Ab initio study of optimal control of ammonia molecular vibrational wavepackets: Towards molecular quantum computing

Optimization two-qubit quantum gate by two optical control methods in molecular pendular states

Implementation of quantum gates are important for quantum computations in physical system made of polar molecules. We investigate the feasibility of implementing gates based on pendular states of the molecular system by two different quantum optical control methods. Firstly, the Multi-Target optimal control theory and the Multi-Constraint optimal control theory are described for optimizing control fields and accomplish the optimization of quantum gates. Numerical results show that the controlled NOT gate (CNOT) can be realized under the control of above methods with high fidelities (0.975 and 0.999) respectively. In addition, in order to examine the dependence of the fidelity on energy difference in the same molecular system, the SWAP gate in the molecular system is also optimized with high fidelity (0.999) by the Multi-Constraint optimal control theory with the zero-area and constant-fluence constraints.

Quantum Computation using Arrays of N Polar Molecules in Pendular States

We investigate several aspects of realizing quantum computation using entangled polar molecules in pendular states. Quantum algorithms typically start from a product state |00⋯0⟩ and we show that up to a negligible error, the ground states of polar molecule arrays can be considered as the unentangled qubit basis state |00⋯0⟩ . This state can be prepared by simply allowing the system to reach thermal equilibrium at low temperature (<1 mK). We also evaluate entanglement, characterized by concurrence of pendular state qubits in dipole arrays as governed by the external electric field, dipole-dipole coupling and number N of molecules in the array. In the parameter regime that we consider for quantum computing, we find that qubit entanglement is modest, typically no greater than 10^-4 , confirming the negligible entanglement in the ground state. We discuss methods for realizing quantum computation in the gate model, measurement-based model, instantaneous quantum polynomial time circuits and the adiabatic model using polar molecules in pendular states.

Complexity and simplicity of optimal control theory pulses shaped for controlling vibrational qubits

In the context of molecular quantum computation the optimal control theory (OCT) is used to obtain shaped laser pulses for high-fidelity control of vibrational qubits. Optimization is done in time domain and the OCT algorithm varies values of electric field in each time step independently, tuning hundreds of thousands of parameters to find one optimal solution. Such flexibility is not available in experiments, where pulse shaping is done in frequency domain and the number of "tuning knobs" is much smaller. The question of possible experimental interpretations of theoretically found OCT solutions arises. In this work we analyze very accurate optimal pulse that we obtained for implementing quantum gate CNOT for the two-qubit system encoded into the exited vibrational states of thiophosgene molecule. Next, we try to alter this pulse by reducing the number of available frequency channels and intentionally introducing systematic and random errors (in frequency domain, by modifying the values of amplitudes and phases of different frequency components). We conclude that a very limited number of frequency components (only 32 in the model of thiophosgene) are really necessary for accurate control of the vibrational two-qubit system, and such pulses can be readily constructed using OCT. If the amplitude and phase errors of different frequency components do not exceed ±3% of the optimal values, one can still achieve accurate transformations of the vibrational two-qubit system, with gate fidelity of CNOT exceeding 0.99.

Frequency-based Quantum Computers from a Chemist's Perspective

Quantum computer elements are often designed and tested using molecular or nanoscopic components that form registers of qubits in which memory is stored and information processed. Often such registers are probed and manipulated using frequency-based techniques such as nuclear-magnetic resonance spectroscopy. A major challenge is to design molecules to act as these registers. We provide a basis for rational molecular design through consideration of the generic spectroscopic properties required for quantum computing, bypassing the need for intricate knowledge of the way these molecules are used spectroscopically. Designs in which two-qubit gate times scale similarly to those for one-qubit gates are presented. The specified spectroscopic requirements are largely independent of the type of spectroscopy used (e.g. magnetic resonance or vibrational) and are often independent of technical details of the application (e.g. broadband or high-resolution spectroscopy). This should allow the design of much larger quantum registers than have currently been demonstrated.

Controlled full adder-subtractor by vibrational computing

The implementation of a quantum-controlled full adder-subtractor of two binary digits and of a "carry in" or a "borrow in" is simulated by encoding four qubits in the vibrational eigenstates of a tetra-atomic molecule (trans-HONO). The laser field of the gate is computed using optimal control theory by treating dynamics in full dimensionality. A controlled qubit enforces the addition or the subtraction. The global unitary transformation that connects the inputs to the outputs is driven by a single laser pulse. This decreases the duration of the operation and allows for a better use of the optical resources and for an improvement of the fidelity (>97%). Initialization and reading out are discussed. The timescale of the sequence initialization, gate and read out is<100 ps.

On the role of vibrational anharmonicities in a two-qubit system

Anharmonicities of vibrational modes allow controlling state-to-state transitions and encoding quantum information bits into vibrational eigenstates. Resonances between different transitions hinder the control. In this paper all resonances that can occur in a two-qubit system are identified and studied. Computational experiments are carried out using optimal control theory and numerical propagation of vibrational wave packets. Useful insight is obtained into the mechanisms of negative effect of these resonances onto the control tasks. A set of general criteria is derived for evaluation of candidate molecules for practical realization of vibrational two-qubit system. Several simple strategies for avoiding such resonances by choosing suitable vibrational characteristics (mode frequencies and anharmonicities) are proposed.

Wave-Packet and Coherent Control Dynamics

This review summarizes progress in coherent control as well as relevant recent achievements, highlighting, among several different schemes of coherent control, wave-packet interferometry (WPI). WPI is a fundamental and versatile scenario used to control a variety of quantum systems with a sequence of short laser pulses whose relative phase is finely adjusted to control the interference of electronic or nuclear wave packets (WPs). It is also useful in retrieving quantum information such as the amplitudes and phases of eigenfunctions superposed to generate a WP. Experimental and theoretical efforts to retrieve both the amplitude and phase information are recounted. This review also discusses information processing based on the eigenfunctions of atoms and molecules as one of the modern and future applications of coherent control. The ultrafast coherent control of ultracold atoms and molecules and the coherent control of complex systems are briefly discussed as future perspectives.

Free-time and fixed end-point optimal control theory in quantum mechanics: application to entanglement generation

We have constructed free-time and fixed end-point optimal control theory for quantum systems and applied it to entanglement generation between rotational modes of two polar molecules coupled by dipole-dipole interaction. The motivation of the present work is to solve optimal control problems more flexibly by extending the popular fixed time and fixed end-point optimal control theory for quantum systems to free-time and fixed end-point optimal control theory. As a demonstration, the theory that we have constructed in this paper will be applied to entanglement generation in rotational modes of NaCl-NaBr polar molecular systems that are sensitive to the strength of entangling interactions. Our method will significantly be useful for the quantum control of nonlocal interaction such as entangling interaction, which depends crucially on the strength of the interaction or the distance between the two molecules, and other general quantum dynamics, chemical reactions, and so on.

Computational investigation and experimental considerations for the classical implementation of a full adder on SO2 by optical pump-probe schemes

Following the scheme recently proposed by Remacle and Levine [Phys. Rev. A 73, 033820 (2006)], we investigate the concrete implementation of a classical full adder on two electronic states (X 1A1 and C 1B2) of the SO2 molecule by optical pump-probe laser pulses using intuitive and counterintuitive (stimulated Raman adiabatic passage) excitation schemes. The resources needed for providing the inputs and reading out are discussed, as well as the conditions for achieving robustness in both the intuitive and counterintuitive pump-dump sequences. The fidelity of the scheme is analyzed with respect to experimental noise and two kinds of perturbations: The coupling to the neighboring rovibrational states and a finite rotational temperature that leads to a mixture for the initial state. It is shown that the logic processing of a full addition cycle can be realistically experimentally implemented on a picosecond time scale while the readout takes a few nanoseconds.

Vibrational computing: simulation of a full adder by optimal control

Within the context of vibrational molecular quantum computing, we investigate the implementation of a full addition of two binary digits and a carry that provides the sum and the carry out. Four qubits are necessary and they are encoded into four different normal vibrational modes of a molecule. We choose the bromoacetyl chloride molecule because it possesses four bright infrared active modes. The ground and first excited states of each mode form the one-qubit computational basis set. Two approaches are proposed for the realization of the full addition. In the first one, we optimize a pulse that implements directly the entire addition by a single unitary transformation. In the second one, we decompose the full addition in elementary quantum gates, following a scheme proposed by Vedral et al. [Phys. Rev. A 54, 147 (1996)]. Four elementary quantum gates are necessary, two two-qubit CNOT gates (controlled NOT) and two three-qubit TOFFOLI gates (controlled-controlled NOT). All the logic operations consist in one-qubit flip. The logic implementation is therefore quasiclassical and the readout is based on a population analysis of the vibrational modes that does not take the phases into account. The fields are optimized by the multitarget extension of the optimal control theory involving all the transformations among the 2(4) qubit states. A single cycle of addition without considering the preparation or the measure or copy of the result can be carried out in a very competitive time, on a picosecond time scale.

Quantum computing based on vibrational eigenstates: pulse area theorem analysis

In a recent paper [D. Babikov, J. Chem. Phys. 121, 7577 (2004)], quantum optimal control theory was applied to analyze the accuracy of quantum gates in a quantum computer based on molecular vibrational eigenstates. The effects of the anharmonicity parameter of the molecule, the target time of the pulse, and the penalty function on the accuracy of the qubit transformations were investigated. We demonstrate that the effects of all the molecular and laser-pulse parameters can be explained utilizing the analytical pulse area theorem, which originates from the standard two-level model. Moreover, by analyzing the difference between the optimal control theory results and those obtained using the pulse area theorem, it is shown that extremely high quantum gate fidelity can be achieved for a qubit system based on vibrational eigenstates.

Optimal control simulation of the Deutsch-Jozsa algorithm in a two-dimensional double well coupled to an environment

The quantum Deutsch-Jozsa algorithm is implemented by using vibrational modes of a two-dimensional double well. The laser fields realizing the different gates (NOT, CNOT, and HADAMARD) on the two-qubit space are computed by the multitarget optimal control theory. The stability of the performance index is checked by coupling the system to an environment. Firstly, the two-dimensional subspace is coupled to a small number Nb of oscillators in order to simulate intramolecular vibrational energy redistribution. The complete (2+Nb)D problem is solved by the coupled harmonic adiabatic channel method which allows including coupled modes up to Nb=5. Secondly, the computational subspace is coupled to a continuous bath of oscillators in order to simulate a confined environment expected to be favorable to achieve molecular computing, for instance, molecules confined in matrices or in a fullerene. The spectral density of the bath is approximated by an Ohmic law with a cutoff for some hundreds of cm(-1). The time scale of the bath dynamics (of the order of 10 fs) is then smaller than the relaxation time and the controlled dynamics (2 ps) so that Markovian dissipative dynamics is used.

Anharmonic properties of the vibrational quantum computer

We developed an efficient approach to study the coherent control of vibrational state-to-state transitions. The approximations employed in our model are valid in the regime of the low vibrational excitation specific to the vibrational quantum computer. Using this approach we explored how the vibrational properties of a two-qubit system affect the accuracy of subpicosecond quantum gates. The optimal control theory and numerical propagation of laser-driven vibrational wave packets were employed. The focus was on understanding the effect of the three anharmonicity parameters of the system. In the three-dimensional anharmonicity parameter space we identified several spots of high fidelity separated by low fidelity planar regions. The seemingly complicated picture is explained in terms of interferences between different state-to-state transitions. Very general analytic relationships between the anharmonicity parameters and the frequencies are derived to describe the observed features. Geometrically, these expressions represent planes in the three-dimensional anharmonicity parameter space. Results of this work should help to choose a suitable candidate molecule for the practical implementation of the vibrational two-qubit system.

Analytic approach for controlling quantum states in complex systems

We examine random matrix systems driven by an external field in view of optimal control theory (OCT). By numerically solving OCT equations, we can show that there exists a smooth transition between two states called "moving bases" which are dynamically related to initial and final states. In our previous work [J. Phys. Soc. Jpn. 73, 3215 (2004); Adv. Chem. Phys. 130A, 435 (2005)], they were assumed to be orthogonal, but in this paper, we introduce orthogonal moving bases. We can construct a Rabi-oscillation-like representation of a wave packet using such moving bases, and derive an analytic optimal field as a solution of the OCT equations. We also numerically show that the newly obtained optimal field outperforms the previous one.

Phase control in the vibrational qubit

In order to use molecular vibrations for quantum information processing one should be able to shape infrared laser pulses so that they can play the role of accurate quantum gates and drive the required vibrational transitions. In this paper we studied theoretically how the relative phase of the optimized transitions affects accuracy of the quantum gates in such a system. Optimal control theory and numerical propagation of laser-driven vibrational wave packets were employed. The dependencies observed for one-qubit gates NOT, pi-rotation, and Hadamard transform are qualitatively similar to each other. The results of the numerical tests agree well with the analytical predictions.

Quantum optimal control: Hessian analysis of the control landscape

Seeking an effective quantum control entails searching over a landscape defined as the objective as a functional of the control field. This paper considers the problem of driving a state-to-state transition in a finite level quantum system, and analyzes the local topology of the landscape of the final transition probability in terms of the variables specifying the control field. Numerical calculation of the eigenvalues of the Hessian of the transition probability with respect to the control field variables reveals systematic structure in the spectra reflecting the existence of a generic and simple control landscape topology. An illustration shows that the number of nonzero Hessian eigenvalues is determined by the number of quantum states in the system. The Hessian eigenvectors associated with its nonzero eigenvalues are shown to give insight into the cooperative roles of the control variables. The practical consequences of these findings for quantum control are discussed.

Pulse shaping for optimal control of molecular processes

In this paper, a new method is proposed to design optimized control fields with desired temporal and/or spectral properties. The method is based on penalizing the difference between an optimized field obtained from an iterative scheme and a reference field with desired temporal and/or spectral properties. Compared with the standard optimal control theory, the current method allows a simple, experimentally accessible field be found on the fly; while compared with parameter space searching optimization, the iterative nature of this method allows automatic exploration of the intrinsic mechanism of the population transfer. The method is illustrated by examing the optimal control of vibrational excitation of the Cl-O bond with both temporally and spectrally restricted pulses.

Implementation of quantum gate operations in molecules with weak laser fields

We numerically propose a way to perform quantum computations by combining an ensemble of molecular states and weak laser pulses. A logical input state is expressed as a superposition state (a wave packet) of molecular states, which is initially prepared by a designed femtosecond laser pulse. The free propagation of the wave packet for a specified time interval leads to the specified change in the relative phases among the molecular basis states, which corresponds to a computational result. The computational results are retrieved by means of quantum interferometry. Numerical tests are implemented in the vibrational states of the B state of I2 employing controlled-NOT gate, and 2 and 3 qubits Fourier transforms. All the steps involved in the computational scheme, i.e., the initial preparation, gate operation, and detection steps, are achieved with extremely high precision.

Manganese pentacarbonyl bromide as candidate for a molecular qubit system operated in the infrared regime

Our concept for a quantum computational system is based on qubits encoded in vibrational normal modes of polyatomic molecules. The quantum gates are implemented by shaped femtosecond laser pulses. We adopt this concept to the new species manganese pentacarbonyl bromide [MnBr(CO)5] and show that it is a promising candidate in the mid-infrared (IR) frequency range to connect theory and experiment. As direct reference for the ab initio calculations we evaluated experimentally the absorption bands of MnBr(CO)5 in the mid-IR as well as the related transition dipole moments. The two-dimensional potential-energy surface spanned by the two strongest IR active modes and the dipole vector surfaces are calculated with density-functional theory. The vibrational eigenstates representing the qubit system are determined. Laser pulses are optimized by multitarget optimal control theory to form a set of global quantum gates: NOT, CNOT, Pi, and Hadamard. For all of them simply structured pulses with low pulse energies around 1 microJ could be obtained. Exemplarily for the CNOT gate we investigated the possible transfer to experimental shaping, based on the mask function for pulse shaping in the frequency regime as well as decomposition into a train of subpulses.