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Implementation of CNOT and Toffoli gates with higher-dimensional spaces

by Wen-Qiang Liu, Hai-Rui Wei, Leong-Chuan Kwek

Submission summary

As Contributors: Hai-Rui Wei
Arxiv Link: https://arxiv.org/abs/2105.10631v2 (pdf)
Date submitted: 2021-07-20 04:41
Submitted by: Wei, Hai-Rui
Submitted to: SciPost Physics
Academic field: Physics
Specialties:
  • Quantum Physics
Approaches: Theoretical, Computational

Abstract

Minimizing the number of necessary two-qubit gates is an important task in quantum information processing. By introducing non-computational quantum states in auxiliary spaces, we construct effective circuits for the controlled-NOT (CNOT) gate and the n-control-qubit Toffoli gate with (2n-1) qubit-qudit gates and (2n-2) single-qudit gates. We propose the polarization CNOT and Toffoli gates based on the designed quantum circuits in linear optics by operating on the spatial-mode degree of freedom of photons. Our optical schemes can be achieved with a higher success probability and no extra auxiliary photons are needed.

Current status:
Editor-in-charge assigned


List of changes

1. We have clarified our method is inspiredby Refs. [20,21] and gave more explicit statements and comparisons. [see the paragraph 4 in Section 1, andline 3 of the paragraph 1 below Section 4]
2. We rewrote the implementations of P-SWAP, CNOT, and Toffoli gates in terms of the creation operations. [see Eqs. (11-45)]
3. We added input-output modes inFigs.5 and 6, and fixed Tab.1. [see Figs. 5 and 6, and Tab.1]
4. We explained the joining of output modes of P-SWAP gate. [see the paragraphs belowEqs. (22), below (28),below(33), below(35),below(37)]
5. We clarified the calculation of success probability of P-SWAP, CNOT and Toffoli gates. [see the paragraphs below Eq. (21), below Eq. (32), and below Eq. (45)]
6. We explained the realization of feed-forward operationsexplicitly [see the paragraphs below Eq. (21), below Eqs. (31) and (32), and below Eqs. (44) and (45)]
7. We have added the definition of the cost of quantum circuits and removed all inappropriate comparisons between our gate cost and previous cost in qubit system. [see paragraph 1 in Introduction]
8. We clarified the implementation of CNOT and Toffoli gates is as the form:(Q. gate) – (Q. gate) – (detector). [see subsections 3.2 and 3.3]
9. We added subsections title 3.1, 3.2, and 3.3, and changed the title of our paper, according to the suggestion of Reviewer 2.
10. We have revised n-qubit Toffoli asn-control-qubit Toffoli, and changed the required gate number as(2n-1) qubit-quditgates and (2n-2) single-qudit gates.[see all text in the revised version]
11. We have fixed the global factors in Eqs. (23-32) and (36-45).
12. We have added Eqs.(42-45).
13. We added some interesting references [8], [27], [28], [30-32], [34], [61] and [62].
14. We have polished the English of this paper.

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