Loop Quantum Photonic Chip for Coherent Multi-Time-Step Evolution

Simulating the evolution of quantum systems [1, 2] represents one of the most promising applications of programmable quantum processors [3, 4], providing practical means to explore and understand complex quantum systems that remain intractable for classical computers. A notable testbed for studying complex quantum evolution is the spin-boson model [5, 6] that describes the dynamics of a spin coupled to a bath. This model plays a crucial role in understanding light-matter interactions [7], quantum chemistry [8], charge transfer [9], exciton transport [10], macroscopic quantum tunneling in superconducting systems [11], polarons [12], and the nonadiabatic dynamics of molecules [13].

Recently, there has been significant progress in the development of different quantum processor platforms, including ultracold gases [14], trapped ions [15], superconducting qubits [16], and photonic [17, 18], even for the simulation of spin-boson-like systems [19, 20, 21]. The development of photonic systems has led to the introduction of on-chip quantum processors with the integration of photon generation and detection. These photonic chips  [22, 22, 23, 24] are solid state solutions, which work at room temperature, are reprogrammable, and whose realization is compatible with existing semiconductor fabrication technology. Importantly, they can be used for fault-tolerant computing, variational approaches, and analogue quantum simulation [25].

However, implementing quantum evolution on photonic platforms still presents significant challenges. Current approaches primarily rely on sequentially modifying chip parameters to perform time evolution for different time durations [26]. Additionally, these methods require reloading the computed unitary parameters [27, 28] or input states [22] onto the chip at each time step, resulting in significant overhead from repeated digital-to-analog and analog-to-digital conversions, as well as increased computational costs for recalculating the unitary parameters. To address these issues, multiplexing techniques such as wavelength-division multiplexing [29] and time-division multiplexing [30] have been explored. However, while these approaches can alleviate some hardware constraints, they often introduce additional challenges, including increased chip size and system noise. Furthermore, they do not eliminate the need for repeated quantum state tomography measurements, or repeated re-encoding of the output signal into a new input signal for the photonic chip.

To overcome these limitations, we propose a loop quantum photonic chip (Loop-QPC) that enables efficient quantum evolution by reusing a single chip for multiple time steps. The core of our design lies in a recirculating loop structure, which directs the output from one time step back into the system as input for the subsequent step, thereby eliminating the need for quantum state tomography and repeated parameter configurations. We thus program our Loop-QPC to execute the quantum evolution of the spin-boson model for time step $\Delta T$. By leveraging the low-loss properties of Silicon Nitride (SiN) and the high detection efficiency of Superconducting Nanowire Single Photon Detectors (SNSPDs), we are able to use our photonic chip to simulate the quantum evolution for three distinct time steps $\Delta T$ with high fidelity, while being able to take measurements at each of the time steps.

We consider the spin-boson model with a single spin coupled to a harmonic oscillator as depicted in Fig. 1a. In the strong coupling regime, its evolution in time is given by the following Hamiltonian [31],

where $\hbar$ is Planck’s constant, $\hat{a}^{\dagger}$ and $\hat{a}$ are the creation and annihilation operators for the harmonic oscillator, respectively, while the operators $\hat{\sigma}^{x}=\hat{\sigma}^{+}+\hat{\sigma}^{-}$ and $\hat{\sigma}^{z}$ are Pauli operators acting on the spin. The parameter $h$ represents the local magnetic field in the $z-$direction while $\epsilon$ is the field in the $x-$direction. The coupling strength between the spins and the harmonic oscillator is denoted by $\lambda$ and $\omega$ is the frequency of the oscillator. In the following, we will consider two main regimes with fundamentally different dynamics. One in which the field, the coupling strength and the frequency are identical $\epsilon=\lambda=\hbar\omega=h$, and the case in which they are similar in magnitude but different in value, i.e. $\epsilon=0.5h,\lambda=0.8h,\hbar\omega=1.2h$, so as to demonstrate the generalization of our setting.

To experimentally simulate the unitary time evolution generated by $\hat{\mathcal{H}}_{SB}$ over a time $T$, we decompose the evolution over $N_{T}$ time steps of duration $\Delta T=T/N_{T}$ and use

where $\hat{\mathcal{U}}_{\Delta T}$ is thus the evolution operator for a time $\Delta T$.

To represent the spin-boson model state and its evolution, we use the unary encoding (more detail on the unary encoding in App. A). In particular, we consider both levels for the spin$-1/2$ and the first three levels for the harmonic oscillator, thus resulting in a total of 6 levels, one per channel of our photonic chip (similarly to [32]). Our design of the Loop-QPC, shown in Figs. 1b and c, is composed of seven main modules: the photon generation module that produces one photon for the computation and one idler photon, the data encoding module, the ratio splitter module, the quantum unitary computation module $\hat{\mathcal{U}}_{\Delta T}$, the loop module $\mathcal{L}$, and the measurement module.

To facilitate the understanding of the process, we describe the system through three stages, input stage $\boldsymbol{X}$ corresponding to the initial input in the Loop-QPC, intermediate stages $\boldsymbol{I}_{n}$ that are the signal after going through the unitary portion $n$ times, and output stages $\boldsymbol{Y}_{n}$ that go to the measurement devices after passing $n$ times through the unitary. We use $\hat{\mathcal{S}}_{\mu}=2/3\;\hat{1}$ and $\tilde{\mathcal{S}}_{\mu}=1/3\;\hat{1}$ for the two splitting ratios of the ratio splitters (derivation in App. D), where $\hat{1}$ is the identity, $\mu=i,o$ indicates whether the ratio splitter is at the input ($i$) or output ($o$) of the unitary module, and we denote by $\mathcal{L}$ the loop portion. We can thus write the evolution as

For the experimental realization, we employ an integrated quantum photonic architecture implemented on a SiN chip as our experimental platform, as shown in Fig. 1c. This architecture minimizes the propagation path length and ensures uniform propagation loss across all paths. The structure consists of 15 unit cells. Each unit is constructed by a Mach-Zehnder interferometer (MZI) composed of two symmetric (50/50) beam splitters and an internal/external relative phase shift $\theta$/$\phi$. This configuration allows for the realization of an arbitrary 6-mode unitary matrix. The fundamental transformation of each unit cell is defined as,

Our compact phase shifter design shortens each MZI unit cell, effectively reducing the total propagation loss. Thanks to high-quality SiN fabrication, the waveguides achieve an optical transmission ($\alpha$) around 0.6 dB/cm. The MZI architecture also facilitates the input/output ratio splitting. In the experiment, the initial values for $\theta$ and $\phi$ are computed first using Clements et al. decomposition [27] and further refined through machine learning techniques to enhance the fidelity. The detailed decomposition of the arbitrary $N$-mode unitary matrix for the spin-boson model can be found in App. B, with machine learning optimization methods detailed in [33] and App. C. The loop module incorporates a low-loss SiN delay line in a spiral structure. This specially designed delay line ensures a fixed temporal separation between photons from adjacent loops, thereby preventing interference between different loop photons, which maintains the independence of the computations at each time step. To guarantee this independence, the loop length must exceed a critical threshold of $\approx 4$ cm computed from the speed of light in SiN and the temporal resolution of the SNSPD used for detecting photons. The low-loss SiN waveguides and high-efficiency SNSPDs are thus crucial for determining the large observable evolution steps in the Loop-QPC.

The whole Loop-QPC computing process is as follows: A pump laser operating at a 500 MHz repetition rate generates input light, resulting in signal peaks at 2 $\mu$s intervals. The pump light initiates a pair of signal and idler photons in the long waveguide via the spontaneous four-wave mixing process. By regulating the pump power, we balance the photon generation rate and the probability of generating multi-pair terms. The signal photons are used for computation, while the idler photons are used as a heralding to mark the timing of the signal photon. Since the signal and idler photons are generated simultaneously, coincidence measurements enable precise determination of the arrival times of signal photons. Output photons are coupled to fibers through a grating array, and photon detection is performed using seven SNSPDs, with six dedicated to signal photons and one for idler photons. This configuration allows the Loop-QPC to collect multiple outputs from different loops on a single chip in a single run, thereby efficiently realizing quantum dynamic evolution.

In this study, we have experimentally demonstrated the implementation of the unitary evolution operator for the Hamiltonian of the spin-boson model and its corresponding quantum dynamics on our Loop-QPC platform. The integrated design of a photonic circuit with loops, and the measurement method, allow us to simulate different time steps of the evolution with the same chip. This results in the avoidance of recalibrating the chip to simulate different time steps, or having to perform a state tomography to prepare states at intermediate times, which would require a significant amount of resources.

Furthermore, the approach is independent of the type of the unitary used nor of the encoding. It thus opens the way to the efficient, and scalable solution for quantum simulations, for example using binary encoding, advancing the potential of photonic quantum technologies. In the future, further reduction of the losses and improved tuning of the chip parameters could result in simulations for longer time steps and lower errors. This could open a way to simulate open dynamics by engineering, either via collisional models [36] or finite-size baths based on typicality [37].