Nuclear-Magnetic-Resonance-Based Physical Realization of Quantum Toffoli Gate

Yonggang Peng

doi:10.3788/LOP220821

Journals >Laser & Optoelectronics Progress >Volume 60 >Issue 7 >Page 0727002 > Article

Laser & Optoelectronics Progress
Vol. 60, Issue 7, 0727002 (2023)

Nuclear-Magnetic-Resonance-Based Physical Realization of Quantum Toffoli Gate

Yonggang Peng^*

Author Affiliations

Department of Applied Physics, College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, Jiangsu, China

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DOI: 10.3788/LOP220821 Cite this Article Set citation alerts

Yonggang Peng. Nuclear-Magnetic-Resonance-Based Physical Realization of Quantum Toffoli Gate[J]. Laser & Optoelectronics Progress, 2023, 60(7): 0727002 Copy Citation Text

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Abstract

The three-qubit Toffoli gate is equivalent to the combination of two two-qubit controlled NOT gates and three two-qubit controlled phase-shift gates. The controlled NOT and phase-shift gates respectively comprise a combination of nuclear-magnetic-resonance pulse sequences and a free development operator of two nuclear spins with time. According to the order in which the controlled NOT and phase-shift gates act on the nuclear spin system, the nuclear-magnetic-resonance-based physical realization of a quantum Toffoli gate is achieved. The time-dependent Schr?dinger equation is numerically solved by the Suzuki formula to confirm the correctness and feasibility of the realization of the quantum Toffoli gate based on nuclear magnetic resonance.

Keywords

free development operator nuclear-magnetic resonance quantum optics quantum Toffoli gate Suzuki product formula

H = - J_{123}^{z} S_{1}^{z} S_{2}^{z} S_{3}^{z} - J_{12}^{z} S_{1}^{z} S_{2}^{z} - J_{13}^{z} S_{1}^{z} S_{3}^{z} - J_{23}^{z} S_{2}^{z} S_{3}^{z} - h_{1}^{z} S_{1}^{z} - h_{2}^{z} S_{2}^{z} - h_{3}^{z} S_{3}^{z} - \sum_{j = 1}^{3} \sum_{α = x, y} {\tilde{h}}_{j}^{α} s i n (2 π f_{j}^{α} t + φ_{j}^{α})

，（1）

|0〉 = |↑〉 = (\begin{matrix} 1 \\ 0 \end{matrix})

，

|1〉 = |↓〉 = (\begin{matrix} 0 \\ 1 \end{matrix})

。（2）

X = e^{i π S_{j}^{x} / 2} = \frac{1}{\sqrt[]{2}} (\begin{matrix} 1 & i \\ i & 1 \end{matrix})

，

Y = e^{i π S_{j}^{y} / 2} = \frac{1}{\sqrt[]{2}} (\begin{matrix} 1 & 1 \\ - 1 & 1 \end{matrix})

。（3）

\bar{X} = e^{- i π S_{j}^{x} / 2} = \frac{1}{\sqrt[]{2}} (\begin{matrix} 1 & - i \\ - i & 1 \end{matrix})

，

\bar{Y} = e^{- i π S_{j}^{y} / 2} = \frac{1}{\sqrt[]{2}} (\begin{matrix} 1 & - 1 \\ 1 & 1 \end{matrix})

。（4）

i ℏ \frac{\partial}{\partial t} |ψ (t)〉 = - [h_{j}^{z} S_{j}^{z} + {\tilde{h}}_{j}^{x} s i n (ω t)] |ψ (t)〉

。（5）

τ = \frac{π}{{\tilde{h}}_{j}^{x}}

。（6）

V = \frac{e^{- i π / 4}}{\sqrt[]{2}} (\begin{matrix} 1 & i \\ i & 1 \end{matrix})

。（7）

V_{23} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & e^{- i π / 4} / \sqrt[]{2} & i e^{- i π / 4} / \sqrt[]{2} \\ 0 & 0 & i e^{- i π / 4} / \sqrt[]{2} & e^{- i π / 4} / \sqrt[]{2} \end{matrix}) = {\bar{Y}}_{3} (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & e^{- i π / 2} \end{matrix}) Y_{3} = e^{- i α} {\bar{Y}}_{3} U_{23} Y_{3}

，（8）

Y_{3} = \frac{1}{\sqrt[]{2}} (\begin{matrix} 1 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & e^{- i π / 2} \end{matrix})

。（9）

U_{23} = e^{- i τ H_{23}} = (\begin{matrix} e^{i ϕ_{00}} & 0 & 0 & 0 \\ 0 & e^{i ϕ_{01}} & 0 & 0 \\ 0 & 0 & e^{i ϕ_{10}} & 0 \\ 0 & 0 & 0 & e^{i ϕ_{11}} \end{matrix}) = e^{i α} (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & e^{- i π} \end{matrix})

，（10）

H_{23} = - J_{23}^{z} S_{2}^{z} S_{3}^{z} - h (S_{2}^{z} + S_{3}^{z})

，

h_{2}^{z} = h_{3}^{z} = h

。（11）

ϕ_{00} = ϕ_{01} = ϕ_{10} = α

，

ϕ_{11} = α - \frac{π}{2}

。（12）

τ (\frac{J_{23}}{4} + h) = - τ \frac{J_{23}}{4} = α

，

τ (\frac{J_{23}}{4} - h) = α - \frac{π}{2}

。（13）

h = - \frac{J_{23}}{2}

，

τ J_{23} = - \frac{π}{2}

，（14）

C_{N O T} = (\begin{matrix} I_{1} & 0 \\ 0 & 2 S_{2}^{z} \end{matrix}) = (\begin{matrix} I_{1} & 0 \\ 0 & 2 {\bar{Y}}_{2} S_{2}^{z} Y_{2} \end{matrix}) =

{\bar{Y}}_{2} (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix}) Y_{2} = e^{- i β} {\bar{Y}}_{2} I_{12} Y_{2}

，（15）

I_{12} = e^{- i τ H_{12}} = (\begin{matrix} e^{i φ_{00}} & 0 & 0 & 0 \\ 0 & e^{i φ_{01}} & 0 & 0 \\ 0 & 0 & e^{i φ_{10}} & 0 \\ 0 & 0 & 0 & e^{i φ_{11}} \end{matrix}) = e^{i β} (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & - 1 \end{matrix})

，（16）

H_{12} = - J_{12}^{z} S_{1}^{z} S_{2}^{z} - h (S_{1}^{z} + S_{2}^{z})

，

h_{1}^{z} = h_{2}^{z} = h

，（17）

φ_{00} = φ_{01} = φ_{10} = β

，

φ_{11} = β - π

。（18）

τ (\frac{J_{12}}{4} + h) = - τ \frac{J_{12}}{4} = β

，

τ (\frac{J_{12}}{4} - h) = β - π

。（19）

h = - \frac{J_{12}}{2}

，

τ J_{12} = - π

，（20）

{\bar{Y}}_{3} U_{13} Y_{3} {\bar{Y}}_{2} I_{12} Y_{2} {\bar{Y}}_{3} {\bar{U}}_{23} Y_{3} {\bar{Y}}_{2} I_{12} Y_{2} {\bar{Y}}_{3} U_{23} Y_{3}

。（21）

2 π h_{1}^{z} = 2 π h_{2}^{z} = 2 π h_{3}^{z} = 500 M H z

，

(1)

J_{12}^{z} = J_{13}^{z} = J_{23}^{z} = - 215 H z

，

(1)

2 π {\tilde{h}}_{1}^{y} = 2 π {\tilde{h}}_{2}^{y} = 2 π {\tilde{h}}_{3}^{y} = 125 M H z

，

(1)

2 π f_{1}^{x} = 2 π f_{2}^{x} = 2 π f_{3}^{x} = 500 M H z

。（22）

{\tilde{h}}_{1}^{y} = {\tilde{h}}_{2}^{y} = {\tilde{h}}_{3}^{y} = 0.25 M H z

，

h_{1}^{z} = h_{2}^{z} = h_{3}^{z} = 1 M H z

，

(1)

f_{1}^{x} = f_{2}^{x} = f_{3}^{x} = 1 M H z

，

J_{12}^{z} = J_{13}^{z} = J_{23}^{z} = - 0.43 \times 10^{- 6} M H z

。（23）

i \frac{\partial}{\partial t} |Φ (t)〉 = H (t) |Φ (t)〉

。（24）

Q_{j} = Q_{j}^{z} = \frac{1}{2} - 〈Φ (t)| I_{j}^{z} |Φ (t)〉

，（25）

U (t + τ) = e x p_{+} (- i \int_{t}^{t + τ} H (u) d u)

（26）

U (t + τ, t) = U [t + m δ, t + (m - 1) δ], \dots, U (t + δ, t)

，（27）

U (δ) = e^{- i δ H_{x} / 2} e^{- i δ H_{y} / 2} e^{- i δ H_{x}} e^{- i δ H_{y} / 2} e^{- i δ H_{z} / 2}

，（28）

\begin{matrix} U [t + (m + 1), t + m δ] = e^{- i δ H_{z} [t + (m + 1 / 2) δ] / 2} e^{- i δ H_{y} [t + (m + 1 / 2) δ] / 2} e^{- i δ H_{x} [t + (m + 1 / 2) δ]} \cdot \\ e^{- i δ H_{y} [t + (m + 1 / 2) δ] / 2} e^{- i δ H_{z} [t + (m + 1 / 2) δ] / 2} \end{matrix}

，（29）

Yonggang Peng. Nuclear-Magnetic-Resonance-Based Physical Realization of Quantum Toffoli Gate[J]. Laser & Optoelectronics Progress, 2023, 60(7): 0727002

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