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Abstract
We present a systematic approach to implementation of basic quantum logic gates operating on polar molecules in pendular states as qubits for a quantum computer. A static electric field prevents quenching of the dipole moments by rotation, thereby creating the pendular states; also, the field gradient enables distinguishing among qubit sites. Multitarget optimal control theory is used as a means of optimizing the initialtotarget transition probability via a laser field. We give detailed calculations for the SrO molecule, a favorite candidate for proposed quantum computers. Our simulation results indicate that NOT, Hadamard and CNOT gates can be realized with high fidelity, as high as 0.985, for such pendular qubit states.
For useful discussions, Jing Zhu thanks Ross Hoehn and Siwei Wei, and appreciates correspondence with Dr. Philippe Pellegrini. For support of this work at Purdue, we are grateful to the National Science Foundation CCI center, “Quantum Information for Quantum Chemistry (QIQC),” Award No. CHE1037992, and to the Army Research Office. At Texas A&M, support was provided by the Institute for Quantum Science and Engineering, as well as the Office of Naval Research and National Science Foundation (NSF) Award No. CHE0809651.
I. INTRODUCTION
II. THEORY
A. Eigenstates for polar molecules in pendular states
B. Multitarget optical control theory
III. SIMULATION RESULTS FOR POLAR DIATOMIC MOLECULES
IV. DISCUSSION
Key Topics
 Qubits
 30.0
 Quantum computing
 20.0
 Logic elements
 9.0
 Electric dipole moments
 8.0
 Electric fields
 8.0
Figures
The configuration of two polar molecules. μ _{ 1 } and μ _{ 2 } are permanent dipole moments for molecules 1 and 2; r _{ 12 } is the distance vector from molecules 1 to 2. ε is the external electric field and α is the angle between r _{ 12 } and the external field. θ_{1} (θ_{2}) is the angle between μ _{ 1 } ( μ _{ 2 }) and ε .
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The configuration of two polar molecules. μ _{ 1 } and μ _{ 2 } are permanent dipole moments for molecules 1 and 2; r _{ 12 } is the distance vector from molecules 1 to 2. ε is the external electric field and α is the angle between r _{ 12 } and the external field. θ_{1} (θ_{2}) is the angle between μ _{ 1 } ( μ _{ 2 }) and ε .
Ratio of frequency shift, Δω, to the dipoledipole interaction parameter, , as a function of reduced field strength, x = μ ε /B and x′ = μ ε ′/B at sites of the two dipoles.
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Ratio of frequency shift, Δω, to the dipoledipole interaction parameter, , as a function of reduced field strength, x = μ ε /B and x′ = μ ε ′/B at sites of the two dipoles.
Converged laser pulses for NOT gate. The upper panel is the laser pulses for realizing the NOT gate for dipole 1, on the left and dipole 2 on the right. The initial and target states are listed in Table I . The lower panel shows the evolution of all populations driven by NOT pulses. The left panel exhibits the population evolution under the NOT pulse for dipole 1. The initial state is and the final populations are 0.733 for 00⟩ and 0.265 for 11⟩. The right panel shows the population evolution via the NOT pulse for dipole 2. It has the same initial condition as the left panel and the converged population is 0.216 for 00⟩ and 0.781 for 11⟩. Both pulses perform the corresponding NOT gates nicely.
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Converged laser pulses for NOT gate. The upper panel is the laser pulses for realizing the NOT gate for dipole 1, on the left and dipole 2 on the right. The initial and target states are listed in Table I . The lower panel shows the evolution of all populations driven by NOT pulses. The left panel exhibits the population evolution under the NOT pulse for dipole 1. The initial state is and the final populations are 0.733 for 00⟩ and 0.265 for 11⟩. The right panel shows the population evolution via the NOT pulse for dipole 2. It has the same initial condition as the left panel and the converged population is 0.216 for 00⟩ and 0.781 for 11⟩. Both pulses perform the corresponding NOT gates nicely.
Optimized laser pulses for Hadamard gates. The left panel pertains to performing the Hadamard gate on dipole 1 and the right one to performing that on dipole 2. The lower panels show the population evolution under the Hadamard laser pulse. In order to exhibit the curves clearly, we chose 01⟩ as the initial state; the final populations of 01⟩ and 11⟩ are 0.429 and 0.566, respectively. For the case of the Hadamard gate on dipole 2, the 00⟩ qubit was chosen as the initial state. The populations at the end of the evolution are 0.528 for 00⟩ and 0.470 for 01⟩. The functions of these gates are realized very well by these two pulses.
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Optimized laser pulses for Hadamard gates. The left panel pertains to performing the Hadamard gate on dipole 1 and the right one to performing that on dipole 2. The lower panels show the population evolution under the Hadamard laser pulse. In order to exhibit the curves clearly, we chose 01⟩ as the initial state; the final populations of 01⟩ and 11⟩ are 0.429 and 0.566, respectively. For the case of the Hadamard gate on dipole 2, the 00⟩ qubit was chosen as the initial state. The populations at the end of the evolution are 0.528 for 00⟩ and 0.470 for 01⟩. The functions of these gates are realized very well by these two pulses.
Optimized laser pulse for realizing the CNOT gate. The initial and target states are listed in Table III . In the lower panel, the population evolution is driven by the CNOT pulse. The initial state is . After population oscillations due to the effect of the laser pulse, the final qubit populations are 0.00363 (00⟩), 0.25807 (01⟩), 0.00273 (10⟩), and 0.73546 (11⟩). The populations of state 10⟩ and 11⟩ are switched, which confirms the correctness of the converged laser pulse.
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Optimized laser pulse for realizing the CNOT gate. The initial and target states are listed in Table III . In the lower panel, the population evolution is driven by the CNOT pulse. The initial state is . After population oscillations due to the effect of the laser pulse, the final qubit populations are 0.00363 (00⟩), 0.25807 (01⟩), 0.00273 (10⟩), and 0.73546 (11⟩). The populations of state 10⟩ and 11⟩ are switched, which confirms the correctness of the converged laser pulse.
Optimized laser pulse for realizing the CNOT gate when the distance between two dipoles is 75 nm. The initial and target states are listed in Table III . The optimized laser pulse, which is shown in the upper panel, lasts 110 ns. The lower panel shows the population evolution driven by the pulse. The initial state is the same as Fig. 5 . The final converged populations for state 01⟩ and 10⟩ are 0.22 and 0.73, respectively.
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Optimized laser pulse for realizing the CNOT gate when the distance between two dipoles is 75 nm. The initial and target states are listed in Table III . The optimized laser pulse, which is shown in the upper panel, lasts 110 ns. The lower panel shows the population evolution driven by the pulse. The initial state is the same as Fig. 5 . The final converged populations for state 01⟩ and 10⟩ are 0.22 and 0.73, respectively.
Tables
The initial and target states of the NOT gate. The final fidelities for both NOT gates are 0.967 and 0.985, respectively.
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The initial and target states of the NOT gate. The final fidelities for both NOT gates are 0.967 and 0.985, respectively.
The initial and target states of the Hadmard gate. The yield fidelities are 0.944 and 0.902.
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The initial and target states of the Hadmard gate. The yield fidelities are 0.944 and 0.902.
The initial and target states of the CNOT gate. The converged fidelity is 0.975.
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The initial and target states of the CNOT gate. The converged fidelity is 0.975.
Comparison with other polar molecular systems.
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Comparison with other polar molecular systems.
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Abstract
We present a systematic approach to implementation of basic quantum logic gates operating on polar molecules in pendular states as qubits for a quantum computer. A static electric field prevents quenching of the dipole moments by rotation, thereby creating the pendular states; also, the field gradient enables distinguishing among qubit sites. Multitarget optimal control theory is used as a means of optimizing the initialtotarget transition probability via a laser field. We give detailed calculations for the SrO molecule, a favorite candidate for proposed quantum computers. Our simulation results indicate that NOT, Hadamard and CNOT gates can be realized with high fidelity, as high as 0.985, for such pendular qubit states.
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