Modern Physics Letters A

THE TRAPPED-ION QUBIT: COHERENT CONTROL IN INFINITE-DIMENSIONAL QUANTUM SYSTEMS

Chitra Rangan

Published Oct 20, 2009• 0 citations

Abstract

Theories of quantum control have, until recently, made the assumption that the Hilbert space of a quantum system can be truncated to finite dimensions. Such truncations, which can be achieved for most quantum systems via bandwidth restrictions, have enabled the development of a rich variety of quantum control and optimal control schemes. Recent studies in quantum information processing have addressed the control of infinite-dimensional quantum systems such as the quantum states of a trapped-ion. Controllability in an infinite-dimensional quantum system is hard to prove with conventional methods, and infinite-dimensional systems provide unique challenges in designing control fields. In this paper, we will discuss the control of a popular system for quantum computing the trapped-ion qubit. This system, modeled by a spin-half particle coupled to a quantized harmonic oscillator, is an example for a surprisingly rich variety of control problems. We will show how this infinite-dimensional quantum system can be examined via the lens of the Finite Controllability Theorem, two-color STIRAP, the generalized Heisenberg system, etc. These results are important from the viewpoint of developing more efficient quantum control protocols, particularly in quantum computing systems. This work shows how one can expand the scope of quantum control research to beyond that of finite-dimensional quantum systems.

DOI: 10.1142/s0217732309032022

Related papers

Control of finite-dimensional quantum systems: Application to a spin-12 particle coupled with a finite quantum harmonic oscillator

Chitra Rangan, Anthony M. Bloch

In this paper, we consider the problem of the controllability of a finite-dimensional quantum system in both the Schrödinger and interaction pictures. Introducing a Quantum Transfer Graph, we elucidate the role of Lie algebra rank conditions and the complex nature of the control matrices. We analyze the example of a sequentially coupled N-level system: a spin-12 particle coupled to a finite quantum harmonic oscillator. This models an important physical paradigm of quantum computers—the trapped ion. We describe the control of the finite model obtained, under the right conditions, from the original infinite-dimensional system.

Journal of Mathematical PhysicsFeb 8, 20059 citations

Quantum computation and simulation with vibrational modes of trapped ions

Wentao Chen, Jaren Gan, Jing-Ning Zhang +2 more

Vibrational degrees of freedom in trapped-ion systems have recently been gaining attention as a quantum resource, beyond the role as a mediator for entangling quantum operations on internal degrees of freedom, because of the large available Hilbert space. The vibrational modes can be represented as quantum harmonic oscillators and thus offer a Hilbert space with infinite dimensions. Here we review recent theoretical and experimental progress in the coherent manipulation of the vibrational modes, including bosonic encoding schemes in quantum information, reliable and efficient measurement techniques, and quantum operations that allow various quantum simulations and quantum computation algorithms. We describe experiments using the vibrational modes, including the preparation of non-classical states, molecular vibronic sampling, and applications in quantum thermodynamics. We finally discuss the potential prospects and challenges of trapped-ion vibrational-mode quantum information processing.

arXiv:2103.14299Chinese Physics BMay 17, 202138 citations

Robust Quantum Control & Verification & Quantum Simulation with Trapped Ions

Harrison Ball

This PhD thesis reports on the development of a wide variety of theoretical and experimental capabilities in the field of quantum information processing using trapped ions. The presentation is organized in two distinct parts. The first part presents a series of published articles containing theoretical results, and their experimental validation, advancing a range of tools and techniques in error-resilient quantum control and robust verification procedures. One of the fundamental challenges in realizing a general purpose quantum computer is the management of error in fragile quantum systems. A familiar aspect of this challenge is decoherence, a process by which the system looses its stored quantum information, rendering it useless for subsequent computation. This collection of works focuses on reducing sources of quantum decoherence and precisely quantifying residual errors in real quantum systems dominated by realistic temporally-correlated noise processes, of central importance in any future architecture for quantum computing. The second part focuses on the design, construction and characterization of a complex experimental system capable of supporting a broad calls of quantum simulation experiments with hundreds of spin qubits using 9Be+ ions in a Penning trap. The trap design is optimized to suppress non-harmonic trapping potentials while providing wide optical access for engineering enhanced spin-motion coupling across large ion crystals. A superconducting magnet, equipped with a fully closed-cycle dual LN2/LHe recondensor, provides the magnetic field for trapping and produces the 55-GHz Zeeman-split qubit states suitable for microwave control. A custom UV laser system for driving the relevant transitions in neutral and ionized beryllium, as well as UV-compatible fibers and in-bore optomechanics for delivering microwave and laser beams to the trap, is presented. The thesis concludes with a brief outlook towards future theoretical and experimental studies, and extensions of the experimental setup

Feb 21, 20180 citations

Control of a quantum model for two trapped ions

Esteban Paduro, Mario Sigalotti

We prove approximate controllability for a bilinear quantum system modeling a pair of trapped ions. The proof is based on a spectral decoupling technique, which exploits finite-dimensional considerations to tackle the infinite-dimensional controllability problem.

IEEE Conference on Decision and Control (CDC)Dec 1, 20153 citations

Preparing encoded states in an oscillator

B. C. Travaglione, Gerard J Milburn

Recently a scheme has been proposed for constructing quantum error-correcting codes that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. One of the difficult steps in this scheme is the preparation of the encoded states. We show how these states can be generated by coupling a continuous quantum variable to a single qubit. An ion trap quantum computer provides a natural setting for a continuous system coupled to a qubit. We discuss how encoded states may be generated in an ion trap.

arXiv:quant-ph/0205114Physical Review ANov 22, 200268 citations

Control and Measurement of a Single-Ion Quantum Harmonic Oscillator

Katherine C. McCormick

A trapped-ion harmonic oscillator represents a rich and well-controlled platform for tests of quantum physics, with applications to quantum simulations, quantum sensing, and quantum information processing. Owing to its net charge, the ion's motion is affected by changes in the electric potential. Consequently, electric field and potential curvature noise will limit the coherence of the ion's harmonic oscillation. While in some respects, this serves as a limitation in the performance of the trapped ion system (for example, in the fidelity of two-qubit gates or quantum simulations with motional quanta), it also opens an opportunity to use the ion as a precise field sensor and to use engineered quantum states of motion to enhance the sensitivity to these fields beyond the standard quantum limit (SQL). This thesis describes the implementation of such applications on a single beryllium ion confined in a Paul trap. In the thesis, I present the theory of an ideal harmonic oscillator and the implementation of this system using the motion of a single trapped ion, including producing special quantum states of motion; these states include the energy eigenstates of the harmonic oscillator, called number states, up to n=100, superpositions of number states of the form (1/√2)(|0> + |n>), with n up to 18, and coherently displaced number states with an average occupation up to approximately 300. I describe how we use these quantum states of motion for precise sensing of fluctuations of the harmonic oscillator frequency. The sensitivity of the (1/√2)(|0> + |n>) state ideally follows the 1/n Heisenberg scaling for frequency sensitivity. Additionally, I describe investigations of the spectrum of motional frequency noise using a series of coherent displacements of the motion of the ion, with features similar to Ramsey and dynamical decoupling sequences for two-level systems. While these displaced states don't improve sensitivity over the SQL, they can be simply and rapidly implemented with trapped ions. I discuss extensions to all of these experiments, including to multiple ions in a new trap designed to confine 3 or 4 ions in individually tunable potential wells.

Jan 1, 20191 citation