Circuit Complexity and Spread Complexity Linked by Unitary Gate Synthesis.

Research demonstrates spread complexity arises as a specific case within a circuit complexity framework utilising time evolution and superposition. This approach, employing unitary gates and beam splitting, provides a physical interpretation of spread complexity and offers a viable alternative to conventional methods like the Lanczos algorithm, particularly when dealing with non-perturbative or divergent return amplitudes.

The quantification of quantum chaos remains a significant challenge in theoretical physics. Recent research explores a connection between ‘spread complexity’ – a measure of how rapidly quantum information disseminates within a system – and the more established concept of ‘circuit complexity’, which assesses the resources needed to construct a quantum state via a series of logical gates. A new study by Beetara et al. establishes a formal link between these two measures, demonstrating that spread complexity arises naturally within a specific circuit complexity framework. The research, conducted by Cameron Beetara, Jeff Murugana, and Hendrik J R Van Zyl from The Laboratory for Quantum Gravity & Strings at the University of Cape Town, in collaboration with Eric Lobalo Graef and Horatiu Nastase from the Instituto de Fisica Teorica, UNESP-Universidade Estadual Paulista, is detailed in their article, ‘A Quantum Computational Perspective on Spread Complexity’. Their approach utilises unitary gates and beam-splitting operations to generate quantum states, offering a potentially more robust method for calculating complexity, particularly in systems where conventional techniques falter.

A Novel Metric Correlates Spread and Circuit Complexity in Quantum Systems

Recent research demonstrates a quantifiable relationship between ‘spread complexity’ and ‘circuit complexity’ within quantum systems, offering a new approach to understanding the resources required to characterise quantum states. The work establishes spread complexity as a specific case within the broader framework of circuit complexity, potentially circumventing limitations inherent in existing computational techniques.

Circuit complexity is assessed by determining the minimum number of elementary quantum gates – in this instance, unitary gates and beam-splitting operations – required to construct a given quantum state. This ‘cost’ of construction serves as the metric for complexity. Researchers validated this framework using the SU(2) symmetry group, a fundamental concept in quantum mechanics describing rotational symmetry.

The significance of this approach lies in its potential to address challenges faced by traditional methods, such as the Lanczos algorithm. The Lanczos algorithm, a common technique for finding eigenvalues and eigenvectors of large matrices, can struggle with systems exhibiting non-perturbative behaviour or divergent characteristics. This new framework remains robust in such scenarios.

Crucially, the research links abstract mathematical measures of complexity to the concrete physical resources – the quantum gates – needed to build a specific quantum state. This connection provides a more intuitive understanding of what complexity represents in a physical context.

The researchers propose extending this framework to investigate phenomena in other areas of quantum physics, including many-body localisation – where quantum systems become trapped in specific states – and thermalisation – the process by which a system reaches equilibrium. This suggests a versatile tool for analysing a range of complex quantum behaviours.