Calculating strongly correlated ground states from the non-Markovian dissipative dynamics of Gaussian fermions

The gist: The mapping between interacting fermionic Hamiltonians and non-equilibrium steady states of dissipative open quantum systems allows for studying strongly correlated ground states by unraveling non-Markovian dynamics, revealing that while individual trajectories are Gaussian, the required number of trajectories scales exponentially with system size due to the sign problem associated with negative dissipative rates.

Mapping and Representation

The paper introduces a mapping between the ground state of interacting fermionic Hamiltonians and the non-equilibrium steady state of a purely dissipative open quantum system. Within this framework, the Fermi-Hubbard Hamiltonian is mapped onto Lindblad jump operators acting on Majorana fermions. Remarkably, both hopping and interaction terms map onto jump operators that preserve the Gaussianity of Majorana fermions along individual quantum trajectories. As a result, the dynamics can be unravelled and each trajectory can be simulated efficiently using only two-point correlation functions, with a computational cost that scales polynomially with system size.

Dissipative Dynamics and Non-Markovianity

The Hamiltonian maps to a Lindbladian Hˆ ↔ −L such that Hˆ ψ⟩ ↔ −Lρ. The density-density interaction terms map into jump operators that are quadratic in the Majorana operators. It is interesting that correlations in space in the interacting fermionic systems map to correlations in time in the OQS. Furthermore, positive values of µk are required to have a non-zero number of particles in the correlated system, resulting in a negative dissipation rate γ = −µ/2 < 0.

Quantum Trajectory Unravelling

The Monte Carlo Quantum Jumps (MCQJ) method unravels the Lindblad dynamics into stochastic quantum trajectories. The solution to the Lindblad equation is then recovered by averaging over a statistically significant number Ntr of trajectories. For non-Markovian dynamics, the protocol involves tracking the evolution of pure quantum states along a trajectory a, ρa = ψa⟩⟨ψa, subject to the random application of the jump operators appearing in the master equation, subject to the random application of the jump operators appearing in the master equation>. The evolution is governed by stochastic protocol involving quantum jumps or non-Hermitian evolution.

Computational Complexity and Sign Problem

The computational cost of simulating one trajectory is reduced to a polynomial scaling with system size. However, the number of trajectories needed to simulate the master equation dynamics scales exponentially with system size. This is because fluctuations of the classical bit s(t) grow exponentially with time. Specifically, one finds that s(t + dt) = s(t) (C2). This implies that Ntr ∼ e L is required for convergence, leading to a complexity of C ∼ L 2e L.

Gaussian Evolution and Expectation Values

A crucial observation is that all jump operators in Table (I) preserve the Gaussianity of a quantum state. This property allows the expectation value of an operator Oˆ to be calculated via Eq. (22). While the trajectory formulation does not eliminate the exponential complexity associated with finding correlated ground states, it reveals how this complexity is transferred from the representation of the quantum state to the statistical sampling required by the non-Markovian trajectory unravelling. In particular, individual trajectories remain Gaussian and can be simulated efficiently using only two-point correlation functions. However, the computational bottleneck emerges due to the exponentially large number of trajectories required to overcome the sign problem associated with negative dissipative rates.

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The gist: The mapping between interacting fermionic Hamiltonians and non-equilibrium steady states of dissipative open quantum systems allows for studying strongly correlated ground states by unraveling non-Markovian dynamics, revealing that while individual trajectories are Gaussian, the required number of trajectories scales exponentially with system size due to the sign problem associated with negative dissipative rates.

1. textbf Discovery of Strongly Correlated Ground States via Dissipative Dynamics Mapping: The improved system can find ground states of interacting fermionic Hamiltonians by mapping them to non-equilibrium steady states (NESS) of a purely dissipative open quantum system, as shown by the statement: we map the Fermi-Hubbard Hamiltonian onto Lindblad jump operators acting on Majorana fermions.

2. textbf Efficient Trajectory Simulation via Gaussian Representation: The system can simulate individual quantum trajectories efficiently using only two-point correlation functions, achieving a computational cost that scales polynomially with system size, exploiting the fact that the resulting dissipative dynamics preserves the Gaussianity of the Majorana modes along each trajectory.

3. textbf Handling Non-Markovian Complexity: The improved system can handle intrinsically non-Markovian dynamics, such as those arising from finite particle number requirements, by employing a stochastic protocol that tracks both quantum jumps and classical bits to manage negative dissipation rates, avoiding the sign problem in a standard sense by using the protocol described in Eq. (13).

4. textbf Scalable Expectation Value Calculation: The system can calculate expectation values of observables without exponential cost via Gaussian evolution, using Eq. (22) which reduces the calculation to the calculation of partition functions over trajectories, where expectation values are expressed as ⟨Oˆ ⟩ = P a,b sasb Z(A a⊕A′b) / (Z(Aa)Z(Ab)).

We introduce a mapping between the ground state of interacting fermionic Hamiltonians and the non-equilibrium steady state of a purely dissipative open quantum system. Within the framework of third quantization, we map the Fermi-Hubbard Hamiltonian onto Lindblad jump operators acting on Majorana fermions. Remarkably, both hopping and interaction terms map onto jump operators that preserve the Gaussianity of Majorana fermions along individual quantum trajectories. As a result, the dynamics can be unravelled and each trajectory can be simulated efficiently using only two-point correlation functions, with a computational cost that scales polynomially with system size. We further show that finite particle number requires negative dissipative rates, leading to an intrinsically non-Markovian dynamics. The corresponding trajectory unravelling involves both positive and negative stochastic weights and exhibits a sign problem and large fluctuations, so that convergence requires an exponentially large number of trajectories. The overall computational cost remains exponential in system size despite the efficient Gaussian representation of individual trajectories, but is crucially dependent on the computational complexity of the non-Markovian unravelling, motivating further studies on the efficiency of such unravellings.