qopt

Welcome to qopt. In this documentation you will find everything you need to know about qopt to simulate qubits and apply optimal control techniques. You can find the source code and install instructions on https://github.com/qutech/qopt.

A complementary publication about the software can be found at Phys. Rev. Applied https://doi.org/10.1103/PhysRevApplied.17.034036 and an older version on the Arxiv https://arxiv.org/abs/2110.05873. This paper gives a sound introduction to the topic of qubit simulation and quantum optimal control. It also provides a systematic theoretical introduction of simulation methods to give a profound understanding of each method’s capabilities.

Abstract

Realistic modelling of qubit systems including noise and constraints imposed by control hardware is required for performance prediction and control optimization of quantum processors. We introduce qopt, a software framework for simulating qubit dynamics and robust quantum optimal control considering common experimental situations. To this end, we model open and closed qubit systems with a focus on the simulation of realistic noise characteristics and experimental constraints. Specifically, the influence of noise can be calculated using Monte Carlo methods, effective master equations or with the filter function formalism, enabling the investigation and mitigation of auto-correlated noise. In addition, limitations of control electronics including finite bandwidth effects can be considered. The calculation of gradients based on analytic results is implemented to facilitate the efficient optimization of control pulses. The software is published under an open source license, well-tested and features a detailed documentation.

Summary

This python package is designed to facilitate the simulation of state-of-the-art quantum bits (qubits) including realistic experimental operation conditions, and for the optimization of noise-robust control pulses. For this purpose, an extensive set of noise simulation tools is included and complemented by methods to describe the limitations posed by the electronics steering the quantum operations. Compared to other available simulation packages qopt stands out by the ability to efficiently simulate the effects of fast Non-Markovian noise, while providing a general interface to define control pulses for any qubit type making qopt platform-independent. The simulation interfaces to optimization algorithms to apply optimal quantum control theory, the field of study which optimizes the accuracy of quantum operations by intelligent steering methods.

The functionalities can be coarsely divided into simulation and optimization of quantum operations. Various cost functions can be evaluated on the simulated evolution of the qubits such as an error rate, a gate or state fidelity or a leakage rate. Since gradient-based optimization algorithms perform extremely well in minimization problems, we implemented the derivatives of the cost functions by the optimization parameters based on analytical calculations.

Simulation

The qopt package simulates closed or open quantum systems on a pulse level by solving the corresponding partial differential equations being the Schroedinger equation or a Lindblad master equation. Thereby, pulses are discretized in time and the differential equations are solved using matrix exponentials. The total propagator is then available for every time step to analyse the dynamics of the qubit system.

Noise

The realistic simulation of noise is one of qopt’s key features. The various methods are therefore mentioned in more detail, and in a brief overview is given stating the advantages and requirements of each method.

Monte Carlo Simulations

The most forward way to simulate noise is to draw samples from the noise distribution and repeat the simulation for each of those noise realizations. Any cost function is then averaged over the repetitions. The sampling can be based on pseudo random number generators. Monte Carlo simulations are universally applicable but computationally expensive for high-frequency noise.

Lindblad Master Equation

In order to include dissipation effects in the simulation, the qubit and its environment can be modeled as open quantum system, described by a master equation in Lindblad form. The solution of the master equation is in the general case not unitary unlike the propagators calculated from Schroedinger’s equation, such that it can also describe the loss of energy or information into the environment. This approach is numerically efficient but only applicable to systems subjected to Markovian noise.

Filter Functions

The filter function formalism is a mathematical approach which allows the estimation of fidelities in the presence of universal classical noise. It is numerically very efficient for low numbers of qubits and widely applicable. This package interfaces to the open source filter function package (https://github.com/qutech/filter_functions) written by Tobias Hangleiter.

Leakage

Leakage occurs when the qubit leaves the computational space spanned by the computational states. To take this kind of error into account, the Hilbert space must be expanded as vector space sum by the leakage levels. The simulation is then performed on the larger Hilbert space and needs to be truncated to the computational states for evaluation. The Leakage rate or transition rate into the leakage states can be used to quantify the error rate caused to leakage.

Pulse Parametrization

The pulse parameterization translates a mathematically described pulse function into discrete-time control amplitudes that appear in the Hamiltonian describing the qubit model. This comprises sampling a potentially continuously defined control pulse, evaluating the physical function that describes the relation between pulse values and the control amplitudes, and including the hardware limitations of the control electronics that generate the control pulse.

Amplitude Functions

The amplitude functions encode a differential relationship between the optimization parameters, which describe the pulse, and the control amplitudes, which appear in the Hamiltonian and describe the dynamics of the quantum system. An example would be a sinusoidal pulse that drives resonant excitations of a qubit. The optimization parameters would be the pulse length, the pulse frequency and the amplitude. The amplitude function samples the continuous pulse and maps the voltage values to energy values from the Hamiltonian, which are the control amplitudes in this example.

Transfer Functions

To model realistic control electronics the package includes transfer functions mapping the ideal pulse to the actual provided voltages. This can include for example exponential saturation to consider finite voltage rise times in pulse generators, Gaussian smoothing of pulses to mimic bandwidth limitations on arbitrary waveform generators, linear transformations or even the measured response of an arbitrary waveform generator to a set of input voltages. The transfer functions then map the ideal pulse to the actually generated pulse.

Optimization

To leverage a given noisy quantum computer to its full potential, optimal control techniques can be applied to mitigate the detrimental effects of noise. The package allows the use of different optimization algorithms by a strong modularity in the implementation.

Analytical Derivatives

Gradient based optimization algorithms such as GRAPE have proven to be versatile and reliable for the application in pulse optimization. For the efficient calculation of gradients, the package implements analytical derivatives for the solution of the Schroedinger equation, the master equation in Lindblad form and all calculations used to estimate fidelities.

Documentation

The documentation is structured in the three parts ‘Features’, ‘Example Applications’ and the ‘API Documentation’.

qopt Features

The first part introduces the qopt functionalities step by step. Refer to this chapter for an introduction to the simulation package.

Example Applications

The ‘Example Applications’ combines an educational introduction to physical phenomena with simulation techniques and theoretical background information. They demonstrate how the package is used and treat FAQs on the way. They can also serve as blueprints for applications.

API Documentation

You can find the full API Documentation in the last section. Each class is implemented in a single module and each module is a subsection in the auto-generated API documentation. These subsections start with a general description of the purpose of the respective class. If you want to gain a quick overview of the class structure, I advise you to read through these descriptions. During the implementation of a simulation using qopt, you can frequently jump to the classes and functions your are using to look up the signatures.

Citing

If you are using qopt for your work then please cite the [qopt paper](https://doi.org/10.1103/PhysRevApplied.17.034036), as the funding of the development depends on the public impact.