Simulation of bilayer Hamiltonians based on monitored quantum trajectories

Abstract

In the study of open quantum systems it is often useful to treat mixed states as pure states of a fictitious doubled system. In this work we explore the opposite approach: mapping isolated bilayer systems to open monolayer systems. Specifically, we show that arbitrary bilayer Hamiltonians possessing an antiunitary layer exchange symmetry, and subject to a constraint on the sign of interlayer couplings, can be mapped to Lindbladians on a monolayer system with some of the jump operators postselected on a fixed outcome ("monitored"). Low-energy states of the bilayer Hamiltonian then correspond to late-time states of the monolayer dynamics. Simulating the latter by quantum trajectory methods has the potential of substantially reducing the computational cost of estimating low-energy observables in the bilayer Hamiltonian by effectively halving the system size. The overhead due to sampling quantum trajectories can be controlled by a suitable importance sampling scheme. We show that, when the quantum trajectories exhibit free fermion dynamics, our approach reduces to the auxiliary field quantum Monte Carlo (AFQMC) method. This provides a physically transparent interpretation of the AFQMC sign-free criteria in terms of properties of quantum dynamics. Finally, we benchmark our approach on the 1D quantum Ashkin-Teller model.

Figures (4)

• Figure 1: Schematic of the mapping between bilayer Hamiltonians (left) and monolayer dynamics (right). The two layers $l$, $r$ are viewed as the "ket" and "bra" sides of a density matrix $\rho$. The bilayer Hamiltonian $\mathcal{H}$ comprises intralayer terms $h_{i,l}$, $\bar{h}_{i,r}$ and interlayer couplings $J_i O_{i,l} \bar{O}_{i,r}$, with $J_i > 0$ and the bar denoting an antiunitary transformation. The monolayer dynamics $\mathcal{L}$ comprises dissipation (jump operators $L_i$) and monitoring (jump operators $\Tilde{L}_i$), see Eq. (\ref{['eq:Li_def']},\ref{['eq:tildeLi_def']}).
• Figure 2: Illustration of the qubit trajectories discussed in Sec. \ref{['subsec: dimer example']}. The trajectories $s$ and $s'$ can be combined into a single trajectory $\sigma$ with the initial and final states being $\ket{0}$. At each time step, the state $\ket{\psi_s}$ is proportional to either $\ket{0}$ or $\ket{1}$. At the middle point, we have $z(s)=z(s')$
• Figure 3: The Hamiltonian of 1D quantum Ashkin-Teller model consists of two copies of transverse-field Ising models represented as blue and red dots respectively, with an interlayer coupling tuned by $\lambda_J$ and $\lambda_h$ represented as dashed rungs.
• Figure 4: Numerical benchmark of importance-sampled quantum trajectory method. We simulate a 1D quantum Ashkin-Teller model with $L = 8$, open boundary conditions, and parameters $J=1$, $h=0.3$, $\lambda_J = \lambda_h = \lambda$ (see legend). (a) Intralayer correlator $C^{(1)} = \langle Z_{1,l} Z_{2,l}\rangle$ and (b) interlayer correlator $C^{(2)} = \langle Z_{1,l} Z_{2,l} Z_{1,r} Z_{2,r}\rangle$ vs inverse temperature $\beta$. Empty diamonds show results of quantum trajectory simulation, averaging 128 runs with $2\times 10^5$ Metropolis updates each. Solid lines show exact Krylov imaginary time evolution under $\mathcal{H}$ (both approaches are Trotterized with $\Delta\beta = 0.1$). Error bars are mostly invisible, with the exception of $\beta=0.5$ and $\lambda = 1$ which displays very large uncertainty. This is explained by the near-zero value of $C^{(1)}$ at the same point (see main text). Inset: statistical uncertainties on $C^{(2)}$ as a function of $|C^{(1)}|$, the dashed line shows inverse proportionality for reference.