Feynman Path Sum Simulation Enables Efficient Calculation of Linear Optics Probability Amplitudes

The simulation of quantum optical circuits presents a significant computational challenge, demanding efficient methods for calculating the probabilities of different outcomes. Wagner F. Balthazar from Instituto Federal do Rio de Janeiro, Quinn M. B. Palmer, Alex. E. Jones, Jake F. F. Bulmer, and Ernesto. F. Galvão from the University of Bristol and Universidade Federal Fluminense have developed a novel approach to this problem, adapting the Feynman path integral formalism, a technique originally used in quantum physics, for classical simulation. This method allows researchers to model linear boson sampling experiments, which are crucial for validating quantum computing technologies, with improved efficiency in both runtime and memory usage. By implementing their Linear-Feynman Path simulator in open-source C code and optimising it with tensor contraction techniques, the team provides a valuable tool for the broader quantum optics community and advances the possibilities for simulating complex quantum systems.

Efficient Amplitude Calculation for Boson Sampling

Scientists have developed a new method for calculating the probability amplitudes crucial for photonic quantum computation, specifically focusing on Gaussian Boson Sampling and related setups. This work addresses the challenge of efficiently determining the likelihood of different outcomes in these complex quantum processes. The method centres on exploiting the inherent structure of the mathematical problems arising in GBS, allowing for streamlined calculations and paving the way for simulating larger and more complex quantum systems. The probability amplitudes in GBS are directly related to the permanent of a matrix, which is notoriously difficult to calculate as the system grows.

This research overcomes this limitation by recognising that the matrices arising from beam splitters, the core components of GBS, possess a unique structure with only a few distinct elements and many repeated rows and columns, allowing for significant computational savings. The team carefully accounts for the number of ways to obtain a particular result, using combinatorial factors to ensure accuracy. The newly developed algorithm calculates the permanent of the beam splitter matrix in a time that scales linearly with the number of photons, a significant improvement over traditional methods. This is achieved by leveraging the matrix’s specific structure and introducing a parameter that limits the number of calculations required.

This optimisation makes it possible to simulate larger GBS circuits, accelerating research in this promising area of quantum computation. This advancement in computational efficiency has significant implications for the field of GBS. By reducing the computational cost of simulating these quantum circuits, scientists can explore more complex designs and investigate the potential of this approach to quantum computation. This work addresses the challenge of accurately calculating probability amplitudes in these experiments, which involve photons evolving through complex networks of beam splitters. Researchers implemented this approach by representing the quantum process as a network of weighted paths, each corresponding to a possible trajectory of photons through the interferometer.

They used tensor contraction techniques to represent and calculate the contributions of each path to the overall probability amplitude, enabling efficient computation on classical hardware. These comparisons were performed using simulations of low-depth linear circuits, allowing for a direct assessment of performance. The team demonstrated that their method offers advantages in both runtime and memory efficiency for these circuits, particularly when compared to algorithms that struggle with the sparsity inherent in shallow interferometer designs.

Researchers further refined the technique by exploring different tensor contraction sequences, allowing for trade-offs between runtime and memory usage. This work centres on simulating how photons behave within complex interferometers, utilising a technique that efficiently manages computational resources and allows for parallel processing. The core of this achievement lies in a new approach to enumerating possible photon paths, significantly reducing the memory requirements. The research team implemented this simulation technique in open-source C code, further optimising performance through tensor contraction techniques.

Experiments demonstrate the method’s effectiveness for low-depth circuits, revealing advantages in both runtime and memory efficiency. A key innovation involves identifying and eliminating invalid photon paths, those that violate fundamental conservation rules, thereby streamlining the calculation process. The method strategically defines upper bounds for photon occupation numbers, allowing iteration over only valid configurations and uniquely determining the occupation of other areas based on these values. The researchers demonstrated that the number of certain areas increases linearly with the number of modes, but the runtime exhibits exponential growth with both the number of modes and the depth of the interferometer.