Rahel Lea Baumgartner of the University of Geneva and Pietro Pelliconi of Princeton University propose a new approach to simulating notoriously difficult all-to-all interacting and disordered many-body systems. The researchers detail how a Trotterization scheme can be used to actually densify the disorder within a model, a counterintuitive technique given that interactions often reduce disorder in quantum simulations.

Their work centers on an implementation using a single-mode cavity QED platform to model the complex Sachdev-Ye-Kitaev model, with detailed analysis of effective couplings, interacting sites and quantum chaos probes. The study concludes with an investigation into the robustness of these findings against dissipation.

Trotterization Densifies Disorder for Quantum Simulation

A technique using Trotterization, a standard method for approximating quantum evolution, is now being applied to increase disorder within a quantum model, a counterintuitive step given that interactions typically reduce disorder in such simulations. This approach aims to create a more accurate representation of all-to-all interacting and disordered systems, notoriously difficult to simulate on quantum platforms. The work details a protocol for transforming a low-rank disordered model into a full-rank one by summing different realizations of the disorder through incremental Trotter steps.

This enhancement helps achieve simulations that accurately reflect the target system’s complexity. Analysis extends beyond simply achieving increased disorder. Researchers also investigated the impact of Trotterization errors on both time evolution and observable dynamical properties. They analytically determined that the Trotter error does not increase with system size, and numerical results demonstrate convergence toward the fully disordered model as the Trotter step size decreases.

This detailed examination of error sources is essential for validating the reliability of the simulation and understanding its limitations. The study also focuses on features of the resulting effective model, including the distribution of effective couplings, the number of interacting sites and the behavior of quantum chaos probes. A key distinction between the models lies in the disordered couplings. A rank-q tensor is expressed as the outer product of a rank-q/2 tensor with itself, potentially introducing spurious correlations.

The researchers state, outlining the core objective of their work. The team’s findings suggest that summing over different realizations can be efficiently implemented using the Trotterization scheme.

Single-Mode Cavity QED Platform for SYK Models

A single-mode optical cavity QED platform offers a pathway to simulate the complex Sachdev-Ye-Kitaev model, addressing a longstanding challenge in quantum many-body physics. This approach uses a strategy of sequentially implementing sparse versions of the model, a technique designed to enhance the inherent disorder essential for accurately representing the target system. This densification relies on digitally summing random interactions over time, rather than attempting to create fully connected interactions directly.

The cavity QED platform can realize each sparse Hamiltonian one after the other, approximating the evolution under the target Hamiltonian with a chosen accuracy threshold. This digital summation avoids the need for complex multi-qubit gates, instead using existing capabilities for single-shot analog simulation of sparse-coupling Hamiltonians. The team’s method involves cycling through R steps, effectively exposing the system to a combined Hamiltonian after n cycles of duration Rδt.

The proposed implementation centers on a quantum degenerate Fermi gas within a high-finesse optical cavity, a system increasingly recognized for its potential in realizing disordered many-body systems. Optical cavities naturally create long-range one-body interactions, a feature particularly advantageous for simulating models like the Sachdev-Ye-Kitaev model.

“Thanks to numerous recent advances in this field, optical cavities have emerged as a highly promising playground for the realization of disordered many–body systems,” notes a reference cited in the work. This platform’s ability to natively implement sparse Hamiltonians is central to the proposed simulation strategy, offering a practical route toward modeling notoriously difficult all-to-all interacting systems.

All-to-All Interactions in Disordered Systems

This approach moves beyond spatially structured interactions to tackle systems with random couplings, mirroring those found in spin glasses and spin liquids, and potentially revealing insights into holographic phases of matter. A key innovation lies in the analog-digital strategy employed, expanding the range of models accessible through cavity platforms. We propose an implementation via a single–mode cavity QED platform of the complex Sachdev–Ye–Kitaev model.

This method’s computational complexity scales favorably with system size, representing an improvement over previous attempts to simulate the SYK model using sparse interactions. The favorable scaling suggests the possibility of simulating larger and more complex disordered systems than previously feasible.

Complex Sachdev-Ye-Kitaev (cSYK) Model Implementation

This approach uses a Trotterization scheme, not merely for computational efficiency, but to actively ‘densify’ the disorder inherent in the model being simulated, enabling a more accurate representation of complex physical phenomena. The proposed cavity QED scheme builds upon prior theoretical work and aims to realize the cSYK model with four-fermion interactions. Numerical simulations of out-of-time-ordered correlators (OTOCs) and the spectral form factor (SFF) were performed on the effective Hamiltonian derived from the experimental setup, revealing strong agreement with theoretical predictions for an ideal SYK system.

These calculations suggest the feasibility of using this platform to accurately model the complex dynamics of disordered quantum systems. The team’s analysis of the effective Hamiltonian includes the distribution of effective couplings, offering insight into the characteristics of the simulated system.

Sparse Hamiltonian Realizations via Speckled Light

Combining multiple sparse, randomly generated Hamiltonians allows the physics of the resulting aggregate to approach that of a fully disordered system, effectively eliminating sparsity as the number of realizations increases. This strategy, detailed in recent work, uses a time-based implementation where a cavity QED platform cycles through each Hamiltonian realization sequentially, a process known as Trotterization. The resulting evolution approximates a target Hamiltonian over a defined time period, achieving a specified accuracy threshold.

This Trotterization scheme achieves chaos, a sensitive dependence on initial conditions, even with a low ratio of realizations to the number of quantum systems involved. The primary purpose of this approach is not computational efficiency, but to eliminate statistical correlations inherent in sparse disorder, representing a notable improvement in simulation methodology.

Trotter Errors and Time Evolution Accuracy

Trotter errors do not increase with system size in this new approach, a result aligning with prior findings on localized errors. This favorable characteristic stems from the specific implementation of a Trotterization scheme designed to densify the disorder within the modeled system, a departure from typical methods where interactions often diminish inherent disorder. The team demonstrated that the time evolution achieved through this process converges uniformly to that of a fully disordered model as the time step decreases, minimizing inaccuracies introduced by the approximation.

The preparation of sparse random disorder models relies on a single-mode optical cavity, where light-matter couplings between atoms and the cavity mode are randomized using a projected light-shifting beam. This technique allows for the creation of the initial Hamiltonian, an important step in simulating complex many-body systems like the Sachdev-Ye-Kitaev (SYK) model.

Analysis of the resulting effective Hamiltonian reveals that increasing the number of Trotter steps improves the agreement between the effective model and the target SYK model, as observed in evaluations of the time-evolution operator’s trace. The team’s work builds on the Trotter-Suzuki formula, a standard method for approximating the time evolution of a sum of Hamiltonians, but applies it with a focus on maintaining and even enhancing disorder. This approach allows for a uniform approximation of the dynamics, offering a robust method for simulating complex quantum systems and probing their behavior over time.

Convergence of Effective Hamiltonian to Dense Model

Increasing the number of distinct random realizations within a quantum simulation effectively drives the system toward a fully disordered state, mirroring the target Hamiltonian despite initial sparsity. This convergence, observed through the effective Hamiltonian, occurs as the number of realizations, denoted as R, grows large, effectively densifying the disorder present in the model. The team established a threshold for uniform convergence between the Trotterized and exact dense models, dependent on the number of Trotter steps and the timescale of the simulation.

This means that the simulation’s accuracy is predictable and controllable, with errors quantifiable based on the chosen parameters. Their work confirms that a Trotterized experimental simulation will closely approximate the exact dense effective model, providing a pathway to simulate complex systems with controlled errors.

Distribution of Effective Couplings in Simulation

Analysis of the effective model’s couplings reveals a convergence toward independent and identically distributed values, a key indicator of simulation accuracy. This consistency, detailed in accompanying visualizations, supports the claim that the Trotterized simulation effectively approximates the target complex Sachdev-Ye-Kitaev model. The team further refined their analysis by shifting the ‘diagonal’ couplings by their means, a technique used to more clearly identify and compare the effective and dense models.

This adjustment allowed for a direct assessment of how well the simulation captures the underlying statistical properties of the system. The resulting distribution of effective couplings, as shown in accompanying panels, aligns with theoretical predictions, bolstering confidence in the simulation’s fidelity.

The study assumes all couplings are independent and identically distributed for each realization, using a Gaussian distribution with a defined variance to model their behavior. Their findings suggest that the proposed single-mode cavity QED platform provides a viable route toward simulating all-to-all interacting and disordered many-body systems, offering a means to explore complex phenomena previously inaccessible to conventional computational methods.

Robustness to Dissipation in Cavity QED Setup

The statistical independence of dissipative and coherent aspects within the model extends to corrections of order 1/R, where R represents the number of speckle patterns employed per Trotter step. This independence is achieved through the densification of random terms using the Trotter scheme, a technique that effectively manages the interplay between disorder and simulation accuracy. Analysis further reveals a method for dynamically determining the effective number of interacting sites, denoted as N, within the cavity itself.

State preparation, using product states as initial conditions, was also investigated as a key component of the simulation process. Section 4 of the study focuses on the impact of dissipation within the cavity platform, specifically examining photon leakage through the mirrors and photon scattering, random photon emission, as primary sources affecting coherent evolution.

Operating in a regime where photon scattering, represented by Γ, is minimized while maintaining a finite photon leakage rate, κ, is a key consideration. The analysis demonstrates that the effect of dissipation relative to coherent interaction strength depends on the ratio γ = R/N. This large-N analysis, detailed in an appendix, focuses specifically on photons leaking from the cavity through the mirrors, described by the jump operator. The speckle pattern introduces a random component to these jump operators, influencing the overall simulation.

👉 More information
🗞 Quantum simulation using Trotterized disorder Hamiltonians in a single-mode optical cavity
✍️ Rahel Lea Baumgartner, Pietro Pelliconi, Soumik Bandyopadhyay, Francesca Orsi, Philipp Hauke, Jean-Philippe Brantut and Julian Sonner
🧠 DOI: http://link.aps.org/doi/10.1103/74wt-dwdz

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