Photonic scheme of quantum phase estimation for quantum algorithms via cross-Kerr nonlinearities under decoherence effect

Changho Hong; Jino Heo; Min-Sung Kang; Min-Sung Kang; Jingak Jang; Hyun-Jin Yang; Daesung Kwon

doi:10.1364/OE.27.031023

1. Introduction

Quantum phase estimation (QPE) [1–14] plays a significant role, or is directly applicable, to accomplishing the quantum information processing needed in various quantum algorithms needed for such as factoring problems [15–20], simulation of molecular properties [21–25], discrete logarithm problems [5,15,26,27], and hidden subgroup problems [28–30]. Therefore, the implementation of a QPE scheme is a critical subroutine for quantum algorithms and quantum computations. After proposal of the iterative phase estimation algorithm by [1], other researchers [3–6] proposed adaptive versions that utilize proof-of-principle realization of QPE. Moreover, these iterative sequential forms of QPE algorithms employed controlled-unitary gates with a single ancillary qubit [7,10] or direct measurement on a register system [11].

However, when QPE algorithms have been experimentally implemented in quantum devices, these are exposed to the currently noisy intermediate scale of quantum computing [31] and are vulnerable to experimental noise [6,24,32,33]. Moreover, the long coherence (strongly coherence-limited) of quantum devices is required for reliable performance of QPE algorithms in quantum circuits [6,24,32,33]. This means that conventional QPE circuits may be impractical in the absence of full error correction or error mitigation.

For reliable performance of QPE circuit (scheme) with high efficiency, the designed controlled-unitary gates using cross-Kerr nonlinearities (XKNLs) are good candidates to implement practical QPE algorithm. In the Kerr medium, the interaction of XKNLs between photons (quantum systems) and probe beams (coherent states) can generate a conditional phase shift (controlled-phase shift) on probe beams to accomplish a quantum non-demolition measurement, which is the indirect measurement of quantum systems, for the realization of multi-qubit controlled operation. Thus, many quantum information processing schemes have employed XKNL interactions, such as quantum controlled gates [34–39], quantum computation [40–45], quantum entanglement [46–50], experiment of delayed-choice [51], preparation of quantum state [49,50,52], and entanglement concentration and purification [47,48,53–55]. However, in the Kerr medium, the controlled-unitary gates or quantum information schemes using XKNLs cannot avoid photon loss and dephasing of coherent parameters by the decoherence effect. In optical fibers, these affections occur to evolve a pure quantum state to a mixed state that gives rise to inefficiency and unreliable performance) [37,39,40,43,46,47,56]. Recently (and fortunately), by applying photon-number-resolving (PNR) measurements [37,39,40,43,46,47,57–60], and a displacement operator [40,43,46] or quantum bus (qubus) beams [37,39,47,49,50] with increasing (stronger) amplitude of the coherent state (probe beam), the decoherence effect (photon loss and dephasing) can be made arbitrarily small [37,39,40,43,46,47,56].

In this paper, we propose a controlled-unitary gate that consists of a controlled-path (CP) gate and a merging-path (MP) gate using XKNLs. The proposed controlled-unitary gate can be directly utilized to implement the circuit of the QPE scheme for the QPE algorithm. Moreover, we can consider the controlled-unitary gate as the basic module of a multi-qubit QPE scheme for scalability. When controlled-unitary gates having different unitary operators (i.e., ${\textrm{U}^{{2^{j - 1}}}}\textrm{, }{\textrm{U}^{{2^{j - 2}}}}\textrm{, }\textrm{. }\textrm{. }\textrm{. , }{\textrm{U}^{{2^{0}}}}$) are arranged simply, we can obtain expansion of the multi-qubit circuit for the QPE scheme. Subsequently, through our analysis by master equation, we demonstrate using a strong coherent state, the condition needed to reduce the decoherence effect (photon loss and dephasing) in our controlled-unitary gate using XKNLs, qubus beams, and PNR measurement. Consequently, a QPE scheme, controlled-unitary gates via XKNLs, can be experimentally implemented with scalability and also could be made robust against the decoherence effect.

2. Theoretical circuit of a quantum phase estimation algorithm

The procedure of quantum phase estimation [1–14] is not a complete quantum algorithm in its own right. QPE should be considered as a kind of subroutine, which can be utilized to perform quantum computational task. Thus, for various quantum algorithms [15–30], the specific goal of QPE algorithm (or structure) [1–14] is to estimate the phase, $\varphi$ (unknown value), of an eigenvalue, ${e^{2\pi i(\varphi )}}$, from available black box when the relation is given as

(1)$$\textrm{U}|u \rangle = {e^{2\pi i(\varphi )}}|u \rangle ,\textrm{ }\because \textrm{ }0 \le \varphi < 1$$

where an arbitrary unitary operator $\textrm{U}$ has the eigenvalue, ${e^{2\pi i\textrm{ }(\varphi )}}$, corresponding to an eigenstate $|u \rangle$. Using binary representation, the phase, $\varphi$, can be written as

(2)$$\varphi \textrm{ } \to \textrm{ }0.\textrm{ }{\varphi _1}\textrm{ }{\varphi _2} \cdot{\cdot} \cdot{\cdot} \textrm{ }{\varphi _{j - 1}}\textrm{ }{\varphi _j},\quad\because \textrm{ }0.\textrm{ }{\varphi _1}\textrm{ } \cdot{\cdot} \cdot{\cdot} \textrm{ }{\varphi _j} \equiv \textrm{ }{\varphi _1} \times {2^{\textrm{ } - 1}} + \textrm{ }{\varphi _2} \times {2^{\textrm{ } - 2}} + \textrm{ } \cdot{\cdot} \cdot{\cdot} + \textrm{ }{\varphi _j} \times {2^{\textrm{ } - j}}$$

where ${\varphi _n} \in \{{0,\textrm{ }1} \}$. To show this goal, the estimation of unknown phase (${\varphi _1}\textrm{, }{\varphi _2}\textrm{, }\textrm{. }\textrm{. }\textrm{. , }{\varphi _{j - 1}}\textrm{, }{\varphi _j}$: binary numeral), and the basic procedure of QPE, we introduce the theoretical circuit of QPE (the standard model) in Fig. 1, which is described in [5,6,9–11].

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) [43,61–66] on j qubits is performed for estimation of the phase (QPE algorithm).

Download Full Size | PDF

And then, based on this structure, we design the critical components of QPE structure using nonlinearly optical resource (cross-Kerr nonlinearities: XKNLs) with feasibility and scalability. In Fig. 1, for the estimation of unknown phase (${\varphi _1}\textrm{, }{\varphi _2}\textrm{, }\textrm{. }\textrm{. }\textrm{. , }{\varphi _{j - 1}}\textrm{, }{\varphi _j}$: binary numeral), we first prepare the input state as ${|+ \rangle _1} \otimes {|+ \rangle _2} \otimes{\cdot} \cdot{\cdot} \cdot{\otimes} {|+ \rangle _j} \otimes |u \rangle$ (j qubits and an eigenstate of $\textrm{U}$), where $|\pm \rangle \equiv ({|0 \rangle \pm |1 \rangle } )/\sqrt 2$. The 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 are applied sequentially to two systems of a qubit (control) and the eigenstate (target), as follows:

(3)$$\begin{array}{l} \boxed{{1\textrm{ st}}}\mathop \to \limits^{{\textrm{U}^{{2^{j - 1}}}}} \frac{1}{{\sqrt 2 }}{\left|+ \right\rangle _1} \otimes{\cdot} \cdot{\cdot} \cdot{\otimes} {\left|+ \right\rangle _{j - 1}} \otimes \left({{{|0 \rangle }_j} + {e^{2\pi i({0.{\varphi_{j}}} )}}{{|1 \rangle }_j}} \right)\otimes |u \rangle ,\\ \boxed{{\textrm{2 nd}}}\mathop \to \limits^{{\textrm{U}^{{2^{j - 2}}}}} \frac{1}{{\sqrt {{2^2}} }}{|+ \rangle _1} \otimes{\cdot} \cdot{\cdot} \cdot{\otimes} \left({{{|0 \rangle }_j} + {e^{2\pi i({0.{\varphi_{j}}} )}}{{|1 \rangle }_j}} \right)\otimes \left({{{|0 \rangle }_{j - 1}} + {e^{2\pi i({0.{\varphi_{j - 1}}{\varphi_{j}}} )}}{{|1 \rangle }_{j - 1}}} \right)\otimes |u \rangle ,\\ \vdots \vdots \\ \boxed{{j - 1\textrm{ th}}}\mathop \to \limits^{{\textrm{U}^{{2^1}}}} \frac{1}{{\sqrt {{2^{j - 1}}} }}\left({{{|0 \rangle }_j} + {e^{2\pi i({0.{\varphi_{j}}} )}}{{|1 \rangle }_j}} \right)\otimes{\cdot} \cdot{\cdot} \cdot{\otimes} {|+ \rangle _1} \otimes \left({{{|0 \rangle }_2} + {e^{2\pi i({0.{\varphi_2} \cdot \cdot \cdot {\varphi_{j - 1}}{\varphi_{j}}} )}}{{|1 \rangle }_2}} \right)\otimes |u \rangle ,\\ \boxed{{j\textrm{ th}}}\mathop \to \limits^{{\textrm{U}^{{2^0}}}} \frac{1}{{\sqrt {{2^{j}}} }}\left({{{|0 \rangle }_j} + {e^{2\pi i({0.{\varphi_{j}}} )}}{{|1 \rangle }_j}} \right)\otimes \left({{{|0 \rangle }_{j - 1}} + {e^{2\pi i({0.{\varphi_{j - 1}}{\varphi_{j}}} )}}{{|1 \rangle }_{j - 1}}} \right)\otimes{\cdot} \cdot{\cdot} \cdot \\ \qquad\qquad\qquad \cdot{\cdot} \cdot{\cdot} \otimes \left({{{|0 \rangle }_2} + {e^{2\pi i({0.{\varphi_2} \cdot \cdot \cdot {\varphi_{j - 1}}{\varphi_{j}}} )}}{{|1 \rangle }_2}} \right)\otimes \left({{{|0 \rangle }_1} + {e^{2\pi i({0.{\varphi_1}{\varphi_2} \cdot \cdot \cdot {\varphi_{j - 1}}{\varphi_{j}}} )}}{{|1 \rangle }_1}} \right)\otimes |u \rangle , \end{array}$$

where the controlled-unitary operation between the qubit and the eigenstate is represented as in Fig. 1. For example, after the controlled-unitary (${\textrm{U}^{\textrm{ }{2^{\textrm{ }j - 1}}}}$) operation in the first step, the calculation for the output state of the qubit, ${|+ \rangle _j}$, and the eigenstate, $|u \rangle$, is given by

(4)$$\begin{array}{l} {\textrm{U}^{{2^{j - 1}}}}\left[{{{|+ \rangle }_j} \otimes |u \rangle } \right] = {\textrm{U}^{{2^{j - 2}}}}\left[{\textrm{U}\left({{{|+ \rangle }_j} \otimes |u \rangle } \right)} \right] \to {\textrm{U}^{{2^{j - 2}}}}\left[ {\frac{1}{{\sqrt 2 }}\left({{{|0 \rangle }_j} \otimes |u \rangle + {e^{2\pi i({0.{\varphi_1} \cdot \cdot \cdot {\varphi_{j}}} )}}{{|1 \rangle }_j} \otimes |u \rangle } \right)} \right]\\ \to \frac{1}{{\sqrt 2 }}\left({{{|0 \rangle }_j} \otimes |u \rangle + {e^{2\pi i({{2^{j - 1}}} )({0.{\varphi_1} \cdot \cdot \cdot {\varphi_{j}}} )}}{{|1 \rangle }_j} \otimes |u \rangle } \right) = \frac{1}{{\sqrt 2 }}\left({{{|0 \rangle }_j} + {e^{2\pi i({0.{\varphi_{j}}} )}}{{|1 \rangle }_j}} \right)\otimes |u \rangle , \end{array}$$

where $\textrm{U}|u \rangle = {e^{2\pi i\textrm{ }({0.\textrm{ }{\varphi_1} \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ }{\varphi_{\textrm{ }j}}} )}}|u \rangle$ and ${e^{2\pi i\textrm{ }({{\varphi_1} \times {2^{\textrm{ }j - 2}}} )}} ={\cdot}{\cdot} \cdot{\cdot} = {e^{2\pi i\textrm{ }({{\varphi_{j - 1}} \times {2^{\textrm{ }0}}} )}} = 1$ for ${\varphi _n} \in \{{0,\textrm{ }1} \}$. After the operation of IDQFT (on j qubits) into the state of the post- j th step, Eq. (3), as shown in Fig. 1, the final state will be transformed as

(5)$$\frac{1}{{\sqrt {{2^{\textrm{ }j}}} }}({{{|0 \rangle }_j} + {e^{2\pi i\textrm{ }({0.\textrm{ }{\varphi_{\textrm{ }j}}} )}}{{|1 \rangle }_j}} )\otimes{\cdot} \cdot{\cdot} \cdot{\otimes} ({{{|0 \rangle }_1} + {e^{2\pi i\textrm{ }({0.\textrm{ }{\varphi_1}\textrm{ } \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ }{\varphi_{\textrm{ }j}}} )}}{{|1 \rangle }_1}} )\otimes |u \rangle \textrm{ }\mathop \to \limits^{\textrm{IDQFT}} \textrm{ }{|{{\varphi_1}} \rangle _1} \otimes{\cdot} \cdot{\cdot} \cdot{\otimes} {|{{\varphi_{\textrm{ }j}}} \rangle _j} \otimes |u \rangle ,$$

where the detailed explanation of IDQFT [43,61–66] is in the APPENDIX. Consequently, when the single-qubit measurement on j qubits $({{{|{{\varphi_1}} \rangle }_1} \otimes {{|{{\varphi_2}} \rangle }_2} \otimes{\cdot} \cdot{\cdot} \cdot{\otimes} {{|{{\varphi_{j - 1}}} \rangle }_{j - 1}} \otimes {{|{{\varphi_{j}}} \rangle }_j}} )$ is performed, we can estimate the unknown phase (${\varphi _1}\textrm{, }{\varphi _2}\textrm{, }\textrm{. }\textrm{. }\textrm{. , }{\varphi _{j - 1}}\textrm{, }{\varphi _j}$: binary numeral) of the eigenvalue ${e^{2\pi i({0.{\varphi_1}{\varphi_2} \cdot \cdot \cdot \cdot {\varphi_{j - 1}}{\varphi_j}} )}}$ corresponding to the eigenstate $|u \rangle$ of the arbitrary unitary operator $\textrm{U}$. In the theoretical circuit of QPE, the main components are the controlled-unitary operations, which perform the unitary operations (${\textrm{U}^{{2^{j - 1}}}}\textrm{, }{\textrm{U}^{{2^{j - 2}}}}\textrm{, }\textrm{. }\textrm{. }\textrm{. , }{\textrm{U}^{{2^{0}}}}$) onto the target (eigenstate: $|u \rangle$) in accordance with the control qubit ($|0 \rangle$: no operation, and $|1 \rangle$: operation). Consequently, to experimentally implement controlled-unitary operations with feasibility is a critical issue for high efficiency and reliable performance of QPE algorithm having the scalability of a multi-qubit QPE scheme.

3. Design of controlled-unitary gate via XKNLs for the QPE algorithm

In this section, we propose a design for a controlled-unitary gate that consists of a controlled-path (CP) gate and a merging-path (MP) gate using XKNLs, qubus (coherent) beams, and PNR measurements, for a controlled-unitary operation to be applied to the optical QPE scheme.

Before the proposition for implementing a controlled-unitary gate, we introduce the XKNL effect between the photon (${|n \rangle _1}$: photon-number state) and coherent state (${|\alpha \rangle _\textrm{P}} = {e^{ - {{|\alpha |}^2}/2}}$ $\sum\nolimits_{n = 0}^\infty {\left( {{\alpha^n}/\sqrt {n!} } \right)|n \rangle }$: probe system). In the Kerr medium, XKNL’s Hamiltonian [34–44,46–50,56–60] is given as ${H_{Kerr}} = \hbar \chi {N_1}{N_\textrm{p}}$, where ${N_i}$ and $\chi$ are the photon-number operator and the strength of nonlinearity. Thus, we can calculate the interaction of XKNL, the photon-coherent state, as ${\textrm{U}_{Kerr}}{|n \rangle _1}{|\alpha \rangle _\textrm{2}} = {e^{i({{H_{Kerr}}} )t/\hbar }}{|n \rangle _1}{|\alpha \rangle _\textrm{2}} = {e^{i\theta {N_1}{N_\textrm{2}}}}{|n \rangle _1}{|\alpha \rangle _\textrm{2}} = {|n \rangle _1}{|{\alpha {e^{in\theta }}} \rangle _\textrm{2}}$, where t is the interaction time, and $\theta ({ = \chi t} )$ is the magnitude of the conditional phase shift.

Figure 2 shows the optical scheme of controlled-unitary (CP and MP) gate with an arbitrary unitary operator $\textrm{U}$, using XKNLs to implement the controlled-unitary operations for the theoretical circuit of the QPE algorithm as shown in Fig. 1.

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.

Download Full Size | PDF

Here, we assume the definition of binary representation in the linear-polarization of the photon is $\{{|H \rangle ,|V \rangle } \}\equiv \{{|0 \rangle ,|1 \rangle } \}$ ($|H \rangle$: horizontal and $|V \rangle$: vertical). Moreover, we define the eigenstate $|u \rangle$, which corresponds to the eigenvalue of ${e^{2\pi i\textrm{ }({0.\textrm{ }{\varphi_1}\textrm{ }{\varphi_2}\textrm{ } \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ }{\varphi_{j - 1}}\textrm{ }{\varphi_j}} )}}$ with the unknown phase (${\varphi _1}\textrm{, }{\varphi _2}\textrm{, }\textrm{. }\textrm{. }\textrm{. , }{\varphi _{j - 1}}\textrm{, }{\varphi _j}$: binary numeral), of the arbitrary unitary operator $\textrm{U}$ as $|u \rangle \equiv \alpha |H \rangle + \beta |V \rangle$ for ${|\alpha |^2} + {|\beta |^2} = 1$. Also we suppose an input state of $|{{\psi_{\textrm{in}}}} \rangle = ({{\textrm{x}_1}{{|H \rangle }^1} + {\textrm{x}_2}{{|V \rangle }^1}} )\otimes {|u \rangle ^1}$ for description of the operation of a controlled-unitary gate.

In the CP gate, as described in Fig. 2, after the input state, $|{{\psi_{\textrm{in}}}} \rangle$, and qubus beam, ${|\alpha \rangle _\textrm{P}}$, pass through a PBS (photon) and BSs (eigenstate and coherent state), the input state $|{{\psi_{\textrm{in}}}} \rangle$, is transformed to

(6)$$\begin{aligned}|{{\psi_{\textrm{in}}}} \rangle \textrm{ }\mathop \to \limits^{\textrm{PBS , BSs}} \textrm{ }|{{\psi_\textrm{1}}} \rangle &= \frac{1}{{\sqrt 2 }}({{\textrm{x}_1}{{|H \rangle }^1} \otimes {{|u \rangle }^1} + {\textrm{x}_1}{{|H \rangle }^1} \otimes {{|u \rangle }^2} + {\textrm{x}_2}{{|V \rangle }^2} \otimes {{|u \rangle }^1} + {\textrm{x}_2}{{|V \rangle }^2} \otimes {{|u \rangle }^2}} )\\ & \quad \otimes \left|{\alpha /\sqrt 2 } \right\rangle _\textrm{P}^\textrm{a}\left|{\alpha /\sqrt 2 } \right\rangle _\textrm{P}^\textrm{b},\end{aligned}$$

where the operations of PBS and BS are as illustrated in Fig. 2. Then, the interactions of the XKNLs are applied to the state $|{{\psi_\textrm{1}}} \rangle$ (photon-eigenstate-coherent state). After these operations and a BS (on qubus), the state $|{{\psi_\textrm{2}}} \rangle$, (pre-measurement: PNR) is given by

(7)$$\begin{array}{l} |{{\psi_\textrm{2}}} \rangle = \frac{1}{{\sqrt 2 }}[{\textrm{ }\{{\textrm{ }{\textrm{x}_1}{{|H \rangle }^1} \otimes {{|u \rangle }^1} + {\textrm{x}_2}{{|V \rangle }^1} \otimes {{|u \rangle }^2}} \}\otimes |\alpha \rangle_\textrm{P}^\textrm{a}|0 \rangle_\textrm{P}^\textrm{b}} \\ \textrm{ }\left. {\textrm{ } + \textrm{ }{e^{ - \textrm{ }\frac{{{{({\alpha \sin \theta } )}^2}}}{2}}}\sum\limits_{n = 0}^\infty {\frac{{{{({i\alpha \sin \theta } )}^n}}}{{\sqrt {n!} }}\{{\textrm{ }{\textrm{x}_1}{{|H \rangle }^1} \otimes {{|u \rangle }^2} + {{({ - 1} )}^n}{\textrm{x}_2}{{|V \rangle }^1} \otimes {{|u \rangle }^1}} \}\otimes |{\alpha \cos \theta } \rangle_\textrm{P}^\textrm{a}|n \rangle_\textrm{P}^\textrm{b}} } \right], \end{array}$$

where $|{ \pm i\alpha \sin \theta } \rangle = {e^{ - \textrm{ }\frac{{{{({\alpha \sin \theta } )}^2}}}{2}}}\sum\nolimits_{n = 0}^\infty {\left[ {{{({ \pm i\alpha \sin \theta } )}^n}/\sqrt {n!} } \right]|n \rangle }$ for $\alpha \in {\mathbb R}$. When the PNR measurement is applied to path b of the coherent state (qubus beam), if the result is in $|0 \rangle _\textrm{P}^\textrm{b}$ (dark detection), the output state $|{{\psi_\textrm{3}}} \rangle$, of the CP gate can be obtained to provide

(8)$$|{{\psi_\textrm{3}}} \rangle = \textrm{ }{\textrm{x}_1}{|H \rangle ^1} \otimes {|u \rangle ^1} + {\textrm{x}_2}{|V \rangle ^1} \otimes {|u \rangle ^2}.$$

Moreover, even if the result is in $|n \rangle _\textrm{P}^\textrm{b}\textrm{ }({n \ne 0} )$, another output state [${\textrm{x}_1}{|H \rangle ^1} \otimes {|u \rangle ^2} + {({ - 1} )^n}{\textrm{x}_2}{|V \rangle ^1} \otimes {|u \rangle ^1}$ from Eq. (7)] can be transformed to the state $|{{\psi_\textrm{3}}} \rangle$ in Eq. (8), by feed-forward (a PS, ${{\boldsymbol {\varphi} }^n}$, and a path switch, ${\textrm{S}_\textrm{1}}$), as described in Fig. 2. Thus, we can acquire the output state of CP gate as the state $|{{\psi_\textrm{3}}} \rangle$, which has divided paths (1 and 2) of eigenstate with regard to the polarizations (H and V) of photon.

Subsequently, in Fig. 2, the arbitrary unitary operator $\textrm{U} = {\textrm{U}^{{2^0}}}$, is applied to path 2 of the eigenstate, as follows:

(9)$$|{{\psi_\textrm{3}}} \rangle \textrm{ }\mathop \to \limits^\textrm{U} \textrm{ }|{{\psi_\textrm{4}}} \rangle = \textrm{ }{\textrm{x}_1}{|H \rangle ^1} \otimes {|u \rangle ^1} + {\textrm{x}_2}{|V \rangle ^1} \otimes \textrm{U}{|u \rangle ^2}.$$

In the MP gate, as described in Fig. 2, after operation (a BS and the interactions of XKNLs) of the MP gate in the state $|{{\psi_\textrm{4}}} \rangle$ in Eq. (9), the state $|{{\psi_\textrm{5}}} \rangle$, (pre-measurement: PNR) will be expressed as

(10)$$\begin{array}{l} |{{\psi_\textrm{5}}} \rangle \textrm{ } = \textrm{ }\frac{{ - 1}}{{\sqrt 2 }}[{\{{\textrm{ }{\textrm{x}_1}{{|H \rangle }^1} \otimes {{|u \rangle }^1} + {\textrm{x}_2}{{|V \rangle }^1} \otimes \textrm{U}{{|u \rangle }^1}} \}\otimes |\alpha \rangle_\textrm{P}^\textrm{a}|0 \rangle_\textrm{P}^\textrm{b}} \\ \textrm{ }\left. {\textrm{ } - \textrm{ }{e^{ - \textrm{ }\frac{{{{({\alpha \sin \theta } )}^2}}}{2}}}\sum\limits_{n = 0}^\infty {\frac{{{{({i\alpha \sin \theta } )}^n}}}{{\sqrt {n!} }}\{{\textrm{ }{\textrm{x}_1}{{|H \rangle }^1} \otimes {{|u \rangle }^2} + {\textrm{x}_2}{{|V \rangle }^1} \otimes \textrm{U}{{|u \rangle }^2}} \}\otimes |{\alpha \cos \theta } \rangle_\textrm{P}^\textrm{a}|n \rangle_\textrm{P}^\textrm{b}} } \right]. \end{array}$$

Then, the PNR measurement is performed on path b of the qubus beam (coherent state). If the outcome is in $|0 \rangle _\textrm{P}^\textrm{b}$ (dark detection), we can achieve the final state $|{{\psi_{\textrm{fin}}}} \rangle$, as $|{{\psi_{\textrm{fin}}}} \rangle = {\textrm{x}_1}{|H \rangle ^1} \otimes {|u \rangle ^1} + {\textrm{x}_2}{|V \rangle ^1} \otimes \textrm{U}{|u \rangle ^1}$ without feed-forward. Otherwise [$|n \rangle _\textrm{P}^\textrm{b}\textrm{ }({n \ne 0} )$], the state $|{\psi_{\textrm{fin}}^ \ast } \rangle$, will collapse to $|{\psi_{\textrm{fin}}^ \ast } \rangle = {\textrm{x}_1}{|H \rangle ^1} \otimes {|u \rangle ^2} + {\textrm{x}_2}{|V \rangle ^1} \otimes \textrm{U}{|u \rangle ^2}$. Then, by feed-forward (a path switch, ${\textrm{S}_\textrm{2}}$), the state $|{\psi_{\textrm{fin}}^ \ast } \rangle$, can be transformed to the final state $|{{\psi_{\textrm{fin}}}} \rangle$, as follows:

(11)$$\begin{aligned}|{\psi_{\textrm{fin}}^ \ast } \rangle \textrm{ }\mathop \to \limits^{\textrm{Feed} - \textrm{forward: }{\textrm{S}_\textrm{2}}} \textrm{ }|{{\psi_{\textrm{fin}}}} \rangle \textrm{ } &= \textrm{ }{\textrm{x}_1}{|H \rangle ^1} \otimes {|u \rangle ^1} + {\textrm{x}_2}{|V \rangle ^1} \otimes \textrm{U}{|u \rangle ^1}\\ & = \textrm{ }({{\textrm{x}_1}{{|H \rangle }^1} + {\textrm{x}_2}{e^{2\pi i\textrm{ }({0.\textrm{ }{\varphi_1}\textrm{ }{\varphi_2}\textrm{ } \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ }{\varphi_{j - 1}}\textrm{ }{\varphi_j}} )}}{{|V \rangle }^1}} )\otimes {|u \rangle ^1}\\ \because \textrm{U}|u \rangle &= {e^{2\pi i\textrm{ }({0.\textrm{ }{\varphi_1}\textrm{ }{\varphi_2}\textrm{ } \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ }{\varphi_{j - 1}}\textrm{ }{\varphi_j}} )}}|u \rangle. \end{aligned}$$

Consequently, the final state, $|{{\psi_{\textrm{fin}}}} \rangle$ in Eq. (11), which is applied to a controlled-unitary gate consisting of a CP gate and a MP gate via XKNLs, is the same as the final state of the controlled-unitary operation (of the j th step) in Fig. 1.

Furthermore, we exhibit the implementation of a three-photon QPE scheme using the controlled-unitary gates (as the basic modules) in Fig. 2 to demonstrate the scalability and experimental realization of the theoretical circuit of a three-qubit QPE algorithm. Figure 3 shows a simple example by which we can implement the theoretical circuit of a three-qubit QPE algorithm into a three-photon QPE scheme utilizing controlled-unitary (${\Omega ^{{2^2}}}\textrm{, }{\Omega ^{{2^1}}}\textrm{, }{\Omega ^{{2^0}}}$) gates via the configuration of our controlled-unitary gates: a CP gate, an arbitrary unitary operation, and a MP gate, as shown in Fig. 3. Then, we can obtain the unknown phase (${\phi _1}\textrm{, }{\phi _2}\textrm{, }{\phi _3}$: binary numeral) of eigenvalue, ${e^{2\pi i\textrm{ }({0.\textrm{ }{\phi_1}\textrm{ }{\phi_2}\textrm{ }{\phi_3}} )}}$, of the unitary operator $\Omega $, from the measurement outcomes on photon 1, 2, and 3 after the operation of IDQFT (on three photons), as described in Fig. 1, in the state of the third step. Therefore, by the arrangement of controlled-unitary gates having various unitary operators, we can expand to a multi-photon QPE scheme for scalability of the optical QPE scheme.

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.

Download Full Size | PDF

In this section, we propose a design of the controlled-unitary gate (in Fig. 2), using XKNLs for the feasible implementation of the QPE algorithm, and show a method for the expansion (see Fig. 3) of the multi-photon QPE scheme for scalability. Our controlled-unitary gate (CP gate and MP gate) via XKNLs is the critical component in the optical QPE scheme. However, when the controlled-unitary gate is experimentally realized for the optical QPE scheme, the decoherence effect (photon loss and dephasing), which can induce the degradation of a quantum pure state to a mixed state, occurs in nonlinearly optical (CP and MP) gates. For high efficiency and reliable performance of the optical QPE scheme, we will analyze the affection of decoherence effect (photon loss and dephasing) by master equation [37,39,40,45,46,49,50,61–63], and also demonstrate a method by which to reduce the decoherence effect in nonlinearly optical (CP and MP) gates, in the next section.

4. Analysis of the decoherence effect in a controlled-path gate and merging-path gate using XKNLs

In our controlled-unitary gate, the nonlinearly optical (CP and MP) gates, which utilize the interactions of XKNLs (conditional phase shift), qubus beams (coherent state), and PNR measurements, are the main elements needed to perform the controlled-unitary operations for the optical QPE scheme. When CP gate and MP gate are experimentally realized in optical fibers [67,68], the photon loss in the qubus beams and the dephasing coherent parameters in the photon-probe system due to the decoherence effect, inevitably affect the efficiency (lowered success probability) and performance (decreased fidelity) [40,46,56,69] of a controlled-unitary gate. Therefore, according to analysis [37,39,40,46,47,50] of the decoherence effect using the master equation, we should obtain a method by which to reduce the affections (photon loss and dephasing) of the decoherence effect for the nonlinearly optical (CP and MP) gates to provide high efficiency and reliable performance.

For analysis [37,39,40,46,47,50] of the decoherence effect in the Kerr medium, the solution of the master equation [70], which can describe the open quantum system, is given by

(12)$$\frac{{\partial \rho (t )}}{{\partial t}} ={-} \frac{i}{\hbar }[{{H_{Kerr}}\textrm{, }\rho } ]+ \gamma \left( {a\rho {a^ + } + \frac{1}{2}({{a^ + }a\rho + \rho {a^ + }a} )} \right),\textrm{ }\because \hat{J}\rho = \gamma a\rho {a^ + },\textrm{ }\hat{L}\rho ={-} \frac{\gamma }{2}({{a^ + }a\rho + \rho {a^ + }a} )$$

where $\gamma$ and ${a^ + }$ ($a$) are the energy decay rate, and the creation (annihilation) operator. The solution of Eq. (12) can be written as $\rho (t )= \exp [{({\hat{J} + \hat{L}} )t} ]\rho (0 )$ where $t\textrm{ }({ = \theta /\chi } )$ is the interaction time. Here, we represent the process model [37,39,40,46,47,50], based on solution of the master equation, of the interaction of XKNL including the decoherence effect, to analyze the efficiency and performance of the CP gate and MP gate (in Sec. 3). When the probe beam, $|\alpha \rangle$ (coherent state), passes a nonlinear (Kerr) medium, the probe beam is given as $|{{\Lambda _t}\alpha } \rangle$ from photon loss, where ${\Lambda _t} = {e^{ - \gamma t/2}}$ (the rate of remaining photons) for the time t. This affection of the decoherence effect, ${\tilde{D}_t}$, in the probe beam, $|\alpha \rangle \left\langle \beta \right|$, can be expressed as

(13)$${\tilde{D}_t}\left( {|\alpha \rangle \left\langle \beta \right|} \right) = \exp [{ - ({1 - {e^{ - \gamma t}}} )\{{ - \alpha {\beta^ \ast } + ({{{|\alpha |}^2} + {{|\beta |}^2}} )/2} \}} ]|{{\Lambda _t}\alpha } \rangle \left\langle {{\Lambda _t}\beta } \right|.$$

In addition, the interaction of XKNL (conditional phase shift: ${\tilde{X}_t}$) between a photon and the probe beam (coherent state) is induced by the Hamiltonian ${H_{Kerr}}$ in the Kerr medium. Thus, this process (decoherence, ${\tilde{D}_t}$, and XKNL, ${\tilde{X}_t}$) can be modeled from Eq. (12) and ${\textrm{U}_{Kerr}}$, as follows:

(14)$$\begin{aligned}{({{{\tilde{D}}_{\Delta t}}{{\tilde{X}}_{\Delta t}}} )^N}|{{1_\textrm{p}}} \rangle \left\langle {{0_\textrm{p}}} \right|\otimes |\alpha \rangle \left\langle \alpha \right|&= \exp \left[ { - \textrm{ }{\alpha^2}({1 - {e^{ - \gamma \Delta t}}} )\sum\limits_{n\textrm{ } = \textrm{ }1}^N {{e^{ - \gamma \Delta t({n - 1} )}}} ({1 - {e^{in\Delta \theta }}} )} \right]\\ & \quad \times |{{1_\textrm{p}}} \rangle \left\langle {{0_\textrm{p}}} \right|\otimes |{{\Lambda _t}\alpha {e^{i\theta }}} \rangle \left\langle {{\Lambda _t}\alpha } \right|,\end{aligned}$$

where the total interaction time t is divided into the arbitrarily small, $\Delta t$, ($t = N\Delta t$) for a good approximation of this process model, and ${\tilde{D}_t}{\tilde{X}_t} = {({{{\tilde{D}}_{\Delta t}}{{\tilde{X}}_{\Delta t}}} )^N}$ for time $t = N\Delta t$. In this process, we also assumed the interaction of XKNL, ${\tilde{X}_t}$, as ${\tilde{X}_{\Delta t}}\left( {|{{1_\textrm{p}}} \rangle \left\langle {{0_\textrm{p}}} \right|\otimes |\alpha \rangle \left\langle \alpha \right|} \right) \to \textrm{ }|{{1_\textrm{p}}} \rangle \left\langle {{0_\textrm{p}}} \right|\otimes |{\alpha {e^{i\Delta \theta }}} \rangle \left\langle \alpha \right|$ (${1_\textrm{p}}$: a single photon and ${0_\textrm{p}}$: no photon) for time $\Delta t$ ($t = N\Delta t$) with $\theta = \chi t = \chi N\Delta t = N\Delta \theta$ and $\alpha \in {\mathbb R}$. In Eq. (14), we consider that a coherent parameter is a coefficient, $\exp \left[ { - \textrm{ }{\alpha^2}({1 - {e^{ - \gamma \Delta t}}} )\sum\nolimits_{n\textrm{ } = \textrm{ }1}^N {{e^{ - \gamma \Delta t({n - 1} )}}({1 - {e^{in\Delta \theta }}} )} } \right]$, to quantify the magnitude of dephasing by the decoherence effect. Moreover, the features of an optical fiber should also be considered when we analyze the nonlinearly optical gates (CP and MP) under the decoherence effect. It is known that an optical fiber of about 3000 km is needed for phase shift $\theta = \pi$ of the XKNL [67,68]. Thus, for analysis of the efficiency and performance of nonlinearly optical gates, based on the process model [Eqs. (12) and (14)], we take commercial fibers [67,68] with a signal loss of $0.364\textrm{ dB/km }({\chi /\gamma = 0.0125} )$, and pure silica core fibers [68] with a signal loss of $0.15\textrm{ dB/km }({\chi /\gamma = 0.0303} )$, to represent current technology.

[Analysis of efficiency]: In the ideal case (without considering the decoherence effect in Sec. 3) of the CP gate and MP gate, the error probabilities ($\textrm{P}_{\textrm{err}}^{\textrm{C - I}}$: CP gate and $\textrm{P}_{\textrm{err}}^{\textrm{M - I}}$: MP gate) can be respectively calculated from the probabilities to measure $|0 \rangle _\textrm{P}^\textrm{b}$ (dark detection) in $|{ \pm i\alpha \sin \theta } \rangle _\textrm{P}^\textrm{b}$ on path b of the qubus beams [Eqs. (7) and (10)], as follows: $\textrm{P}_{\textrm{err}}^{\textrm{C - I}} = \textrm{P}_{\textrm{err}}^{\textrm{M - I}} = [{\exp ({ - \textrm{ }{\alpha^2}{{\sin }^2}\theta } )} ]\textrm{/2} {\approx} [{\exp ({ - \textrm{ }{\alpha^2}{\theta^2}} )} ]\textrm{/2}$ for ${\alpha ^2}{\sin ^2}\theta \approx {\alpha ^2}{\theta ^2}$ with $\alpha > > 1$ and $\theta < < 1$. When the parameters ($\alpha$: amplitude of the coherent state and $\theta$: magnitude of the conditional phase shift) are fixed as $\alpha \theta = 2.5$, the error probabilities ($\textrm{P}_{\textrm{err}}^{\textrm{C - I}}$ and $\textrm{P}_{\textrm{err}}^{\textrm{M - I}}$) can be acquired ($\textrm{P}_{\textrm{err}}^{\textrm{C - I}} = \textrm{P}_{\textrm{err}}^{\textrm{M - I}} < {10^{ - 3}}$). Moreover, if we increase the amplitude of the coherent state or magnitude of the conditional phase shift in this fixed parameter ($\alpha \theta = 2.5$), the error probabilities ($\textrm{P}_{\textrm{err}}^{\textrm{C - I}}$ and $\textrm{P}_{\textrm{err}}^{\textrm{M - I}}$) can approach zero. However, in a practical case, we should consider the photon loss, Eq. (13), of the decoherence effect (without dephasing). When photon loss occurs in the probe beam as $|{{\Lambda _t}\alpha } \rangle \textrm{ }({{\Lambda _t} = {e^{ - \gamma t/2}}} )$, the output states [$|{{\psi_\textrm{2}}} \rangle$ of Eq. (7) and $|{{\psi_\textrm{5}}} \rangle$ of Eq. (10)], pre-measurement of the CP gate and MP gate, should be modified to include photon loss, as follows:

(15)$$\begin{aligned} |{\psi_2^\textrm{P}} \rangle &= \frac{1}{{\sqrt 2 }}({\textrm{ }{\textrm{x}_1}{{|H \rangle }^1} \otimes {{|u \rangle }^1} + {\textrm{x}_2}{{|V \rangle }^1} \otimes {{|u \rangle }^2}} )\otimes |{\Lambda _t^2\alpha } \rangle _\textrm{P}^\textrm{a}|0 \rangle _\textrm{P}^\textrm{b}\\ &\quad+ \textrm{ }\frac{1}{{\sqrt 2 }}({\textrm{x}_1}{{|H \rangle }^1} \otimes {{|u \rangle }^2} \otimes |{\Lambda _t^2\alpha \cos \theta } \rangle_\textrm{P}^\textrm{a}|{i\Lambda _t^2\alpha \sin \theta } \rangle_\textrm{P}^\textrm{b} + \textrm{ }{\textrm{x}_2}{{|V \rangle }^1} \otimes {{|u \rangle }^1} \otimes |{\Lambda _t^2\alpha \cos \theta } \rangle_\textrm{P}^\textrm{a}\\ & \quad \times |{ - i\Lambda _t^2\alpha \sin \theta } \rangle_\textrm{P}^\textrm{b} ),\\ |{\psi_5^\textrm{P}} \rangle &= \frac{{ - 1}}{{\sqrt 2 }}({\textrm{ }{\textrm{x}_1}{{|H \rangle }^1} \otimes {{|u \rangle }^1} + {\textrm{x}_2}{{|V \rangle }^1} \otimes \textrm{U}{{|u \rangle }^1}} )\otimes |{{\Lambda _t}\alpha } \rangle _\textrm{P}^\textrm{a}|0 \rangle _\textrm{P}^\textrm{b}\\ &\quad + \textrm{ }\frac{1}{{\sqrt 2 }}({\textrm{ }{\textrm{x}_1}{{|H \rangle }^1} \otimes {{|u \rangle }^2} + {\textrm{x}_2}{{|V \rangle }^1} \otimes \textrm{U}{{|u \rangle }^2}} )\otimes |{{\Lambda _t}\alpha \cos \theta } \rangle _\textrm{P}^\textrm{a}|{i{\Lambda _t}\alpha \sin \theta } \rangle _\textrm{P}^\textrm{b}, \end{aligned}$$

where ${\Lambda _t} = {e^{ - \gamma t/2}}$ is the rate of remaining photons. Then, we can recalculate two error probabilities, $\textrm{P}_{\textrm{err}}^{\textrm{C - P}}$ and $\textrm{P}_{\textrm{err}}^{\textrm{M - P}}$, (with photon loss) of the CP gate and MP gate with the fixed parameter ($\alpha \theta = \alpha \chi t = 2.5$), as follows:

(16)$$\begin{aligned}\textrm{P}_{\textrm{err}}^{\textrm{C - P}} &\approx \frac{1}{2}\exp ({ - \textrm{ }\Lambda _t^4{\alpha^2}{\theta^2}} )= \frac{1}{2}\exp ({ - \textrm{ }{e^{ - 2\gamma t}} \cdot {{({2.5} )}^2}} ),\\ \textrm{ P}_{\textrm{err}}^{\textrm{M - P}} & \approx \frac{1}{2}\exp ({ - \textrm{ }\Lambda _t^2{\alpha^2}{\theta^2}} )= \frac{1}{2}\exp ({ - \textrm{ }{e^{ - \gamma t}} \cdot {{({2.5} )}^2}} ),\end{aligned}$$

where these (CP and MP) gates are operated in optical fibers having signal loss of $0.364\textrm{ dB/km }({\chi /\gamma = 0.0125} )$ [67] and $0.15\textrm{ dB/km }({\chi /\gamma = 0.0303} )$ [68] with $\chi t = 2.5/\alpha$ ($\alpha \theta = \alpha \chi t = 2.5$). Figure 4 shows 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 in the probe beams of the CP gate and MP gate in the optical fibers, due to the signal loss rates under the decoherence effect. As described in Fig. 4, when the amplitude of the coherent state (probe beam: $\alpha$) increases, the error probabilities ($\textrm{P}_{\textrm{err}}^{\textrm{C - P}}$ and $\textrm{P}_{\textrm{err}}^{\textrm{M - P}}$) can be decreased, and also the rates ($\Lambda _t^4$ and $\Lambda _t^2$) of remaining photons can approach 1 with the fixed parameter ($\alpha \theta = \alpha \chi t = 2.5$). Moreover, the values of the rates of remaining photons and the error probabilities, according to the signal loss rates ($\chi /\gamma = 0.0125\textrm{ and }0.0303$) of optical fibers and the amplitude of coherent states ($\alpha = 10,\textrm{ }1000,\textrm{ and }100000$), are listed in the table of Fig. 4. Consequently, through our analysis [considering only photon loss in Eq. (13)], the values in the table of Fig. 4, obviously show that two nonlinearly optical gates (CP and MP) can acquire high efficiencies, $\textrm{P}_{\textrm{err}}^{\textrm{C - P}} \approx \textrm{P}_{\textrm{err}}^{\textrm{M - P}} < {10^{ - 3}}$, and the high rates of remaining photons, $\Lambda _t^4 \approx \Lambda _t^2 \to 1$, with fixed $\alpha \theta = \alpha \chi t = 2.5$ in optical fibers, when we employ a strong probe beam (increasing the amplitude of the coherent state, $\alpha > > 10$) under the decoherence effect.

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 $\alpha \theta = 2.5$.

Download Full Size | PDF

[Analysis of performance]: In practice, we should analyze the dephasing coherent parameters to quantify the performance of CP gate and MP gate, which are operated in optical fibers [67,68], besides photon loss [for efficiency in Eq. (16)] under the decoherence effect.

For this analysis (dephasing), the output states [$|{\psi_2^\textrm{P}} \rangle$ and $|{\psi_5^\textrm{P}} \rangle$ in Eq. (15)] of the CP gate and MP gate can be written in the forms of density matrices (${\rho _2}$ and ${\rho _5}$) with consideration of the affection of dephasing, as follows:

(17)$$\begin{aligned}{\rho _{\textrm{ 2}}} &= \frac{\textrm{1}}{\textrm{2}}\left( {\begin{array}{cccc} {{{|{{\textrm{x}_1}} |}^2}}&{{{|{\textrm{KC}} |}^\textrm{2}} \cdot {\textrm{x}_1}\textrm{x}_2^ \ast }&{{{|\textrm{L} |}^\textrm{2}} \cdot {{|{{\textrm{x}_1}} |}^2}}&{{{|{\textrm{OC}} |}^\textrm{2}} \cdot {\textrm{x}_1}\textrm{x}_2^ \ast }\\ {{{|{\textrm{KC}} |}^\textrm{2}} \cdot \textrm{x}_1^ \ast {\textrm{x}_2}}&{{{|{{\textrm{x}_2}} |}^2}}&{{{|{\textrm{OC}} |}^\textrm{2}} \cdot \textrm{x}_1^ \ast {\textrm{x}_2}}&{{{|\textrm{L} |}^\textrm{2}} \cdot {{|{{\textrm{x}_2}} |}^2}}\\ {{{|\textrm{L} |}^\textrm{2}} \cdot {{|{{\textrm{x}_1}} |}^2}}&{{{|{\textrm{OC}} |}^\textrm{2}} \cdot {\textrm{x}_1}\textrm{x}_2^ \ast }&{{{|{{\textrm{x}_1}} |}^2}}&{{{|{\textrm{MC}} |}^\textrm{2}} \cdot {\textrm{x}_1}\textrm{x}_2^ \ast }\\ {{{|{\textrm{OC}} |}^\textrm{2}} \cdot \textrm{x}_1^ \ast {\textrm{x}_2}}&{{{|\textrm{L} |}^\textrm{2}} \cdot {{|{{\textrm{x}_2}} |}^2}}&{{{|{\textrm{MC}} |}^\textrm{2}} \cdot \textrm{x}_1^ \ast {\textrm{x}_2}}&{{{|{{\textrm{x}_2}} |}^2}} \end{array}} \right)\textrm{,}\\ {\rho _{5}} &= \frac{\textrm{1}}{\textrm{2}}\left( {\begin{array}{cc} 1&{ - {{|\textrm{C} |}^\textrm{2}}}\\ { - {{|\textrm{C} |}^\textrm{2}}}&1 \end{array}} \right)\textrm{,}\end{aligned}$$

where the basis states of density matrices (${\rho _2}$ and ${\rho _5}$: photon-probe system) are defined as ${|H \rangle ^1}{|u \rangle ^1}|{\Lambda _t^2\alpha } \rangle _\textrm{P}^\textrm{a}|0 \rangle _\textrm{P}^\textrm{b}$, ${|V \rangle ^1}{|u \rangle ^2}|{\Lambda _t^2\alpha } \rangle _\textrm{P}^\textrm{a}|0 \rangle _\textrm{P}^\textrm{b}$, ${|H \rangle ^1}{|u \rangle ^2}|{\Lambda _t^2\alpha \cos \theta } \rangle _\textrm{P}^\textrm{a}|{i\Lambda _t^2\alpha \sin \theta } \rangle _\textrm{P}^\textrm{b}$, and ${|V \rangle ^1}{|u \rangle ^1}|{\Lambda _t^2\alpha \cos \theta } \rangle _\textrm{P}^\textrm{a}|{ - i\Lambda _t^2\alpha \sin \theta } \rangle _\textrm{P}^\textrm{b}$ (${\rho _2}$), and also are defined as $({{\textrm{x}_1}{{|H \rangle }^1}{{|u \rangle }^1} + {\textrm{x}_2}{{|V \rangle }^1}}$ $\textrm{U}{{|u \rangle }^1} )|{{\Lambda _t}\alpha } \rangle _\textrm{P}^\textrm{a}|0 \rangle _\textrm{P}^\textrm{b}$, and $({{\textrm{x}_1}{{|H \rangle }^1}{{|u \rangle }^2} + {\textrm{x}_2}{{|V \rangle }^1}\textrm{U}{{|u \rangle }^2}} )|{{\Lambda _t}\alpha \cos \theta } \rangle _\textrm{P}^\textrm{a}|{i{\Lambda _t}\alpha \sin \theta } \rangle _\textrm{P}^\textrm{b}$ (${\rho _5}$) from left to right and top to bottom. Here, we can calculate the coherent parameters ($\textrm{C}$,$\textrm{K}$,$\textrm{O}$, $\textrm{L}$, and $\textrm{M}$: off-diagonal terms) from the process model (decoherence, ${\tilde{D}_t}$, and XKNL, ${\tilde{X}_t}$) in Eqs. (12) and (14), as follows:

(18)$$\begin{aligned}\textrm{C} &= \exp \left[ { - \textrm{ }\frac{{{\alpha^2}}}{2}({1 - {e^{ - \gamma \Delta t}}} )\sum\limits_{n\textrm{ } = \textrm{ }1}^N {{e^{ - \gamma \Delta t({n - 1} )}}({1 - {e^{in\Delta \theta }}} )} } \right],\\ \textrm{ K} &= \exp \left[ { - \textrm{ }\frac{{{\alpha^2}}}{2}{e^{ - \gamma t}}({1 - {e^{ - \gamma \Delta t}}} )\sum\limits_{n\textrm{ } = \textrm{ }1}^N {{e^{ - \gamma \Delta t({n - 1} )}}({1 - {e^{i\textrm{ }({\theta \textrm{ } - \textrm{ }n\Delta \theta } )}}} )} } \right],\\ \textrm{O} &= \exp \left[ { - \textrm{ }\frac{{{\alpha^2}}}{2}{e^{ - \gamma t}}({1 - {e^{ - \gamma \Delta t}}} )({1 - {e^{i\theta }}} )\sum\limits_{n\textrm{ } = \textrm{ }1}^N {{e^{ - \gamma \Delta t({n - 1} )}}} } \right],\\ \textrm{ L} &= \exp \left[ { - \textrm{ }\frac{{{\alpha^2}}}{2}{e^{ - \gamma t}}({1 - {e^{ - \gamma \Delta t}}} )\sum\limits_{n\textrm{ } = \textrm{ }1}^N {{e^{ - \gamma \Delta t({n - 1} )}}({1 - {e^{in\Delta \theta }}} )} } \right] \textrm{, }\\ \textrm{M} &= \exp \left[ { - \textrm{ }\frac{{{\alpha^2}}}{2}{e^{ - \gamma t}}({1 - {e^{ - \gamma \Delta t}}} )\sum\limits_{n\textrm{ } = \textrm{ }1}^N {{e^{ - \gamma \Delta t({n - 1} )}}({1 - {e^{i\textrm{ }({\theta \textrm{ } + \textrm{ }n\Delta \theta } )}}} )} } \right], \end{aligned}$$

where ${\tilde{D}_t}{\tilde{X}_t} = {({{{\tilde{D}}_{\Delta t}}{{\tilde{X}}_{\Delta t}}} )^N}$, and $\theta = \chi t = \chi N\Delta t = N\Delta \theta$ for small time $\Delta t$ ($t = N\Delta t$) with $\alpha \in {\mathbb R}$. From the coherent parameters in Eq. (18), it is possible to quantify the amount of dephasing, which can generate the evolution from a pure state to a mixed state, in the output state, ${\rho _2}$ (CP) and ${\rho _5}$ (MP). According to Eq. (18), Fig. 5 shows the distributions of values of coherent parameters (off-diagonal terms) in density matrices (output states) of ${\rho _2}$ (CP) and ${\rho _5}$ (MP), in Eq. (17), for the amplitude of the probe beam ($\alpha$) and $\chi /\gamma$ (the signal loss rate of the optical fibers) with $\alpha \theta = \alpha \chi t = 2.5$ and $N = {10^3}$. For the containment of evolution to a mixed state (the preservation of a pure state) by the decoherence effect, the values of off-diagonal terms (coherent parameters) in the density matrices (output states) of ${\rho _2}$ (CP) and ${\rho _5}$ (MP), in Eq. (17) should be maintained to one (${|{\textrm{KC}} |^\textrm{2}} \approx {|\textrm{L} |^\textrm{2}} \approx {|{\textrm{OC}} |^\textrm{2}} \approx {|{\textrm{MC}} |^\textrm{2}} \approx {|\textrm{C} |^\textrm{2}} \to \textrm{ }\textrm{1}$: reduction of dephasing). As described in Fig. 5, we can confirm that the values of coherent parameters in ${\rho _2}$ and ${\rho _5}$ increase (approaching 1) when the strong amplitude $\alpha$, of the coherent state is utilized for the whole range of $\chi /\gamma$ (the rate of signal loss in optical fibers) with $\alpha \theta = \alpha \chi t = 2.5$ and $N = {10^3}$. Thus, in Fig. 5, if we employ the strong coherent state (probe beam) in our gates (CP and MP), we can reduce the affection of dephasing the coherent parameters under the decoherence effect.

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 $\alpha \theta = 2.5$.

Download Full Size | PDF

Then, from this result, we can quantify the performance of the CP gate and MP gate using fidelities (${\textrm{F}_{\textrm{CP}}}$: CP, and ${\textrm{F}_{\textrm{MP}}}$: MP). Before, we assumed the pure (output) states were $|{\psi_2^\textrm{P}} \rangle$ and $|{\psi_5^\textrm{P}} \rangle$ in Eq. (15), which were modified due only to photon loss of our gates without dephasing. Thus, for the performance including the photon loss and dephasing, we should calculate the fidelities (${\textrm{F}_{\textrm{CP}}}$ and ${\textrm{F}_{\textrm{MP}}}$) between Eq. (15), without dephasing, and (17), with photon loss and dephasing, as follows:

(19)$${\small \begin{aligned}{\textrm{F}_{\textrm{CP}}} &= \frac{1}{{2\pi }}\int\limits_0^{2\pi } {\left|{\sqrt {\left\langle {\psi_2^\textrm{P}} \right|{\rho_2}|{\psi_2^\textrm{P}} \rangle } } \right|d\omega } \\ & = \frac{{\sqrt 2 }}{{8\pi }}\int\limits_0^{2\pi } {\left|{\sqrt {3\left(\! {1\, {+}\, {{|\textrm{L} |}^\textrm{2}}\, {+}\, \frac{{{{|{\textrm{OC}} |}^\textrm{2}}}}{3}\, {+}\, \frac{{{{|{\textrm{KC}} |}^\textrm{2}}\, {+}\, {{|{\textrm{MC}} |}^\textrm{2}}}}{3}} \!\right)\, {+}\, \cos ({4\omega } )\left(\! {1 \, {+}\, {{|\textrm{L} |}^\textrm{2}} \, {-}\, {{|{\textrm{OC}} |}^\textrm{2}} \, {-}\, \frac{{{{|{\textrm{KC}} |}^\textrm{2}} \, {+}\, {{|{\textrm{MC}} |}^\textrm{2}}}}{2}} \!\right)} } \right|d\omega } ,\\ {\textrm{F}_{\textrm{MP}}} &= \left|{\sqrt {\left\langle {\psi_5^\textrm{P}} \right|{\rho_5}|{\psi_5^\textrm{P}} \rangle } } \right| = \frac{1}{{\sqrt 2 }}\left|{\sqrt {1 + {{|{\textrm{OC}} |}^\textrm{2}}} } \right|, \end{aligned}}$$

where the coefficients ${\textrm{x}_1}$ and ${\textrm{x}_2}$ of the photon state are defined as $\textrm{cos}\omega$ and $\sin \omega$ ($\textrm{co}{\textrm{s}^2}\omega + \textrm{si}{\textrm{n}^2}\omega = 1$). In Table 1, according to the difference in the amplitude ($\alpha$) of the coherent state and signal loss rates, we calculated and list the values of the fidelities (${\textrm{F}_{\textrm{CP}}}$ and ${\textrm{F}_{\textrm{MP}}}$) and error probabilities ($\textrm{P}_{\textrm{err}}^{\textrm{C - P}}$: CP and $\textrm{P}_{\textrm{err}}^{\textrm{M - P}}$: MP) from Eqs. (19) and (16) with $\alpha \theta = \alpha \chi t = 2.5$ and $N = {10^3}$. To reduce dephasing of the coherent parameters, we need to figure out two features from Table 1, as follows: (a) the usage of a weak coherent state, $\alpha < 10$, and (b) the usage of a strong coherent state, $\alpha > {10^4}$. In both cases, we can acquire reliable performance (high fidelity: approaching 1) of the CP gate and MP gate against the affection of dephasing induced by the decoherence effect, as listed in Table 1. However, in the case of (a), the error probabilities of the CP gate and MP gate over the whole range of the signal loss rate are high (0.5).

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 ($\alpha = 10,\textrm{ }50,\textrm{ 1}{\textrm{0}^2}\textrm{, 1}{\textrm{0}^3}\textrm{, 1}{\textrm{0}^4}\textrm{, and 1}{\textrm{0}^5}$).

View Table | View all tables in this article

This means that we cannot obtain high efficiency (low success probabilities) while we can reduce the affection of dephasing (reliable performance) using a weak coherent state (i.e., when $\alpha = 10$, $\chi /\gamma = 0.0125$ and $0.0303$, ${\textrm{F}_{\textrm{CP}}} \approx {\textrm{F}_{\textrm{MP}}} \approx 1$, and $\textrm{P}_{\textrm{err}}^{\textrm{C - P}} \approx \textrm{P}_{\textrm{err}}^{\textrm{M - P}} \approx 0.5$ in Table 1). Thus, for high efficiency and reliable performance of the CP gate and MP gate, we should utilize a strong coherent state (probe beam), in case (b), under the decoherence effect (i.e., when $\alpha = {10^5}$, $\chi /\gamma = 0.0125$ and $0.0303$, ${\textrm{F}_{\textrm{CP}}} \approx {\textrm{F}_{\textrm{MP}}} \approx 1$, and $\textrm{P}_{\textrm{err}}^{\textrm{C - P}} \approx \textrm{P}_{\textrm{err}}^{\textrm{M - P}} < {10^{ - 3}}$ in Table 1).

So far, we have demonstrated a method to utilize a strong (increasing amplitude) coherent state for the establishment of high efficiency and reliable performance in our nonlinearly optical gates (CP and MP) under the decoherence effect. This is according to our analysis [the process model, in Eq. (14), based on the master equation, in Eq. (12)]. Furthermore, though the large strength of XKNL is not necessary in our QPE scheme, the recent experimental progresses in XKNL for the sufficient large strength of XKNL have been showed using various physical resources, such as electromagnetically induced transparency (EIT) [71], circuit electromechanics [72], an artificial atom [73], and three-dimensional circuit quantum electrodynamic architecture [74]. And Friedler et al. [75] showed the large nonlinear interactions between ultraslow-light pulses or two stopped light pulses [76] in the regime of EIT to realize the strong phase shift.

Consequently, the proposed QPE scheme, which consists of controlled-unitary gates, can be experimentally feasible with scalability of the multi-qubit QPE scheme because the CP gate and MP gate, which are the main components in a controlled-unitary gate, can obtain high efficiency and reliable performance by employing a strong coherent state under the decoherence effect.

5. Conclusions

In this paper, we proposed to design a controlled-unitary gate (consisting of CP gate and MP gate) using XKNLs, in Sec. 3, for the controlled-unitary operations in a QPE algorithm. The proposed controlled-unitary gate can be directly implemented an optical QPE scheme for a theoretical circuit of the QPE algorithm of Fig. 1. Moreover, we acquired scalability of a multi-photon QPE scheme, in Sec. 3, to arrange the controlled-unitary gates (applied as the basic modules) for the multi-qubit QPE algorithm. Before, the various methods [1,3–7] were proposed to perform the phase estimation using iterative process [1], proof-of-principle, [3–6], and optimal measurement [7] in the only theoretical range. And then, to set up the linearly optical devices as optical interferometers [8,9] and direct measurement [10,11] without QFT, the probabilistic performance of QPE could be acquired in the experimental implementation. Recently, in [12–14], the structure of QPE is embedded on small-scale photonic chip [12,14], or is enhanced by using a single artificial atom [13]. Here, we designed our QPE scheme using controlled-unitary gates (consisting of CP gate and MP gate) via XKNLs, and can obtain the deterministic performance and scalability of QPE, compared with the previous works [1,3–14]. And also, because our scheme based on the standard model with DQFT, the advantage of QPE scheme is to easily apply to various quantum algorithms. For this, therefore, because the most important components in our QPE scheme are the CP gate and MP gate (main elements in a controlled-unitary gate) via XKNLs, qubus beams, and PNR measurements, we emphasize the advantages of these gates, as follows:

(1) QPE algorithm plays the main role as subroutine in various quantum algorithms and quantum computations. Thus, our controlled-unitary gate (CP and MP) can be considered a basic module for constructing QPE scheme with feasibility for the implementation of a QPE algorithm. Moreover, by the simple arrangement of controlled-unitary gates having unitary operators, we obtained scalability for the multi-photon QPE scheme, as described in Sec. 3.
(2) In Sec. 4, by our analysis using the process model, we demonstrated that nonlinearly optical (CP and MP) gates should employ a strong coherent state (probe beam) to achieve high efficiency (low error probability) and reliable performance (high fidelity). In the previous works [35,58,77], which have proposed the various nonlinearly optical gates (including to CP and MP gates), for quantum information processing schemes, the affection of the decoherence effect, in practice, have been overlooked. Compared with these works [35,58,77], we analyzed the decoherence effect by master equation, and derived the method, using strong coherent state, to reduce photon loss and dephasing (decoherence) in Sec. 4. Thus, when our controlled-unitary gate for the optical QPE scheme can be experimentally implemented, it will be robust against the decoherence effect (photon loss and dephasing).
(3) In nature, the magnitude of the conditional phase shift by Kerr media is tiny, $\theta = \chi t \approx {10^{ - 18}}$ [78]. Also, it is difficult to increase the magnitude of a conditional phase shift by electromagnetically induced transparency to $\theta \approx {10^{ - 2}}$ [57,79]. However, when increasing the amplitude of the coherent state with $\alpha \theta = \alpha \chi t = 2.5$ for the high efficiency and reliable performance of our gates (CP and MP), we need only the small magnitude of a conditional phase shift. For example, if $\alpha = {10^5}$ with $\chi /\gamma = 0.0303$ and $\alpha \theta = 2.5$ in the CP gate and MP gate, we can acquire a small magnitude conditional phase shift, $\theta = 2.5 \times {10^{ - 5}}$ (${10^5} \cdot \theta = 2.5$) in addition to the high efficiency ($\textrm{P}_{\textrm{err}}^{\textrm{C - P}} \approx \textrm{P}_{\textrm{err}}^{\textrm{M - P}} < {10^{ - 3}}$) and reliable performance (${\textrm{F}_{\textrm{CP}}} \approx {\textrm{F}_{\textrm{MP}}} \approx 1$) as described in Table 1. Therefore, when we employ the strong coherent state (probe beam), the CP gate and MP gate are feasible for realizing this approach experimentally owing to the small conditional phase shift.
(4) For the optical QPE scheme, the designed controlled-unitary gate employs the strategy of PNR measurement with XKNLs and qubus beams. Thus, we can utilize only positive conditional phase shifts ($\theta$) by XKNL. In Ref. [80–82], to change the sign of the conditional phase shift ($\theta \to - \theta$) is extremely challenging using current technology. Our nonlinearly optical gates (CP and MP) using only positive conditional phase shifts ($\theta$), are more feasible than other nonlinearly optical gates [37,40,46,59,83] that use the negative and positive conditional phase shifts ($- \theta$ and $\theta$).

Consequently, the proposed QPE scheme, which consists of controlled-unitary gates via XKNLs, can be experimentally feasible, with scalability of the multi-qubit QPE scheme as well, because the CP gate and MP gate using the strong coherent state can obtain high efficiency and reliable performance, and the robustness against the decoherence effect.

Appendix

In the QPE algorithm in Fig. 1, the inverse discrete quantum Fourier transform (IDQFT), as in Eq. (5), should be performed to estimate the unknown phase (${\varphi _1}\textrm{, }{\varphi _2}\textrm{, }\textrm{. }\textrm{. }\textrm{. , }{\varphi _{j - 1}}\textrm{, }{\varphi _j}$: binary numeral) of the eigenvalue ${e^{2\pi i\textrm{ }({0.\textrm{ }{\varphi_1}\textrm{ }{\varphi_2}\textrm{ } \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ }{\varphi_{j - 1}}\textrm{ }{\varphi_j}} )}}$ corresponding to the eigenstate $|u \rangle$ of the arbitrary unitary operator $\textrm{U}$, before utilizing the single-qubit measurement on j qubits. Thus, we introduce the theoretical circuit of IDQFT, and the propositions to implement IDQFT schemes in the previous works [43,61–66,84–86]. Figure 6 shows that the theoretical circuit of IDQFT can transform from the input states of the post- j th step in Eq. (3) to ${|{{\varphi_1}} \rangle _1} \otimes{\cdot} \cdot{\cdot} \cdot{\otimes} {|{{\varphi_{\textrm{ }j}}} \rangle _j}$ ($\because {\varphi _n} \in \{{0,\textrm{ }1} \}$) for the IDQFT operation. The main components in IDQFT scheme are controlled-rotation k [CRk ($k = 2,\textrm{ }.\textrm{ }.\textrm{ }.\textrm{ },\textrm{ }j$)] operations, ${\textrm{U}_{\textrm{CRk}}}$, between two qubits. When the input state is in the state $|{{\psi_{\textrm{in}}}} \rangle = {c_1}{|0 \rangle _\textrm{A}}{|0 \rangle _\textrm{B}} + {c_2}{|0 \rangle _\textrm{A}}{|1 \rangle _\textrm{B}} + {c_3}{|1 \rangle _\textrm{A}}{|0 \rangle _\textrm{B}} + {c_4}{|1 \rangle _\textrm{A}}{|1 \rangle _\textrm{B}}$ for $\sum\nolimits_{i = 1}^4 {{{|{{c_i}} |}^2} = 1}$, the CRk operation can transform as

(20)$$|{{\psi_{\textrm{in}}}} \rangle \textrm{ }\mathop \to \limits^{{\textrm{U}_{\textrm{CRk}}}} \textrm{ }|{{\psi_{\textrm{out}}}} \rangle = {c_1}{|0 \rangle _\textrm{B}}{|0 \rangle _\textrm{A}} + {c_2}{|1 \rangle _\textrm{B}}{|0 \rangle _\textrm{A}} + {c_3}{|0 \rangle _\textrm{B}}{|1 \rangle _\textrm{A}} + {c_4}{e^{2\pi i\textrm{ }/{2^k}}}{|1 \rangle _\textrm{B}}{|1 \rangle _\textrm{A}}.$$

According to the CRk operation (red-box in Fig. 6), the paths of qubits (A and B) are swapped, and the operation of rotation k, ${\boldsymbol {R}}_{\textrm{ }k}^{\textrm{ } + } = |0 \rangle \left\langle 0 \right|+ {e^{ - 2\pi i\textrm{ }/{2^k}}}|1 \rangle \left\langle 1 \right|$, is applied to the target qubit, A, when a control qubit B, is in the state ${|1 \rangle _\textrm{B}}$.

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.

Download Full Size | PDF

Thus, we can obtain a theoretical circuit for the IDQFT scheme with scalability by the simple arrangement of CRk ($k = 2,\textrm{ }.\textrm{ }.\textrm{ }.\textrm{ },\textrm{ }j$) operations and Hadamard operations, as described in Fig. 6. Through this scheme (IDQFT in Fig. 6), we can calculate the final state in Eq. (5), which is represented by the product state (${\varphi _1}\textrm{, }{\varphi _2}\textrm{, }\textrm{. }\textrm{. }\textrm{. , }{\varphi _{j - 1}}\textrm{, }{\varphi _j}$: binary numeral) of j qubits, for estimation of the unknown phase of eigenvalue ${e^{2\pi i\textrm{ }({0.\textrm{ }{\varphi_1}\textrm{ }{\varphi_2}\textrm{ } \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ } \cdot \textrm{ }{\varphi_{j - 1}}\textrm{ }{\varphi_j}} )}}$.

In Fig. 7, for the simple example, two-qubit IDQFT is used to show the detailed process of the IDQFT, we assume the initial state, $|{{{\psi }_{\textrm{in}}}} \rangle$, as follows:

(21)$$|{{{\psi }_{\textrm{in}}}} \rangle = \frac{1}{{\sqrt {{2^{\textrm{ }2}}} }}({{{|0 \rangle }_2} + {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }2}}} )}}{{|1 \rangle }_2}} )\otimes ({{{|0 \rangle }_1} + {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }1}}\textrm{ }{m_{\textrm{ }2}}} )}}{{|1 \rangle }_1}} ),$$

where ${m_1}$ and ${m_2}$ are arbitrarily binary numbers for ${m_n} \in \{{0,\textrm{ }1} \}$. After the Hadamard operation (qubit 2) and CR2 operation were applied to state $|{{{\psi }_{\textrm{in}}}} \rangle$, state $|{{{\psi }_\textrm{1}}} \rangle$ is given by

(22)$$\begin{aligned}|{{{\psi }_{\textrm{in}}}} \rangle \textrm{ }\mathop \to \limits^\textrm{H} &\frac{1}{{\sqrt {{2^{\textrm{ }2}}} }}({{{|+ \rangle }_2} + {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }2}}} )}}{{|- \rangle }_2}} )\otimes ({{{|0 \rangle }_1} + {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }1}}\textrm{ }{m_{\textrm{ }2}}} )}}{{|1 \rangle }_1}} )\textrm{ }\mathop \to \limits^{\textrm{CR2}}\\ \to \textrm{ }|{{{\psi }_\textrm{1}}} \rangle &= \frac{1}{{\sqrt {{2^{\textrm{ }3}}} }}[{\textrm{ }({1 + {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }2}}} )}}} ){{|0 \rangle }_2} \otimes ({{{|0 \rangle }_1} + {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }1}}\textrm{ }{m_{\textrm{ }2}}} )}}{{|1 \rangle }_1}} )\textrm{ }} \\& \quad { + \textrm{ }({1 - {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }2}}} )}}} ){{|1 \rangle }_2} \otimes ({{{|0 \rangle }_1} + {e^{ - 2\pi i\textrm{ }({0.\textrm{ }0\textrm{ }1} )}} \cdot {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }1}}\textrm{ }{m_{\textrm{ }2}}} )}}{{|1 \rangle }_1}} )} ]\\ &= \frac{1}{{\sqrt {{2^{\textrm{ }3}}} }}[({1 + {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }2}}} )}}} ){{|0 \rangle }_2} \otimes ({{|0 \rangle }_1}\\ & \quad + {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }1}}\textrm{ }{m_{\textrm{ }2}}} )}}{{|1 \rangle }_1} )+ ({1 - {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }2}}} )}}} ){{|1 \rangle }_2} \otimes ({{{|0 \rangle }_1} - i \cdot {e^{2\pi i\textrm{ }({0.\textrm{ }{m_{\textrm{ }1}}\textrm{ }{m_{\textrm{ }2}}} )}}{{|1 \rangle }_1}} ) ], \end{aligned}$$

where ${e^{ - 2\pi i\textrm{ }({0.\textrm{ }0\textrm{ }1} )}} = {e^{ - 2\pi i\textrm{ }/{2^2}}} ={-} i$ in the rotation 2 operator ${\boldsymbol {R}}_{\textrm{ }2}^{\textrm{ } + }$, as described in Fig. 7. The operations of Hadamard are expressed as

(23)$$\textrm{H}|\textrm{0} \rangle \textrm{ } \to \textrm{ }\frac{1}{{\sqrt 2 }}({|0 \rangle + {e^{ - 2\pi i\textrm{ }({0.\textrm{ }0} )}}|1 \rangle } )\textrm{ } = \textrm{ }|+ \rangle \textrm{ },\textrm{ H}|\textrm{1} \rangle \textrm{ } \to \textrm{ }\frac{1}{{\sqrt 2 }}({|0 \rangle + {e^{ - 2\pi i\textrm{ }({0.\textrm{ }1} )}}|1 \rangle } )\textrm{ } = \textrm{ }|- \rangle ,$$

where ${e^{ - 2\pi i\textrm{ }({0.\textrm{ }0} )}} = 1$ and ${e^{ - 2\pi i\textrm{ }({0.\textrm{ }1} )}} ={-} 1$. Finally, we can obtain the output state (the equation on the right-side, as described in Fig. 7) after the crossing-paths of two qubits and Hadamard operation (qubit 1) is applied to state $|{{{\psi }_\textrm{1}}} \rangle$. In Table 2, we can check to see what the output state has transformed to, according to ${m_{\textrm{ }1}}$ and ${m_{\textrm{ }2}}$ (binary numeral).

Fig. 7. Theoretical circuit of two-qubit IDQFT, based on a CR2 operation and two Hadamard operations.

Download Full Size | PDF

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).

View Table | View all tables in this article

Let us assume that if the binary number (unknown phase: $0.{m_{\textrm{ }1}}{m_{\textrm{ }2}}$) is 0.01 (1/4 = 0.25: decimal), the initial state, $|{{{\psi }_{\textrm{in (01)}}}} \rangle$ (from the post-second step in QPE, Fig. 1), before IDQFT is given by

(24)$$|{{{\psi }_{\textrm{in (01)}}}} \rangle = \frac{1}{{\sqrt {{2^{\textrm{ }2}}} }}({{{|0 \rangle }_2} + {e^{2\pi i\textrm{ }({0.\textrm{ }1} )}}{{|1 \rangle }_2}} )\otimes ({{{|0 \rangle }_1} + {e^{2\pi i\textrm{ }({0.\textrm{ }0\textrm{ }1} )}}{{|1 \rangle }_1}} )= \frac{1}{{\sqrt {{2^{\textrm{ }2}}} }}({{{|0 \rangle }_2} - {{|1 \rangle }_2}} )\otimes ({{{|0 \rangle }_1} + i{{|1 \rangle }_1}} ).$$

After this state, $|{{{\psi }_{\textrm{in (01)}}}} \rangle$, passes through the two-qubit IDQFT scheme in Fig. 7, the output state will transform to

(25)$$|{{{\psi }_{\textrm{in (01)}}}} \rangle \textrm{ }\mathop \to \limits^{\textrm{two - qubit IDQFT}} \textrm{ }{|0 \rangle _1} \otimes {|1 \rangle _2}\textrm{ }({ = \textrm{ }{{|{{m_{\textrm{ }1}}} \rangle }_1} \otimes {{|{{m_{\textrm{ }2}}} \rangle }_2}} ).$$

Then we can confirm this output state from Fig. 7 and Table 2. Consequently, we can estimate the unknown phase ($0.{m_{\textrm{ }1}}{m_{\textrm{ }2}} = 0.01$: our assumption) of the eigenvalue ${e^{2\pi i\textrm{ }({0.\textrm{ }{m_1}\textrm{ }{m_2}} )}} = {e^{2\pi i\textrm{ }({0.\textrm{ }0\textrm{ }1} )}}$, due to the results of measurement on the output state in Eq. (25).

For the quantum information processing schemes (including quantum algorithms), many researchers [43,61–66,84–86] have proposed to implement an IDQFT scheme theoretically and experimentally. Recently, the methods needed to realize QFT and IDQFT schemes have been proposed by utilizing the XKNLs in [43,66]. In addition, for feasibility and robustness against the decoherence effect, quantum controlled operations for DQFT were designed using quantum optical resources, such as optical cavities or cavity QED [61,63,65,85,86], and quantum linearly optical devices [62,64,84]. Consequently, we employed the IDQFT process for the previous step to estimate the unknown phase of the eigenvalue by the measurement in our QPE scheme.

Funding

Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education (2019R1I1A1A01042699); R&D Convergence Program, National Research Council of Science and Technology of the Republic of Korea (CAP-18-08-KRISS).

References

1. A. Kitaev, “Quantum measurements and the Abelian stabilizer problem,” Electron. Colloq. Comput. Complex. 3, 1 (1996).

2. A. Luis and J. Peřina, “Optimum phase-shift estimation and the quantum description of the phase difference,” Phys. Rev. A 54(5), 4564–4570 (1996). [CrossRef]

3. R. B. Griffiths and C. S. Niu, “Semiclassical Fourier transform for quantum computation,” Phys. Rev. Lett. 76(17), 3228–3231 (1996). [CrossRef]

4. R. Cleve, A. Ekert, C. Macchiavello, and M. Mosca, “Quantum algorithms revisited,” Proc. R. Soc. London, Ser. A 454(1969), 339–354 (1998). [CrossRef]

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

6. 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). [CrossRef]

7. E. Knill, G. Ortiz, and R. D. Somma, “Optimal quantum measurements of expectation values of observables,” Phys. Rev. A 75(1), 012328 (2007). [CrossRef]

8. U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal Quantum Phase Estimation,” Phys. Rev. Lett. 102(4), 040403 (2009). [CrossRef]

9. R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80(1), 013825 (2009). [CrossRef]

10. K. M Svore, M. B. Hastings, and M. Freedman, “Faster phase estimation,” Quant. Inf. Comput. 14, 306 (2013).

11. 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). [CrossRef]

12. 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]

13. 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]

14. 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]

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

16. 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). [CrossRef]

17. B. P. Lanyon, T. J. Weinhold, N. K. Langford, M. Barbieri, D. F. V. James, A. Gilchrist, and A. G. White, “Experimental Demonstration of a Compiled Version of Shor’s Algorithm with Quantum Entanglement,” Phys. Rev. Lett. 99(25), 250505 (2007). [CrossRef]

18. E. Martin-Lopez, A. Laing, T. Lawson, R. Alvarez, X.-Q. Zhou, and J. L. O’Brien, “Experimental realization of Shor's quantum factoring algorithm using qubit recycling,” Nat. Photonics 6(11), 773–776 (2012). [CrossRef]

19. T. Monz, D. Nigg, E. A. Martinez, M. F. Brandl, P. Schindler, R. Rines, S. X. Wang, I. L. Chuang, and R. Blatt, “Realization of a scalable Shor algorithm,” Science 351(6277), 1068–1070 (2016). [CrossRef]

20. 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]

21. A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head-Gordon, “Simulated quantum computation of molecular energies,” Science 309(5741), 1704–1707 (2005). [CrossRef]

22. B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, “Towards quantum chemistry on a quantum computer,” Nat. Chem. 2(2), 106–111 (2010). [CrossRef]

23. J. Du, N. Xu, X. Peng, P. Wang, S. Wu, and D. Lu, “NMR Implementation of a Molecular Hydrogen Quantum Simulation with Adiabatic State Preparation,” Phys. Rev. Lett. 104(3), 030502 (2010). [CrossRef]

24. 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]

25. 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]

26. M. Mosca and C. Zalka, “Exact quantum Fourier transforms and discrete logarithm algorithms,” Int. J. Quantum Inf. 02(01), 91–100 (2004). [CrossRef]

27. S. Y. Song, “Quantum Computing for Discrete Logarithms,” Quantum Computational Number Theory. Springer, Cham, 121 (2015).

28. 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).

29. R. Jozsa, “Quantum factoring, discrete logarithms, and the hidden subgroup problem,” Comput. Sci. Eng. 3(2), 34–43 (2001). [CrossRef]

30. 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]

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

32. M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A 75(5), 053805 (2007). [CrossRef]

33. N. Wiebe and C. Granade, “Efficient Bayesian Phase Estimation,” Phys. Rev. Lett. 117(1), 010503 (2016). [CrossRef]

34. K. Nemoto and W. J. Munro, “Nearly Deterministic Linear Optical Controlled-NOT Gate,” Phys. Rev. Lett. 93(25), 250502 (2004). [CrossRef]

35. Q. Lin and J. Li, “Quantum control gates with weak cross-Kerr nonlinearity,” Phys. Rev. A 79(2), 022301 (2009). [CrossRef]

36. L. Dong, Y. F. Lin, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, Y. P. Ren, X. M. Xiu, Y. J. Gao, and C. H. Oh, “Nearly deterministic Fredkin gate based on weak cross-Kerr nonlinearities,” J. Opt. Soc. Am. B 33(2), 253 (2016). [CrossRef]

37. 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]

38. 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]

39. 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]

40. H. Jeong, “Using weak nonlinearity under decoherence for macroscopic entanglement generation and quantum computation,” Phys. Rev. A 72(3), 034305 (2005). [CrossRef]

41. Q. Lin and B. He, “Highly Efficient Processing of Multi-photon States,” Sci. Rep. 5(1), 12792 (2015). [CrossRef]

42. 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]

43. J. Heo, M. S. Kang, C. H. Hong, H. Yang, and S. G. Choi, “Discrete quantum Fourier transform using weak cross-Kerr nonlinearity and displacement operator and photon-number-resolving measurement under the decoherence effect,” Quantum Inf. Process. 15(12), 4955–4971 (2016). [CrossRef]

44. W. J. Munro, K. Nemoto, and T. P. Spiller, “Weak nonlinearities: a new route to optical quantum computation,” New J. Phys. 7, 137 (2005). [CrossRef]

45. 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]

46. H. Jeong, “Quantum computation using weak nonlinearities: Robustness against decoherence,” Phys. Rev. A 73(5), 052320 (2006). [CrossRef]

47. 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]

48. C. Cao, C. Wang, L. Y. He, and R. Zhang, “Polarization-Entanglement Purification for Ideal Sources Using Weak Cross-Kerr Nonlinearity,” Int. J. Theor. Phys. 52(4), 1265–1273 (2013). [CrossRef]

49. 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]

50. 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]

51. L. Dong, Y. F. Lin, Q. Y. Li, X. M. Xiu, H. K. Dong, and Y. J. Gao, “Optical proposals for controlled delayed-choice experiment based on weak cross-Kerr nonlinearities,” Quantum Inf. Process. 16(5), 122 (2017). [CrossRef]

52. Li Dong, Jun-Xi Wang, Qing-Yang Li, Hong-Zhi Shen, Hai-Kuan Dong, Xiao-Ming Xiu, Ya-Jun Gao, and Choo Hiap Oh, “Nearly deterministic preparation of the perfect W state with weak cross-Kerr nonlinearities,” Phys. Rev. A 93(1), 012308 (2016). [CrossRef]

53. L. L. Sun, H. F. Wang, S. Zhang, and K. H. Yeon, “Entanglement concentration of partially entangled three-photon W states with weak cross-Kerr nonlinearity,” J. Opt. Soc. Am. B 29(4), 630 (2012). [CrossRef]

54. 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). [CrossRef]

55. 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]

56. S. D. Barrett and G. J. Milburn, “Quantum-information processing via a lossy bus,” Phys. Rev. A 74(6), 060302 (2006). [CrossRef]

57. M. D. Lukin and A. Imamoğlu, “Nonlinear Optics and Quantum Entanglement of Ultraslow Single Photons,” Phys. Rev. Lett. 84(7), 1419–1422 (2000). [CrossRef]

58. Q. Lin and B. He, “Single-photon logic gates using minimal resources,” Phys. Rev. A 80(4), 042310 (2009). [CrossRef]

59. X. M. Xiu, Q. Y. Li, Y. F. Lin, H. K. Dong, L. Dong, and Y. J. Gao, “Preparation of four-photon polarization-entangled decoherence-free states employing weak cross-Kerr nonlinearities,” Phys. Rev. A 94(4), 042321 (2016). [CrossRef]

60. 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]

61. M. Scully and M. Zubairy, “Cavity QED implementation of the discrete quantum Fourier transform,” Phys. Rev. A 65(5), 052324 (2002). [CrossRef]

62. M. Mohseni, J. Lundeen, K. Resch, and A. Steinberg, “Experimental Application of Decoherence-Free Subspaces in an Optical Quantum-Computing Algorithm,” Phys. Rev. Lett. 91(18), 187903 (2003). [CrossRef]

63. H. F. Wang, S. Zhang, and K. H. Yeon, “Implementing Quantum Discrete Fourier Transform by Using Cavity Quantum Electrodynamics,” J. Korean Phys. Soc. 53, 1787 (2008). [CrossRef]

64. P. V. Loock, W. J. Munro, K. Nemoto, T. P. Spiller, T. D. Ladd, S. L. Braunstein, and G. L. Milburn, “Hybrid quantum computation in quantum optics,” Phys. Rev. A 78(2), 022303 (2008). [CrossRef]

65. 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(5), 1078 (2012). [CrossRef]

66. 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). [CrossRef]

67. H. Kanamori, H. Yokota, G. Tanaka, M. Watanabe, Y. Ishiguro, I. Yoshida, T. Kakii, S. Itoh, Y. Asano, and S. Tanaka, “Transmission characteristics and reliability of pure-silica-core single-mode fibers,” J. Lightwave Technol. 4(8), 1144–1150 (1986). [CrossRef]

68. K. Nagayama, M. Matsui, M. Kakui, T. Saitoh, K. Kawasaki, H. Takamizawa, Y. Ooga, I. Tsuchiya, and Y. Chigusa, “Ultra low loss (0.1484 dB/km) pure silica core fiber,” SEI Tech. Rev. 57, 3 (2004).

69. C. Wittmann, U. L. Andersen, M. Takeoka, D. Sych, and G. Leuchs, “Discrimination of binary coherent states using a homodyne detector and a photon number resolving detector,” Phys. Rev. A 81(6), 062338 (2010). [CrossRef]

70. S. J. D. Phoenix, “Wave-packet evolution in the damped oscillator,” Phys. Rev. A 41(9), 5132–5138 (1990). [CrossRef]

71. T. Kampschulte, W. Alt, S. Brakhane, M. Eckstein, R. Reimann, A. Widera, and D. Meschede, “Optical Control of the Refractive Index of a Single Atom,” Phys. Rev. Lett. 105(15), 153603 (2010). [CrossRef]

72. Z. Y. Xue, L. N. Yang, and J. Zhou, “Circuit electromechanics with single photon strong coupling,” Appl. Phys. Lett. 107(2), 023102 (2015). [CrossRef]

73. I. C. Hoi, A. F. Kockum, T. Palomaki, T. M. Stace, B. Fan, L. Tornberg, S. R. Sathyamoorthy, G. Johansson, P. Delsing, and C. M. Wilson, “Giant Cross–Kerr Effect for Propagating Microwaves Induced by an Artificial Atom,” Phys. Rev. Lett. 111(5), 053601 (2013). [CrossRef]

74. G. Kirchmair, B. Vlastakis, Z. Leghtas, S. E. Nigg, H. Paik, E. Ginossar, M. Mirrahimi, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Observation of quantum state collapse and revival due to the single-photon Kerr effect,” Nature 495(7440), 205–209 (2013). [CrossRef]

75. I. Friedler, D. Petrosyan, M. Fleischhauer, and G. Kurizki, “Long-range interactions and entanglement of slowsingle-photon pulses,” Phys. Rev. A 72(4), 043803 (2005). [CrossRef]

76. 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(17), 173603 (2012). [CrossRef]

77. 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). [CrossRef]

78. P. W. Kok, J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79(1), 135–174 (2007). [CrossRef]

79. M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature 413(6853), 273–276 (2001). [CrossRef]

80. P. Kok, “Effects of self-phase-modulation on weak nonlinear optical quantum gates,” Phys. Rev. A 77(1), 013808 (2008). [CrossRef]

81. B. He, Y. Ren, and J. A. Bergou, “Creation of high-quality long-distance entanglement with flexible resources,” Phys. Rev. A 79(5), 052323 (2009). [CrossRef]

82. B. He, Y. Ren, and J. A. Bergou, “Universal entangler with photon pairs in arbitrary states,” J. Phys. B 43(2), 025502 (2010). [CrossRef]

83. L. Dong, J. X. Wang, Q. Y. Li, H. Z. Shen, H. K. Dong, X. M. Xiu, and Y. J. Gao, “Single logical qubit information encoding scheme with the minimal optical decoherence-free subsystem,” Opt. Lett. 41(5), 1030 (2016). [CrossRef]

84. Y. Weinstein, M. Pravia, and E. Fortunato, “Implementation of the quantum Fourier transform,” Phys. Rev. Lett. 86(9), 1889–1891 (2001). [CrossRef]

85. H. F. Wang, S. Zhang, and K. H. Yeon, “Implementing quantum discrete Fourier transform by using cavity quantum electrodynamics,” J. Korean Phys. Soc. 53(4), 1787–1790 (2008). [CrossRef]

86. 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]

$m_{1}, m_{2}$ : binary	0, 0	0, 1	1, 0	1, 1
$e^{2 π i (0. m_{2})} = e^{2 π i (m_{2} / 2)}$	1	-1	1	-1
$e^{2 π i (0. m_{1} m_{2})} = e^{2 π i (m_{1} / 2)} \cdot e^{2 π i (m_{2} / 4)}$	1	i	-1	-i

$m_{1}, m_{2}$ : binary	0, 0	0, 1	1, 0	1, 1
$e^{2 π i (0. m_{2})} = e^{2 π i (m_{2} / 2)}$	1	-1	1	-1
$e^{2 π i (0. m_{1} m_{2})} = e^{2 π i (m_{1} / 2)} \cdot e^{2 π i (m_{2} / 4)}$	1	i	-1	-i

Photonic scheme of quantum phase estimation for quantum algorithms via cross-Kerr nonlinearities under decoherence effect

Abstract

1. Introduction

2. Theoretical circuit of a quantum phase estimation algorithm

3. Design of controlled-unitary gate via XKNLs for the QPE algorithm

4. Analysis of the decoherence effect in a controlled-path gate and merging-path gate using XKNLs

5. Conclusions

Appendix

Funding

References

Cited By

Figures (7)

Tables (2)

Equations (25)

Optics Express