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Physical Review A

Rovibrational wave-packet manipulation using shaped midinfrared femtosecond pulses toward quantum computation: Optimization of pulse shape by a genetic algorithm

Masaaki Tsubouchi, Takamasa Momose

Published May 20, 2008• 57 citations
Publisher

Abstract

We have numerically examined optimization of the experimentally achievable shapes of mid-infrared femtosecond laser pulses to construct elementary quantum gates using molecular rovibrational states of a single molecule. Instead of optimizing the electric field by optimal control theory, the method generally used in theoretical studies of coherent control, the transmittance and the phase shift of conventional pulse shapers were optimized in the frequency domain by a genetic algorithm. The target molecular system we examined was the rotation-vibration states of CO in the $X({^{1}\ensuremath{\Sigma}}^{+})$ ground electronic state. Although the existence of the rotational degrees of freedom makes the quantum gate operation complicated, high fidelity of over 0.95 was achieved for one-rovibrational-qubit systems, which indicates that a molecular quantum computer may be feasible for at least one-qubit calculations on a diatomic molecule. On the other hand, two-qubit calculations are more difficult to achieve in a single CO molecule using rotational and vibrational degrees of freedom as different qubits. The importance of the control of the rotational in addition to the vibrational wave packet in order to realize a molecular quantum computer is discussed.

DOI
10.1103/physreva.77.052326

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We have numerically examined optimization of the experimentally achievable shapes of mid-infrared femtosecond laser pulses to construct elementary quantum gates using molecular rovibrational states of a single molecule. Instead of optimizing the electric field by optimal control theory, the method generally used in theoretical studies of coherent control, the transmittance and the phase shift of conventional pulse shapers were optimized in the frequency domain by a genetic algorithm. The target molecular system we examined was the rotation-vibration states of CO in the X(1\ensuremathΣ+)X({^{1}\ensuremath{\Sigma}}^{+})X(1\ensuremathΣ+) ground electronic state. Although the existence of the rotational degrees of freedom makes the quantum gate operation complicated, high fidelity of over 0.95 was achieved for one-rovibrational-qubit systems, which indicates that a molecular quantum computer may be feasible for at least one-qubit calculations on a diatomic molecule. On the other hand, two-qubit calculations are more difficult to achieve in a single CO molecule using rotational and vibrational degrees of freedom as different qubits. The importance of the control of the rotational in addition to the vibrational wave packet in order to realize a molecular quantum computer is discussed.
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The importance of the ro-vibrational state energies on the ability to produce high fidelity binary shaped laser pulses for quantum logic gates is investigated. The single frequency 2-qubit ACNOT(1) and double frequency 2-qubit NOT(2) quantum gates are used as test cases to examine this behaviour. A range of diatomics is sampled. The laser pulses are optimized using a genetic algorithm for binary (two amplitude and two phase parameter) variation on a discretized frequency spectrum. The resulting trends in the fidelities were attributed to the intrinsic molecular properties and not the choice of method: a discretized frequency spectrum with genetic algorithm optimization. This is verified by using other common laser pulse optimization methods (including iterative optimal control theory), which result in the same qualitative trends in fidelity. The results differ from other studies that used vibrational state energies only. Moreover, appropriate choice of diatomic (relative ro-vibrational state arrangement) is critical for producing high fidelity optimized quantum logic gates. It is also suggested that global phase alignment imposes a significant restriction on obtaining high fidelity regions within the parameter search space. Overall, this indicates a complexity in the ability to provide appropriate binary laser pulse control of diatomics for molecular quantum computing.
Accurate rotation–vibration energy levels and transition dipoles of the molecule thiophosgene are used to model the execution of quantum gates with shaped laser pulses. Qubits are encoded in 2 n vibrational computing states on the ground electronic surface of the molecule. Computations are carried out by cycling amplitude between these computing states and a gateway state with a shaped laser pulse. The shaped pulse that performs the computation is represented by a physical model of a 128–1024 channel pulse shaper. Pulse shapes are optimized with a standard genetic algorithm, yielding experimentally realizable computing pulses. The robustness of optimization is studied as a function of the vibrational states selected, rotational level structure, additional vibrational levels not assigned to the computation, and compensation for laser power variation across a molecular ensemble.
My thesis focuses on theoretical studies in the molecular quantum computing with vibrational qubits. We studied how the molecular vibrational properties affect the fidelity of the logic gates. Our approach is to encode states of the quantum information register into the vibrational eigenstates of a molecule and use the shaped laser pulse (femtosecond, infrared) to control the vibrational state-to-state transition. By using the Optimal Control Theory to shape a femtosecond laser and numerical propagation of laser-driven vibrational wave packets, we analyzed how the vibrational properties of a two-qubit system affect the accuracy of the quantum gates and how to effectively control the vibrational state-to-state transitions. From these studies, we found the anharmonicities are very important for the control of the vibrational state-to-state transitions and an intricate interplay between the frequencies and anharmonicities in the two-qubit system leads to the occurrence of resonances between different transitions in the vibrational manifold. Such resonances cause "leakage" of population into other vibrational states and hinder the control. From theoretical analysis of resonances, we formulate several criteria for selection of molecule for implementation of the vibrational two-qubit system. These criteria can help experimentalists to choose the best molecules to achieve accurate qubit transformations in the experiment.
Frequency domain shaped binary laser pulses were optimized to perform 2 qubit quantum gate operations in (12)C(16)O. The qubit rovibrational state representation was chosen so that all gate operations consisted of one-photon transitions. The amplitude and phase varied binary pulses were determined using a genetic algorithm optimization routine. Binary pulses have two possible amplitudes, 0 or 1, and two phases, 0 or pi, for each frequency component of the pulse. Binary pulses are the simplest to shape experimentally and provide a minimum fidelity limit for amplitude and phase shaped pulses. With the current choice of qubit representation and using optimized binary pulses, fidelities of 0.80 and as high as 0.97 were achieved for the controlled-NOT and alternative controlled-NOT quantum gates. This indicates that with a judicious choice of qubits, most of the required control can be obtained with a binary pulse. Limited control was observed for 2 qubit NOT and Hadamard gates due to the need to control multiple excitations. The current choice of qubit representation produces pulses with decreased energies and superior fidelities when compared with rovibrational qubit representations consisting of two-photon transitions. The choice of input pulse energy is important and applying pulses of increased energy does not necessarily lead to a better fidelity.
Implementation of quantum controlled-NOT (CNOT) gates in realistic molecular systems is studied using stimulated Raman adiabatic passage (STIRAP) techniques optimized in the time domain by genetic algorithms or coupled with optimal control theory. In the first case, with an adiabatic solution (a series of STIRAP processes) as starting point, we optimize in the time domain different parameters of the pulses to obtain a high fidelity in two realistic cases under consideration. A two-qubit CNOT gate constructed from different assignments in rovibrational states is considered in diatomic (NaCs) or polyatomic (SCCl2)({\text{SCCl}}_{2})(SCCl2​) molecules. The difficulty of encoding logical states in pure rotational states with STIRAP processes is illustrated. In such circumstances, the gate can be implemented by optimal control theory and the STIRAP sequence can then be used as an interesting trial field. We discuss the relative merits of the two methods for rovibrational computing (structure of the control field, duration of the control, and efficiency of the optimization).