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Quantum Computation in Large Ion Crystals

Yukai Wu

Published Jan 1, 2019• 3 citations

Abstract

Ion trap is one of the most promising candidates for quantum computing. High-fidelity gates have been demonstrated in small ion crystals and schemes like ion shuttling have been proposed for larger systems. This thesis discusses the possibility of direct quantum computing on a large ion crystal in a Paul trap, without any shuttling of the ions. We first review a scheme to entangle two ions in a small ion crystal mediated by the collective phonon modes and analyze the gate errors. The generalization to larger systems is divided into three parts. (1) We present numerical methods to solve all the normal modes of the ion crystal, including the micromotion, up to arbitrary precision. The stability of the crystal under infinitesimal perturbation is ensured when all the normal modes have real frequencies. For finite disturbance, direct molecular dynamics simulation will be needed; after discussing some potential problems in the simulation, we give a rule of thumb for the ion crystal to be stable at a given temperature. (2) We show that when designing an entangling gate between two nearby ions, all the ions far away can be neglected, so that only a small number of normal modes are relevant. Similarly, distant entangling gates can be applied in parallel and the crosstalk is shown to decay cubically with the distance between these gates. (3) Then we present numerical methods to include the solved micromotion into the design of the entangling gates efficiently, again up to arbitrary precision. Thus we conclude that the design and the implementing of entangling gates are scalable in a large ion crystal. Finally we consider a near-term application to simulate an all-to-all coupled Ising model in a small to medium-sized ion crystal.

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Ion trap is one of the most promising candidates for quantum computing. High-fidelity gates have been demonstrated in small ion crystals and schemes like ion shuttling have been proposed for larger systems. This thesis discusses the possibility of direct quantum computing on a large ion crystal in a Paul trap, without any shuttling of the ions. We first review a scheme to entangle two ions in a small ion crystal mediated by the collective phonon modes and analyze the gate errors. The generalization to larger systems is divided into three parts. (1) We present numerical methods to solve all the normal modes of the ion crystal, including the micromotion, up to arbitrary precision. The stability of the crystal under infinitesimal perturbation is ensured when all the normal modes have real frequencies. For finite disturbance, direct molecular dynamics simulation will be needed; after discussing some potential problems in the simulation, we give a rule of thumb for the ion crystal to be stable at a given temperature. (2) We show that when designing an entangling gate between two nearby ions, all the ions far away can be neglected, so that only a small number of normal modes are relevant. Similarly, distant entangling gates can be applied in parallel and the crosstalk is shown to decay cubically with the distance between these gates. (3) Then we present numerical methods to include the solved micromotion into the design of the entangling gates efficiently, again up to arbitrary precision. Thus we conclude that the design and the implementing of entangling gates are scalable in a large ion crystal. Finally we consider a near-term application to simulate an all-to-all coupled Ising model in a small to medium-sized ion crystal.

An ion trap is one of the most promising candidates for quantum computing. Current schemes mainly focus on a linear chain of up to about 100 ions in a Paul trap. To further scale up the qubit number, one possible direction is to use 2D or 3D ion crystals (Wigner crystals). In these systems, ions are generally subjected to large micromotion due to the strong fast-oscillating electric field, which can significantly influence the performance of entangling gates. In this paper, we develop an efficient numerical method to design high-fidelity entangling gates in a general 3D ion crystal. We present numerical algorithms to solve the equilibrium configuration of the ions and their collective normal modes. We then give a mathematical description of the micromotion and use it to generalize the gate scheme for linear ion chains into a general 3D crystal. The involved time integral of highly oscillatory functions is expanded into a fast-converging series for accurate and efficient evaluation and optimization. As a numerical example, we show a high-fidelity entangling gate design between two ions in a 100-ion crystal, with a theoretical fidelity above 99.9%.

Ion trap quantum computers are based on modulating the Coulomb interaction between atomic ion qubits using external forces. However, the spectral crowding of collective motional modes could pose a challenge to the control of such interactions for large numbers of qubits. Here, we show that high-fidelity quantum gate operations are still possible with very large trapped ion crystals by using a small and fixed number of motional modes, simplifying the scaling of ion trap quantum computers. We present analytical work that shows that gate operations need not couple to the motion of distant ions, allowing parallel entangling gates with a crosstalk error that falls off as the inverse cube of the distance between the pairs. We also experimentally demonstrate high-fidelity entangling gates on a fully connected set of seventeen

^{171}\mathrm{Yb}^{+}

qubits using simple laser pulse shapes that primarily couple to just a few modes.

We propose a scheme to realize scalable quantum computation in a planar ion crystal confined by a Paul trap. We show that the inevitable in-plane micromotion affects the gate design via three separate effects: renormalization of the equilibrium positions, coupling to the transverse motional modes, and amplitude modulation in the addressing beam. We demonstrate that all of these effects can be taken into account and high-fidelity gates are possible in the presence of micromotion. This proposal opens the prospect to realize large-scale fault-tolerant quantum computation within a single Paul trap.

Abstract Trapped-ion has shown great advantages in building quantum computers. While high fidelity entangling-gate has been realized for a few ions, how to maintain the high fidelity for large scale trapped-ions remains an open problem. Here, we present an analysis of arbitrary scale ion chains and focus on motional-related errors, reported as one of the leading error sources in state-of-the-art experiments. We theoretically analyze two-qubit entangling-gate infidelity in a large ion crystal under the phase insensitive configuration. To verify our result, we develop an efficient numerical simulation algorithm that avoids exponential increases in of the Hilbert space dimension. For the motional heating error, we derive a much tighter bound of gate infidelity than previously estimated O ( N Γ τ ) , and we give an intuitive understanding from the trajectories in the phase space of motional modes. Our discoveries may inspire the scheme of pulse design against incoherent errors and shed light on the way toward constructing scalable quantum computers with large ion crystals.

Two- or three-dimensional Paul traps can confine a large number of ions forming a Wigner crystal, which would provide an ideal architecture for scalable quantum computation except for the micromotion, an issue that is commonly believed to be the obstacle for any high-fidelity quantum gate. Here we show that the obstacle of micromotion can be overcome with current technology, even though the magnitude of the micromotion is way outside the Lamb-Dicke region. Through exact solution of the quantum Mathieu equations, we demonstrate the principle of the gate design under micromotion using two ions in a quadrupole Paul trap as an example. The proposed micromotion quantum gates can be extended to the many-ion case and may pave a new way for scalable trapped-ion quantum computation.

Fast entangling gate operations are a fundamental prerequisite for quantum simulation and computation. We propose an entangling scheme for arbitrary pairs of ions in a linear crystal, harnessing the high electric polarizability of highly excited Rydberg states. An all-to-all quantum gate connectivity is based on an initialization of a pair of ions to a superposition of ground and Rydberg states by laser excitation, followed by the entangling gate operation, which relies on a state-dependent frequency shift of collective vibrational modes of the crystal. This gate operation requires applying an electric waveform to trap electrodes. Employing transverse collective modes of oscillation, we reveal operation times on the order of microseconds within any of the qubit pairs in a small crystal. In our calculation, we take into account realistic experimental conditions and feasible electric field ramps. The proposed gate operation is ready to be combined with a scalable processor architecture to reconfigure the qubit register, either by shuttling ions or by dynamically controlling optical tweezer potentials.