Quantum Ising model on two-dimensional anti–de Sitter space

This paper investigates the transverse Ising model on a discretization of two-dimensional anti–de Sitter space. We use classical and quantum algorithms to simulate real-time evolution and measure out-of-time-ordered correlators (OTOC). The latter can probe thermalization and scrambling of quantum information under time evolution. We compared tensor network-based methods both with simulation on gate-based superconducting quantum devices and analog quantum simulation using Rydberg arrays. While studying this system’s thermalization properties, we observed different regimes depending on the radius of curvature of the space. In particular, we find a region of parameter space where the thermalization time depends only logarithmically on the number of degrees of freedom.

Figure 1

von Neumann Entropy versus $J$ for $N=37$, $l_{\max }=3.0,h=3.0,m=0.25$.

Figure 2

Magnetic susceptibility versus $J$ for $N=37$, $l_{\max }=3.0,h=3.0,m=0.25$.

Figure 3

$⟨S^z⟩$ for a lattice with $N=37$ spins and parameters set at $J=2.0,h=2.0,m=0.25,l_{\max }=3.0$.

Figure 4

Trotter evolution circuit for the first Trotter step for a 5-Qubit spin chain in the hyperbolic lattice. Here, ${\theta }_i=-\frac{J\delta t}{4}({\eta }_i+{\eta }_{i+1})$ and ${\varphi }_i=h\delta t{\eta }_i$.

Figure 5

Local magnetization $⟨S_z^i(t)⟩$ at site $i$ for TFI model on hyperbolic lattice chain with 13 lattice sites. Parameters: $J=2.0$, $h=1.05$, $l_{\max }=3.0$. We took advantage of the symmetry of the lattice Hamiltonian to find the average magnetization: $⟨S_i^z⟩\to (⟨S_i^z⟩+⟨S_{N-i}^z⟩)/2$. This played an important role since some of the physical qubits of the Guadalupe machine have smaller energy relaxation ($T1$) and dephasing($T2$) times.

Figure 6

Trotter evolution of local magnetization $⟨S_i^z(t)⟩$ with exact diagonalization. Parameters: $N=13$, $J=2.0$, $h=1.05$, $l_{\max }=3.0$.

Figure 7

Trotter evolution of local magnetization $⟨S_i^z(t)⟩$ computed using guadalupe quantum processing unit (QPU). Parameters: $N=13$, $J=2.0$, $h=1.05$, $l_{\max }=3.0$. Magnetization data on the edges of the lattice chain are omitted due to the large Trotter-error associated with it. Note that the deformation strength is stronger on the edges.

Figure 8

Heisenberg time evolution for a local operator.

Figure 9

Heisenberg time evolution for an operator $W(t)$.

Figure 10

OTOC [$O_i(t)$] for the planar Ising model can be obtained by setting the deformation scale $l_{\max }$ to zero. Parameters: $J=6.0$, $h=3.05$, $m=0.25$.

Figure 11

OTOC [$O_i(t)$] in the hyperbolic Ising model. Parameters: $l_{\max }=3.0$, $J=6.0$, $h=3.05$, $m=0.25$.

Figure 12

Site averaged OTOC for $J=6,h=3.05,m=0.25,l_{\max }=3.0$.

Figure 13

von Neumann entropy for $J=6,h=3.05,l_{\max }=3.0$.

Figure 14

OTOC phase diagram for $N=37$ lattice spins.

Figure 15

Curvature dependence of the propagation behavior of OTOC for $N=37,J=6.0,h=3.05,m=0.25$.

Figure 16

$XX$-OTOC for $N=37,J=6.0,h=3.05,m=0.25$.

Figure 17

Schematic circuit diagram of OTOC using the definition at Eq. (12).

Figure 18

Modified OTOCs are computed from the correlation of the measurement of two different operators (a) $⟨W(t)⟩$ and (b) $⟨V^{†}W(t)V⟩$. The same set of unitaries are required to find the correlation between the measurements. The process is repeated for many different sets of unitaries.

Figure 19

Change in correlation of the operators $⟨W(t)⟩=⟨{\sigma }_3^z(t)⟩$ and $⟨VW(t)V⟩=⟨{\sigma }_2^z{\sigma }_3^z(t){\sigma }_2^z⟩$ over time. Parameters: $l_{\max }=3.0,J=-0.5$, $h=-0.525$, $W(t)={\sigma }_3^z(t)$, and $V={\sigma }_2^z$.

Figure 20

Modified OTOC of the zeroth order, $O_0(t)$ for $l_{\max }=3.0,J=-0.5$, $h=-0.525$, $W(t)={\sigma }_3^z(t)$, and $V={\sigma }_2^z$.

Figure 21

Modified OTOC as the position $i$ of the $V$ operator varies. Parameters: $l_{\max }=3.0,J=-0.5$, $h=-0.525$, $W(t)={\sigma }_3^z(t)$, and $V={\sigma }_i^z$.

Figure 22

Shot noise analysis at Guadalupe machine is presented with local magnetization data. Shot noise associated for each trotter step is demonstrated in the bottom panel for the better visualization. The number in the labels denote the number of shots applied for measurements. Gap between the corresponding classical TEBD simulation results and QPU results indicate the presence of other coherent and incoherent sources of noise. Parameters: $J=2.0$, $h=1.05$, $l_{\max }=3.0$.

Figure 23

Comparison of Trotter evolution of magnetization results with different operator ordering. Parameters: $J=2.0$, $h=1.05$, $l_{\max }=3.0$.

Figure 24

Comparison of magnetization results with different mitigation techniques and their combinations. Parameters: $J=2.0$, $h=1.05$, $l_{\max }=3.0$.

Figure 25

Comparison of Trotter evolution of magnetization results in different noise scaled circuits. Noise $\mathrm{scale}=n$ indicates $n$-fold noise compared to the original circuit for the Trotter evolution of the local magnetization. Parameters: $J=2.0$, $h=1.05$, $l_{\max }=3.0$.

Figure 26

(a–b) Example of the extraction of the zero noise extrapolated data (red cross) at the second and the sixth trotter step, obtained from the measurements of the noise-scaled-circuits at Guadalupe machine. (c) Extrapolated values are obtained for all trotter steps to plot ZNE data of the local magnetization. Parameters: $J=2.0$, $h=1.05$, $l_{\max }=3.0$, $\delta t=0.2$.

Figure 27

Time evolution of the Rydberg density.

Figure 28

Choice of the Trotter step $\delta t\sim 0.5$ seems a good choice for the OTOC computation with our choice of parameters.

Figure 29

OTOC computed with the protocol with global unitaries match with traced data of products of operators.

Figure 30

$ZZ$ OTOC for $J=6,h=3.05,m=0.25,l=2.0$.

Figure 31

$ZZ$ OTOC for $J=6,h=3.05,m=0.25,l=5.0$.