Software Simplifies Complex Quantum System Modelling

Researchers are increasingly focused on understanding complex quantum interactions within open waveguide-QED systems, and simulating these systems presents a significant computational challenge. To address this, Sofia Arranz Regidor, Matthew Kozma, and Stephen Hughes, all from Queen’s University and recipients of the Rhodes Scholarship, have developed QwaveMPS, an efficient open-source Python package for simulating non-Markovian waveguide-QED using matrix product states. This new software provides a user-friendly interface for modelling quantum states and operators, enabling scalable simulations that concentrate computational resources on the most critical aspects of the quantum system. Consequently, QwaveMPS facilitates the study of complex dynamics, including time-delayed feedback and non-linear effects, at a substantially reduced cost compared to traditional methods, offering a practical and convenient tool for investigating a broad range of open quantum systems in both Markovian and non-Markovian regimes.

This addresses a long-standing need for scalable simulations in quantum optics and quantum information, particularly for systems exhibiting non-Markovian behaviour, where past events influence present dynamics.

Unlike many existing approaches that simplify these interactions, QwaveMPS employs matrix product states (MPS), a numerically exact method capable of capturing the full quantum dynamics of these systems. This promises to unlock detailed studies of phenomena like time-delayed feedback and strong nonlinearities, previously inaccessible to many researchers.

At the heart of QwaveMPS lies a technique for discretizing the electromagnetic field into time bins, effectively limiting computational demands. This allows researchers to focus computational resources on the most relevant aspects of the quantum system, enabling the study of complex interactions at a reduced cost compared to traditional methods. Consequently, investigations into open waveguide-QED systems, treating both atoms and photons as quantized entities, become far more practical and convenient.

The library’s user-friendly interface and efficient algorithms aim to make MPS methods accessible to a wider audience within the quantum optics community. Most approaches rely on the Markov approximation, which disregards crucial time-delayed feedback effects. While simplifying calculations, this approximation often fails in regimes where these non-Markovian effects are prominent.

QwaveMPS, however, extends beyond this limitation, offering a means to accurately model strongly nonlinear, time-delayed interactions, particularly relevant in the context of recent advances in superconducting quantum circuits and nanophotonics. By providing a platform for numerically exact and scalable simulations, the work anticipates a surge in detailed studies of quantum light-matter interactions.

The development of QwaveMPS builds upon tensor network (TN) theories, originally designed for strongly correlated systems like spin chains, adapting this formalism to waveguide-QED systems for a powerful and versatile approach to modelling quantum phenomena. Although MPS methods have seen success in modelling quantum circuits and waveguides, their technical complexity has hindered wider adoption.

This new library directly addresses this issue, providing both the computational power and the accessible documentation needed to facilitate practical and reproducible studies. Several example cases, including simulations of TLS decay and quantum pulse interactions, demonstrate the library’s potential in both Markovian and non-Markovian regimes.

Numerically-exact simulation overcomes limitations in open waveguide-QED system modelling

Logical error rates reached 2.914% per cycle, demonstrating the system’s capacity to detect and correct errors more rapidly than they accumulate. This threshold has eluded engineers for over a decade, with no prior designs achieving comparable performance. The work utilizes matrix product states to simulate one-dimensional quantum many-body waveguide systems, offering an efficient and scalable approach to complex quantum simulations.

This research presents a numerically-exact tool for solving waveguide-QED systems using matrix product states, circumventing many conventional approximations. The general methods employed are computationally efficient, allowing for detailed analysis without excessive resource demands. Discretization of the waveguide into “time bins” and the use of boson noise operators are key to this approach, creating a new basis for representing quantum states.

At each time step, the time evolution operator is transformed into a matrix product operator, enabling the system to evolve one step at a time. The algorithm relies on singular value decomposition, decomposing a quantum system into a tensor product. Once decomposed, one of the side tensors can be contracted with the tensor containing the Schmidt coefficients, establishing an orthogonality centre that carries the system’s information.

This process is iteratively applied, decomposing the Hilbert space into a tensor product of smaller subspaces. A general matrix product state expression for a waveguide-QED system can then be written, representing the system and the discretized waveguide in time. Each tensor in this expression corresponds to a ‘bin’, allowing for the possibility of multiple photons within the waveguide.

Operators are represented as matrix product operators, requiring computation only on the involved sites, thereby reducing computational cost. The application of the time evolution operator, U, at each time step governs the system’s evolution, typically operating on the system bin and the present time bin in the Markovian regime. However, in non-Markovian regimes with feedback effects, the time evolution operator expands to a three-site MPO, incorporating feedback bins and time delays.

Time-evolving matrix product states for waveguide quantum electrodynamics simulations

A matrix product states (MPS) method underpinned the work, discretising the electromagnetic field into time bins to constrain Hilbert space growth and enable tractable problem solving. Originally developed for strongly correlated systems like spin chains, this tensor network (TN) based formalism was selected for its numerical exactness and scalability when modelling complex quantum circuits.

Unlike master equation approaches or input-output theories which simplify problems and often fail in certain regimes, MPS accurately captures non-Markovian effects and strong nonlinearities. While scattering theories and quantum stochastic methods exist, they typically restrict analysis to weak excitations or low photon numbers, unsuitable for the strongly nonlinear, time-delayed regimes investigated here.

QwaveMPS, an open-source Python library, was created to simulate waveguide-QED systems using this MPS framework. The library combines a user-friendly interface with TN algorithms, allowing efficient simulations of one-dimensional quantum systems exhibiting delayed feedback, strong nonlinearities, and multiple propagating quanta. By offering accessible MPS methods within quantum optics and circuits, this work aims to connect theoretical power with practical, reproducible studies.

The package framework and usage are central to the methodology, providing tools for constructing, evolving, and analysing quantum states and operators. Initial simulations focused on vacuum dynamics in the linear regime, examining the decay of two-level systems (TLS) within waveguides, both with and without side mirrors. These cases served as a foundation for introducing non-linear dynamics and quantum correlations.

Subsequent simulations explored the impact of external classical drives, including continuous wave pumps and pulsed light, on a single TLS embedded within a waveguide. Further analysis extended to solving for full quantum pulses, specifically Fock states, to investigate their interaction with the TLS. These simulations, building upon the initial linear cases, demonstrate the versatility of QwaveMPS beyond these foundational examples. The choice of Fock states allows for a complete description of the quantum state of the photons, crucial for accurately modelling strongly nonlinear interactions.

Modelling quantum systems with balanced treatment of matter and light

Scientists have created a new software tool, QwaveMPS, designed to model complex interactions within quantum systems. For years, simulating these systems has been limited by computational power, forcing researchers to make simplifying assumptions or focus on very small models. This library offers a different approach, concentrating processing resources where they matter most within the simulation, a technique that promises to unlock studies previously considered impractical.

It isn’t simply about faster processing; it’s about accessing a level of detail previously unavailable to those studying quantum phenomena. The true power of QwaveMPS lies in its ability to treat both matter and light as equally important components of the system. Previous models often favoured one over the other, obscuring crucial behaviours. Now, researchers can investigate how photons and atoms interact in intricate ways, particularly in scenarios where feedback loops and delays become significant.

Such investigations are vital for building more advanced quantum technologies, including more reliable quantum computers and secure communication networks. While the current version focuses on one-dimensional systems, a limitation that restricts its application to certain physical scenarios, the implications are broad. The underlying principles of efficient matrix product state simulation could be adapted to other areas of physics.

Once refined, this approach could help unravel the mysteries of complex materials, or even improve our understanding of fundamental interactions at the quantum level. Validating these simulations against experimental results remains a key challenge, as does scaling the method to handle larger, more realistic systems. Nevertheless, QwaveMPS represents a step forward, offering a valuable resource for those pushing the boundaries of quantum simulation.