Quantum theory governs the behaviour of fundamental particles, atoms, and molecules, i.e., of nature at the smallest scale. Intriguingly and counter-intuitively, quantum systems can simultaneously ‘be’ in combinations of physical states that are mutually incompatible according to classical physics. Examples we can name are the (in)famous Schrödinger’s Cat thought experiment and Einstein’s objection to “spooky actions at a distance”. Intense research over recent decades has established that these quantum phenomena really exist, and much current research effort is focused on harnessing them to technological advantage. 

 

However, a major challenge for realising quantum technologies is the unavoidable interaction of all quantum systems with their surrounding physical environment. This often spoils their “quantumness” and therefore the advantage of quantum devices over their classical counterparts. Even one of the simplest quantum systems, a single excited atom in vacuum, eventually sees its excitation emitted as a photon. Early pioneers of quantum mechanics, like Einstein, modelled such processes phenomenologically, assuming a decay with a rate depending on the particular atom. Later, Wigner and Weisskopf established a theory that traces photon emission back to interaction with the electromagnetic field modes of the vacuum. We can show that Einstein's result is recovered under the assumption that this interaction is weak and that the environment is essentially static. 

 

However, many real-world quantum devices are coupled more strongly to more than one or richer environments. In solid-state devices as well as in molecular systems, vibrations of atoms constitute the dominant source ofenvironmental interaction. Quantum memories based on spins, magnetic moments of electrons or nuclei, are dominantly coupled to other magnetic moments in their surroundings. In any of these systems, our expectation is this system-environment interaction can be strong enough to disturb the environment such that this disturbance later acts back on the system. This time-delayed feedback makes predictions of strongly coupled open quantum systems particularly challenging because it requires the dynamics of the environment itself to be treated on a quantum level. 

 

We understand that the same effect responsible for the quantum advantage of quantum computers over classical computers — the exponential scaling of the information capacity of quantum systems with the number of degrees of freedom — is also the reason why the classical simulation of the full quantum dynamics of all environmental degrees of freedom is extremely challenging. Therefore,modelling the dynamics of quantum systems typically relies on simplifying assumptions and approximations. Identifying these not only takes significant skill and effort but may also affect the accuracy of the resulting model. The stronger the environmental interaction, the more sophisticated a model needs to be derived and implemented, rendering research into strongly-coupled quantum systems a laborious and time-consuming undertaking. 

 

For certain special environments so-called numerically exact techniques exist. These provide a systematic way to trade off computational resources against accuracy without the need to manually derive equations representing a specific model. As this allows outsourcing a large part of the researcher’s work to computers, we realize much could be gained if numerically exact methods were available for arbitrary open quantum systems. 

 

This challenge has motivated our novel algorithm dubbedAutomated Compression of Environments (ACE). Here, environmental influences on quantum systems are encapsulated into the so-called process tensor, which captures the flow of information from the system to the environment, its propagation, and its back-action on the system at a later point in time. Building on the theory of tensor trains in computer science and matrix product states in the context of many-body quantum physics, we employ existing efficient and well-established techniques for compressing large process tensors into a compact and manageable form. Physically, this compression corresponds to a reduction of the very many environment degrees of freedom to the most relevant ones. 

 

Harnessing this approach, ACE achieves a numerically exact description of quantum behaviour thatstarts directly from the knowledge of the microscopic interactions, without requiring any bespoke, problem-specific derivation. Through our research, we showcase the simulation of quantum systems with a broad variety of environments including those comprised of electrons, photons, vibrations, spins, and importantly also combinations of multiple environments — all within a single computer code. 

 

As such, we conclude that ACE constitutes a proof-of-principle and a significant step towards an ambitious vision: a one-size-fits-all universal tool for modellingso-called open quantum systems. The process tensor formalism seems to be a promising platform to build on to achieve this long-term goal. However, already today, ACE and related methods support researchers like us in simulating, assessing, and designing novel state-of-the-art quantum devices with unprecedented accuracy.