Linear-optical implementations of the iSWAP and controlled NOT gates based on conventional detectors

Monika Bartkowiak; Adam Miranowicz

doi:10.1364/JOSAB.27.002369

Journal of the Optical Society of America B
Vol. 27,
Issue 11,
pp. 2369-2377
(2010)
•https://doi.org/10.1364/JOSAB.27.002369

Linear-optical implementations of the iSWAP and controlled NOT gates based on conventional detectors

Monika Bartkowiak and Adam Miranowicz

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Monika Bartkowiak and Adam Miranowicz, "Linear-optical implementations of the iSWAP and controlled NOT gates based on conventional detectors," J. Opt. Soc. Am. B 27, 2369-2377 (2010)

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Abstract

The majority of linear-optical nondestructive implementations of universal quantum gates are based on single-photon resolving detectors. We propose two implementations, which are nondestructive (i.e., destroying only ancilla states) and work with conventional detectors (i.e., those which do not resolve a number of photons). Moreover, we analyze a recently proposed scheme of Wang et al. [J. Opt. Soc. Am. B 27, 27 (2010)] of an optical iSWAP gate based on two ancillae in Bell’s states, classical feedforward, and conventional detectors with the total probability of success equal to $η^{4} / 32$ , where η is detector’s efficiency. By observing that the iSWAP gate can be replaced with the controlled NOT gate with additional deterministic gates, we list various possible linear-optical implementations of the iSWAP gate: (i) assuming various ancilla states (unentangled, two-photon, and multiphoton-entangled states) or no ancillae at all, (ii) with or without classical feedforward, (iii) destructive or nondestructive schemes, and (iv) using conventional or single-photon detectors. In particular, we show how the nondestructive iSWAP gate can be implemented with the success probability of $η^{4} / 8$ assuming the same ancillae, classical feedforward, and a fewer number of conventional detectors than those in the scheme of Wang et al. We discuss other schemes of the nondestructive universal gates using conventional detectors and entangled ancillae in a cluster state, and Greenberger–Horne–Zeilinger and Bell’s states giving the success probabilities of $η^{4} / 4$ , $η^{6} / 8$ , and $η^{4} / 8$ , respectively. In the latter scheme, we analyze how detector imperfections (dark counts in addition to finite efficiency and no photon-number resolution) and imperfect sources of ancilla states deteriorate the quantum gate operation.

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Equations (14)

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No.	Author	E/T	Comment	P	Feedforward	Entangled Ancillae	Destructive	Conventional Detector
I. Unentangled ancillae
1	KLM [2]	T		$\frac{1}{16}$	No	0	No	No
2	Ralph et al. [47]	T	Simplified No. 1	$\frac{1}{16}$	No	0	No	No
3	Knill [29]	T	Improved No. 1	$\frac{2}{27}$	No	0	No	No
4	Pittman et al. [34]	E		$\frac{1}{8}$	No	0	Yes^a	No
5	Pittman et al. [34]	T	Modified No. 4	$\frac{1}{4}$	Yes	0	Yes ^a	No
6	Giorgi et al. [32]	T	Modified No. 16	$\frac{1}{8}$	Yes	0	No	No
7	Bao et al. [33]	E	Modified No. 13	$\frac{1}{8}$	Yes	0	No	No
II. Entangled ancillae
8	KLM [2]	T		$\frac{1}{4}$	Yes	EPR	No	No
9	KYI [3]	T		$\frac{1}{16}$	Yes	EPR	No	No
10	KYI [3]	T	Modified No. 9	$\frac{1}{4}$ ^b	Yes	$3 \times EPR$	No	No
11	KYI [3]	T	Modified No. 9	$\frac{1}{4}$	Yes	$5 \times EPR$	No	No
12	Pittman et al. [7]	T		$\frac{1}{16}$	No	EPR	No	No
13	Pittman et al. [7]	T	Modified No. 12	$\frac{1}{4}$	Yes	EPR	No	No
14	Gasparoni et al. [5]	E	Realization of No. 12	$\frac{1}{16}$	No	EPR	Yes^a	Yes
15	Zhao et al. [6]	E	Realization of No. 12	$\frac{1}{16}$	No	EPR	Yes^a	Yes
16	Giorgi et al. [32]	T	Related to No. 12	$\frac{1}{4}$	Yes	EPR	No	No
17	Zou et al. [4]	T	Related to No. 12	$\frac{1}{8}$	Yes	$2 \times EPR$	No	Yes
18	Gottesman and Chuang [10]	T		—	Yes	$\| χ ⟩$	No	—
19	Pittman et al. [7]	T	Based on No. 18	$\frac{1}{4}$	Yes	$\| χ ⟩$	No	No
III. Without ancillae
20	Pittman et al. [7]	T		$\frac{1}{4}$	No	0	Yes	No
21	Pittman et al. [7]	T	Modified No. 20	$\frac{1}{2}$	Yes	0	Yes	No
22	Pittman et al. [48]	E	Realization of No. 20	$\frac{1}{4}$	No	0	Yes	No
23	Giorgi et al. [32]	T	Related to No. 20	$\frac{1}{4}$	No	0	Yes	No
24	Giorgi et al. [32]	T	Modified No. 23	$\frac{1}{2}$	Yes	0	Yes	No
25	Hofmann and Takeuchi [35]	T		$\frac{1}{9}$	No	0	Yes^a	No
26	Ralph et al. [36]	T	Equivalent to No. 25	$\frac{1}{9}$	No	0	Yes^a	No
27	O’Brien [49]	E	Realization of No. 25, No. 26	$\frac{1}{9}$	No	0	Yes^a	No
28	Okamoto et al. [50]	E	Realization of No. 25, No. 26	$\frac{1}{9}$	No	0	Yes^a	No
29	Kiesel et al. [51]	E	Simplified No. 25, No. 26	$\frac{1}{9}$	No	0	Yes^a	No
30	Langford et al. [52]	E	Equivalent to No. 29	$\frac{1}{9}$	No	0	Yes^a	No

$D_{3 H}$		$D_{3 V}$		$D_{4 H}$		$D_{4 V}$		$U^{j}$
1		0		1		0		I
0		1		0		1		I
1		0		0		1		$σ_{z}^{(5)} \otimes σ_{z}^{(6)}$
0		1		1		0		$σ_{z}^{(5)} \otimes σ_{z}^{(6)}$
$D_{c H}$	$D_{c V}$	$D_{1 H}$	$D_{1 V}$	$D_{6 H}$	$D_{6 V}$	$D_{t H}$	$D_{t V}$	$V^{k l}$
1	0	1	0	1	0	1	0	I
1	0	1	0	0	1	0	1	I
0	1	0	1	1	0	1	0	I
0	1	0	1	0	1	0	1	I
1	0	0	1	1	0	1	0	$σ_{z}^{(2)}$
1	0	0	1	0	1	0	1	$σ_{z}^{(2)}$
0	1	1	0	1	0	1	0	$σ_{z}^{(2)}$
0	1	1	0	0	1	0	1	$σ_{z}^{(2)}$
1	0	0	1	1	0	0	1	$σ_{z}^{(5)}$
1	0	0	1	0	1	1	0	$σ_{z}^{(5)}$
0	1	1	0	1	0	0	1	$σ_{z}^{(5)}$
0	1	1	0	0	1	1	0	$σ_{z}^{(5)}$
1	0	1	0	1	0	0	1	$σ_{z}^{(2)} \otimes σ_{z}^{(5)}$
1	0	1	0	0	1	1	0	$σ_{z}^{(2)} \otimes σ_{z}^{(5)}$
0	1	0	1	1	0	0	1	$σ_{z}^{(2)} \otimes σ_{z}^{(5)}$
0	1	0	1	0	1	1	0	$σ_{z}^{(2)} \otimes σ_{z}^{(5)}$

$D_{2 H}$	$D_{2 V}$	$D_{3 H}$	$D_{3 V}$	$U^{j}$
1	0	1	0	I
0	1	0	1	I
1	0	0	1	$σ_{z}^{(2)}$
0	1	1	0	$σ_{z}^{(2)}$
$D_{c H}$	$D_{c V}$	$D_{t H}$	$D_{t V}$	$V^{k l}$
1	0	1	0	I
0	1	1	0	$σ_{z}^{(1)}$
1	0	0	1	$σ_{z}^{(4)}$
0	1	0	1	$σ_{z}^{(1)} \otimes σ_{z}^{(4)}$

No.	Author	E/T	Comment	P	Feedforward	Entangled Ancillae	Destructive	Conventional Detector
I. Unentangled ancillae
1	KLM [2]	T		$\frac{1}{16}$	No	0	No	No
2	Ralph et al. [47]	T	Simplified No. 1	$\frac{1}{16}$	No	0	No	No
3	Knill [29]	T	Improved No. 1	$\frac{2}{27}$	No	0	No	No
4	Pittman et al. [34]	E		$\frac{1}{8}$	No	0	Yes^a	No
5	Pittman et al. [34]	T	Modified No. 4	$\frac{1}{4}$	Yes	0	Yes ^a	No
6	Giorgi et al. [32]	T	Modified No. 16	$\frac{1}{8}$	Yes	0	No	No
7	Bao et al. [33]	E	Modified No. 13	$\frac{1}{8}$	Yes	0	No	No
II. Entangled ancillae
8	KLM [2]	T		$\frac{1}{4}$	Yes	EPR	No	No
9	KYI [3]	T		$\frac{1}{16}$	Yes	EPR	No	No
10	KYI [3]	T	Modified No. 9	$\frac{1}{4}$ ^b	Yes	$3 \times EPR$	No	No
11	KYI [3]	T	Modified No. 9	$\frac{1}{4}$	Yes	$5 \times EPR$	No	No
12	Pittman et al. [7]	T		$\frac{1}{16}$	No	EPR	No	No
13	Pittman et al. [7]	T	Modified No. 12	$\frac{1}{4}$	Yes	EPR	No	No
14	Gasparoni et al. [5]	E	Realization of No. 12	$\frac{1}{16}$	No	EPR	Yes^a	Yes
15	Zhao et al. [6]	E	Realization of No. 12	$\frac{1}{16}$	No	EPR	Yes^a	Yes
16	Giorgi et al. [32]	T	Related to No. 12	$\frac{1}{4}$	Yes	EPR	No	No
17	Zou et al. [4]	T	Related to No. 12	$\frac{1}{8}$	Yes	$2 \times EPR$	No	Yes
18	Gottesman and Chuang [10]	T		—	Yes	$\| χ ⟩$	No	—
19	Pittman et al. [7]	T	Based on No. 18	$\frac{1}{4}$	Yes	$\| χ ⟩$	No	No
III. Without ancillae
20	Pittman et al. [7]	T		$\frac{1}{4}$	No	0	Yes	No
21	Pittman et al. [7]	T	Modified No. 20	$\frac{1}{2}$	Yes	0	Yes	No
22	Pittman et al. [48]	E	Realization of No. 20	$\frac{1}{4}$	No	0	Yes	No
23	Giorgi et al. [32]	T	Related to No. 20	$\frac{1}{4}$	No	0	Yes	No
24	Giorgi et al. [32]	T	Modified No. 23	$\frac{1}{2}$	Yes	0	Yes	No
25	Hofmann and Takeuchi [35]	T		$\frac{1}{9}$	No	0	Yes^a	No
26	Ralph et al. [36]	T	Equivalent to No. 25	$\frac{1}{9}$	No	0	Yes^a	No
27	O’Brien [49]	E	Realization of No. 25, No. 26	$\frac{1}{9}$	No	0	Yes^a	No
28	Okamoto et al. [50]	E	Realization of No. 25, No. 26	$\frac{1}{9}$	No	0	Yes^a	No
29	Kiesel et al. [51]	E	Simplified No. 25, No. 26	$\frac{1}{9}$	No	0	Yes^a	No
30	Langford et al. [52]	E	Equivalent to No. 29	$\frac{1}{9}$	No	0	Yes^a	No

$D_{3 H}$		$D_{3 V}$		$D_{4 H}$		$D_{4 V}$		$U^{j}$
1		0		1		0		I
0		1		0		1		I
1		0		0		1		$σ_{z}^{(5)} \otimes σ_{z}^{(6)}$
0		1		1		0		$σ_{z}^{(5)} \otimes σ_{z}^{(6)}$
$D_{c H}$	$D_{c V}$	$D_{1 H}$	$D_{1 V}$	$D_{6 H}$	$D_{6 V}$	$D_{t H}$	$D_{t V}$	$V^{k l}$
1	0	1	0	1	0	1	0	I
1	0	1	0	0	1	0	1	I
0	1	0	1	1	0	1	0	I
0	1	0	1	0	1	0	1	I
1	0	0	1	1	0	1	0	$σ_{z}^{(2)}$
1	0	0	1	0	1	0	1	$σ_{z}^{(2)}$
0	1	1	0	1	0	1	0	$σ_{z}^{(2)}$
0	1	1	0	0	1	0	1	$σ_{z}^{(2)}$
1	0	0	1	1	0	0	1	$σ_{z}^{(5)}$
1	0	0	1	0	1	1	0	$σ_{z}^{(5)}$
0	1	1	0	1	0	0	1	$σ_{z}^{(5)}$
0	1	1	0	0	1	1	0	$σ_{z}^{(5)}$
1	0	1	0	1	0	0	1	$σ_{z}^{(2)} \otimes σ_{z}^{(5)}$
1	0	1	0	0	1	1	0	$σ_{z}^{(2)} \otimes σ_{z}^{(5)}$
0	1	0	1	1	0	0	1	$σ_{z}^{(2)} \otimes σ_{z}^{(5)}$
0	1	0	1	0	1	1	0	$σ_{z}^{(2)} \otimes σ_{z}^{(5)}$

Abstract

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Equations (14)

Journal of the Optical Society of America B