User:Andydeneris/sandbox/Neutral Atom Quantum Computing: Difference between revisions

 

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In optimal control numerical algorithms are used to select control parameters which maximize fidelity while minimizing gate time. In general, the maximization of speed and fidelity go hand-in-hand with respect to quantum logic gate as a major source of error is the limited coherence times of quantum systems. In the standard TOLP gate, we let all control parameters vary with time in a piecewise fashion. Thus our parameter make up an <math> N</math>-dimensional space of <math> \Omega_j </math>, <math> \phi_j </math> and <math> \Delta_j</math>, where <math> N</math> is the number of time steps and <math> j = 1, 2, …, N </math>. However, under certain conditions, ideal fidelities result when we hold the Rabi frequency constant at its maximum. Furthermore, detuning will vary with phase (as <math>\Delta_j = \frac{d \phi_j}{dt} </math>) and so we can limit our parameter space to be either phase or frequency modulation. From there, quantum control algorithms can be used to find the optimal set of parameters (<math> \phi_j </math> or <math>\Delta_j</math>) which minimize time and infidelity. Since detuning can be highly uncertain in experiments, phase control is common and frequency control is often associated with adiabatic methods [wikilink?] as these are more robust to the varying control parameter. Numerical studies showed sinusoidal phase modulation to be optimal [cite].

In optimal control numerical algorithms are used to select control parameters which maximize fidelity while minimizing gate time. In general, the maximization of speed and fidelity go hand-in-hand with respect to quantum logic gate as a major source of error is the limited coherence times of quantum systems. In the standard TOLP gate, we let all control parameters vary with time in a piecewise fashion. Thus our parameter make up an <math> N</math>-dimensional space of <math> \Omega_j </math>, <math> \phi_j </math> and <math> \Delta_j</math>, where <math> N</math> is the number of time steps and <math> j = 1, 2, …, N </math>. However, under certain conditions, ideal fidelities result when we hold the Rabi frequency constant at its maximum. Furthermore, detuning will vary with phase (as <math>\Delta_j = \frac{d \phi_j}{dt} </math>) and so we can limit our parameter space to be either phase or frequency modulation. From there, quantum control algorithms can be used to find the optimal set of parameters (<math> \phi_j </math> or <math>\Delta_j</math>) which minimize time and infidelity. Since detuning can be highly uncertain in experiments, phase control is common and frequency control is often associated with adiabatic methods [wikilink?] as these are more robust to the varying control parameter. Numerical studies showed sinusoidal phase modulation to be optimal [cite].

Typical algorithms which are used in optimal control include GRAPE (Gradient Ascent Pulse Engineering) [ J. Magn. Reson. ”’172”’, 296 (2005)], Pontryagins Maximum Principle (PMP), and CRAB (Chopped RAndom Basis) [https://arxiv.org/pdf/2104.07687<nowiki>]. GRAPE was originally developed for NMR spectroscopy and has been used in various quantum computing platforms.</nowiki>

Typical algorithms which are used in optimal control include GRAPE (Gradient Ascent Pulse Engineering) [ J. Magn. Reson. ”’172”’, 296 (2005)] and CRAB (Chopped RAndom Basis) [https://arxiv.org/pdf/2104.07687<nowiki>]. GRAPE was originally developed for NMR spectroscopy and has been used in various quantum computing platforms.</nowiki>

==== Robust Control ====

==== Robust Control ====

In robust control, the cost function is expanded to include features to minimize other aspects of the procedure (other than speed) related to specific sources of noise. The robust control of the LP gate identified the dominating sources of error and designed cost functions to address these. Typical sources of noise in these protocols include

In robust control, the cost function is expanded to include features to minimize other aspects procedure (other than speed) to noise. robust control identified the dominating sources of error and designed cost functions to address these. Typical sources of noise in these protocols include

* Calibration of the Rabi frequency

* Calibration of the Rabi frequency

Since the earliest schemes for quantum computing atoms have been proposed as a means to process quantum information. While early demonstrations focused on ions due to their controllability through charge, neutral atoms were also a theoretical platform [Deutsch 99]. However, limited laser power at the time was insufficient for experimental realization. In 2010 the first demonstrations of an entangling quantum logic gate were performed separately by Mark Saffmann and Mikhail Lukin [cite]. Still, the fidelity of such gates was limited by laser power. It was not until 2019 when the Levine-Pichler gate was shown to achieve ~97% fidelity, that the quantum computing community started to take neutral atoms seriously. Despite this achievement, the fidelities have been orders of magnitude worse than in superconducting and trapped ion systems. In —- neutral atoms reached another milestone through the demonstration of scalable error correcting codes and logical qubits. Since then, error correcting codes have had the best performance on neutral atom platforms, making cold atoms a competitor in the race to fault tolerance.

The Time Optimal Levine-Pichler (TOLP) gate protocol uses the same entangling phases and basic principles of the original LP gate, but applies various numerical coherent control techniques to find the specific functions of laser phase, detuning and power which optimize the procedure. This is used to study the best possible implementation of an LP-style gate and has been shown to greatly improve speed with fidelities around 99.5%. In the original LP protocol, the detuning and intensity are kept constant while the phase changes in one discrete jump to create the loops. While this gate is much better in terms of speed and fidelity compared to other protocols of the time, it has since been improved drastically through coherent control (wiki-link) [cite]. In coherent control for quantum computing there is usually two separate goals: optimal control and robust control.

In optimal control numerical algorithms are used to select control parameters which maximize fidelity while minimizing gate time. In general, the maximization of speed and fidelity go hand-in-hand with respect to quantum logic gate as a major source of error is the limited coherence times of quantum systems. In the standard TOLP gate, we let all control parameters vary with time in a piecewise fashion. Thus our parameter make up an ${\displaystyle N}$-dimensional space of ${\displaystyle \Omega _{j}}$, ${\displaystyle \phi _{j}}$ and ${\displaystyle \Delta _{j}}$, where ${\displaystyle N}$ is the number of time steps and ${\displaystyle j=1,2,…,N}$. However, under certain conditions, ideal fidelities result when we hold the Rabi frequency constant at its maximum. Furthermore, detuning will vary with phase (as ${\displaystyle \Delta _{j}={\frac {d\phi _{j}}{dt}}}$) and so we can limit our parameter space to be either phase or frequency modulation. From there, quantum control algorithms can be used to find the optimal set of parameters (${\displaystyle \phi _{j}}$ or ${\displaystyle \Delta _{j}}$) which minimize time and infidelity. Since detuning can be highly uncertain in experiments, phase control is common and frequency control is often associated with adiabatic methods [wikilink?] as these are more robust to the varying control parameter. Numerical studies showed sinusoidal phase modulation to be optimal [cite].

Typical algorithms which are used in optimal control include GRAPE (Gradient Ascent Pulse Engineering) [ J. Magn. Reson. 172, 296 (2005)] and CRAB (Chopped RAndom Basis) [https://arxiv.org/pdf/2104.07687]. GRAPE was originally developed for NMR spectroscopy and has been used in various quantum computing platforms. As the name suggests GRAPE computes the gradient of the fidelity function and scans the search space in the direction of minimization of the cost function. This approach is very common in neutral atom Rydberg systems and scales well with system size. In CRAB we represent the control function in a reduced Fourier-like basis and optimize the coefficients. There are many options for an algorithm to search the coefficient space for an optimal solution and so a gradient based method is not necessary. CRAB can also introduce realistic experimental constraints.

In robust control, the cost function is expanded to include features to minimize other aspects a procedure (other than speed) in order to address and minimize parameters causing specific noise. Several studies in robust control have identified the dominating sources of error in the LP gate and designed cost functions to address these. Typical sources of noise in these protocols include

• Calibration of the Rabi frequency
• Doppler shifts
• Spontaneous emission of Rydberg/intermediate states
• Laser inhomogeneity

A common goal of robust control is to minimize the amount of time that atoms are in the Rydberg and intermediate states, which are highly unstable compared to the computational states. In the work by Pagano. et al. numerical protocols were able to reduce the amount of time in the Rydberg state by 10% [cite]. In order to reduce noise due to uncertainty in detuning and coupling strength, certain robust control aims to minimize the sensitivity of the overall gate to changes in these variables. The derivative method aims to accomplish this by minimizing the rate of change of control parameters with respect to uncertain variables, for example if we are using phase control then procedures which minimize ${\displaystyle {\frac {d\phi }{dv}}}$ (where ${\displaystyle v}$ is any uncalibrated variable such as ${\displaystyle \Delta }$ or ${\displaystyle \Omega }$) are ideal. [PhysRevResearch.5.033052]. In general we also want the control parameters ${\displaystyle \phi (t)}$, ${\displaystyle \Omega (t)}$, and ${\displaystyle \Delta (t)}$ to be smooth functions of time as this is further robust against Doppler shift noise [PhysRevA.94.032306].

Some other approaches to robust control include maximizing population in the dark state and tuning pulses on resonance. Studies have shown that tuning on resonance with the Rydberg state in conjunction with modulating the phases between ${\displaystyle \pm i}$ to be more robust to uncertainty in the coupling strength [Charles Fromonteil,1, 2 Dolev Bluvstein,3 and Hannes Pichler1, 2]. Using the dark state can reduce noise due to scattering [High-fidelity parallel entangling gates on a neutral atom quantum computer]. Studies in robust control have developed various procedures which are resilient to these noise sources. To address uncertainty in the laser power, some studies suggest employing pulses which are resonant with the Rydberg state.

In certain approaches to robust control, we take focus on error sources which might be more detrimental to error correcting codes. Laser inhomogeneity and Doppler effects are particularly harmful to logical qubit systems and the work of Jandura (2023) discusses two separate protocols which are robust to these source respectively. Since error correction is universally seen as a necessary procedure for quantum computing, designing protocols which are robust to these errors are an important area of research [cite].

Overall there are several techniques to mitigate the varying sources of noise in neutral atom systems. The specific tools and pulse shapes implemented depend on the specifics of the system. In this way robust control offers a highly adaptable and thorough set of procedures that can be used to maximize the fidelity of an experiment.

The TOLP gate is demonstrated in Evered. et al. [High-fidelity parallel entangling gates on a neutral atom quantum computer] on a 60 qubit system, achieving 99.5% fidelity. In this work, the authors use a combination of results from time-optimal and robust control as well as advanced experimental techniques to achieve 99.5% fidelity in a 60 qubit system. The robust control that this experiment is based on comes from the work of Pagano. et al. which focuses on limiting time in the Rydberg state, using the dark state, and single smooth pulses. In certain instances the robust control counters results from optimal control with tradeoffs between robustness and speed.

Rather than holding the Rabi frequency constant, this experiment explores the implementation of a so-called time optimal pulse, which keeps \Omega constant at its maximum, compared to that of a “smooth amplitude” pulse with a varying \Omega (t). While this results in slower gate times, it improves robustness to imperfections in \Omega. This does however result in the peak Rabi frequency being quite high which will limit the effect of the Rydberg blockade leading to related sources of noise. These trade-offs do not result in significant changes in fidelity and instead offer flexibility to choose pulses dependent on the strengths and weaknesses of the specific experimental set-up. In order to reduce population in the bright state, the detuning, \delta, on the Rydberg state and the detuning, \Delta, on the intermediate state are chosen such that \Delta\delta < 0. Smooth pulse shapes will also decrease population in the Rydberg state. Pulse shapes are carefully calibrated by tuning the phase as a sinusoidal function “formula” with A, the amplitude, w, the frequency and \delta_0 the initial value.

Outside of optimal and robust control, other developments in experimental techniques make the realized fidelity possible. Enhanced laser cooling and trapping methods reduced the temperature of atoms and limit noise due to thermal motion. Improved lasers realize a more idealized Rydberg blockade.

An important result of the TOLP gate is its ability to be applied on a large dimensional qubit system, which is an important feature for scaling up quantum computing platforms.