Experimental demonstration of quantum learning speedup with classical input data

We consider quantum-classical hybrid machine learning in which large-scale input channels remain classical and small-scale working channels process quantum operations conditioned on classical input data. This does not require the conversion of classical (big) data to a quantum superposed state, in contrast to recently developed approaches for quantum machine learning. We performed optical experiments to illustrate a single-bit universal machine, which can be extended to a large-bit circuit for a binary classification task. Our experimental machine exhibits quantum learning speedup of approximately $36\%$, as compared with the fully classical machine. In addition, it features strong robustness against dephasing noise.

Figure 1

Binary classification circuits. (a) We consider a reversible circuit that consists of two types of channels: one deals with $N$-bit classical input data and the other provides a working bit. This circuit runs $2^N$ gate operations to realize all possible $N$-bit Boolean functions: one single-bit operation and $2^N-1$ operations conditioned by the bits $x_j$ of $\mathbf{x}$. These gate operations are supposed to be either identity ($\mathbb{1}$) or logical-not ($X$). Note that such a circuit can realize all possible ($2^{2^N}$) binary classification functions [32]. We focus on the basic machine of a single-bit circuit, referred to as the universal single-feature circuit. (b) Here, we consider two versions of the basic circuits: one is the classical circuit that operates the probabilistic operations, and the other is the quantum circuit that operates the unitary operations utilizing quantum interference.

Figure 2

Schematic view of our experimental setups. We design two versions of USFC learning experiments [i.e., (a) classical and (b) quantum] in the linear-optical regime, where single-photon polarizations, i.e., horizontal ($H$) and vertical ($V$), are used as single-bit information carriers of the working channel. In quantum USFC learning, the quantum superposition effects involved in single-photon polarization are exploited. On the other hand, in classical USFC learning, the quantum superposition of photon polarization is completely destroyed when passing through the polarization beam splitter (PBS) placed between $g_0$ and $g_1$. In our experiments, differential evolution (DE) [36] is employed as a learning algorithm.

Figure 3

Learning evolution of USFC. Experimental task fidelity $F$ as a function of the iteration $n$. Contour plots show, for a selected trial, the evolution of the training USFC points in the parameter space of $\text{Pr}(g_0\to \mathbb{1})$ and $\text{Pr}(g_1\to \mathbb{1})$ as the learning (iteration) progresses. Here, $M=10$. The red line in the contour plot indicates the boundary of $F=0.99$ (or ${\epsilon }_t=0.01$).

Figure 4

Experimental results for cUSFC and qUSFC learning. (a) Here, the qUSFC learning experiments are performed with three $\mathrm{\Delta }$ settings: $\mathrm{\Delta }=0$, $\frac{\pi }{2}$, and ${\mathrm{\Delta }}_{TI}$. The error tolerance ${\epsilon }_t$ is set to 0.01. The data are created by counting the cases in which learning is completed before a certain iteration $n$, by sampling 200 trials of the learning experiments. We present the learning probabilities $P\left(n\right)$ by fitting the counting data (red points). The data are well fit (blue lines) by the integrated Gaussian ${\int }_{-\infty }^{n}d{n}^{\prime }\rho \left(n^{\prime }\right)$, where $\rho \left(n\right)=\frac{1}{\sqrt{2\pi }{\sigma }_n}exp\left(-\frac{{(n-n_c)}^2}{2{\sigma }_n}\right)$. Here, we also depict $\rho \left(n\right)$ (green lines). (b) The average number $n_c$ of iterations, the center position of $\rho \left(n\right)$, is compared for qUSFC and cUSFC. Here, the blue (red) bar represents the 95% (99%) statistical confidence interval. Our results show quantum learning speedup of $\simeq 36\%$ for both $\mathrm{\Delta }=0$ and ${\mathrm{\Delta }}_{TI}$. All the experimental data are in good agreement with our analytical predictions.

Figure 5

Decoherence effects in qUSFC learning. (a) We assume that decoherence occurs between $g_0$ and $g_1$. (b) The experiments are performed for $\mathrm{\Delta }=0$. The experimental values of $n_c$ (red solid circles) are shown together with numerically simulated data (green dotted curve). One hundred learning trials were used for each decay rate of $\gamma$. The blue line is for the classical approach.

Figure 6

Schematic view of the primitive strategy for larger $N$. (a) Basic structure: a single USFC and training system are casted. The training system includes the learning algorithm and $2^{N-1}$ memory blocks. (b) After completion of learning, the entire $N$-bit classification circuit can be constructed.

Figure 7

By performing numerical simulations, we determined the average number of iterations, $n_c$, by varying the free parameters $W$ and $C_r$, and obtained the density plot of $n_c$ in the two-dimensional space of ($C_r$, $W$). The qUSFC simulations are performed for $\mathrm{\Delta }=0$, $\frac{\pi }{2}$, and ${\mathrm{\Delta }}_{TI}$. Here, the best-optimized parameters ($C_{r,\text{best}}$, $W_{\text{best}}$) are found in each case (depicted by the red points). We also obtained the density plots of the number of failed learning processes out of 1000 trials. For failed learning, the task fidelity could not reach 0.99 for 100 evolution steps, which indicates that the chosen parameter values are practically unavailable.