Abstract

Quantum phase estimation (QPE) is the key procedure in various quantum algorithms. The main aim of the QPE scheme is to estimate the phase of an unknown eigenvalue, corresponding to an eigenstate of an arbitrary unitary operation. The QPE scheme can be applied as a subroutine to design many quantum algorithms. In this paper, we propose the basic structure of a QPE scheme that could be applied in quantum algorithms, with feasibility by utilizing cross-Kerr nonlinearities (controlled-unitary gates) and linearly optical devices. Subsequently, we analyze the efficiency and the performance of the controlled-unitary gate. This gate consists of a controlled-path gate and a merging-path gate via cross-Kerr nonlinearities under the decoherence effect. Also shown in this paper is a method by which to enhance robustness against the decoherence effect to provide a reliable QPE scheme based on controlled-unitary gates.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (7)

T. E. O’Brien, B. Tarasinski, and B. M. Terhal, “Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments,” New J. Phys. 21(2), 023022 (2019).
[Crossref]

W. C. Peng, B. N. Wang, F. Hu, Y. J. Wang, X. J. Fang, X. Y. Chen, and C. Wang, “Factoring larger integers with fewer qubits via quantum annealing with optimized parameters,” Sci. China: Phys., Mech. Astron. 62(6), 60311 (2019).
[Crossref]

X. M. Xiu, X. Geng, S. L. Wang, C. Cui, Q. Y. Li, Y. Q. Ji, and L. Dong, “Construction of a Polarization Multiphoton Controlled One-Photon Unitary Gate Assisted by the Spatial and Temporal Degrees of Freedom,” Adv. Quantum Technol. 2(9), 1900066 (2019).
[Crossref]

M. S. Kang, J. Heo, S. G. Choi, S. Moon, and S. W. Han, “Implementation of SWAP test for two unknown states in photons via cross-Kerr nonlinearities under decoherence effect,” Sci. Rep. 9(1), 6167 (2019).
[Crossref]

W. C. Gao, C. Cao, X. F. Liu, T. J. Wang, and C. Wang, “Implementation of a two-dimensional quantum walk using cross-Kerr nonlinearity,” OSA Continuum 2(5), 1667 (2019).
[Crossref]

F. F. Du, Y. T. Liu, Z. R. Shi, Y. X. Liang, J. Tang, and J. Liu, “Efficient hyperentanglement purification for three-photon systems with the fidelity-robust quantum gates and hyperentanglement link,” Opt. Express 27(19), 27046 (2019).
[Crossref]

J. Heo, K. Won, H. J. Yang, J. P. Hong, and S. G. Choi, “Photonic scheme of discrete quantum Fourier transform for quantum algorithms via quantum dots,” Sci. Rep. 9(1), 12440 (2019).
[Crossref]

2018 (8)

C. H. Hong, J. Heo, M. S. Kang, J. Jang, and H. J. Yang, “Optical scheme for generating hyperentanglement having photonic qubit and time-bin via quantum dot and cross-Kerr nonlinearity,” Sci. Rep. 8(1), 2566 (2018).
[Crossref]

J. Heo, M. S. Kang, C. H. Hong, J. P. Hong, and S. G. Choi, “Preparation of quantum information encoded on three-photon decoherence-free states via cross-Kerr nonlinearities,” Sci. Rep. 8(1), 13843 (2018).
[Crossref]

L. Dong, S. L. Wang, C. Cui, X. Geng, Q. Y. Li, H. K. Dong, X. M. Xiu, and Y. J. Gao, “Polarization Toffoli gate assisted by multiple degrees of freedom,” Opt. Lett. 43(19), 4635 (2018).
[Crossref]

L. Zhang and K. W. C. Chan, “Scalable Generation of Multi-mode NOON States for Quantum Multiple-phase Estimation,” Sci. Rep. 8(1), 11440 (2018).
[Crossref]

J. I. Colless, V. V. Ramasesh, D. Dahlen, M. S. Blok, M. E. Kimchi-Schwartz, J. R. McClean, J. Carter, W. A. de Jong, and I. Siddiqi, “Computation of Molecular Spectra on a Quantum Processor with an Error-Resilient Algorithm,” Phys. Rev. X 8, 011021 (2018).
[Crossref]

R. Santagati, J. Wang, A. A. Gentile, S. Paesani, N. Wiebe, J. R. McClean, S. Morley-Short, P. J. Shadbolt, D. Bonneau, J. W. Silverstone, D. P. Tew, X. Zhou, J. L. O’Brien, and M. G. Thompson, “Witnessing eigenstates for quantum simulation of Hamiltonian spectra,” Sci. Adv. 4(1), eaap9646 (2018).
[Crossref]

J. Preskill, “Quantum Computing in the NISQ era and beyond,” Quantum 2, 79 (2018).
[Crossref]

S. Danilin, A. V. Lebedev, A. Vepsäläinen, G. B. Lesovik, G. Blatter, and G. S. Paraoanu, “Quantum-enhanced magnetometry by phase estimation algorithms with a single artificial atom,” npj Quantum. Inf. 4(1), 29 (2018).
[Crossref]

2017 (5)

S. Paesani, A. A. Gentile, R. Santagati, J. Wang, N. Wiebe, D. P. Tew, J. L. O’Brien, and M. G. Thompson, “Experimental Bayesian Quantum Phase Estimation on a Silicon Photonic Chip,” Phys. Rev. Lett. 118(10), 100503 (2017).
[Crossref]

D. N. Gonçalves, T. D. Fernandes, and C. M. M. Cosme, “An efficient quantum algorithm for the hidden subgroup problem over some non-abelian groups,” Tend. Mat. Apl. Comput. 18(2), 0215 (2017).
[Crossref]

J. Heo, C. H. Hong, H. J. Yang, J. P. Hong, and S. G. Choi, “Analysis of optical parity gates of generating Bell state for quantum information and secure quantum communication via weak cross-Kerr nonlinearity under decoherence effect,” Quantum Inf. Process. 16(1), 10 (2017).
[Crossref]

J. Heo, M. S. Kang, C. H. Hong, H. J. Yang, S. G. Choi, and J. P. Hong, “Distribution of hybrid entanglement and hyperentanglement with time-bin for secure quantum channel under noise via weak cross-Kerr nonlinearity,” Sci. Rep. 7(1), 10208 (2017).
[Crossref]

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2016 (7)

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2015 (3)

S. Kimmel, G. H. Low, and T. J. Yoder, “Robust calibration of a universal single-qubit gate set via robust phase estimation,” Phys. Rev. A 92(6), 062315 (2015).
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Y. Xia, M. Lu, J. Song, P. M. Lu, and H. S. Song, “Effective protocol for preparation of four-photon polarization-entangled decoherence-free states with cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 30(2), 421 (2013).
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L. Dong, X. M. Xiu, H. Z. Shen, Y. J. Gao, and X. X. Yi, “Quantum Fourier transform of polarization photons mediated by weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 30(10), 2765 (2013).
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2012 (4)

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2010 (5)

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Q. Lin, B. He, J. A. Bergou, and Y. Ren, “Processing multiphoton states through operation on a single photon: Methods and applications,” Phys. Rev. A 80(4), 042311 (2009).
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2008 (4)

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2007 (5)

M. Dobšíček, G. Johansson, V. Shumeiko, and G. Wendin, “Arbitrary accuracy iterative quantum phase estimation algorithm using a single ancillary qubit: A two-qubit benchmark,” Phys. Rev. A 76(3), 030306 (2007).
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S. D. Barrett and G. J. Milburn, “Quantum-information processing via a lossy bus,” Phys. Rev. A 74(6), 060302 (2006).
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2004 (3)

M. Mosca and C. Zalka, “Exact quantum Fourier transforms and discrete logarithm algorithms,” Int. J. Quantum Inf. 02(01), 91–100 (2004).
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M. Scully and M. Zubairy, “Cavity QED implementation of the discrete quantum Fourier transform,” Phys. Rev. A 65(5), 052324 (2002).
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2000 (2)

L. M. K. Vandersypen, M. Steffen, G. Breyta, C. S. Yannoni, R. Cleve, and I. L. Chuang, “Experimental Realization of an Order-Finding Algorithm with an NMR Quantum Computer,” Phys. Rev. Lett. 85(25), 5452–5455 (2000).
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S. Y. Song, “Quantum Computing for Discrete Logarithms,” Quantum Computational Number Theory. Springer, Cham, 121 (2015).

M Michele and A. Ekert, “The hidden subgroup problem and eigenvalue estimation on a quantum computer,” Quantum Computing and Quantum Communications. Springer, Berlin, Heidelberg, 174 (1999).

P. W. Shor, “Algorithms for quantum computation: Discrete logarithms and factoring,” in Proceedings of 35th Annual Symposium on Foundations of Computer Science (IEEE, 1994).

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000).

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Figures (7)

Fig. 1.
Fig. 1. Plot presents the theoretical circuit to implement the operation of QPE in order to estimate the phase, $0.\textrm{ }{\varphi _1}\textrm{ }{\varphi _2} \cdot{\cdot} \cdot{\cdot} \textrm{ }{\varphi _{j - 1}}\textrm{ }{\varphi _j}$. This circuit consists of controlled-unitary (${\textrm{U}^{\textrm{ }{2^{\textrm{ }j - 1}}}}\textrm{, }{\textrm{U}^{\textrm{ }{2^{\textrm{ }j - 2}}}}\textrm{, }\textrm{. }\textrm{. }\textrm{. , }{\textrm{U}^{\textrm{ }{2^{\textrm{ }0}}}}$) operations, which are performed between a qubit ($1,\textrm{ }2, \cdot{\cdot} \cdot{\cdot} ,\textrm{ }j$: control) and the eigenstate ($|u \rangle$: target) in sequence. Then an operation of an inverse discrete quantum Fourier transform (IDQFT) [43,6166] on j qubits is performed for estimation of the phase (QPE algorithm).
Fig. 2.
Fig. 2. Controlled-unitary gate: For implementation of the controlled-unitary operation in Fig. 1, this controlled-unitary gate employs two (CP and MP) gates via XKNLs and an arbitrary unitary operator, $\textrm{U}$, with linearly optical devices (PBSs, BSs, and feed-forwards: PS, path switches). In the CP gate, the paths of two systems (photonic state and eigenstate) are conditionally arranged by the interaction of XKNLs. Then the arbitrary unitary operator $\textrm{U}$, is performed on path 2 of the eigenstate. Finally, the MP gate can operate to merge two paths into a single path of eigenstate. Moreover, the proposed controlled-unitary gate can implement various controlled-unitary (${\textrm{U}^{{2^{j - 1}}}}\textrm{, }{\textrm{U}^{{2^{j - 2}}}}\textrm{, }\textrm{. }\textrm{. }\textrm{. , }{\textrm{U}^{{2^{0}}}}$) operations in the QPE algorithm by the alternation of arbitrary unitary operators between the CP gate and MP gate.
Fig. 3.
Fig. 3. Theoretical circuit of a three-qubit QPE based on three controlled-unitary (${\Omega ^{{2^2}}}\textrm{, }{\Omega ^{{2^1}}}\textrm{, }{\Omega ^{{2^0}}}$) operations can be implemented in a three-photon QPE scheme. It consists of three controlled-unitary gates via XKNLs, qubus beams, and PNR measurements. This controlled-unitary gate (shown in Fig. 2) employs the three components as a CP gate, an arbitrary unitary operation, and a MP gate.
Fig. 4.
Fig. 4. Graph represents the recalculated error probabilities, $\textrm{P}_{\textrm{err}}^{\textrm{C - P}}$ and $\textrm{P}_{\textrm{err}}^{\textrm{M - P}}$, and the rates of remaining photons, $\Lambda _t^4$ and $\Lambda _t^2$, in the CP gate and MP gate for $\alpha \theta = 2.5$, with optical fibers having signal losses of $0.364\textrm{ dB/km }({\chi /\gamma = 0.0125} )$ and $0.15\textrm{ dB/km }({\chi /\gamma = 0.0303} )$. The values of the error probabilities and the rates of remaining photons are listed in the lower table for the difference in amplitude of coherent states with $\alpha \theta = 2.5$.
Fig. 5.
Fig. 5. Plots of the coherent parameters (off-diagonal terms) in output states ${\rho _2}$ and ${\rho _5}$ of the two gates using XKNLs for the amplitude of the probe beam ($\alpha$) and the $\chi /\gamma$ rate, due to the signal loss of optical fibers, where $N = {10^3}$ with the fixed parameter $\alpha \theta = 2.5$.
Fig. 6.
Fig. 6. This plot presents the theoretical circuit of IDQFT on j qubits in the QPE algorithm in Fig. 1. This circuit consists of controlled-rotation k [CRk ($k = 2,\textrm{ }.\textrm{ }.\textrm{ }.\textrm{ },\textrm{ }j$)] operations and Hadamard operations for IDQFT on j qubits.
Fig. 7.
Fig. 7. Theoretical circuit of two-qubit IDQFT, based on a CR2 operation and two Hadamard operations.

Tables (2)

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Table 1. When the fixed parameters are $\alpha \theta = \alpha \chi t = 2.5$ and $N = {10^3}$ in optical fibers having signal loss rates of $\chi /\gamma = 0.0125$ and $\chi /\gamma = 0.0303$, the values of fidelity (${\textrm{F}_{\textrm{CP}}}$, ${\textrm{F}_{\textrm{MP}}}$) and error probabilities ($\textrm{P}_{\textrm{err}}^{\textrm{C - P}}$, $\textrm{P}_{\textrm{err}}^{\textrm{M - P}}$), which can be calculated from Eqs. (19) and (16), are summarized for the amplitude of the coherent state ($\alpha = 10,\textrm{ }50,\textrm{ 1}{\textrm{0}^2}\textrm{, 1}{\textrm{0}^3}\textrm{, 1}{\textrm{0}^4}\textrm{, and 1}{\textrm{0}^5}$).

Tables Icon

Table 2. In a two-qubit IDQFT scheme, the relative phases (${e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }2}}} )}}$ and ${e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }1}}\textrm{ }{m_{\textrm{ }2}}} )}}$) in the output state are written in the table, due to the initial binary numbers, ${m_{\textrm{ }1}}$ and ${m_{\textrm{ }2}}$. Then, according to this table, the unknown phase ($0.{m_{\textrm{ }1}}{m_{\textrm{ }2}}$) of the eigenvalue can be estimated from the results of measurement of the output state (two qubits).

Equations (25)

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U | u = e 2 π i ( φ ) | u ,     0 φ < 1
φ     0.   φ 1   φ 2   φ j 1   φ j ,   0.   φ 1     φ j   φ 1 × 2   1 +   φ 2 × 2   2 +   +   φ j × 2   j
1  st U 2 j 1 1 2 | + 1 | + j 1 ( | 0 j + e 2 π i ( 0. φ j ) | 1 j ) | u , 2 nd U 2 j 2 1 2 2 | + 1 ( | 0 j + e 2 π i ( 0. φ j ) | 1 j ) ( | 0 j 1 + e 2 π i ( 0. φ j 1 φ j ) | 1 j 1 ) | u , j 1  th U 2 1 1 2 j 1 ( | 0 j + e 2 π i ( 0. φ j ) | 1 j ) | + 1 ( | 0 2 + e 2 π i ( 0. φ 2 φ j 1 φ j ) | 1 2 ) | u , j  th U 2 0 1 2 j ( | 0 j + e 2 π i ( 0. φ j ) | 1 j ) ( | 0 j 1 + e 2 π i ( 0. φ j 1 φ j ) | 1 j 1 ) ( | 0 2 + e 2 π i ( 0. φ 2 φ j 1 φ j ) | 1 2 ) ( | 0 1 + e 2 π i ( 0. φ 1 φ 2 φ j 1 φ j ) | 1 1 ) | u ,
U 2 j 1 [ | + j | u ] = U 2 j 2 [ U ( | + j | u ) ] U 2 j 2 [ 1 2 ( | 0 j | u + e 2 π i ( 0. φ 1 φ j ) | 1 j | u ) ] 1 2 ( | 0 j | u + e 2 π i ( 2 j 1 ) ( 0. φ 1 φ j ) | 1 j | u ) = 1 2 ( | 0 j + e 2 π i ( 0. φ j ) | 1 j ) | u ,
1 2   j ( | 0 j + e 2 π i   ( 0.   φ   j ) | 1 j ) ( | 0 1 + e 2 π i   ( 0.   φ 1         φ   j ) | 1 1 ) | u   IDQFT   | φ 1 1 | φ   j j | u ,
| ψ in   PBS , BSs   | ψ 1 = 1 2 ( x 1 | H 1 | u 1 + x 1 | H 1 | u 2 + x 2 | V 2 | u 1 + x 2 | V 2 | u 2 ) | α / 2 P a | α / 2 P b ,
| ψ 2 = 1 2 [   {   x 1 | H 1 | u 1 + x 2 | V 1 | u 2 } | α P a | 0 P b     +   e   ( α sin θ ) 2 2 n = 0 ( i α sin θ ) n n ! {   x 1 | H 1 | u 2 + ( 1 ) n x 2 | V 1 | u 1 } | α cos θ P a | n P b ] ,
| ψ 3 =   x 1 | H 1 | u 1 + x 2 | V 1 | u 2 .
| ψ 3   U   | ψ 4 =   x 1 | H 1 | u 1 + x 2 | V 1 U | u 2 .
| ψ 5   =   1 2 [ {   x 1 | H 1 | u 1 + x 2 | V 1 U | u 1 } | α P a | 0 P b       e   ( α sin θ ) 2 2 n = 0 ( i α sin θ ) n n ! {   x 1 | H 1 | u 2 + x 2 | V 1 U | u 2 } | α cos θ P a | n P b ] .
| ψ fin   Feed forward:  S 2   | ψ fin   =   x 1 | H 1 | u 1 + x 2 | V 1 U | u 1 =   ( x 1 | H 1 + x 2 e 2 π i   ( 0.   φ 1   φ 2           φ j 1   φ j ) | V 1 ) | u 1 U | u = e 2 π i   ( 0.   φ 1   φ 2           φ j 1   φ j ) | u .
ρ ( t ) t = i [ H K e r r ρ ] + γ ( a ρ a + + 1 2 ( a + a ρ + ρ a + a ) ) ,   J ^ ρ = γ a ρ a + ,   L ^ ρ = γ 2 ( a + a ρ + ρ a + a )
D ~ t ( | α β | ) = exp [ ( 1 e γ t ) { α β + ( | α | 2 + | β | 2 ) / 2 } ] | Λ t α Λ t β | .
( D ~ Δ t X ~ Δ t ) N | 1 p 0 p | | α α | = exp [   α 2 ( 1 e γ Δ t ) n   =   1 N e γ Δ t ( n 1 ) ( 1 e i n Δ θ ) ] × | 1 p 0 p | | Λ t α e i θ Λ t α | ,
| ψ 2 P = 1 2 (   x 1 | H 1 | u 1 + x 2 | V 1 | u 2 ) | Λ t 2 α P a | 0 P b +   1 2 ( x 1 | H 1 | u 2 | Λ t 2 α cos θ P a | i Λ t 2 α sin θ P b +   x 2 | V 1 | u 1 | Λ t 2 α cos θ P a × | i Λ t 2 α sin θ P b ) , | ψ 5 P = 1 2 (   x 1 | H 1 | u 1 + x 2 | V 1 U | u 1 ) | Λ t α P a | 0 P b +   1 2 (   x 1 | H 1 | u 2 + x 2 | V 1 U | u 2 ) | Λ t α cos θ P a | i Λ t α sin θ P b ,
P err C - P 1 2 exp (   Λ t 4 α 2 θ 2 ) = 1 2 exp (   e 2 γ t ( 2.5 ) 2 ) ,  P err M - P 1 2 exp (   Λ t 2 α 2 θ 2 ) = 1 2 exp (   e γ t ( 2.5 ) 2 ) ,
ρ  2 = 1 2 ( | x 1 | 2 | KC | 2 x 1 x 2 | L | 2 | x 1 | 2 | OC | 2 x 1 x 2 | KC | 2 x 1 x 2 | x 2 | 2 | OC | 2 x 1 x 2 | L | 2 | x 2 | 2 | L | 2 | x 1 | 2 | OC | 2 x 1 x 2 | x 1 | 2 | MC | 2 x 1 x 2 | OC | 2 x 1 x 2 | L | 2 | x 2 | 2 | MC | 2 x 1 x 2 | x 2 | 2 ) , ρ 5 = 1 2 ( 1 | C | 2 | C | 2 1 ) ,
C = exp [   α 2 2 ( 1 e γ Δ t ) n   =   1 N e γ Δ t ( n 1 ) ( 1 e i n Δ θ ) ] ,  K = exp [   α 2 2 e γ t ( 1 e γ Δ t ) n   =   1 N e γ Δ t ( n 1 ) ( 1 e i   ( θ     n Δ θ ) ) ] , O = exp [   α 2 2 e γ t ( 1 e γ Δ t ) ( 1 e i θ ) n   =   1 N e γ Δ t ( n 1 ) ] ,  L = exp [   α 2 2 e γ t ( 1 e γ Δ t ) n   =   1 N e γ Δ t ( n 1 ) ( 1 e i n Δ θ ) ] M = exp [   α 2 2 e γ t ( 1 e γ Δ t ) n   =   1 N e γ Δ t ( n 1 ) ( 1 e i   ( θ   +   n Δ θ ) ) ] ,
F CP = 1 2 π 0 2 π | ψ 2 P | ρ 2 | ψ 2 P | d ω = 2 8 π 0 2 π | 3 ( 1 + | L | 2 + | OC | 2 3 + | KC | 2 + | MC | 2 3 ) + cos ( 4 ω ) ( 1 + | L | 2 | OC | 2 | KC | 2 + | MC | 2 2 ) | d ω , F MP = | ψ 5 P | ρ 5 | ψ 5 P | = 1 2 | 1 + | OC | 2 | ,
| ψ in   U CRk   | ψ out = c 1 | 0 B | 0 A + c 2 | 1 B | 0 A + c 3 | 0 B | 1 A + c 4 e 2 π i   / 2 k | 1 B | 1 A .
| ψ in = 1 2   2 ( | 0 2 + e 2 π i   ( 0.   m   2 ) | 1 2 ) ( | 0 1 + e 2 π i   ( 0.   m   1   m   2 ) | 1 1 ) ,
| ψ in   H 1 2   2 ( | + 2 + e 2 π i   ( 0.   m   2 ) | 2 ) ( | 0 1 + e 2 π i   ( 0.   m   1   m   2 ) | 1 1 )   CR2   | ψ 1 = 1 2   3 [   ( 1 + e 2 π i   ( 0.   m   2 ) ) | 0 2 ( | 0 1 + e 2 π i   ( 0.   m   1   m   2 ) | 1 1 )   +   ( 1 e 2 π i   ( 0.   m   2 ) ) | 1 2 ( | 0 1 + e 2 π i   ( 0.   0   1 ) e 2 π i   ( 0.   m   1   m   2 ) | 1 1 ) ] = 1 2   3 [ ( 1 + e 2 π i   ( 0.   m   2 ) ) | 0 2 ( | 0 1 + e 2 π i   ( 0.   m   1   m   2 ) | 1 1 ) + ( 1 e 2 π i   ( 0.   m   2 ) ) | 1 2 ( | 0 1 i e 2 π i   ( 0.   m   1   m   2 ) | 1 1 ) ] ,
H | 0     1 2 ( | 0 + e 2 π i   ( 0.   0 ) | 1 )   =   | +   ,  H | 1     1 2 ( | 0 + e 2 π i   ( 0.   1 ) | 1 )   =   | ,
| ψ in (01) = 1 2   2 ( | 0 2 + e 2 π i   ( 0.   1 ) | 1 2 ) ( | 0 1 + e 2 π i   ( 0.   0   1 ) | 1 1 ) = 1 2   2 ( | 0 2 | 1 2 ) ( | 0 1 + i | 1 1 ) .
| ψ in (01)   two - qubit IDQFT   | 0 1 | 1 2   ( =   | m   1 1 | m   2 2 ) .

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