Consider a spin-1/2 Ising model with time-dependent transverse field:

$$ H = - \sum_{i<j} J_{i, j} \sigma^z_i \sigma^z_j - \Gamma(t) \sum_i \sigma^x_i$$

Given the initial state $|\psi(0)\rangle$ (usually $|\psi(0)\rangle = \left( \frac{|0\rangle + |1\rangle}{\sqrt{2}} \right)^{\otimes N}$), what are the state-of-the-art methods for numerically calculating the expectation value of the final state with respect to an observable $\sigma^z_i$: $\langle\psi(t')|\sigma^z_i|\psi(t')\rangle$?

My current method

I believe this method is called discrete-time path-integral Monte Carlo method.

Referring to chapter 5.3 of (Binder and Heermann, 2019), by dividing the Hamiltonian into two non-commuting parts and apply Trotterization, we can obtain a corresponding classical Hamiltonian ($z_i^k = \pm 1$) that is one-dimensional higher: $$ H_c = - \sum_{k=1}^P \left( \sum_{i<j} J_{i, j} z_i^k z_j^k + J^\perp(t) \sum_i z_i^k z_i^{k+1} \right)$$ $$ J^\perp(t) = -\frac{PT}{2} \ln \tanh \frac{\Gamma(t)}{PT}$$ and then use a classical Monte Carlo method (e.g. simulated annealing) to sample the classical states at each time step. The expectation value $\langle\psi(t')|\sigma^z_i|\psi(t')\rangle$ is estimated by $\frac{1}{P} \sum_{k=1}^P z_i^k$.

  • $\begingroup$ In how many dimensions are you mainly interested? I believe Matrix-Product States (aka DMRG) are excellent in 1D by now, but much harder for larger dimensions $\endgroup$
    – Wouter
    Oct 27, 2021 at 10:50
  • $\begingroup$ @Wouter Do you mean the number of spins or the connectivity? The number of spins is usually around a few hundreds, sometimes up to ~2,000, with full connectivity between spins (i.e. generally $J_{i, j} \neq 0 \forall i, j$). Simulating 2,000 spins isn't necessary, but I would like to simulate at least 100 spins with full connectivity. $\endgroup$
    – haoyu
    Oct 27, 2021 at 14:38
  • 1
    $\begingroup$ @Wouter I looked into DMRG, unfortunately I need to calculate problems that have denser interactions between spins than even a 3D lattice model. $\endgroup$
    – haoyu
    Oct 27, 2021 at 14:51
  • $\begingroup$ What about simple free fermion simulation? You can turn this problem into a time-dependent Majorana fermion chain and evolve in time by keeping track of the correlation matrix alone. $\endgroup$
    – Zack
    Oct 27, 2021 at 15:07
  • 1
    $\begingroup$ See: arxiv.org/abs/quant-ph/0108010 $\endgroup$
    – Zack
    Oct 27, 2021 at 18:11


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