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

summary

The gist

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

In short

The paper maps interacting fermionic Hamiltonians onto dissipative open quantum systems using Majorana fermions and Lindblad jump operators. This allows studying correlated ground states by unravelling non-Markovian dynamics into stochastic trajectories. While individual trajectories are Gaussian and efficient to simulate, negative dissipation rates introduce a sign problem requiring an exponentially large number of trajectories for convergence, making the overall simulation cost exponential.

Key concepts

Mapping to Majorana Fermions: The paper translates the Fermi-Hubbard Hamiltonian into a system described by Majorana fermions using Lindblad jump operators. This mathematical tool allows complex fermionic interactions to be represented in a way that preserves Gaussian properties along individual quantum paths, simplifying the description of the dynamics.
Non-Markovian Dynamics: This refers to dynamics where the future state depends on the entire history of past states, rather than just the immediate present. In this context, it arises because finite particle numbers necessitate negative dissipative rates in certain regions, leading to a complex temporal correlation that complicates standard Markovian descriptions.
Quantum Trajectory Unravelling: This method breaks down the continuous evolution of the open quantum system into a sequence of discrete stochastic paths. Each path represents a specific realization of quantum jumps dictated by the jump operators. Simulating these individual trajectories is computationally efficient, scaling polynomially with system size.
Sign Problem and Exponential Cost: Because negative dissipative rates are present, the stochastic weights in the trajectory unravelling become mixed (positive and negative). This leads to a sign problem where fluctuations grow exponentially with time, necessitating an exponentially large number of trajectories to obtain a converged average, resulting in an overall exponential computational cost.

Terminology used across episodes

Reinforcement Learning (RL) — explained in 25 episodes
Sample Complexity — explained in 18 episodes
Federated Learning — explained in 17 episodes
Retrieval-Augmented Generation (RAG) — explained in 16 episodes
Interpretability — explained in 15 episodes
Quantum Fisher Information (QFI) — explained in 15 episodes

This episode discusses

The paper

Calculating strongly correlated ground states from the non-Markovian dissipative dynamics of Gaussian fermions · Read on arXiv

Giuliano Chiriaco

Dipartimento di Fisica e Astronomia, University of Catania

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.

DOI: 10.1088/2058-9565/aeb256

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Calculating strongly correlated ground states from the non-Markovian dissipative dynamics of Gaussian fermions".

Mira: 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,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: Okay, so to recap where we are is that this 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. The core thesis is that you can map the Fermi-Hubbard Hamiltonian onto Lindblad jump operators acting on Majorana fermions.

Mira: And they make a remarkable claim about those mappings: both the hopping and interaction terms map onto jump operators that preserve the Gaussianity of those Majorana fermions along individual quantum trajectories. This is really neat because it keeps the individual paths simple to track.

Lev: But then they introduce a crucial finding that finite particle number requires negative dissipative rates, which forces the dynamics to be intrinsically non-Markovian, which is a significant physical detail to pin down.

Kai: So the paper is essentially showing that strongly correlated ground states can be studied by looking at this non-equilibrium steady state of an open system, and even though the individual paths are Gaussian, the required statistical sampling becomes very difficult.

Mira: It matters because it shows a new perspective on the complexity of finding ground states in correlated systems by framing them through this dissipative lens. It opens up directions for research specifically regarding that sign problem and how to unravel those non-Markovian master equations.

Lev: From an engineering standpoint, the implication is that while we can simulate one trajectory efficiently using two-point correlation functions, the practical cost of finding the full ground state seems tied directly to how hard it is to handle this non-Markovian unravelling step.

Conclusion: Kai: So we’ve looked at the whole thing now, "Calculating strongly correlated ground states from the non-Markovian dissipative dynamics of Gaussian fermions." The authors are Chiriac`o, Giuliano one, and it's a mapping between Hamiltonians and open quantum systems.

Mira: What this paper really does is show that we can use this dissipative framework to study strongly correlated ground states. The key insight is the Gaussianity of the trajectories versus the exponential cost imposed by non-Markovian dynamics.

Lev: In simple terms, it means you can represent complex fermionic physics in a way that looks mathematically tractable for individual paths, but then you hit a huge statistical hurdle when trying to reconstruct the full steady state because of those negative rates causing sign problems.

Kai: It’s about showing how complexity shifts: instead of just a hard Hamiltonian, the difficulty becomes managing an exponentially large number of stochastic samples needed to average out the non-Markovian evolution.

Mira: And for future research, it points toward needing new ways to handle that unravelling process because that's where the bottleneck is right now.

Lev: Right, so the focus shifts from just making one trajectory fast to finding a way around that exponential statistical sampling requirement imposed by those negative dissipative rates.

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