Sequential minimal optimization for quantum-classical hybrid algorithms (1903.12166v1)

Abstract: We propose a sequential minimal optimization method for quantum-classical hybrid algorithms, which converges faster, is robust against statistical error, and is hyperparameter-free. Specifically, the optimization problem of the parameterized quantum circuits is divided into solvable subproblems by considering only a subset of the parameters. In fact, if we choose a single parameter, the cost function becomes a simple sine curve with period $2\pi$, and hence we can exactly minimize with respect to the chosen parameter. Furthermore, even in general cases, the cost function is given by a simple sum of trigonometric functions with certain periods and hence can be minimized by using a classical computer. By repeatedly performing this procedure, we can optimize the parameterized quantum circuits so that the cost function becomes as small as possible. We perform numerical simulations and compare the proposed method with existing gradient-free and gradient-based optimization algorithms. We find that the proposed method substantially outperforms the existing optimization algorithms and converges to a solution almost independent of the initial choice of the parameters. This accelerates almost all quantum-classical hybrid algorithms readily and would be a key tool for harnessing near-term quantum devices.

Sequential Minimal Optimization for Quantum-Classical Hybrid Algorithms

In the presented research, the authors propose a sequential minimal optimization method specifically designed to enhance quantum-classical hybrid algorithms that employ parameterized quantum circuits. This method displays faster convergence, robustness against statistical error, and notable efficiency due to its hyperparameter-free characteristic.

Overview of the Method

The core of the technique lies in dividing the optimization problem by selecting a subset of parameters within the quantum circuit, thereby transforming the cost function into a simple sine curve with a period $2\pi$ for each chosen parameter. Taking advantage of this sine function characteristic, exact minimization can be achieved with respect to a single parameter at a time. In general scenarios, this translates the cost function into a sum of trigonometric functions, which can be minimized effectively using classical computing resources. Through iterative updates of each parameter, the quantum circuits are optimized to achieve minimal values in their cost functions.

Comparative Performance and Numerical Simulations

The authors performed extensive numerical simulations comparing their proposed method against existing gradient-free and gradient-based optimization algorithms. Their findings compellingly demonstrate that their method significantly outperforms current optimization strategies. An outstanding feature of the proposed method is its convergence independence from initial parameter selections, offering substantial robustness and reliability. This is particularly notable when operating on noisy intermediate-scale quantum (NISQ) devices, which are characterized by significant statistical errors due to limited quantum error correction capabilities.

Implications and Future Directions

The implications of this research extend beyond accelerating quantum-classical hybrid algorithms. The approach offers a promising pathway to optimizing quantum circuits necessary for NISQ devices, making it crucial in practical quantum computing applications. By reducing dependency on initial conditions and providing a hyperparameter-free setup, this method aligns well with real-world demands in deploying variational algorithms on actual quantum hardware.

Theoretical implications suggest a shift towards optimization methods that leverage unique properties of quantum circuits' parameterized structures rather than traditional optimization that may not fully harness the quantum systems' potential. Future developments might explore expanding this framework to multi-parameter optimizations or even addressing systems beyond quantum.

This method could set a cornerstone for advancing quantum computing practices and aligns with pursuits to operate near-term quantum devices efficiently. Furthermore, integrating this into classical variational models, such as MERA or quantum-inspired neural networks, can pave the way for innovative cross-disciplinary applications.

The authors have emphasized the simplicity and deterministic nature of their optimization routine, highlighting its practicality and ease of implementation. As the field progresses toward usable quantum computing, methods that enhance efficiency and accuracy, like the one discussed, will continue to gain traction and importance.