⚛️ quantum physics

Realistic simulation of universal multiport interferometers with losses within a Gaussian formalism

This paper proposes a Gaussian formalism to realistically simulate universal multiport interferometers with non-uniform optical losses, enabling accurate characterization and prediction of quantum algorithm outcomes on noisy NISQ photonic platforms.

Original authors: Sebastiano Corli, Dario Chemoli, Samuele Altilia, Edoardo Suerra, Dario Tamascelli, Simone Cialdi, Enrico Prati

Published 2026-10-14

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CC BY 4.0

Original authors: Sebastiano Corli, Dario Chemoli, Samuele Altilia, Edoardo Suerra, Dario Tamascelli, Simone Cialdi, Enrico Prati

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the race to build a machine that can solve problems beyond the reach of today's computers, light has emerged as a leading contender. Unlike other approaches that rely on tiny electronic circuits or trapped atoms, these photonic computers use individual particles of light, called photons, to carry information. The advantage lies in how naturally light behaves; it travels fast, generates little heat, and can be manipulated with mirrors and lenses to perform complex calculations. However, this path is fraught with a single, persistent obstacle: loss. As photons travel through the delicate glass channels of a computer chip, some inevitably vanish, absorbed by the material or scattered into the air. This disappearance is not just a minor glitch; it is the primary barrier preventing these machines from reaching their full potential. To understand how close we are to a working device, scientists must be able to simulate these machines on classical computers, but doing so is incredibly difficult when the system is imperfect. If a simulation assumes the light behaves perfectly, it may predict a result that the real, flawed machine can never produce.

A team of researchers has developed a new way to model these imperfect machines, creating a realistic blueprint for how light behaves when it leaks out of the system. Their work focuses on a specific type of optical computer that uses a grid of tiny interferometers—devices that split and recombine light beams—to process information. In an ideal world, these devices would guide every photon exactly where it needs to go. In reality, the glass and the connections between them are imperfect, causing a fraction of the light to be lost at every step. The researchers realized that trying to track every single lost photon and the environment it escapes into would make the computer simulation too heavy to run. Instead, they devised a method to condense all these losses into a manageable mathematical form. By treating the entire network of light paths as a single system interacting with a simplified environment, they could calculate the final state of the light without needing to simulate the infinite possibilities of the lost energy.

The core of their discovery is a technique that allows them to take a complex, messy network of light paths and describe it using a single, unified map of how the light changes. They demonstrated that even when the light is lost unevenly—meaning some paths are dirtier or more damaged than others—the overall effect can be captured by a specific set of numbers that describe the system's "fuzziness" or uncertainty. This approach works for two different ways of building these computers: one where light flows through many parallel wires at once, and another where light travels in loops, bouncing back and forth. The researchers tested their method by running thousands of simulations on a classical computer, mimicking the behavior of these optical devices under various conditions. They found that while the light's behavior is indeed affected by the specific layout of the machine, the new model could accurately predict the outcome of the experiment, including the probability of detecting specific numbers of photons at the end.

One of the most significant findings of this study challenges a common shortcut used by scientists. For years, many researchers have assumed that optical losses are uniform, meaning they happen at the exact same rate in every part of the machine, to make their calculations easier. The new simulations show that this assumption is often incorrect. In one of the common designs, where light travels through a triangular mesh of components, the losses are not uniform; they vary depending on where the light is in the network. The researchers showed that assuming uniform losses can lead to wrong predictions about the machine's output, especially as the machine gets larger. In contrast, their new model, which accounts for the specific, uneven nature of the losses, provides a much more accurate picture of what the machine will actually do. This is crucial because if scientists want to build error-correction systems to fix these mistakes, they first need to know exactly what the mistakes look like.

The team also explored how these losses affect the final result of a calculation known as Gaussian boson sampling, a task designed to prove that quantum computers can outperform classical ones. They simulated the process of sending squeezed light—a special state where the uncertainty of the light is reduced in one direction—through a lossy network. They observed that as the light travels through the imperfect machine, the probability of detecting photons shifts dramatically. The vacuum state, where no photons are detected, becomes much more common in a noisy environment than in a perfect one. Furthermore, they found that the specific pattern of losses changes the likelihood of detecting certain numbers of photons, breaking the symmetry that would exist in a perfect machine. By comparing their realistic model against the simplified uniform-loss assumption, they could see exactly where the two diverge. The difference grows larger as the number of light paths increases and as the amount of loss gets worse, confirming that the simplified view is insufficient for large-scale devices.

This work provides a vital tool for the future of optical quantum computing. It offers a way to take real-world data about how a specific chip loses light and plug it directly into a simulation to see how a quantum algorithm would perform on that exact hardware. This means that instead of guessing how a machine will behave, engineers can now test their designs with a high degree of realism before they are even built. The researchers suggest that this framework can be adapted to any type of continuous-variable optical architecture, whether it uses spatial paths or time-based loops. By accurately characterizing these losses, the scientific community can move closer to developing effective error correction techniques, which are essential for building a fault-tolerant quantum computer. The study does not claim to have solved the problem of light loss, but it has provided a clear, realistic map of the terrain, showing exactly where the pitfalls lie and how to navigate them. This clarity is a necessary step toward turning the theoretical promise of photonic quantum computing into a practical reality.

Title: Realistic simulation of universal multiport interferometers with losses within a Gaussian formalism

Problem Statement
Photonic technologies are leading contenders for demonstrating quantum advantage, yet photon losses represent the primary obstacle to realizing fault-tolerant optical quantum computers. Accurate characterization of these losses is critical for classical simulations of quantum algorithms on noisy backends. Current simulation approaches often rely on the a priori hypothesis that optical losses are uniform across all channels. The authors argue that this assumption may lead to incorrect predictions regarding algorithm outcomes on specific hardware. Furthermore, while error correction codes exist for qubit-based devices, no established technique exists for correcting photon losses in continuous-variable (CV) quantum computing. Existing software libraries for CV simulations in noisy regimes are lacking compared to those for matter-based qubits (e.g., Qiskit, Cirq). Additionally, modeling losses by introducing external environmental modes for every interaction leads to a polynomial increase in the number of modes, rendering simulations of Fock space computationally unfeasible.

Methodology
The authors propose a Gaussian formalism to characterize and simulate optical losses in multiport interferometers, applicable to both spatial-bin and time-bin encoded architectures. The methodology proceeds as follows:

Theoretical Framework: The system is described using the Heisenberg basis (quadratures $\hat{q}$ and $\hat{p}$ ) rather than the Fock basis. The dynamics of Gaussian states are governed by the mean vector and the covariance matrix. Optical losses are modeled as interactions between physical modes and external environmental modes, represented as beam-splitter operations.
Characterization of Imperfect Interferometers: The authors build upon the characterization of imperfect Mach-Zehnder Interferometers (MZIs) described by Qi et al. [52]. They model a lossy interferometer $T_\eta$ by incorporating loss factors at the input, output, and within the beam-splitter itself.
Singular Value Decomposition (SVD): To avoid the computational explosion of tracking numerous external environmental modes, the authors employ SVD. The non-unitary operator $A$ resulting from a mesh of imperfect interferometers is decomposed as $A = U_2 \sqrt{H} U_1$ , where $U_1$ and $U_2$ are unitary matrices and $\sqrt{H}$ is a diagonal matrix containing the dissipative factors (singular values). This condenses all lossy channels into $M$ external modes (where $M$ is the number of physical modes), reducing the system size from a polynomial increase to a manageable $2M \times 2M$ covariance matrix.
Covariance Matrix Derivation: The authors derive an analytical expression for the covariance matrix $\sigma_S$ of a Gaussian Boson Sampling (GBS) process subject to these losses. This matrix is obtained by partially tracing out the environmental modes from the global symplectic transformation.
Numerical Simulation: The authors implement a pipeline using a Gaussian backend (Python library PNN) to simulate GBS. They compare:
- Ideal GBS outcomes.
- Realistic GBS outcomes using the derived non-uniform loss matrix $\sqrt{H}$ .
- GBS outcomes assuming uniform losses (where $\sqrt{H} \propto I$ ).
  They test both Clements and Reck decomposition schemes for the interferometer mesh.

Key Contributions

Analytical Formalism: The paper provides a closed-form expression (Equation 24) for the covariance matrix of a GBS process performed with imperfect interferometers, derived via SVD. This allows for the simulation of noisy GBS and general-dyne based algorithms on classical backends without resorting to the Fock space.
Efficient Mode Reduction: By leveraging SVD, the method reduces the number of auxiliary modes required to model losses from a polynomial dependence on circuit depth to a linear dependence ( $2M$ ), significantly lowering computational costs.
Validation of Uniformity Assumption: The work numerically assesses the validity of assuming uniform optical losses. It demonstrates that this assumption is not universally true and depends heavily on the interferometer decomposition scheme (Clements vs. Reck) and the number of modes.
Device-Specific Characterization: The framework allows for the integration of experimental data regarding specific hardware imperfections (e.g., specific loss rates for individual MZIs) directly into the simulation pipeline.

Results

Distribution Shifts: Numerical simulations show that optical losses increase the population of the vacuum state and allow for the detection of odd photon counts (which are forbidden in ideal squeezed state inputs), altering the probability distribution significantly compared to the ideal case.
Decomposition Dependence: The study finds that the deviation of the loss matrix $\sqrt{H}$ $H$ from the identity matrix (i.e., the non-uniformity of losses) behaves differently depending on the architecture:
- In the Reck scheme (equivalent to loop-based time-bin encoding), the distance from the identity increases with the number of modes $M$ . The authors fit this trend to a power law and a logarithmic function, finding the power law fits slightly better.
- In the Clements scheme (spatial encoding), the distance oscillates and converges to a bounded range after $M=5$ , suggesting a more stable deviation from uniformity.
Hafnian Divergence: When simulating the sampling of the Hafnian of a matrix, the authors observe that the results from a realistic non-uniform loss model diverge more rapidly from the ideal case than the uniform-loss approximation does. This discrepancy becomes more pronounced as optical losses increase (lower transmissivity $\eta$ ) and as the number of modes grows.

Significance and Claims
The authors claim that this work provides a "blueprint" for future characterizations of specific devices and architectures. The proposed framework enables realistic simulations of quantum algorithms on Noisy Intermediate-Scale Quantum (NISQ) optical platforms. By accurately modeling non-uniform losses, the method aims to prevent wrong predictions about algorithm outcomes that might arise from assuming uniform loss distributions. The authors position this work as a foundational step toward integrating realistic noise models with error mitigation techniques and potentially developing future error correction codes for continuous-variable computing. They explicitly state that the work does not propose new error correction codes but rather a simulation tool to better understand and characterize the noise landscape.

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