Entanglement-Based Machine Learning on a Quantum Computer

Machine learning, a branch of artificial intelligence, learns from previous experience to optimize performance, which is ubiquitous in various fields such as computer sciences, financial analysis, robotics, and bioinformatics. A challenge is that machine learning with the rapidly growing “big data” could become intractable for classical computers. Recently, quantum machine learning algorithms [Lloyd, Mohseni, and Rebentrost, arXiv.1307.0411] were proposed which could offer an exponential speedup over classical algorithms. Here, we report the first experimental entanglement-based classification of two-, four-, and eight-dimensional vectors to different clusters using a small-scale photonic quantum computer, which are then used to implement supervised and unsupervised machine learning. The results demonstrate the working principle of using quantum computers to manipulate and classify high-dimensional vectors, the core mathematical routine in machine learning. The method can, in principle, be scaled to larger numbers of qubits, and may provide a new route to accelerate machine learning.

Figure 1

Experimental setup for quantum machine learning with photonic qubits. Ultraviolet laser pulses with a central wavelength of 394 nm, pulse duration of 120 fs, and a repetition rate of 76 MHz pass through two type-II $\beta$-barium borate (BBO) crystals with a thickness of 2 mm to produce two entangled photon pairs. The photons pass through pairs of birefringent compensators consisting of a 1-mm BBO crystal and a HWP to compensate the walk-off between horizontal and vertical polarization, and are prepared in the quantum state: $(|H⟩|V⟩+|V⟩|H⟩)/\sqrt{2}$. Two extra HWPs placed in arm 3 and anc are used to transform the state into $(|H⟩|H⟩+|V⟩|V⟩)/\sqrt{2}$. Two single photons, one from each pair, are temporally and spatially superposed on a PBS to generate a four-photon entangled state: $(|H⟩|H⟩|H⟩|H⟩+|V⟩|V⟩|V⟩|V⟩)/\sqrt{2}$. The photons 1, 2, and 3 are sent to Sagnac-like interferometers, where each single photon splits into two spatial modes by the PBS with regard to its polarization, and recombines on a nonpolarizing beam splitter (NBS). Various vectors are independently encoded into the two spatial modes using HWPs. The specially designed beam splitter cube is half-PBS coated and half-NBS coated. High-precision small-angle prisms are inserted for fine adjustments of the relative delay of the two different paths. The photons are detected by five single-photon detectors (quantum efficiency $>60\%$), and the two four-photon coincidence events, $D_3{D}_2{D}_1{D}_T$ and $D_3{D}_2{D}_1{D}_R$, are simultaneously registered by a homemade FPGA-based coincidence unit.

Figure 2

Theoretical prediction (a) and experimental results (b) for classifying two-dimensional vectors into two clusters. The red and blue cross are reference vectors. The evaluated value of difference of the distances of each tested vector to the two reference vectors is coded as the fill color. The result of classification is coded as the edge color ($\mathrm{blue}=A$, $\mathrm{red}=B$). The gray line is where the distances are the same in theory. The data acquisition time is 1 sec for each vector, collecting about 10000 events. The statistical standard deviation is much smaller than the error caused by the imperfection of the entanglement state; thus, error bars are omitted. The data are represented in polar coordinate.

Figure 3

Demonstration of unsupervised machine learning. (a) Eight gray circles, labeled from $A$ to $H$, are two-dimensional vectors to be classified. (b) A random classification is initialized, where $A$ and $B$ belong to the red group and $C$, $D$, $E$, $F$, $G$, $H$ belong to blue group. (c) We use the entanglement-based method presented in this Letter to experimentally evaluate the distance from each vector to the other vectors within a group. Then the mean distance to both the red and the blue group is calculated, $D_{A\text{- red}}=0.12$, $D_{A\text{- blue}}=0.67$, $D_{B\text{- red}}=0.12$, $D_{B\text{- blue}}=0.65$, $D_{C\text{- red}}=0.41$, $D_{C\text{- blue}}=0.68$, $D_{D\text{- red}}=0.33$, $D_{D\text{- blue}}=0.58$, $D_{E\text{- red}}=0.73$, $D_{E\text{- blue}}=0.59$, $D_{F\text{- red}}=0.84$, $D_{F\text{- blue}}=0.57$, $D_{G\text{- red}}=0.91$, $D_{G\text{- blue}}=0.57$, $D_{H\text{- red}}=0.77$, $D_{H\text{- blue}}=0.50$. It can be seen that $C$ and $D$ are closer to the red group but were wrongly classified into the blue group, whose labels are therefore changed. The labels of the other six vectors remain unchanged. (d) The optimizing process is repeated in the new configuration until there is no change. In this configuration every vector is in the group with a closer mean distance, and the system can confirm the configuration as an ultimate classification result.