Quantum kernels to learn the phases of quantum matter

Classical machine learning has succeeded in the prediction of both classical and quantum phases of matter. Notably, kernel methods stand out for their ability to provide interpretable results, relating the learning process with the physical order parameter explicitly. Here we exploit quantum kernels instead. They are naturally related to the fidelity, and thus it is possible to interpret the learning process with the help of quantum information tools. In particular, we use a support vector machine (with a quantum kernel) to predict and characterize second-order quantum phase transitions. We explain and understand the process of learning when the fidelity per site (rather than the fidelity) is used. The general theory is tested in the Ising chain in transverse field. We show that for small-sized systems, the algorithm gives accurate results, even when trained away from criticality. Besides, for larger sizes we confirm the success of the technique by extracting the correct critical exponent $\nu$. Finally, we present two algorithms, one based on fidelity and one based on the fidelity per site, to classify the phases of matter in a quantum processor.

Figure 1

Critical point from the SVM. $\lambda (J_{\pm },J)$ in the one-dimensional quantum Ising model, see Eq. (10), with $J_{-}=0.73$ and $J_{+}=1.27$ for $N=1000$ spins. $J$ has units of energy ($\hslash =1$). $J_{\pm }$ are represented by the open circles. Inset: Zoom of the intersection. The gray dotted line points out the intersection between the two curves, i.e., the $J$ at which $d=0$, and so the boundary predicted by the SVM, ${\tilde{J}}_c\left(N\right)$. The maximum of the derivative ${\partial }_x\lambda (J_{+},x)$ is marked with the black dashed line, $J_c\left(N\right)$.

Figure 2

Quantum kernels learning phases of matter. (a) $J_c\left(N\right)$ using the kernel $K^{\left(F\right)}$ for the three trainings discussed in this work. Namely, by intervals taking points failing in the $\mathrm{\Delta }$ subsets ${\delta }_1=[0.8,0.9]\cup [1.2,1.3]$, ${\delta }_2=[0.6,0.7]\cup [1.6,1.7]$, and taking $M=133$ points randomly distributed over $\mathrm{\Delta }$ (random). The set $\mathrm{\Delta }$ are 1000 ground states taking $J\mathrm{s}$ equally spaced in $\mathrm{\Delta }=[0.25,1.75]$. $J$ has units of energy ($\hslash =1$). Open circles are SVM results using the class sklearn.svm.SVC from Sklearn [65]. The crosses stand for a finite-size scaling analysis following [56]. The corresponding dashed lines are fittings (see Table 1). (b) Distance function, Eq. (6), for two sizes $N$. The open circles are the SVs and the shaded (gray) zones mark the training interval, ${\delta }_2$. (c, d) Contour plots of the matrix $K_{ij}^{\left(F\right)}$ where $i,j$ run on the total $\mathrm{\Delta }$. The labels mark the (theoretical) phases of the model for $J_i$ ($J_j$). Lower row, panels (e), (f), (g), and (h), are the counterparts but using the kernel $K^{\left(\lambda \right)}$.

Figure 3

Classification algorithms. (a) Trial quantum circuit constructed to estimate the fidelity between quantum stated. This measurements are the key of the classification of the phases in the system of study. (b) Our proposal of a quantum algorithm to estimate the fidelity per site in the chain, minimizing the scaling errors.