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arXiv:quant-ph/0601141 · Physical Review A

Scalable designs for quantum computing with rare-earth-ion-doped crystals

Janus H. Wesenberg, Klaus Mølmer, Lars Rippe, Stefan Kröll

Published Jan 3, 2007• 100 citations
arXivHTMLPDF

Abstract

Due to inhomogeneous broadening, the absorption lines of rare-earth-ion dopants in crystals are many order of magnitudes wider than the homogeneous linewidths. Several ways have been proposed to use ions with different inhomogeneous shifts as qubit registers, and to perform gate operations between such registers by means of the static dipole coupling between the ions. In this paper we show that in order to implement high-fidelity quantum gate operations by means of the static dipole interaction, we require the participating ions to be strongly coupled, and that the density of such strongly coupled registers in general scales poorly with register size. Although this is critical to previous proposals which rely on a high density of functional registers, we describe architectures and preparation strategies that will allow scalable quantum computers based on rare-earth-ion-doped crystals.

DOI
10.1103/physreva.75.012304

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Due to inhomogeneous broadening, the absorption lines of rare-earth-ion dopants in crystals are many order of magnitudes wider than the homogeneous linewidths. Several ways have been proposed to use ions with different inhomogeneous shifts as qubit registers, and to perform gate operations between such registers by means of the static dipole coupling between the ions. In this paper we show that in order to implement high-fidelity quantum gate operations by means of the static dipole interaction, we require the participating ions to be strongly coupled, and that the density of such strongly coupled registers in general scales poorly with register size. Although this is critical to previous proposals which rely on a high density of functional registers, we describe architectures and preparation strategies that will allow scalable quantum computers based on rare-earth-ion-doped crystals.
In the last decade, the solid-state rare-earth-ion system has demonstrated increasing appeal for quantum computation. Despite this progress, two hardware limitations prevent a scalable implementation. These current limitations are the inability to miniaturise and the inability to perform single-ion qubit readout. This thesis addresses both these limitations and demonstrates that neither poses a fundamental restriction on increasing the scale of rare-earth-ion quantum computers. The challenge in miniaturising rare-earth-ion quantum hardware arises from the increases in homogeneous and inhomogeneous broadening that accompany micron-scale architectures. The success of miniaturised architectures, such as waveguides, depends on the properties of bulk ions being preserved within microns of the crystal surface. In addition, bulk ion properties must also be preserved in regions of high residual stress resultant from waveguide fabrication. The inhomogeneous and homogeneous properties of near-surface ions and ions in highly stressed environments in Pr3+:Y2SiO5 were studied via white light interferometry combined with micron resolution fluorescence microscopy. It was found that the bulk ion properties could be preserved both near the surface and in highly stressed regions close to micron-scale surface damage. The observation of excess inhomogeneous and homogeneous broadening was found to be consistent with the damage present at the crystal surface. The main outcome of the study was a set of waveguide fabrication guidelines to ensure that the appealing properties of bulk crystals can be maintained in a miniaturised architecture. The current inability to perform single-ion qubit readout is a consequence of the difficulty in isolating a single ion and lack of cyclicity in rare-earth-ion materials. Two techniques are proposed to form a solution: Stark activation and Zeeman enhanced cyclicity. When combined, these techniques offer direct readout for single-ion frequency-based quantum computing. Stark activation is designed to isolate a single rare-earth ion in a macroscopic crystal. The proposal is based on defining the condition for resonant excitation through a spatially varying electric field. A proof-of-principle experiment successfully created a 10 um absorption region within a millimetre thick crystal. In addition, the signal-to-noise ratio of the technique was characterised in experiments probing Pr3+:Y2SiO5 at the single-ion level. Future improvements to the apparatus should allow the nanometre spatial resolution and the noise level to be reduced to allow single-ion optical detection. High cyclicity is essential for high-fidelity optical readout of a single-ion qubit. Zeeman enhanced cyclicity achieves this by manipulating the hyperfine structure of the resonant crystal field levels to induce strong hyperfine selection rules. The technique is shown to be applicable to even the lowest symmetry sites. The simulated level of cyclicity in Pr3+:Y2SiO5 was greater than 99.99% by applying a 10 T field. The investigation of scalability in the rare-earth-ion system marks a movement away from the traditional ensemble-based methods in macroscopic crystals. This study required an understanding of these materials at a single-ion level and the development of high spatial resolution spectroscopic techniques. These advances extend the ability to engineer rare-earth-ion systems for applications including, but not limited to, quantum computing.
Quantum computers based on rare-earth-ion-doped crystals show promising properties in terms of scalability and connectivity if single ions can be used as qubits. Through simulations, we investigate gate operations on such qubits. We discuss how gate and system parameters affect gate errors, the required frequency bandwidth per qubit, and the risk for instantaneous spectral diffusion (ISD). Furthermore, we examine how uncertainties in the system parameters affect the gate errors, and how precisely the system needs to be known. We find gate errors for arbitrary single-qubit gates of $2.1\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ when ISD is not considered and $3.4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ when we take heed to minimize it. Additionally, we construct two-qubit gates with errors ranging from $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}\ensuremath{\rightarrow}3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$ over a broad range of dipole-dipole interaction strengths.
In this paper initial experiments towards constructing simple quantum gates in a solid state material are presented. Instead of using specially tailored materials, the aim is to select a subset of randomly distributed ions in the material, which have the interaction necessary to control each other and therefore can be used to do quantum logic operations. The experimental results demonstrate that part of an inhomogeneously broadened absorption line can be selected as a qubit and that a subset of ions in the material can control the resonance frequency of other ions. This opens the way for the construction of quantum gates in rare-earth-ion doped crystals.
We present a composite pulse controlled phase gate which, together with a bus architecture, improves the feasibility of a recent quantum computing proposal based on rare-earth-ion-doped crystals. The proposed gate operation is tolerant to variations between ions of coupling strengths, pulse lengths, and frequency shifts. In the absence of decoherence effects, it achieves worst case fidelities above 0.9990.9990.999 with relative variations in coupling strength as high as 1010%10 and frequency shifts up to several percent of the resonant Rabi frequency of the laser used to implement the gate. We outline an experiment to demonstrate the creation and detection of maximally entangled states in the system.
Rare-earth ions doped into a transparent crystal can have optically excited states with coherence times of milliseconds and hyperfine levels with coherence times of hours when cooled down to liquid helium temperatures. This makes them attractive for developing quantum computing hardware. A quantum transistor can be implemented in a rare-earth-ion-doped crystal by utilizing a dipole blockade effect, where one excited ion prevents nearby ions from being excited. By using single rare-earth ions as qubits, the quantum processor is believed to be scalable to several qubits. In this thesis a readout mechanism for qubits represented by single ions (Eu or Pr) in a Y2SiO5 crystal is explored. A single cerium ion, in the same crystal as the qubits, could potentially be used as an indicator ion for non-demolition quantum state detection of the qubits. Cerium has a strong transition with high fluorescence yield which makes it possible to detect single ions. A cerium ion can be controlled by a qubit via the dipole blockade effect, and fluoresce if the qubit is in state |1> but stay dark if the qubit is in state |0>. Relevant spectroscopic properties of Ce:Y2SiO5 are studied, as well as interactions between cerium ions and qubit ions. The inhomogeneous and homogeneous linewidth of the Ce:Y2SiO5 4f-5d transition is measured and fluorescence spectra for ions in site 1 and site 2 are characterized. The change in electric dipole moment as a cerium ion is brought to the excited state is measured. The static dipole interaction between a single cerium ion and a qubit ion is shown to be strong enough for efficient qubit readout. Energy transfer from cerium to qubit ions is investigated. A lower bound on the cerium-qubit distance is found in order to avoid energy transfer. Satellite lines belonging to cerium-qubit pairs are mapped out. A microscope set-up is built to detect single ions in a crystal. Single ion detection is believed to be within reach.
An overview of current status and prospects of the development of quantum computer hardware based on inorganic crystals doped with rare-earth ions are presented. Major parts of the experimental work in this area have been done in two places, Canberra, Australia and Lund, Sweden, and the present description follows more closely the Lund work. Techniques will be described that include optimal filtering of the initially inhomogeneously broadened profile down to well separated and narrow ensembles, as well as the use of advanced pulse-shaping in order to achieve robust arbitrary single-qubit operations with fidelities above 90%, as characterized by quantum state tomography. It is expected that full scalability of these systems will require the ability to determine the state of single rare-earth ions. It has been proposed that this can be done using special readout ions doped into the crystal and an update is given on the work to find and characterize such ions. Finally, a few aspects on possibilities for remote entanglement of ions in separate rare-earth-ion-doped crystals are considered.