Implementation of the quantum Fourier transform on a hybrid qubit–qutrit NMR quantum emulator

Dogra, Shruti; Dorai, Kavita

doi:10.1142/s0219749915500598

“…When d = 3, it is called a qutrit. The implementation of the QFT on a quantum computer working with qudits is studied in [2,13,6]. In the current section, we show that our derivation of the QFT on a qubit system extends to qudit systems in a straightforward manner, and leads to a class of equivalent QFT circuits on a qudit system.…”

Section: Quantum Circuit For Radix-d Qftmentioning

confidence: 79%

“…In section 6, we generalize the radix-2 QFT algorithm to a radix-d QFT algorithm and present the quantum circuit for implementing such a QFT on a d-level quantum computer. The QFT on d-level quantum computers was also studied in [2,13,6].…”

Section: Introductionmentioning

confidence: 99%

Quantum Fourier Transform Revisited

Camps,

Van Beeumen,

Yang

2020

Preprint

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The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank-1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix-2 QFT decomposition to a radix-d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.

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“…When d = 3, it is called a qutrit . The implementation of the QFT on a quantum computer working with qudits is studied in References 10–12. In the current section, we show that our derivation of the QFT on a qubit system extends to qudit systems in a straightforward manner, and leads to a class of equivalent QFT circuits on a qudit system.…”

Section: Generalization To Radix‐d Qftmentioning

confidence: 80%

“…In Section 6, we generalize the radix‐2 QFT algorithm to a radix‐ d QFT algorithm and present the quantum circuit for implementing such a QFT on a d ‐level quantum computer. The QFT on d ‐level quantum computers was also studied in References 10–12.…”

Section: Introductionmentioning

confidence: 99%

Quantum Fourier transform revisited

Camps

¹

,

Beeumen

²

,

Yang

³

2020

Numerical Linear Algebra App

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The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by further decomposing the diagonal factors of the FFT matrix decomposition into products of matrices with Kronecker product structure. We analyze the implication of this Kronecker product structure on the discrete Fourier transform of rank-1 tensors on a classical computer. We also explain why such a structure can take advantage of an important quantum computer feature that enables the QFT algorithm to attain an exponential speedup on a quantum computer over the FFT algorithm on a classical computer. Further, the connection between the matrix decomposition of the DFT matrix and a quantum circuit is made. We also discuss a natural extension of a radix-2 QFT decomposition to a radix-d QFT decomposition. No prior knowledge of quantum computing is required to understand what is presented in this paper. Yet, we believe this paper may help readers to gain some rudimentary understanding of the nature of quantum computing from a matrix computation point of view.

show abstract

“…The phases of the constituent states are taken care of by using the information of overlaps of respective constituent states with the referential state (see Appendix A). This is then followed by Fourier transformation of the qunit, which is in fact the generalization of the Hadamard operation to higher-dimensional states [17]. The resultant state, which is a generalization of the two-qubit state in Eq.…”

Section: Superposition Of Multiple Quditsmentioning

confidence: 99%

Superposing pure quantum states with partial prior information

Dogra

¹

,

Thomas

²

,

Ghosh

³

et al. 2018

Self Cite

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The principle of superposition is an intriguing feature of quantum mechanics, which is regularly exploited in many different circumstances. A recent work [PRL 116, 110403 (2016)] shows that the fundamentals of quantum mechanics restrict the process of superimposing two unknown pure states, even though it is possible to superimpose two quantum states with partial prior knowledge. The prior knowledge imposes geometrical constraints on the choice of input states. We discuss an experimentally feasible protocol to superimpose multiple pure states of a d-dimensional quantum system and carry out an explicit experimental realization for two single-qubit pure states with partial prior information on a two-qubit NMR quantum information processor.

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Implementation of the quantum Fourier transform on a hybrid qubit–qutrit NMR quantum emulator

Cited by 16 publications

References 50 publications

Quantum Fourier Transform Revisited

Quantum Fourier Transform Revisited

Quantum Fourier transform revisited

Superposing pure quantum states with partial prior information

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