Efficiently simulating complex quantum systems has long been a challenge in physics; now, calculations within the time-evolving matrix product operator (TEMPO) method, used to simulate open quantum systems, require fewer computational resources. By reducing dimensions of internal connections in its tensor networks to (2m+1)(m+1), where’m represents coupling channels and remains constant regardless of system size, scaling effects are confined to the final layer. The technique simplifies calculations when simulating complex quantum systems that interact with their surroundings, known as open quantum systems.

This improvement reduces computational demands by limiting how quickly calculation complexity grows alongside larger simulated systems. Researchers achieved this through optimising connections in “tensor networks”, reducing dimensions to (2m+1)(m+1). An artificial intelligence tool assisted in discovering this optimisation process. Calculations within simulations were optimised using building blocks (Legos) connected to represent energy flow over time; these connections are known as “tensor networks”.

Traditionally, calculating interactions becomes exponentially more difficult as pieces and connections increase, but the team reduced network dimensions down to (2m+1)(m+1), where’m represents the number of interaction channels. This simplification enables larger system sizes than previously possible with TEMPO methods while maintaining accuracy equivalent to standard approaches.

Fixed Bond Dimension Enables Scalable Open Quantum System Simulations

Previously, simulating open quantum systems with TEMPO faced rapidly increasing computational demands. Now, an optimisation has reduced internal bond dimensions within its tensor networks from scaling with system size (d²) to 6 for a single interaction channel. This dimensional reduction crosses a key threshold and allows access to complex multichannel problems previously intractable using standard TEMPO methods. The breakthrough came through analytically rewriting equations governing interactions between quantum systems and their environment, exploiting low-rank structure in network gates minimizes redundant calculations.

Results demonstrate the power of this new approach on a challenging multichannel problem unsolvable by traditional TEMPO techniques. The method generalises directly to multiple channels, reducing network dimension to (2m+1)(m+1) monomials where ‘m’ represents independent coupling channels. Validation via benchmark testing with a damped harmonic oscillator revealed that calculations formerly requiring d⁴ dimensional spaces are now possible with substantially fewer computational resources. This improvement opens avenues for investigating decoherence dynamics and environmental effects on quantum systems at parameter regimes and system sizes previously inaccessible.

Second-Order Accuracy Defines Speed and Stability Limits

This breakthrough confines the bulk of computational burden to the final layer, an important step toward simulating larger quantum systems. However, pushing for even greater speed might necessitate accepting some loss in precision or could introduce instability during longer simulation runs which remain unexplored by the team. The claims hold “within the usual second-order time-step error.” Acknowledging only this level of accuracy so far means faster simulations may potentially sacrifice stability; Technische Universität Berlin researchers have fundamentally altered how complex quantum system simulations scale within TEMPO. This technique simulates open quantum systems over time using interconnected calculations called “tensor networks.” By restructuring these internal connections, computational demands now depend on the number of independent interaction channels rather than overall size or complexity.

This research demonstrated a way to reduce the computational cost of simulating open quantum systems with the Time-Evolving Matrix Product Operator (TEMPO) method. The authors achieved this by reorganizing the tensor network used in their calculations such that its dimensions are determined by the number of independent coupling channels (m), scaling as (2m+1)(m+1).

This means simulations previously limited by system size can now be performed with fewer resources and allows access to more complex multichannel problems. Researchers validated this approach using a damped harmonic oscillator, showing it enables investigation of decoherence dynamics at larger scales than before.

👉 More information
🗞 The TEMPO network can be made independent of the system size
✍️ Valentin Link (Technische Universität Berlin)
🧠 ArXiv: https://arxiv.org/abs/2610.37012

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