Finite-size resource scaling for learning quantum phase transitions with fidelity-based support vector machines

Abstract

Quantum kernels offer a valid procedure for learning quantum phase transitions on quantum processing devices, yet issues on the scalability of the learning strategy in connection with the symmetry of the critical model have not been clarified. We derive a link between model symmetry and fidelity-kernel resource scaling. We quantify the measurement resources required to estimate fidelity-based quantum kernels for many-body ground states while preserving the structure of the resulting Gram matrix under finite-shot sampling. Crucially, we show that increasing symmetry in the underlying spin model systematically amplifies these shot requirements. Moving from the $\mathbb{Z}_2$-symmetric Ising/XY regimes to the $U(1)$-symmetric XX (and XXZ) regimes leads to stronger kernel concentration and therefore substantially larger shot costs under the same bounds. We consider a tunable one-dimensional spin-$\tfrac{1}{2}$ Hamiltonian spanning the transverse-field Ising, XY, XX, and XXZ limits, and define the kernel as the ground-state fidelity. Kernel entries are estimated using a SWAP-test estimator with $S$ shots, and we adapt the ensemble spread and concentration-avoidance shot bounds to obtain practical shot requirements in terms of the interquartile range of kernel values and a representative kernel magnitude. For the free-fermion XY/XX family, we use the closed-form Bogoliubov-angle fidelity, while for the interacting XXZ chain we compute fidelities by exact diagonalization and benchmark shot-noise effects. Our symmetry-aware bounds provide a pragmatic procedure for physics-informed quantum machine learning.

Figures (10)

• Figure 1: Schematic overview of the quantum kernel estimation pipeline for many-body spin chain systems. (1)Hamiltonian, phase labeling, and training set construction: the anisotropic spin-chain Hamiltonian encompasses the Ising, XY, XX, XXZ models through the parameters $(\gamma ,\Delta, h)$. Training samples are chosen from two parameter windows on opposite sides of the critical point $\bm{x}_c$, assigning the labels $y_i = -1$ (ordered phase) and $y_i = +1$ (disordered phase). For the Ising universality class, the control parameter is $\bm{x}_i \equiv h_i$ while for XXZ it is $\bm{x}_i \equiv\Delta_i$. (2)Fidelity kernel and Gram matrix: the labeled samples are used to build the fidelity-based kernel matrix (Gram matrix) with entries $k_{i,j}=|\langle \psi(\bm{x}_i)|\psi(\bm{x}_j)\rangle|^2$, where $| \psi(\bm{x}_j)\rangle$ is the ground state for model parameters $\bm{x}_j$. (3)Distribution of the elements of the Gram matrix: the information about the phase transition is encoded in the entries of the Gram matrix. To visualize the distribution of the entries we can use a histogram of their values between $0$ and $1$. The median value is denoted as $k_{\mathrm{repr}}$, and the spread of the distribution is quantified by the inter-quartile range (IQR) $\Delta_{\mathrm{ensemble}} = Q_3-Q_1$, where $Q_1$ and $Q_3$ are the first and third quartiles of the distribution. (4)Finite-size resource bounds: each fidelity $k_{i,j}$ may be estimated experimentally with a SWAP test. Given a finite number of experimental measurements, the estimated fidelity is subject to statistical fluctuations. The Chebyshev's inequality gives us a bound on the minimum number of measurements needed to ensure that the statistical fluctuations are small compared with the natural distribution of the Gram matrix elements.
• Figure 2: Circuit diagram of the SWAP test employed to estimate the kernel entry $K(h_i, h_j) = |\braket{\psi(h_i)}{\psi(h_j)}|^2$. An ancilla qubit controls $N$ parallel SWAP gates, each acting on the corresponding pair of qubits from the two input registers. Measurement of the ancilla qubit yields the expectation value $\langle \sigma^z \rangle$, which is linearly related to the fidelity.
• Figure 3: Fidelity $F(x,x{+}\delta x)$ vs control parameter for $N=14,16,18$. The dip marks the transition: near $h\!\approx\!1$ for (Ising/XX/XY), and near $\Delta\!\approx\!0.5$ for (XXZ, $h=0$, first-order boundary). The BKT boundary at $\Delta\!\approx\!-0.5$ is analyzed in Appendix \ref{['app:xxz_bkt']}.
• Figure 4: The figure shows the signed SVM decision functions and the corresponding kernel matrices, for four Hamiltonian models. The signed SVM decision functions, introduced in Eq. \ref{['eq:DF']}, are shown in the line plots in the first and third columns. The kernel matrices are shown in the heat map plots in the second and fourth columns. In the decision-function panels, the horizontal axis is the control parameter ($h$ for Ising/XY/XX, and $\Delta$ for XXZ), and the vertical axis is the signed decision function value $d(x)$. In the kernel-matrix panels, the horizontal and vertical axes label the pairwise control parameters used to build the Gram matrix ($h,h'$ for Ising/XY/XX, and $\Delta,\Delta'$ for XXZ), and the color scale gives the kernel value. SWAP-test (ST) estimates are benchmarked against numerical exact diagonalization (ED), and against the analytical result, when available (i.e. for the XY/XX cases). The vertical dotted line marks the expected critical point. For each model, the upper panels (first and third rows) show the kernel matrix defined from the global fidelity kernel, while the lower panels correspond to fidelity per site (see Eq. \ref{['eq:fid_per_site']}). For the Ising/XY/XX panels, the SVM is trained on two field windows around the critical point $h_c=1$, namely $h \in [0.7,0.95] \cup [1.05,1.3]$, using 16 training points per side ($M=32$ total). For the XXZ panels (at $h=0$), the SVM is trained on two anisotropy windows around the critical point $\Delta_c=0.5$, namely $\Delta\in[0.35,0.45]\cup[0.55,0.65]$, using 16 training points per side ($M=32$ total). For the XX case, we display the kernel matrix over a broad field range that includes both the gapless region ($h<1$) and the saturated phase ($h>1$), and we choose the training set from the same two windows around $h_c=1$ as in the Ising/XY cases.
• Figure 5: Decision boundary as a geometric midpoint for the per-site fidelity kernel (XY chain, $\gamma=0.5$). We train an SVM with a precomputed per-site kernel $f_N(h,h')=F(h,h')^{1/N}$ on two field windows $h\in[0.76,0.95]\cup[1.05,1.30]$ (16 points per side, $M=32$ total), using the analytical Bogoliubov-angle fidelity at $N=1000$. The solid and dashed curves show the similarities to the inner (dominant) support vectors, plotted as $f_N(h_L,h)$ and $f_N(h_R,h)$, where $h_L=\max\{h_i:\alpha_i>0,y_i=-1\}$ and $h_R=\min\{h_i:\alpha_i>0,y_i=+1\}$. Open circles mark the two dominant support vectors. The dashed vertical line marks the expected critical field $h_c=1$, while the dotted line marks the midpoint estimate $h_{\mathrm{mid}}$ defined by $f_N(h_L,h_{\mathrm{mid}})=f_N(h_R,h_{\mathrm{mid}})$.