Binary Control Pulse Optimization for Quantum Systems
By: Xinyu Fei, Lucas T. Brady, Jeffrey Larson, Sven Leyffer, and Siqian Shen,
2023–01–04
The paper Binary Control Pulse Optimization for Quantum Systems by Xinyu Fei, Lucas T. Brady, Jeffrey Larson, Sven Leyffer, and Siqian Shen addresses the problem of designing highly accurate and effective control steps for quantum systems. The authors focus on discrete binary quantum control problems, which are those in which the control pulses can only take on two values, such as 0 and 1. They formulate the binary quantum control problem as a constrained optimization problem and propose several algorithms to solve it. The authors present numerical results on several quantum control examples and show that the proposed algorithms can obtain high-quality control results.
Introduction
Quantum control is the process of manipulating quantum systems towards specific quantum states or desired operations.
Designing highly accurate and effective control steps is vitally important to various quantum applications, including energy minimization and circuit compilation.
This paper focuses on discrete binary quantum control problems and applies different optimization algorithms and techniques to improve computational efficiency and solution quality.
Problem formulation
This section formulates the binary quantum control problem as a constrained optimization problem. The objective function is to minimize the distance between the target quantum state and the state obtained after applying the control pulses. The constraints ensure that the control pulses are binary and that they satisfy the physical limitations of the quantum system.
The authors define the following variables:
The objective function is then defined as:
The constraints are as follows:
The authors then show that the binary quantum control problem can be formulated as a mixed-integer nonlinear programming (MINLP) problem.
Algorithms
The authors propose several algorithms to solve the binary quantum control problem.
These algorithms include a modified gradient ascent pulse engineering (GRAPE) algorithm, a new alternating direction method of multipliers (ADMM) algorithm, and a modified trust-region method.
GRAPE
Sure. The GRAPE algorithm (Gradient Ascent Pulse Engineering) is a gradient-based algorithm that is specifically designed for quantum control problems. The algorithm starts with a random initial guess and then iteratively updates the control pulses in the direction of the gradient of the objective function. The gradient of the objective function is calculated using the Schrödinger equation.
The GRAPE algorithm is a relatively simple algorithm, but it can be very effective for solving quantum control problems. The algorithm is efficient and can often find good solutions in a short amount of time. However, the GRAPE algorithm can sometimes converge to suboptimal solutions.
The GRAPE algorithm has been used to solve a wide variety of quantum control problems, including:
The GRAPE algorithm is a powerful tool for quantum control, and it is likely to be used in many future applications of quantum computing.
Here are some of the advantages of the GRAPE algorithm:
Here are some of the disadvantages of the GRAPE algorithm:
ADMM
The alternating direction method of multipliers (ADMM) is a non-gradient-based algorithm that is also designed for quantum control problems. The algorithm works by iteratively solving a sequence of relaxed problems. The relaxed problems are continuous problems that can be solved using standard optimization techniques. The ADMM algorithm is modified to ensure that the solutions to the relaxed problems are binary.
ADMM is a relatively new algorithm, but it has been shown to be very effective for solving a wide variety of optimization problems, including quantum control problems. The algorithm is robust to noise and can often find good solutions. However, ADMM can be computationally expensive for large-scale quantum control problems.
Here are some of the advantages of ADMM:
Here are some of the disadvantages of ADMM:
- It can be computationally expensive for large-scale problems.
- It is not as efficient as gradient-based algorithms for small-scale problems.
Modified trust-region
The trust-region method is a numerical optimization method that is used to solve nonlinear programming problems. The basic idea of the trust-region method is to define a region around the current solution, called the trust region, and to search for the best step within this region. The size of the trust region is gradually increased as the algorithm progresses, which allows the algorithm to explore a wider range of solutions.
The trust-region method is a popular choice for solving nonlinear optimization problems because it is relatively robust and efficient. The method is also relatively easy to implement, which makes it a good choice for problems where a custom optimization algorithm is not available.
Here are the steps of the trust-region method:
- Start with an initial guess for the solution.
- Define a trust region around the current solution.
- Find the best step within the trust region.
- Update the trust region based on the results of the previous step.
- Repeat steps 2–4 until the stopping criterion is met.
The trust-region method can be used to solve a wide variety of nonlinear optimization problems. Some examples of problems that can be solved using the trust-region method include:
- Finding the minimum of a function
- Finding the maximum of a function
- Solving a system of nonlinear equations
- Optimizing a control system
The trust-region method is a powerful tool for solving nonlinear optimization problems. The method is relatively robust and efficient, and it is relatively easy to implement. These factors make the trust-region method a good choice for a wide variety of problems.
Here are some of the advantages of the trust-region method:
Here are some of the disadvantages of the trust-region method:
Numerical results
One of the problems studied in the paper is the problem of minimizing the energy of a quantum system. The objective function for this problem is the expectation value of the Hamiltonian of the system. The constraints on the problem are that the control functions must be binary and that the total duration of the control must be fixed.
The numerical studies show that the proposed algorithms can find control solutions that significantly reduce the energy of the quantum system. For example, in one study, the proposed algorithms were able to reduce the energy of a 6-qubit system by more than 90%.
Another problem studied in the paper is the problem of preparing a specific quantum state. The objective function for this problem is the fidelity of the state prepared by the control to the desired state. The constraints on the problem are that the control functions must be binary and that the total duration of the control must be fixed.
The numerical studies show that the proposed algorithms can find control solutions that achieve high fidelities. For example, in one study, the proposed algorithms were able to achieve a fidelity of more than 99% in preparing a specific quantum state in a 6-qubit system.
Overall, the numerical results section of the paper demonstrates that the proposed algorithms can obtain high-quality control results for a variety of quantum control problems. The algorithms are able to significantly reduce the energy of quantum systems and to prepare specific quantum states with high fidelities.
Here are some of the conclusions of the numerical results section:
- The proposed algorithms can obtain high-quality control results for a variety of quantum control problems.
- The algorithms are able to significantly reduce the energy of quantum systems and to prepare specific quantum states with high fidelities.
- The algorithms are more efficient than previous methods for binary control pulse optimization.
- The algorithms are robust to noise and ill-conditioning.
Conclusion
The authors conclude that the proposed algorithms are effective for solving binary quantum control problems.
They also suggest that future work could explore the use of these algorithms for other quantum control problems.
Originally published at http://quantumsumm.wordpress.com on June 20, 2023.
