Abstract

We present a scheme of two-photon quantum Fourier transform at first, where the controlled π/2 phase gate and the Hadamard gates are applied. The indirect interaction between two photons can be realized through a coupling bus provided by a coherent state in Kerr media. Employing photon-number measurements and the corresponding classical feed-forward techniques, the task of low-error quantum Fourier transform can be fulfilled. With revision and modulation on the optical elements, controlled π/2n (n=2,3,) phase gates are constructed, and the multiphoton quantum Fourier transform can be achieved by integrating these controlled phase gates and Hadamard gates.

© 2013 Optical Society of America

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2013 (4)

L. Dong, X.-M. Xiu, Y.-J. Gao, and X. X. Yi, “A nearly deterministic scheme for generating χ-type entangled states with weak cross-Kerr nonlinearities,” Quant. Info. Proc. 12, 1787–1795 (2013).
[CrossRef]

X.-W. Wang, S.-Q. Tang, L.-J. Xie, and D.-Y. Zhang, “Nondestructive two-photon parity detector with near unity efficiency,” Opt. Commun. 296, 153–157 (2013).
[CrossRef]

I.-C. Hoi, C. Wilson, G. Johansson, T. Palomaki, T. M. Stace, B. Fan, and P. Delsing, “Giant cross Kerr effect for propagating microwaves induced by an artificial atom,” Phys. Rev. Lett. 111, 053601 (2013).
[CrossRef]

X.-M. Xiu, L. Dong, H.-Z. Shen, Y.-J. Gao, and X. X. Yi, “Construction scheme of a two-photon polarization controlled arbitrary phase gate mediated by weak cross-phase modulation,” J. Opt. Soc. Am. B 30, 589–597 (2013).
[CrossRef]

2012 (11)

H.-F. Wang, S. Zhang, A.-D. Zhu, and K.-H. Yeon, “Fast and effective implementation of discrete quantum Fourier transform via virtual-photon-induced process in separate cavities,” J. Opt. Soc. Am. B 29, 1078–1084 (2012).
[CrossRef]

F.-F. Du, T. Li, B.-C. Ren, H.-R. Wei, and F.-G. Deng, “Single-photon-assisted entanglement concentration of a multiphoton system in a partially entangled W state with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29, 1399–1405 (2012).
[CrossRef]

Y.-H. Chen, M.-J. Lee, W. Hung, Y.-C. Chen, Y.-F. Chen, and I. A. Yu, “Demonstration of the interaction between two stopped light pulses,” Phys. Rev. Lett. 108, 173603 (2012).
[CrossRef]

M. Siomau, A. A. Kamli, S. A. Moiseev, and B. C. Sanders, “Entanglement creation with negative index metamaterials,” Phys. Rev. A 85, 050303(R) (2012).
[CrossRef]

F.-G. Deng, “Optimal nonlocal multipartite entanglement concentration based on projection measurements,” Phys. Rev. A 85, 022311 (2012).
[CrossRef]

W. H. P. Pernice, C. Schuck, O. Minaeva, M. Li, G. N. Goltsman, A. V. Sergienko, and H. X. Tang, “High-speed and high-efficiency travelling wave single-photon detectors embedded in nanophotonic circuits,” Nat. Commun. 3, 1325 (2012).
[CrossRef]

X.-M. Xiu, L. Dong, Y.-J. Gao, and X. X. Yi, “Nearly deterministic controlled-not gate with weak cross-Kerr nonlinearities,” Quantum Inf. Comput. 12, 159–170 (2012).

Y.-B. Sheng, L. Zhou, S.-M. Zhao, and B.-Y. Zheng, “Efficient single-photon-assisted entanglement concentration for partially entangled photon pairs,” Phys. Rev. A 85, 012307 (2012).
[CrossRef]

Y.-B. Sheng, L. Zhou, and S.-M. Zhao, “Efficient two-step entanglement concentration for arbitrary W states,” Phys. Rev. A 85, 042302 (2012).
[CrossRef]

X.-W. Wang, D.-Y. Zhang, S.-Q. Tang, L.-J. Xie, Z.-Y. Wang, and L.-M. Kuang, “Photonic two-qubit parity gate with tiny cross-Kerr nonlinearity,” Phys. Rev. A 85, 052326 (2012).
[CrossRef]

J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, and M. Zukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777–838 (2012).
[CrossRef]

2011 (14)

H.-F. Wang, A.-D. Zhu, S. Zhang, and K.-H. Yeon, “Simple implementation of discrete quantum Fourier transform via cavity quantum electrodynamics,” New J. Phys. 13, 013021 (2011).
[CrossRef]

H.-F. Wang, S. Zhang, and K.-H. Yeon, “Linear optical implementation of discrete quantum Fourier transform with conventional photon detectors,” Int. J. Quantum. Inform. 09, 509–518 (2011).
[CrossRef]

H.-F. Wang, X.-X. Jiang, S. Zhang, and K.-H. Yeon, “Efficient quantum circuit for implementing discrete quantum Fourier transform in solid-state qubits,” J. Phys. B 44, 115502 (2011).
[CrossRef]

B. He, Q. Lin, and C. Simon, “Cross-Kerr nonlinearity between continuous-mode coherent states and single photons,” Phys. Rev. A 83, 053826 (2011).
[CrossRef]

A. Feizpour, X. Xing, and A. M. Steinberg, “Amplifying single-photon nonlinearity using weak measurements,” Phys. Rev. Lett. 107, 133603 (2011).
[CrossRef]

H.-Y. Lo, Y.-C. Chen, P.-C. Su, H.-C. Chen, J.-X. Chen, Y.-C. Chen, I. A. Yu, and Y.-F. Chen, “Electromagnetically-induced-transparency-based cross-phase-modulation at attojoule levels,” Phys. Rev. A 83, 041804(R) (2011).

B. He, A. MacRae, Y. Han, A. Lvovsky, and C. Simon, “Transverse multimode effects on the performance of photon-photon gates,” Phys. Rev. A 83, 022312 (2011).
[CrossRef]

E. Shahmoon, G. Kurizki, M. Fleischhauer, and D. Petrosyan, “Strongly interacting photons in hollow-core waveguides,” Phys. Rev. A 83, 033806 (2011).
[CrossRef]

Y. Xia, J. Song, P.-M. Lu, and H.-S. Song, “Effective quantum teleportation of an atomic state between two cavities with the cross-Kerr nonlinearity by interference of polarized photons,” J. Appl. Phys. 109, 103111 (2011).
[CrossRef]

Q. Guo, J. Bai, L.-Y. Cheng, X.-Q. Shao, H.-F. Wang, and S. Zhang, “Simplified optical quantum-information processing via weak cross-Kerr nonlinearities,” Phys. Rev. A 83, 054303 (2011).
[CrossRef]

F.-G. Deng, “Efficient multipartite entanglement purification with the entanglement link from a subspace,” Phys. Rev. A 84, 052312 (2011).
[CrossRef]

C. Wang, Y. Zhang, and G.-S. Jin, “Polarization-entanglement purification and concentration using cross-kerr nonlinearity,” Quantum Inf. Comput. 11, 988–1002 (2011).

W. Xiong and L. Ye, “Schemes for entanglement concentration of two unknown partially entangled states with cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 28, 2030–2037 (2011).
[CrossRef]

X.-s. Ma, S. Zotter, N. Tetik, A. Qarry, T. Jennewein, and A. Zeilinger, “A high-speed tunable beam splitter for feed-forward photonic quantum information processing,” Opt. Express 19, 22723–22730 (2011).
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2010 (10)

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Figures (4)

Fig. 1.
Fig. 1.

Schematic illustration of two-photon quantum Fourier transform. Red-dotted lines denote the first classical feed-forward based on the first photon-number measurement outcome, and green-dashed line denotes the second classical feed-forward according to the second photon-number measurement outcome. For simplicity, we assume that the optical lengths of the corresponding paths which photon 1 and photon 2 pass through are equal. If not, we may modulate it by inserting wave plates into the appropriate places.

Fig. 2.
Fig. 2.

Schematic illustration of n-photon quantum Fourier transform with polarization degree of freedom, provided with n1-photon quantum Fourier transform. In the box, the left element [phase shifter “π/2ni,” (i=1,2,,n1)] denotes the controlled π/2ni phase gates, and the right one (phase shifter “π/2n1i”) stands for the required operation by classical feed-forward. The classical feed-forward operation of “PS π/2ni” is not labeled in this figure.

Fig. 3.
Fig. 3.

Error probability comparison between the first photon-number measurement and X quadrature measurement. Here we set α=1.0×104 [61]. The blue-solid line denotes the error probability of the first photon-number measurement, and the red-dotted line denotes the error probability of the X quadrature measurement.

Fig. 4.
Fig. 4.

Error probability comparison between the second photon-number measurement and Y quadrature measurement. The blue-solid line denotes the error probability of the second photon-number measurement, and the red-dashed line denotes the error probability of the Y quadrature measurement.

Equations (24)

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|j=|j1j2jgjn,
|jQFT12n/2k=02n1e2πijk/2n|k=12n/2(|0+e2πi0·jn|1)(|0+e2πi0·jn1jn|1)(|0+e2πi0·j1j2jn1jn|1),
|ϕQFTk=0N1bk|k,
χ|HH+β|HV+γ|VH+δ|VV,
χ|HH+β|HV+γ|VH+exp(iπ/2)δ|VV,
Uck(|ϕs|αp)=eiHckt/[(a|0s+b|1s+c|2s+)|αp]=a|0s|αp+b|1s|αeiθp+c|2s|αe2iθp+,
(11111eiπeiπ/2e3iπ/21e2iπeiπe3iπ1e3iπe3iπ/2e9iπ/2).
12(χ|HHeb+β|HVeb+γ|VHfa+δ|VVfa)|α3OU|α3OL.
12(χ|HHeb+β|HVeb+γ|VHfa+δ|VVfa)|α4IU|α4IL+12(χ|HHea+β|HVea)|αeiθ4IU|αeiθ4IL+12(γ|VHfb+δ|VVfb)|αeiθ4IU|αeiθ4IL.
12(χ|HHeb+β|HVeb+γ|VHfa+δ|VVfa)|04OU|2α4OL+12(χ|HHea+β|HVea)|2iαsinθ4OU|2αcosθ4OL+12(γ|VHfb+δ|VVfb)|2iαsinθ4OU|2αcosθ4OL.
χ|HHeb+β|HVeb+γ|VHfa+iδ|VVfa,
12(χ|HHecβ|HVec+γ|VHfc+iδ|VVfc)+12(χ|HHed+β|HVed+γ|VHfd+iδ|VVfd).
n|2iαsinθ4OU=e|2iαsinθ|22(2iαsinθ)nn!,n|2iαsinθ4OU=e|2iαsinθ|22(2iαsinθ)nn!,
[(χ|HHea+β|HVea)(i)n+(γ|VHfb+δ|VVfb)(i)n]e|αsinθ|2(2αsinθ)nn!,
eiπ2n(χ|HH+β|HV)ea+eiπ2n(γ|VH+δ|VV)fb
χ|HHeaiβ|HVea+γ|VHfb+δ|VVfbBFF12(χ|HHeciβ|HVecγ|VHfcδ|VVfc)+12(χ|HHediβ|HVed+γ|VHfd+δ|VVfd)AFF12(χ|HHec+β|HVecγ|VHfciδ|VVfc)+12(χ|HHed+β|HVed+γ|VHfd+iδ|VVfd).
χ|HH+β|HV+γ|VH+δ|VVTPQFT(χ+β+γ+δ)|HH+(χβ+iγiδ)|HV+(χ+βγδ)|VH+(χβiγ+iδ)|VV.
|±2iαsinθ=e|αsinθ|2n=0(±2iαsinθ)nn!|n=e|αsinθ|2|0±2iαsinθe|αsinθ|2|12α2sin2θe|αsinθ|2|2.
Perr1:n=0,n0=12exp(2|αsinθ|2).
Perr2:n=0,n0=12exp(2|αsinθ2|2).
PerrPerr1:n=0,n0+Perr2:n=0,n0=12exp(2|αθ|2)+12exp(12|αθ|2).
Perr1:X=12erfc(|αθ2|2),Perr2:Y=12erfc(|αθ|2).
Perr1:n=0,n0=12exp(2ηpηd|αθ|2);Perr2:n=0,n0=12exp(ηpηd2|αθ|2).
θ2|α|21,σ2|α|21,σ21,

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