Analog quantum simulation of the Lipkin-Meshkov-Glick model in a transmon qudit

TL;DR

This work demonstrates an analog quantum simulation of the Lipkin-Meshkov-Glick (LMG) model using a single superconducting transmon qudit with up to $d=9$ levels to realize a collective-spin Hamiltonian. By applying simultaneous drives near each transmon transition and moving to a rotating frame, the authors implement the LMG Hamiltonian in a relabeled basis, enabling a direct mapping of the one-axis twisting term onto first off-diagonals and controlled local fields via detunings. They measure five finite-size precursors of quantum criticality—dynamical phase transitions, closing of the energy gap, Kibble-Zurek-like dynamics, statistics of the order parameter, and excited-state quantum phase transitions—using adiabatic ramps and Ramsey-like spectroscopy without prior knowledge of eigenstates. The results establish high-dimensional transmon qudits as a versatile platform for simulating complex many-body physics and pave the way for scaling to more elaborate qudit-based models and multi-qudit architectures, with $H_{ m LMG}/\,Omega = -h J_z - \frac{\gamma_x}{2j} J_x^2$ in the rotating frame and parity-protected dynamics.

Abstract

The simulation of large-scale quantum systems is one of the most sought-after applications of quantum computers. Of particular interest for near-term demonstrations of quantum computational advantage are analog quantum simulations, which employ analog controls instead of digitized gates. Most analog quantum simulations to date, however, have been performed using qubit-based processors, despite the fact that many physical systems are more naturally represented in terms of qudits (i.e., $d$-level systems). Motivated by this, we present an experimental realization of the Lipkin-Meshkov-Glick (LMG) model using an analog simulator based on a single superconducting transmon qudit with up to $d = 9$ levels. This is accomplished by moving to a rotated frame in which evolution under any time-dependent local field and one-axis twisting can be realized by the application of multiple simultaneous drives. Combining this analog drive scheme with universal control and single-shot readout of the qudit state, we provide a detailed study of five finite-size precursors of quantum criticality in the LMG model: dynamical phase transitions, closing of the energy gap, Kibble-Zurek-like dynamics, statistics of the order parameter, and excited-state phase transitions. For each experiment we devise a protocol for extracting the relevant properties which does not require any prior knowledge of the system eigenstates, and can therefore be readily extended to higher dimensions or more complicated models. Our results cement high-dimensional transmon qudits as an exciting path towards simulating many-body physics.

Figures (5)

• Figure 1: Analog quantum simulation of the LMG model. a, High-$E_J / E_C$ transmon potential well and eigenstate wavefunctions. The state labels on the left indicate the physical transmon states, while those on the right indicate the encoded spin states. b, Diagram of the transmon and resonator (top) and the drive spectrum used for control and readout (bottom, not to scale). Multiple simultaneous drive tones are used to simulate the LMG Hamiltonian in a rotating frame, and three simultaneous readout tones enable single-shot measurement of the qudit state. c, Diagram of the state labeling scheme. The even-parity spin states are encoded into the lowest $j + 1$ levels of the transmon, while the odd-parity spin states are encoded into the remaining $j$ levels. d, Nonzero matrix elements of the LMG Hamiltonian in the transmon basis (left) and the $J_z$ basis (right). In the $J_z$ basis the Hamiltonian has nonzero elements on the diagonal and second off-diagonals, while in the transmon basis, it has elements on the diagonal and first off-diagonals. e, Symmetry-broken and normal phases of the LMG model. We show the Husimi $Q$ functions of the LMG ground states for $j = 4$ with $h / \gamma_x = 0$ (top left), $h / \gamma_x = 0.4$ (top middle), and $h / \gamma_x \gg 1$ (top right). The corresponding semiclassical energy surfaces are shown in the bottom row. f, Experimentally measured evolution of the $J_z$ eigenstates $\ket{4, 4}$ (top) and $\ket{4, 3}$ (middle) under the $j = 4$ LMG Hamiltonian with $h / \gamma_x = 0$ (left) and $h / \gamma_x = 2$ (right). Colored points give the measured populations in the initial states, solid lines indicate the theoretical evolution, and gray points give the other measured populations within the same-parity subspaces. The bottom panels show the expectation value of the parity operator in each case, showing that parity is preserved up to $T_1$ decay from the odd subspace into the even subspace.
• Figure 2: Adiabatic protocol and gap measurement. a, Energy spectrum of the LMG model with $j = 4$ as a function of $h / \gamma_x$. The even-parity states are shown in blue, and odd-parity in yellow. In this experiment we focus on the lowest even-odd and even-even energy gaps. These gaps in the thermodynamic limit are shown in red. b, Gap measurement protocol for $h / \gamma_x < 1$. c, Gap measurement protocol for $h / \gamma_x > 1$. Both protocols implement Ramsey-like experiments in which we adiabatically prepare equal superpositions of LMG eigenstates, evolve under the LMG Hamiltonian, and project the resulting state onto the initial state. d, Time trace (left) and Fourier transform (right) for a single gap experiment with $j = 4$ and $h / \gamma_x = 0.6$. This corresponds to the purple line in the top right panel of e, and the peak of the Fourier transform gives the energy indicated by the purple star in the bottom right panel of e. e, Experimental results for the even-even (top) and even-odd (middle) gaps across a range of $h / \gamma_x$ values. The plots are obtained by computing the Fourier transform of each Ramsey signal as shown in d and plotting the logarithm of its magnitude in grayscale. We see strong peaks corresponding to the gap. In the bottom row we extract the frequencies of these peaks (open circles for even-even, filled circles for even-odd) and plot them on top of the theoretically expected values (dashed black line for even-even, solid black line for even-odd). We also show the corresponding gaps in the thermodynamic limit, $j \rightarrow \infty$, in red.
• Figure 3: Transition between adiabatic and impulse regimes. a, Experimental protocol for the measurement. We adiabatically prepare the $h / \gamma_x = 2$ ground state, then ramp $h$ to zero over a variable time $T$. b, Experimental results showing the probability that the system ends in the $h / \gamma_x = 0$ eigenstate as a function of ramp speed. The solid curves are single-parameter fits of numerical simulations to the data, where the the effect of decoherence is modeled as an exponential decay of the measured population. The dotted lines show the overlaps between the initial and final LMG eigenstates. c, Energy spectrum of the even-parity LMG states. When the system is ramped non-adiabatically through the minimum-gap point, it undergoes transitions to the subspace of even-parity excited states.
• Figure 4: Distribution of $J_x$ eigenstates in the LMG ground state. a, Semiclassical energy surface as a function of $\theta$ and $h / \gamma_x$, setting $\phi = 0$. In the symmetry-broken phase ($h / \gamma_x < 1$), the energy surface has two degenerate minima. At the critical point ($h / \gamma_x = 1$) these minima merge, and we have a single minimum for $h / \gamma_x > 1$. b, Experimental protocol for $h / \gamma_x < 1$. c, Experimental protocol for $h / \gamma_x > 1$. In both cases we adiabatically prepare the LMG ground state as described above, then perform a series of pulses mapping one of the $J_x$ eigenstates onto the transmon ground state prior to readout. d, Histograms representing the $J_x$ eigenstate distributions as a function of $h / \gamma_x$ for $j = 1$ through $j = 4$. e, $J_x$ eigenstate distributions for the particular values of $h / \gamma_x$ indicated by the vertical lines in d. f, Histograms representing the $J_x$ eigenstate distributions as a function of $h / \gamma_x$ for $j = 8$, measured by projecting the $J_x$ eigenstates onto the even-parity subspace and renormalizing. As discussed in the text, the yellow curves give the locations of the minima of the classical energy surface when projected onto the $J_x$ axis. g, $J_x$ eigenstate distributions for the particular values of $h / \gamma_x$ indicated by the vertical lines in f.
• Figure 5: Excited-state quantum phase transition (ESQPT). a, LMG energy spectrum as a function of $h / \gamma_x$ for $j = 8$. The red curve indicates the critical energy $E_c(h)$ of the ESQPT, equal to the height of the barrier of the classical energy surface relative to its minimum energy. b, Semiclassical energy surface for $h / \gamma_x = 0.18$ (vertical dashed line in a), with $\phi = 0$. The horizontal lines show the $j = 8$ eigenenergies. For energies near or above $E_c(h)$, we plot even-parity eigenenergies in blue and odd-parity eigenenergies in yellow. c, Experimental protocol. We measure energy gaps between adjacent states of like parity in a manner analogous to the gap measurement in Figure \ref{['fig:gap']}. Here, however, we use a different energy scale $\Omega$ during the ramps in order to more easily satisfy the adiabatic approximation (see Methods). d, Experimental results for the energy difference of each even-odd pair as a function of that pair's average energy, along with the theoretical expectation for $j = 8$, as measured here, and $j \rightarrow \infty$. We also show the semiclassical density of states result in gray.