Finite-size criticality in fully connected spin models on superconducting quantum hardware

The emergence of a collective behavior in a many-body system is responsible for the quantum criticality separating different phases of matter. Interacting spin systems in a magnetic field offer a tantalizing opportunity to test different approaches to study quantum phase transitions. In this work, we exploit the new resources offered by quantum algorithms to detect the quantum critical behavior of fully connected spin-1/2 models. We define a suitable Hamiltonian depending on an internal anisotropy parameter $\gamma$ that allows us to examine three paradigmatic examples of spin models, whose lattice is a fully connected graph. We propose a method based on variational algorithms run on superconducting transmon qubits to detect the critical behavior for systems of finite size. We evaluate the energy gap between the first excited state and the ground state, the magnetization along the easy axis of the system, and the spin-spin correlations. We finally report a discussion about the feasibility of scaling such approach on a real quantum device for a system having a dimension such that classical simulations start requiring significant resources.

Figure 1

Representation of the chosen ansatz to design the parametric function $\psi \left(\theta \right)$, namely the Hardware efficient ansatz, composed of $R_y\left(\theta \right)$ rotations and CNOTs between neighboring qubits executed in parallel. We sketch here a single layer of the circuit, corresponding to $D=1,$ where higher values of $D$ correspond to sequential repetitions of such a circuit.

Figure 2

We report the energies of the ground state $E_{\mathrm{GS}}$ and first excited one $E_{1\mathrm{st}}$ of the Hamiltonian in Eq. (7) for $N=4$ and for value of the anisotropy parameter $\gamma =0.81$ as a function of the magnetic field $B$. Solid lines represent the values obtained via exact classical diagonalization, while discrete points are the results obtained via VQE. The results for $E_{\mathrm{GS}}$ are marked in light blue with tri-down markers (state vector simulation) and filled squares (QASM noiseless simulation), while results for $E_{1\mathrm{st}}$ are in orange with crosses marking the state vector simulation and filled circles the QASM noiseless outputs. Vertical dotted lines show crossing points for values given in Eq. (9). The inset makes it possible to better appreciate the accuracy of energy estimates as a function of the number of shoots, where the error bar correspond to one standard deviation.

Figure 3

We plot the difference between the energy of the first-excited-state $E_{1\mathrm{st}}$ and the ground-state energy $E_{\mathrm{GS}}$ as a function of the tuning field $B$. The results are shown for a system with $N=5$ lattice sites, focusing on different values of the anisotropy parameter $\gamma =0.36,0.81$. The results obtained via VQE are marked by blue crosses ($\gamma =0.36$) and orange dots ($\gamma =0.81$). The blue dashed line ($\gamma =0.36$) and the orange dotted line ($\gamma =0.81$) represent the exact diagonalization values of $E_{1\mathrm{st}}-E_{\mathrm{GS}}$. Vertical lines correspond to values of the critical magnetic field such that Eq. (9) holds, where the (I) refers to $\gamma =0.36$ and (II) to $\gamma =0.81$.

Figure 4

We report the energy difference $E_{1\mathrm{st}}-E_{\mathrm{GS}}$ as a function of the magnetic field $B$. The results are computed at $\gamma =0.44$ for different lattice sizes $N=4$ (blue), $N=5$ (orange), and $N=6$ (red). The results obtained via VQE are represented by a blue cross ($N=4$), an orange dot ($N=5$), and a red plus ($N=6$), while the blue dashed line ($N=4$), the orange dotted line ($N=5$), and a red dot-dashed line ($N=6$) represent the values obtained by exact diagonalization of the Hamiltonian $H$. The vertical lines correspond to the values of the magnetic field for which by we observe a crossing points between the two lowest state energies [see Eq. (9)]; they are labeled with (I) for $N=4$, (II) for $N=5$, and (III) for $N=6$ sites.

Figure 5

We report the three relevant quantities to detect the criticality in a fully connected spin system. In (a) we show the energy gap $E_{1\mathrm{st}}-E_{\mathrm{GS}}$. In (b) we plot the correlation function $\langle S_x^2\rangle$ and in (c) the mean magnetization along the direction of the magnetic field $\langle S_z\rangle$. The results are shown for $N=5$ spins for particular values of the anisotropy identifying different classes of models $\gamma =0$ (fully connected Ising model), $\gamma =0.49$ (LMG model), and $\gamma =1$ (Lipkin-Dicke model). In all the three plots, the results obtained via the quantum variational algorithm are represented by blue crosses ($\gamma =0$), orange dots ($\gamma =0.49$), and red plusses ($\gamma =1$) and the benchmarking is given by the corresponding exact diagonalization values, shown as a blue dashed line ($\gamma =0$), an orange dotted line ($\gamma =0.49$), and a red dot-dashed line ($\gamma =1$). The results are in agreement with the critical values in Eq. (9) and such values are reported as vertical lines [(I) denotes the only trivial value for $\gamma =0$, (II) denotes critical values for $\gamma =0.49$, and (III) denotes critical values for $\gamma =1$].

Figure 6

Topology of the superconducting quantum device ibmq_kolkata with color map representation. Color associated to each qubit represents the readout error at the time of calibration while the color of the connection between two qubits represents the CNOT error rate. Image taken from the IBM Q Lab [83].

Figure 7

We report the three relevant quantities to detect the criticality in a fully connected spin system, using the VQE algorithm run on the superconducting device ibmq_kolkata. The energy gap $E_{1\mathrm{st}}-E_{\mathrm{GS}}$ is in (a). The mean value of two magnetic observables: In (b) the correlation function along an axis transverse to the magnetic field $\langle S_x^2\rangle$, and in (c) the mean magnetization along the direction of the magnetic field $\langle S_z\rangle$. The results are done for a lattice size of $N=5$ spins at $\gamma =0.49$. Points are obtained on the superconducting device ibmq_kolkata with (red crosses) and without (blue tri-left markers) error mitigation. The experimental values are compared to noiseless simulation (black dots) and the final benchmark is given by the exact diagonalization values (solid line). The inset in (a) shows the extrapolation to the zero noise regime, both with an exponential and linear fit.

Figure 8

Representation of various excited states for $N=4$ and $\gamma =1$. VQE results are represented by a blue cross (ground state), an orange dot (first excited state), a red plus (second excited state), and a black triangle (third excited state). The blue dashed line (ground state), the orange dotted line (first excited state), the red dot-dashed line (second excited state), and the black solid lines (third, fourth, fifth, and sixth excited states, some are degenerate) represent the classical diagonalization values. Vertical lines indicate crossing points.

Figure 9

We plot the energies $E_{2\mathrm{nd}}$ and $E_{3\mathrm{rd}}$ of the two excited states, beyond the first one, for the Hamiltonian in Eq. (7) with $N=4$ at the value anisotropy $\gamma =0.67$. In order to make the comparison clear, we report also the values of the energies of the ground state $E_{\mathrm{GS}}$ and of the first one $E_{1\mathrm{st}}$. The markers of the VQE results are the following: blue cross for the ground-state energy, an orange dot for the first excited state, a red plus for second excited state, and a black triangle for third excited state. Exact diagonalization results are shown as a blue dashed line (ground state), an orange dotted line (first excited state), and a red dot-dashed line (second excited state), while the black solid lines are for the energies of degenerated states (ranging from the third excited and up to the sixth). Vertical lines indicate crossing points [see Eq. (9)].

Figure 10

We plot the SLSQP loss curve for the two lowest-energy states for different values of the magnetic field. Namely, in (a) we set $B=0.01$ and in (b) $B=0.5$ for $N=4$ spins at $\gamma =0.49$. The blue crosses refers to the ground state (GS) and the green filled circles to the first excited state (1st).