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Implementation of CNOT and Toffoli gates with higherdimensional spaces
by WenQiang Liu, HaiRui Wei, LeongChuan Kwek
Submission summary
As Contributors:  HaiRui Wei 
Arxiv Link:  https://arxiv.org/abs/2105.10631v2 (pdf) 
Date submitted:  20210720 04:41 
Submitted by:  Wei, HaiRui 
Submitted to:  SciPost Physics 
Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Minimizing the number of necessary twoqubit gates is an important task in quantum information processing. By introducing noncomputational quantum states in auxiliary spaces, we construct effective circuits for the controlledNOT (CNOT) gate and the ncontrolqubit Toffoli gate with (2n1) qubitqudit gates and (2n2) singlequdit gates. We propose the polarization CNOT and Toffoli gates based on the designed quantum circuits in linear optics by operating on the spatialmode 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:
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 PSWAP, CNOT, and Toffoli gates in terms of the creation operations. [see Eqs. (1145)]
3. We added inputoutput 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 PSWAP gate. [see the paragraphs belowEqs. (22), below (28),below(33), below(35),below(37)]
5. We clarified the calculation of success probability of PSWAP, CNOT and Toffoli gates. [see the paragraphs below Eq. (21), below Eq. (32), and below Eq. (45)]
6. We explained the realization of feedforward 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 nqubit Toffoli asncontrolqubit Toffoli, and changed the required gate number as(2n1) qubitquditgates and (2n2) singlequdit gates.[see all text in the revised version]
11. We have fixed the global factors in Eqs. (2332) and (3645).
12. We have added Eqs.(4245).
13. We added some interesting references [8], [27], [28], [3032], [34], [61] and [62].
14. We have polished the English of this paper.