Accuracy of gates in a quantum computer based on vibrational eigenstates

A model is developed to study the properties of a quantum computer that uses vibrational eigenstates of molecules to implement the quantum information bits and shaped laser pulses to apply the quantum logic gates. Particular emphasis of this study is on understanding how the different factors, such as properties of the molecule and of the pulse, can be used to affect the accuracy of quantum gates in such a system. Optimal control theory and numerical time-propagation of vibrational wave packets are employed to obtain the shaped pulses for the gates NOT and Hadamard transform. The effects of the anharmonicity parameter of the molecule, the target time of the pulse and of the penalty function are investigated. Influence of all these parameters on the accuracy of qubit transformations is observed and explained. It is shown that when all these parameters are carefully chosen the accuracy of quantum gates reaches 99.9%.     

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.     

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.     

The operations of quantum logic gates with pure and mixed initial states

The implementations of quantum logic gates realized by the rovibrational states of a C(12)O(16) molecule in the X((1)Σ(+)) electronic ground state are investigated. Optimal laser fields are obtained by using the modified multitarget optimal theory (MTOCT) which combines the maxima of the cost functional and the fidelity for state and quantum process. The projection operator technique together with modified MTOCT is used to get optimal laser fields. If initial states of the quantum gate are pure states, states at target time approach well to ideal target states. However, if the initial states are mixed states, the target states do not approach well to ideal ones. The process fidelity is introduced to investigate the reliability of the quantum gate operation driven by the optimal laser field. We found that the quantum gates operate reliably whether the initial states are pure or mixed.     

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

Optimal control theory is applied to a molecular vibrational system in light of its possible application to quantum computing (QC). We present the numerical results of an ammonia molecular vibrational model system with two modes: a bending mode and an asymmetric stretching mode. We demonstrate logic gates fundamental to QC algorithms, namely Hadamard and controlled-NOT gates. Our results show that averages of population transfers at each gate are above 93% high fidelity. A mode that has a double-well structured potential is found to have many transfer pathways, which facilitates obtaining optimal laser pulses.     

Design of an infrared laser pulse to control the multiphoton dissociation of the Fe-CO bond in CO-heme compounds

Optimal control theory is used to design a laser pulse for the multiphoton dissociation of the Fe-CO bond in the CO-heme compounds. The study uses a hexacoordinated iron-porphyrin-imidazole-CO complex in its ground electronic state as a model for CO liganded to the heme group. The potential energy and dipole moment surfaces for the interaction of the CO ligand with the heme group are calculated using density functional theory. Optimal control theory, combined with a time-dependent quantum dynamical treatment of the laser-molecule interaction, is then used to design a laser pulse capable of efficiently dissociating the CO-heme complex model. The genetic algorithm method is used within the mathematical framework of optimal control theory to perform the optimization process. This method provides good control over the parameters of the laser pulse, allowing optimized pulses with simple time and frequency structures to be designed. The dependence of photodissociation yield on the choice of initial vibrational state and of initial laser field parameters is also investigated. The current work uses a reduced dimensionality model in which only the Fe-C and C-O stretching coordinates are explicitly taken into account in the time-dependent quantum dynamical calculations. The limitations arising from this are discussed in Sec. IV.     

An implementation of the Deutsch–Jozsa algorithm on molecular vibronic coherences through four-wave mixing: a theoretical study

Time-frequency resolved coherent anti-Stokes Raman scattering (TFRCARS) was recently proposed as a means to implement quantum logic using the molecular ro-vibrational manifold as a quantum register [R. Zadoyan et al., Chem. Phys. 266 (2001) 323]. We give a concrete example of how this can be accomplished through an illustrative algorithm that solves the Deutsch–Jozsa (DJ) problem. We use realistic molecular parameters to recognize that, as the problem size expands, shaped pulses must be tailored to maintain fidelity of the algorithm.     

Mechanisms of local and global molecular quantum gates and their implementation prospects

We explore how the globality of quantum logic operations is ensured in the context of optimal control theory when qubits are encoded in vibrational eigenstates of different normal modes and specially shaped laser fields act as quantum logic operations. In a two-qubit model system, transition mechanisms for optimized laser fields generating single qubit flips, local NOT and global NOT and controlled-NOT (CNOT) gates are investigated and compared. We evaluate the participation of vibrational eigenstates beyond the qubit basis in the global gate mechanisms and how different features of CNOT and NOT gates relate to the characteristics of the vibrational manifold. When a non-qubit normal mode interacting via anharmonic resonances is introduced, neither the global gate mechanisms nor the optimized laser fields show a significant increase in complexity. Similar features of the global quantum gates in both model systems indicate a generality of the deduced principles. Finally, a primary concept for a realization of global quantum gates in an actual experiment referring to state-of-the-art techniques is presented. The possible reconstruction of optimized laser fields with sequences of simple Gaussian subpulses is demonstrated and some critical parameters are deduced.     

Simulations of quantum computation with a molecular ion

The capabilities of a new system for coherent control—intense terahertz light pulses acting on trapped, gas phase D₃O⁺ molecules—are investigated using realistic molecular and pulse parameters. Computer simulations show that a set of three shaped pulses can be used to perform four level (two qubit) quantum computational gates on the inversion–rotation energy levels and read-out the result using degenerate four-wave mixing. Two pulse shaping techniques are employed, one directly shaping a terahertz pulse, and another shaping a visible laser pulse that is rectified by a terahertz antenna. Both are found to be effective for control. Methods for initializing the inversion–rotation wavepacket, making the pulse robust against power variations, maximizing its fidelity to a unitary gate transformation and addressing limitations of energy level connectivity are discussed.     

On the control of product yields in the photofragmentation of deuteriumchlorid ions (DCl+)

The prospect of controlling the photofragmentation of deuterium chloride ions (DCl+) via strong ultrashort IR laser pulses has been investigated by a numerical solution of coupled Schrodinger equations. The calculations provide evidence that the ratio of product ion yields Cl+ versus D+ can be manipulated by an appropriate choice of laser pulse parameters, in particular, central laser frequency, pulse duration, intensity, and chirp. The analysis of time-dependent populations reveals competition between intra- and interelectronic state excitations, enabling the understanding of quantum control at the molecular level.     

A simple method for the preparation of pseudopure states in nuclear magnetic resonance quantum information processing

The use of nuclear magnetic resonance (NMR) to carry out quantum information processing (QIP) often requires the preparation, transformation, and detection of pseudopure states. In our previous work, it was shown that the use of pairs of pseudopure states (POPS) as a basis for QIP is very convenient because of the simplicity in experimental execution. It is now further demonstrated that the product of the NMR spectra corresponding to two sets of POPS that share a common pseudopure state has the same peak frequencies as those of the common (single) pseudopure state. Examples of applying two different quantum logic gates to a 5-qubit system are given.     

Coherent control of the population transfer in complex solvated molecules at weak excitation. An experimental study

We performed a series of successful experiments for the optimization of the population transfer from the ground to the first excited state in a complex solvated molecule (rhodamine 101 in methanol) using shaped excitation pulses at very low intensities (1 absorbed photon per 100-500 molecules per pulse). We found that the population transfer can be controlled and significantly enhanced by applying excitation laser pulses with crafted pulse shapes. The optimal shape was found in feedback-controlled experiments using a genetic search algorithm. The temporal profile of the optimal excitation pulse corresponds to a comb of subpulses regularly spaced by approximately 150 fs, whereas its spectrum consists of a series of well-resolved peaks spaced apart by approximately 6.5 nm corresponding to a frequency of 220 cm(-1). This frequency matches very well with the frequency modulation of the population kinetics (period of approximately 150 fs), observed by excitation with a short (approximately 20 fs) transform-limited laser pulse directly after excitation. In addition, an antioptimization experiment was performed under the same conditions. The difference in the population of the excited state for the optimal and antioptimal pulses reaches approximately 30% even at very weak excitation. The results of optimization are reproducible and have clear physical meaning.     

Open-loop and closed-loop control of dissociative ionization of ethanol in intense laser fields

The relative yield of the C-O bond breaking with respect to the C-C bond breaking in ethanol cation C2H5OH+ is maximized in intense laser fields (10(13)-10(15) Wcm2) by open-loop and closed-loop optimization procedures. In the open-loop optimization, a train of intense laser pulses are synthesized so that the temporal separation between the first and last pulses becomes 800 fs, and the number and width of the pulses within a train are systematically varied. When the duration of 800 fs is filled with laser fields by increasing the number of pulses or by stretching all pulses in a triple pulse train, the relative yield of the C-O bond breaking becomes significantly large. In the closed-loop optimization using a self-learning algorithm, the four dispersion coefficients or the phases of 128 frequency components of an intense laser pulse are adopted as optimized parameters. From these optimization experiments it is revealed that the yield ratio of the C-O bond breaking is maximized as far as the total duration of the intense laser field reaches as long as approximately 1 ps and that the intermittent disappearance of the laser field within a pulse does not affect the relative yields of the bond breaking pathways.     

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.     

Optical control of excited-state vibrational coherences of a molecule in solution: The influence of the excitation pulse spectrum and phase in LD690

Spectral and phase shaping of femtosecond laser pulses is used to selectively excite vibrational wave packets on the ground (S0) and excited (S1) electronic states in the laser dye LD690. The transient absorption signals observed following excitation near the peak of the ground-state absorption spectrum are characterized by a dominant 586 cm(-1) vibrational mode. This vibration is assigned to a wave packet on the S0 potential energy surface. When the excitation pulse is tuned to the blue wing of the absorption spectrum, a lower frequency 568 cm(-1) vibration dominates the response. This lower frequency mode is assigned to a vibrational wave packet on the S1 electronic state. The spectrum and phase of the excitation pulse also influence both the dephasing of the vibrational wave packet and the amplitude profiles of the oscillations as a function of probe wavelength. Excitation by blue-tuned, positively chirped pulses slows the apparent dephasing of the vibrational coherences compared with a transform-limited pulse having the same spectrum. Blue-tuned negatively chirped excitation pulses suppress the observation of coherent oscillations in the ground state.     

Quantum control of molecular vibrational and rotational excitations in a homonuclear diatomic molecule: a full three-dimensional treatment with polarization forces

The optimal control of the vibrational excitation of the hydrogen molecule [Balint-Kurti et al., J. Chem. Phys. 122, 084110 (2005)] utilizing polarization forces is extended to three dimensions. The polarizability of the molecule, to first and higher orders, is accounted for using explicit ab initio calculations of the molecular electronic energy in the presence of an electric field. Optimal control theory is then used to design infrared laser pulses that selectively excite the molecule to preselected vibrational-rotational states. The amplitude of the electric field of the optimized pulses is restricted so that there is no significant ionization during the process, and a new frequency sifting method is used to simplify the frequency spectrum of the pulse. The frequency spectra of the optimized laser pulses for processes involving rotational excitation are more complex than those relating to processes involving only vibrational excitation.     

Chemoselective quantum control of carbonyl bonds in Grignard reactions using shaped laser pulses

Grignard reactants like methylmagnesium chloride are not selective with respect to different carbonyl bonds. We present a theoretical study where shaped laser pulses are utilized to prefer specific bonds in a mixture of more than one carbonyl reactant. A mixture of cyclohexanone and cyclopentanone has been chosen as a representative example. The light pulse is supposed to provide the activation energy and to adopt the function of a protecting group. The control aim is to stretch exclusively the C-O bond of one compound to the length required in the Grignard transition state. To guarantee an experimentally realizable bandwidth for the unshaped pulse, we use our recently developed optimal control theory algorithm, which allows the simultaneous optimization of the light field in the time and frequency domain. Highly selective picosecond control pulses could be optimized in the infrared regime suggesting that laser assisted chemoselectivity is possible to a large extent. To obtain control not only on the final product but also on the excitation mechanism, various initial conditions and frequency restrictions were investigated.     

Use of the Gerchberg–Saxton algorithm in optimal coherent anti-Stokes Raman spectroscopy

We are utilizing recent advances in ultrafast laser technology and recent discoveries in optimal shaping of laser pulses to significantly enhance the stand-off detection of explosives via control of molecular processes at the quantum level. Optimal dynamic detection of explosives is a method whereby the selectivity and sensitivity of any of a number of nonlinear spectroscopic methods are enhanced using optimal shaping of ultrafast laser pulses. We have recently investigated the Gerchberg–Saxton algorithm as a method to very quickly estimate the optimal spectral phase for a given analyte from its spontaneous Raman spectrum and the ultrafast laser pulse spectrum. Results for obtaining selective coherent anti-Stokes Raman spectra (CARS) for an analyte in a mixture, while suppressing the CARS signals from the other mixture components, are compared for the Gerchberg–Saxton method versus previously obtained results from closed-loop machine-learning optimization using evolutionary strategies.     

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.     

Molecular wave packet interferometry and quantum entanglement

We study wave packet interferometry (WPI) considering the laser pulse fields both classical and quantum mechanically. WPI occurs in a molecule after subjecting it to the interaction with a sequence of phase-locked ultrashort laser pulses. Typically, the measured quantity is the fluorescence of the molecule from an excited electronic state. This signal has imprinted the interference of the vibrational wave packets prepared by the different laser pulses of the sequence. The consideration of the pulses as quantum entities in the analysis allows us to study the entanglement of the laser pulse states with the molecular states. With a simple model for the molecular system, plus several justified approximations, we solve for the fully quantum mechanical molecule-electromagnetic field state. We then study the reduced density matrices of the molecule and the laser pulses separately. We calculate measurable corrections to the case where the fields are treated classically.