Quantum circuit learning

We propose a classical-quantum hybrid algorithm for machine learning on near-term quantum processors, which we call quantum circuit learning. A quantum circuit driven by our framework learns a given task by tuning parameters implemented on it. The iterative optimization of the parameters allows us to circumvent the high-depth circuit. Theoretical investigation shows that a quantum circuit can approximate nonlinear functions, which is further confirmed by numerical simulations. Hybridizing a low-depth quantum circuit and a classical computer for machine learning, the proposed framework paves the way toward applications of near-term quantum devices for quantum machine learning.

Figure 1

Comparison of the QVE/QAOA, QRC, and presented QCL frameworks. In the QVE, the output of the quantum circuit is directly minimized. The QRC and QCL both optimize the output to the teacher $f\left(\mathit{x}\right)$. QRC optimization is done by tuning the linear weight $\mathit{w}$, as opposed to the QCL approach, which tunes the circuit parameter $\mathbf{\theta }$.

Figure 2

Quantum circuit used in numerical simulations. The parameters $\mathbf{\theta }$ of single-qubit arbitrary unitaries $U\left({\theta }_j^{\left(i\right)}\right)$ are optimized to minimize the cost function. $D$ denotes the depth of the circuit.

Figure 3

Demonstration of QCL performance to represent functions. “Initial” indicates the output of the quantum circuit with randomly chosen $\theta$; “final,” the output from the optimized quantum circuit. Fitting of (a) $x^2$, (b) $e^x$, (c) $sinx$, and (d) $\left|x\right|$.

Figure 4

Demonstration of a simple nonlinear classification task. (a) Teacher data. Blue dots indicate class 0; red dots, class 1. (b) Optimized output from the first qubit (after softmax transformation). The threshold for classification is 0.5; less than 0.5 and greater than 0.5 indicate that the point is classified as class 0 and class 1, respectively.

Figure 5

Demonstration of fitting quantum many-body dynamics. The partial dynamics of the 10-spin system can be well approximated by a six-qubit circuit.

Figure 6

A simple example of the avoidance of the overfitting resulting from unitarity. (a), (c) Fitting result of noise-added $sinx$ and $x^2$ using QCL. (b), (d) Fitting result of noise-added $sinx$ and $x^2$ using the classical regression with same basis functions as used in QCL.