.
10.1007/s11128-015-0923-2
Publisher site
See Article on Publisher Site

### Abstract

While it is important to investigate the negative effects of decoherence on the performance of quantum information processors, Landauer was one of the first to point out that an equally basic problem, i.e., the effects of unavoidable hardware flaws in the real-world implementations of quantum gates, needs to be investigated as well. Following Landauer’s suggestion, we investigated the structural stability of the quantum Fourier transform (QFT) via significantly changing the analytical form of its controlled rotation gates, thus modeling structural flaws in the Hamiltonian of the QFT. Three types of modified rotation gates were investigated, numerically and analytically, changing the exact QFT rotation angles $$\pi /2^j$$ π / 2 j to (1) $$\pi /\alpha ^j$$ π / α j , (2) $$\pi /2 j^{\beta }$$ π / 2 j β , and (3) $$\pi /\log _{\gamma }(j+1)$$ π / log γ ( j + 1 ) , where $$\alpha$$ α , $$\beta$$ β , and $$\gamma$$ γ are constants and $$j$$ j is the integer distance between QFT qubits. Surprisingly good performance is observed in all the three cases for a wide range of $$\alpha$$ α , $$\beta$$ β , and $$\gamma$$ γ . This demonstrates the structural stability of the QFT Hamiltonian. Our results also demonstrate that the precise implementation of QFT rotation angles is not critical as long as the angles (roughly) observe a monotonic decrease in $$j$$ j (hierarchy). This result is important since it indicates that stringent tolerances do not need to be imposed in the actual manufacturing process of quantum information hardware components.

### Journal

Quantum Information ProcessingSpringer Journals

Published: Jan 28, 2015

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