Computational speed-up with a single qudit

Abstract

Quantum algorithms are known for providing more efficient solutions to certain computational tasks than any corresponding classical algorithm. Here we show that a single qudit is sufficient to implement an oracle based quantum algorithm, which can solve a black-box problem faster than any classical algorithm. For 2d permutation functions defined on a set of d elements, deciding whether a given permutation is even or odd, requires evaluation of the function for at least two elements. We demonstrate that a quantum circuit with a single qudit can determine the parity of the permutation with only one evaluation of the function. Our algorithm provides an example for quantum computation without entanglement since it makes use of the pure state of a qudit. We also present an experimental realization of the proposed quantum algorithm with a quadrupolar nuclear magnetic resonance using a single four-level quantum system, i.e., a ququart.

Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates

“…A qutrit is the smallest system that exhibits inherent quantum features such as contextuality [28][29][30], which has been conjectured to be a resource for quantum computing [31][32][33][34][35]. NMR qudit quantum computing can be performed by using nuclei with spin s > 1 2 or can be modeled by two or * shrutidogra@iisermohali.ac.in † kavita@iisermohali.ac.in ‡ arvind@iisermohali.ac.in more coupled spin-1 2 nuclei [36,37].…”

Quantum Genetic Algorithm on Multilevel Quantum Systems

“…Recently much attention has been devoted to studies of possibility of using of the quantum systems for realization of quantum algorithms [8][9][10]. In contradiction to qubit, qudit is a quantum unit of information, which may be in any of basis states |0⟩, |1⟩, |2⟩, .…”

Spin Quantum Computing with Endohedral Fullerenes

“…34 The four levels of the high = 3 2 electron spin can in principle be used to encode either two qubits 33,35 or a ququart (a four-level generalization of the two-level qubit), which is useful in some quantum computing schemes and has been studied for analogous = 3 2 nuclear spins in the NMR domain. 36,37 In both cases, however, transition-selective addressing is mandatory, requiring a finite 'quadrupolar' splitting Hamiltonian of the form ! !…”