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The Journal of Chemical Physics

Feasibility of encoding Shor's algorithm into the motional states of an ion in the anharmonic trap

Lei Wang, Dmitri Babikov

Published Aug 8, 2012• 6 citations
Publisher

Abstract

We demonstrate theoretically that it may be possible to encode states of a multi-qubit system into the progression of quantized motional∕vibrational levels of an ion trapped in a weakly anharmonic potential. Control over such register of quantum information is achieved by applying oscillatory radio-frequency fields shaped optimally for excitation of the desired state-to-state transitions. Anharmonicity of the vibrational spectrum plays a key role in this approach to the control and quantum computation, since it allows resolving different state-to-state transitions and addressing them selectively. Optimal control theory is used to derive pulses for implementing the four-qubit version of Shor's algorithm in a single step. Accuracy of the qubit-state transformations, reached in the numerical simulations, is around 0.999. Very detailed insight is obtained by analysis of the time-evolution of state populations and by spectral analysis of the optimized pulse.

DOI
10.1063/1.4742309

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The Journal of Chemical Physics•Jul 16, 2009•13 citations
We demonstrate theoretically that it may be possible to encode states of a multi-qubit system into the progression of quantized motional∕vibrational levels of an ion trapped in a weakly anharmonic potential. Control over such register of quantum information is achieved by applying oscillatory radio-frequency fields shaped optimally for excitation of the desired state-to-state transitions. Anharmonicity of the vibrational spectrum plays a key role in this approach to the control and quantum computation, since it allows resolving different state-to-state transitions and addressing them selectively. Optimal control theory is used to derive pulses for implementing the four-qubit version of Shor's algorithm in a single step. Accuracy of the qubit-state transformations, reached in the numerical simulations, is around 0.999. Very detailed insight is obtained by analysis of the time-evolution of state populations and by spectral analysis of the optimized pulse.
A method for encoding a multiqubit system into the quantized motional states of ion string in an anharmonic linear trap is proposed. Control over this system is achieved by applying oscillatory electric fields (rf) shaped optimally for desired state-to-state transitions. Anharmonicity of the vibrational spectrum of the system plays a key role in this approach to the control and quantum computation, since it allows resolving different state-to-state transitions and addressing them selectively. The anharmonic trap architecture proposed earlier [Phys. Rev. A 83, 022305 (2011)] is explored here and the optimal control theory is used to derive pulses for a set of universal quantum gates. An accurate choice of pulse parameters allows deriving gates that are both accurate and simple. A practical realization of this approach seems to be within the reach of today's technology.
A string of cold ions confined in a linear trap represents a man-made quantum object with a broad range of applications in atomic and molecular spectroscopy, such as high-precision measurement of atomic properties. An efficient isolation from the environment guarantees excellent coherent properties of such systems and makes them suitable for practical realization of the quantum information processing. The pioneering theoretical [Cirac and Zoller] and experimental [Wineland and Monroe] work in 1990s resulted in the explosive expansion of this field during the last decade. In this dissertation an alternative new method for controlling the quantized motional/vibrational states of ions in a trap is explored theoretically. It is proposed to create small anharmonicity in the trapping potential which would modify the spectrum of states and allow addressing the state-to-state transitions selectively. In this approach all ions remain in the ground electronic state and their motion is controlled adiabatically and coherently by applying the optimally shaped electric fields (RF). The optimal control theory, accurate numerical calculation of the energies and wavefunctions, and numerical propagation of wave packets are employed. Two sources of vibrational anharmonicity are studied: the intrinsic Coulomb anharmonicity due to the ion-ion interactions and the external anharmonicity of the trapping potential. It is shown that the magnitude of Coulomb anharmonicity is insufficient for the control. In contrast, anharmonicity of the trapping potential allows controlling the motion of ions very accurately. It is demonstrated that one ion in a slightly anharmonic trap can be easily controlled and used to represent one qubit. A multi-qubit system can be created by employing a long progression of states of a single ion, or by trapping multiple ions and controlling several normal vibration modes of the ion string. Up to four qubits are modeled in this work and accurate pulses are optimized for a set of universal quantum gates: NOT, conditional NOT (CNOT) and Hadamard transformation. The control field for Shor's algorithm (quantum algorithm for factorization onto prime numbers) is also obtained. It is demonstrated that a careful choice of system properties allows achieving very high accuracy of qubit transformations, up to 0.999.
Anharmonicity of the quantized motional states of ions in a Paul trap can be utilized to address the state-to-state transitions selectively and control the motional modes of trapped ions coherently and adiabatically [Zhao and Babikov, Phys. Rev. A 77, 012338 (2008)]. In this paper we study two sources of the vibrational anharmonicity in the ion traps: the intrinsic Coulomb anharmonicity due to ion-ion interactions and the external anharmonicity of the trapping potential. An accurate numerical approach is used to compute energies and wave functions of vibrational eigenstates. The magnitude of the Coulomb anharmonicity is determined and shown to be insufficient for successful control. In contrast, anharmonicity of the trapping potential allows one to control the motion of ions very efficiently using the time-varying electric fields. Optimal control theory is used to derive the control pulses. One ion in a slightly anharmonic trap can be easily controlled. In the two- and three-ion systems the symmetric stretching mode is dark and cannot be controlled at all. The other two normal modes of the three-ion system can be controlled and used, for example, to encode a two-qubit system into the motional states of ions. A trap architecture that allows the necessary amount of vibrational anharmonicity to be achieved is proposed.
It is widely believed that Shor’s factoring algorithm provides a driving force to boost the quantum computing research. However, a serious obstacle to its binary implementation is the large number of quantum gates. Non-binary quantum computing is an efficient way to reduce the required number of elemental gates. Here, we propose optimization schemes for Shor’s algorithm implementation and take a ternary version for factorizing 21 as an example. The optimized factorization is achieved by a two-qutrit quantum circuit, which consists of only two single qutrit gates and one ternary controlled-NOT gate. This two-qutrit quantum circuit is then encoded into the nine lower vibrational states of an ion trapped in a weakly anharmonic potential. Optimal control theory (OCT) is employed to derive the manipulation electric field for transferring the encoded states. The ternary Shor’s algorithm can be implemented in one single step. Numerical simulation results show that the accuracy of the state transformations is about 0.9919.
Following a recent proposal of L. Wang and D. Babikov [J. Chem. Phys. 137, 064301 (2012)], we theoretically illustrate the possibility of using the motional states of a Cd(+) ion trapped in a slightly anharmonic potential to simulate the single-particle time-dependent Schrödinger equation. The simulated wave packet is discretized on a spatial grid and the grid points are mapped on the ion motional states which define the qubit network. The localization probability at each grid point is obtained from the population in the corresponding motional state. The quantum gate is the elementary evolution operator corresponding to the time-dependent Schrödinger equation of the simulated system. The corresponding matrix can be estimated by any numerical algorithm. The radio-frequency field which is able to drive this unitary transformation among the qubit states of the ion is obtained by multi-target optimal control theory. The ion is assumed to be cooled in the ground motional state, and the preliminary step consists in initializing the qubits with the amplitudes of the initial simulated wave packet. The time evolution of the localization probability at the grids points is then obtained by successive applications of the gate and reading out the motional state population. The gate field is always identical for a given simulated potential, only the field preparing the initial wave packet has to be optimized for different simulations. We check the stability of the simulation against decoherence due to fluctuating electric fields in the trap electrodes by applying dissipative Lindblad dynamics.
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.