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arXiv:2212.03367 · Physical Review A

Comparison of spontaneous emission in trapped-ion multiqubit gates at high magnetic fields

Allison L. Carter, Sean R. Muleady, Athreya Shankar, Jennifer Lilieholm, Bryce B. Bullock, M. Affolter, Ana María Rey, J. J. Bollinger

Published Apr 21, 2023• 7 citations
arXivHTMLPDF

Abstract

Penning traps have been used for performing quantum simulations and sensing with hundreds of ions and provide a promising route toward scaling up trapped ion quantum platforms because of the ability to trap and control hundreds or thousands of ions in two- and three-dimensional crystals. In both Penning traps and the more common radiofrequency Paul traps, lasers are often used to drive multiqubit entangling operations. A leading source of decoherence in these operations is off-resonant spontaneous emission. While many trapped ion quantum computers or simulators utilize clock qubits, other systems, especially those with high magnetic fields such as Penning traps, rely on Zeeman qubits, which require a more complex calculation of this decoherence. We therefore examine theoretically the impacts of spontaneous emission on quantum gates performed with trapped-ion ground-state Zeeman qubits in a high magnetic field. In particular, we consider two types of gates---light-shift $({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{i}^{z}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{j}^{z})$ gates and M\o{}lmer-S\o{}rensen $({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{i}^{x}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{j}^{x})$ gates---obtained with laser beams directed approximately perpendicular to the magnetic field (the quantization axis) and compare the decoherence errors in each. Within each gate type, we also compare different operating points with regard to the detunings, polarizations, and required intensity of the laser beams used to drive the gates. We show that both gates can have similar performance at their optimal operating conditions at high magnetic fields and examine the experimental feasibility of various operating points. By examining the magnetic field dependence of each gate, we demonstrate that, when the $P$ state fine-structure splitting is large compared to the Zeeman splittings, the theoretical performance of the M\o{}lmer-S\o{}rensen gate is significantly better than that of the light-shift gate. Additionally, for the light-shift gate, we make an approximate comparison between the fidelities that can be achieved at high fields with the fidelities of state-of-the-art two-qubit trapped ion quantum gates. We show that, with regard to spontaneous emission, the achievable infidelity with our current configuration is about an order of magnitude larger than that of the best low-field gates, but we also discuss several alternative configurations with potential error rates that are comparable with those for state-of-the-art trapped ion gates.

DOI
10.1103/physreva.107.042618

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Penning traps have been used for performing quantum simulations and sensing with hundreds of ions and provide a promising route toward scaling up trapped ion quantum platforms because of the ability to trap and control hundreds or thousands of ions in two- and three-dimensional crystals. In both Penning traps and the more common radiofrequency Paul traps, lasers are often used to drive multiqubit entangling operations. A leading source of decoherence in these operations is off-resonant spontaneous emission. While many trapped ion quantum computers or simulators utilize clock qubits, other systems, especially those with high magnetic fields such as Penning traps, rely on Zeeman qubits, which require a more complex calculation of this decoherence. We therefore examine theoretically the impacts of spontaneous emission on quantum gates performed with trapped-ion ground-state Zeeman qubits in a high magnetic field. In particular, we consider two types of gates---light-shift (\ensuremathσ\ifmmode^\elseˆ\fiiz\ensuremathσ\ifmmode^\elseˆ\fijz)({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{i}^{z}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{j}^{z})(\ensuremathσ\ifmmode^\elseˆ\fi​iz​\ensuremathσ\ifmmode^\elseˆ\fi​jz​) gates and M\o{}lmer-S\o{}rensen (\ensuremathσ\ifmmode^\elseˆ\fiix\ensuremathσ\ifmmode^\elseˆ\fijx)({\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{i}^{x}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}_{j}^{x})(\ensuremathσ\ifmmode^\elseˆ\fi​ix​\ensuremathσ\ifmmode^\elseˆ\fi​jx​) gates---obtained with laser beams directed approximately perpendicular to the magnetic field (the quantization axis) and compare the decoherence errors in each. Within each gate type, we also compare different operating points with regard to the detunings, polarizations, and required intensity of the laser beams used to drive the gates. We show that both gates can have similar performance at their optimal operating conditions at high magnetic fields and examine the experimental feasibility of various operating points. By examining the magnetic field dependence of each gate, we demonstrate that, when the PPP state fine-structure splitting is large compared to the Zeeman splittings, the theoretical performance of the M\o{}lmer-S\o{}rensen gate is significantly better than that of the light-shift gate. Additionally, for the light-shift gate, we make an approximate comparison between the fidelities that can be achieved at high fields with the fidelities of state-of-the-art two-qubit trapped ion quantum gates. We show that, with regard to spontaneous emission, the achievable infidelity with our current configuration is about an order of magnitude larger than that of the best low-field gates, but we also discuss several alternative configurations with potential error rates that are comparable with those for state-of-the-art trapped ion gates.
High-fidelity and parallel realization in scalable platforms of the two-qubit entangling gates fundamental to universal quantum computing constitutes one of the largest challenges in implementing fault-tolerant quantum computation. Integrated optical addressing of trapped-ion qubits offers routes to scaling the high-fidelity optical control demonstrated to date in small systems. Here we show that in addition to scaling, capabilities practically enabled by integrated optics can substantially alleviate laser powers required for both light-shift (LS) and Molmer-Sorensen (MS) geometric phase gates acting on long-lived ground-state qubit encodings in a broad range of ion species. In the proposed gate schemes utilizing carrier nulling via ion positioning at phase-stable standing-wave (SW) nodes, our calculations suggest that suppressed spontaneous photon scattering at the SW node allows for gate drives operating at smaller Raman detunings, resulting in approximately an order-of-magnitude reduction in power requirement (and significantly larger in certain parameter regimes) for gates of a given duration and scattering-limited fidelity as compared to conventional running wave (RW)-based approaches. The SW schemes have the additional benefit of simultaneously eliminating undesired coherent couplings that typically limit gate speeds. Our work quantifies power requirements for multiple ion species and enhancements to be expected from carrier-nulled configurations practically enabled by integrated delivery, and informs experiments and systems for realization of fast and power-efficient laser-based entangling gates in scalable platforms.
We provide an introduction to the use of ion crystals in a Penning trap for experiments in quantum information. Macroscopic Penning traps allow for the containment of a few to a few million atomic ions whose internal states may be used in quantum information experiments. Ions are laser Doppler cooled, and the mutual Coulomb repulsion of the ions leads to the formation of crystalline arrays. The structure and dimensionality of the resulting ion crystals may be tuned using a combination of control laser beams and external potentials. We discuss the use of two-dimensional 9^{9}9Be+^{+}+ ion crystals for experimental tests of quantum control techniques. Our primary qubit is the 124 GHz ground-state electron spin flip transition, which we drive using microwaves. An ion crystal represents a spatial ensemble of qubits, but the effects of inhomogeneities across a typical crystal are small, and as such we treat the ensemble as a single effective spin. We are able to initialize the qubits in a simple state and perform a projective measurement on the system. We demonstrate full control of the qubit Bloch vector, performing arbitrary high-fidelity rotations ($\tau_{\pi}\sim$200 μ\muμs). Randomized Benchmarking demonstrates an error per gate (a Pauli-randomized π/2\pi/2π/2 and π\piπ pulse pair) of 8±1×10−48\pm1\times10^{-4}8±1×10−4. Ramsey interferometry and spin-locking measurements are used to elucidate the limits of qubit coherence in the system, yielding a typical free-induction decay coherence time of $T_{2}\sim$2 ms, and a limiting $T_{1\rho}\sim$688 ms. These experimental specifications make ion crystals in a Penning trap ideal candidates for novel experiments in quantum control. As such, we briefly describe recent efforts aimed at studying the error-suppressing capabilities of dynamical decoupling pulse sequences, demonstrating an ability to extend qubit coherence and suppress phase errors. We conclude with a discussion of future avenues for experimental exploration, including the use of additional nuclear-spin-flip transitions for effective multiqubit protocols, and the potential for Coulomb crystals to form a useful testbed for studies of large-scale entanglement.
To date, the highest-fidelity quantum logic gates between two qubits have been achieved using the geometric-phase gate in trapped ions, with the two leading variants being the M\o{}lmer-S\o{}rensen gate and the light-shift (LS) gate. Both of these approaches have their respective advantages and challenges. For example, the latter is technically simpler and is natively insensitive to optical phases, but it has not been made to work directly on a clock-state qubit. We present a technique for implementing the LS gate that combines the best features of these two approaches: By using a small ($\ensuremath{\sim}\mathrm{MHz}$) detuning from a narrow (dipole-forbidden) optical transition, we are able to operate an LS gate directly on hyperfine clock states, achieving gate fidelities of 99.74(4)99.74(4)%99.74(4) using modest laser power at visible wavelengths. Current gate infidelities appear to be dominated by technical noise, and theoretical modeling suggests a path toward gate fidelity above 99.9999.99%99.99.
We consider the implementation of quantum logic gates in trapped ions using tightly focused optical tweezers. Strong polarization gradients near the tweezer focus lead to qubit-state dependent forces on the ion. We show that these may be used to implement quantum logic gates on pairs of ion qubits in a crystal. The qubit-state dependent forces generated by this effect live on the plane perpendicular to the direction of propagation of the laser beams opening new ways of coupling to motional modes of an ion crystal. The proposed gate does not require ground state cooling of the ions and does not rely on the Lamb-Dicke approximation, although the waist of the tightly focused beam needs to be comparable with its wavelength in order to achieve the needed field curvature. Furthermore, the gate can be performed on both ground state and magnetic field insensitive clock state qubits without the need for counter-propagating laser fields. This simplifies the setup and eliminates errors due to phase instabilities between the gate laser beams. Finally, we show that imperfections in the gate execution, in particular pointing errors <30<30<30 nm in the tweezers reduce the gate fidelity from F≳0.99998\mathcal F\gtrsim 0.99998F≳0.99998 to ≳0.999\gtrsim 0.999≳0.999.
We consider the implementation of quantum logic gates in trapped ions using tightly focused optical tweezers. Strong polarization gradients near the tweezer focus lead to qubit-state-dependent forces on the ion. We show that these may be used to implement quantum logic gates on pairs of ion qubits in a crystal. The qubit-state-dependent forces generated by this effect are located on the plane perpendicular to the direction of propagation of the laser beams opening alternate ways of coupling to motional modes of an ion crystal. The proposed gate does not require ground-state cooling of the ions although the waist of the tightly focused beam needs to be comparable to its wavelength in order to achieve the needed field curvature. Furthermore, the gate can be implemented on both ground-state and magnetic-field-insensitive clock-state qubits without the need for counterpropagating laser fields. This simplifies the setup and eliminates errors due to phase instabilities between the gate laser beams. Finally, we show that imperfections in the gate execution, in particular, a 30-nm tweezer alignment error, lead to an infidelity of $\ensuremath{\sim}{10}^{\ensuremath{-}3}$. In the absence of experimental imperfections and within the limits of the gate model explored in this paper the fidelity is predicted to be $\ensuremath{\sim}0.999\phantom{\rule{0.16em}{0ex}}88$ when using a Laguerre-Gaussian beam to suppress photon scattering errors.
We propose the implementation of a geometric-phase gate on magnetic-field-insensitive qubits with ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\sigma}}}^{z}$-dependent forces for trapped ion quantum computing. The force is exerted by two laser beams in a Raman configuration. Qubit-state dependency is achieved by a small frequency detuning from the virtually excited state. Ion species with excited states of long radiative lifetimes are used to reduce the chance of a spontaneous photon emission to less than ${10}^{\ensuremath{-}8}\phantom{\rule{0.3em}{0ex}}\mathrm{per}$ gate run. This eliminates the main source of gate infidelity in previous implementations. With this scheme it seems possible to reach the fault-tolerant threshold.