Photonic Simulation of Localization Phenomena Using Boson Sampling

TL;DR

The study demonstrates that photonic boson sampling can act as a compact, room-temperature platform to simulate localization dynamics in non-interacting tight-binding systems. By mapping the Hamiltonian evolution to photonic interferometers via a Lie–Trotter decomposition, it reproduces Bloch oscillations, dynamical localization under periodic driving, and the Aubry–André–Harper phase transition, with results matching conventional numerical calculations. The work also analyzes how sampling depth (shots) affects accuracy and outlines pathways to extend to many-body and Gaussian boson sampling, highlighting potential practical advantages and open scalability questions for photonic quantum simulation. This approach offers a promising route to study complex quantum dynamics without full state tomography, leveraging continuous-variable photonics and room-temperature operation.

Abstract

Quantum simulation in its current state faces experimental overhead in terms of physical space and cooling. We propose boson sampling as an alternative compact synthetic platform performing at room temperature. Identifying the capability of estimating matrix permanents, we explore the applicability of boson sampling for tackling the dynamics of quantum systems without having access to information about the full state vector. By mapping the time-evolution unitary of a Hamiltonian onto an interferometer via continuous-variable gate decompositions, we present proof-of-principle results of localization characteristics of a single particle. We study the dynamics of one-dimensional tight-binding systems in the clean and quasiperiodic-disordered limits to observe Bloch oscillations and dynamical localization, and the delocalization-to-localization phase transition in the Aubry- Andre-Harper model respectively. Our computational results obtained using boson sampling are in complete agreement with the dynamical and static results of non-interacting tight-binding systems obtained using conventional numerical calculations. Additionally, our study highlights the role of number of sampling measurements or shots for simulation accuracy.

Figures (11)

• Figure 1: An illustration of the methodology to study quantum dyamics using boson sampling. The steps enclosed within the black box denote the actions to be implemented at each timestep to be considered. The steps enclosed within the red box may be performed for a high number of 'shots' for an increasingly accurate reconstruction of the probability distribution upon averaging over the samples.
• Figure 2: An example of a boson sampling circuit with $M=7, N=2$. The probability of finding the photons (Eq. (\ref{['perm']})) in a particular configuration depends on the permanent of a $2\times2$ sub-matrix of the $7\times7$ unitary matrix characterizing the interferometer. The columns of the sub-matrix are determined based on the output state, and the rows of the sub-matrix based on the input state.
• Figure 3: (a) An example of the decomposition of a time-evolution unitary of an arbitrary Hamiltonian (Eq. (\ref{['eq:arbitraryhamiltonian']})) into continuous-variable quantum gates (Eq. (\ref{['unitarytransformation']})). The system size here is $M = 5$ and a single photon ($N=1$), with $l=3$ as per Eq. (\ref{['unitarytransformation']}). (b) An example of the boson sampling circuit ($M=7$) for simulation the dynamics of the static case of Eq. (\ref{['hamiltonian_bloch']}) upto the first timestep. This interferometer has been characterized according to a rectangular decompositionclements with $M(M-1)/2 = 21$ beam-splitters using the class.
• Figure 4: Static Field (a): Evolution of onsite populations $P_i(t)$ in the presence of a static electric field, obtained via boson sampling (first row) and numerical computations (second row). The $F=1$ case illustrates boundary effects in a device of size smaller than the Bloch length, whereas Bloch oscillations are clearly visible in the $F=2$ case.
• Figure 5: Dynamics of the system subjected to static electric field (a), (b): Comparison of results between numerics and boson sampling for mean squared displacement as a function of time for field strengths $F = 1$ and $F = 2$ respectively. Parameters considered are $M = 7$, $J= 1$, $t = 3T_b$.