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.
Highlights
Quantum computing devices with almost a hundred qubits are within reach in the near future [1,2,3]
In the present numerical simulations, we sampled the outcome 1024 times for the estimation of the cost function, which determine the amount of the statistical error
We proposed an efficient optimization method for quantum-classical hybrid algorithms using parameterized quantum circuits
Summary
Quantum computing devices with almost a hundred qubits are within reach in the near future [1,2,3]. Term devices, has been proposed in various machine learning settings such as supervised learning [18,19,20], unsupervised learning [21], generative model [22], generative adversarial model [23], and so on All these major quantum-classical hybrid algorithms have a common structure: a parameterized quantum circuit and its optimization with respect to an observed cost function. The proposed optimization method has several good properties: hyperparameter-free, faster convergence, less dependence on the initial choice of the parameters, and robust against the statistical error To this end, we use the fact that the cost function, as a function of a parameter θ , behaves very as a sine curve with period 2π , when the parameterized quantum circuit consists of a unitary gate exp(.
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