Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator

Abstract

A quantum simulator is a type of quantum computer that controls the interactions between quantum bits (or qubits) in a way that can be mapped to certain quantum many-body problems. As it becomes possible to exert more control over larger numbers of qubits, such simulators will be able to tackle a wider range of problems, such as materials design and molecular modelling, with the ultimate limit being a universal quantum computer that can solve general classes of hard problems. Here we use a quantum simulator composed of up to 53 qubits to study non-equilibrium dynamics in the transverse-field Ising model with long-range interactions. We observe a dynamical phase transition after a sudden change of the Hamiltonian, in a regime in which conventional statistical mechanics does not apply. The qubits are represented by the spins of trapped ions, which can be prepared in various initial pure states. We apply a global long-range Ising interaction with controllable strength and range, and measure each individual qubit with an efficiency of nearly 99 per cent. Such high efficiency means that arbitrary many-body correlations between qubits can be measured in a single shot, enabling the dynamical phase transition to be probed directly and revealing computationally intractable features that rely on the long-range interactions and high connectivity between qubits.

Extended Data Figure 1:. Distributions of the largest domain size.

Statistics of the largest domain size in each experimental shot (200 experiments for each of the last 5 time steps). Considering only the largest domains of each shot eliminates undesirable biasing toward small domain sizes present in Fig. 4a. Domain sizes are related to many-body correlators, where a domain size of N corresponds to an N-body correlator. Dashed lines are fits to a two parameter Gamma distribution proportional to e^−x/β x^α−1, with shape parameter α and scale parameter β.

Extended Data Figure 2:. Domain size observable for 16 spins.

Mean of maximum domain sizes as a function of the (Kac normalized [40]) transverse field for 16 spins. Experimental data is analyzed exactly as Fig. 4 (b) of the main text. Dashed line is a numerical simulation of the Hamiltonian determined from the experimental parameters.

Extended Data Figure 3:. Theoretical calculations of the correlations.

The spatially and long-time averaged correlation $\overline{C_2}(\infty )\equiv {\lim }_{T\to \infty }\frac{1}{T}{\int }_0^T\langle {({\Sigma }^x(t)∕N)}^2\rangle \mathit{dt}$ calculated as a function of the ratio ${\tilde{B}}_z∕J_0$ for the case of α = 0. The finite N curves are calculated using exact diagonalization, and the N = ∞ curve is calculated analytically from Eq. (6).

Figure 1:. Illustration of the DPT from a quantum quench.

We subject a system of interacting spins to a sudden change of the Hamiltonian and study the resulting quantum dynamics. (a) An isolated spin system is prepared in a product state, and an Ising spin-spin interaction is suddenly turned on, along with a tunable transverse magnetic field (see text for details). At the end of the evolution, we measure the spin magnetizations along the initial spin orientation direction. (b) A Bloch-sphere representation [20] of the average spin magnetization. Spins are initially fully polarized along the longitudinal x direction of the Bloch sphere, and evolve with Ising interactions along x competing with the transverse field along z, resulting in oscillations and relaxations. Blue curves illustrate the quench dynamics with a low transverse field; green curves indicate the dynamics with a large transverse field across criticality.

Figure 2:. Real-time spin dynamics after a quantum quench of 16 spins in an Ising chain.

(a) Polarized spins evolve under the long-range Ising Hamiltonian with a small transverse field (${\tilde{B}}_z∕J_0=0.6$). The broken symmetry given by the initial polarized state is preserved during the evolution. (b) When the transverse field is increased (${\tilde{B}}_z∕J_0=0.8$), the dynamics shows a faster initial relaxation, before settling to a non-zero plateau. (c) Under larger transverse fields (${\tilde{B}}_z∕J_0=1.6$), the Larmor precession takes over, and the spins oscillate and relax to zero average magnetization. The dashed lines are numerical simulations based on exact diagonalization. Insets: cumulative time-averages of the spin magnetization, smoothing out temporal fluctuations and showing the plateaus. Each point is the average of 200 experimental repetitions. Error bars are statistical, computed from quantum projection noise and detection infidelities as described in Methods.

Figure 3:. Two-body Correlations.

Long-time averaged values of the two-body correlations C₂ over all pairs of spins versus transverse field, for different numbers of spins in the chain. The final evolution times correspond to 2πJ₀t = (10.3, 5.3, 4.8, 6.5) for 8, 12, 16 and 53 spins, respectively. Statistical error bars are ± one standard deviation from measurements covering 21 different time steps. Solid lines in (a)-(c) are exact numerical solutions to the Schrödinger equation, and the shaded regions take into account uncertainties from experimental Stark shift calibration errors. Dashed lines in (a) and (b) are calculations using a canonical (thermal) ensemble with an effective temperature corresponding to the initial energy density. For N=53 spins in (d), the correlations are uniformly degraded from residual Stark shifts across the ion chain, so in this case we normalize to the maximum correlation at small field (see Methods). Exact diagonalization for N=53 spins is out of reach, so we instead fit the experimental data to a Lorentzian function with linear background, shown by the dashed line.

Figure 4:. Domain statistics and reconstructed single shot images of 53 spins.

(a) Top and bottom: reconstructed images based on binary detection of spin state (see Methods). The top image shows a chain of 53 ions in bright spin states. The other three images show 53 ions in combinations of bright and dark spin states. Center: statistics of the sizes of domains, or blocks with spins pointing along the same direction. Histograms are plotted on a logarithmic scale, to visualize the rare regions with large domains. Dashed lines are fits to exponential functions, which would be expected for a thermal state of the spins and could thus characterize defects such as imperfect preparation and measurement of the qubits. Long tails of deviations are clearly visible, and vary depending on ${\tilde{B}}_z∕J_0$. (b) Mean of the largest domain sizes in each single experimental shot. Error bars are the standard deviation of the mean (see Methods). Dashed lines represent a piecewise linear fit, from which we extract the transition point (see text). The green, yellow, and red data points correspond to the transverse fields shown in the domain statistics data on the left.