Trainable Quantum Circuits Remain Viable with Linear Combination of Unitaries.

The optimisation of quantum circuits, a critical step in realising the potential of quantum computation, frequently encounters ‘barren plateaus’, regions where the gradient used to refine circuit parameters diminishes rapidly, hindering effective training. Researchers now demonstrate that combining parametrised quantum circuits using a technique called the linear combination of unitaries (LCU) does not necessarily exacerbate this problem, and may even offer computational advantages. Nikhil Khatri, Stefan Zohren, from the Machine Learning Research Group at the University of Oxford, and Gabriel Matos from Quantinuum, present analytical evidence, detailed in their article ‘Trainability of Parametrised Linear Combinations of Unitaries’, that these LCU circuits retain trainability, even when considering the effects of ‘postselection’, a technique where only certain measurement outcomes are considered. Their findings, supported by numerical simulations using fermionic Gaussian unitaries – a specific type of quantum circuit – suggest a pathway to constructing more expressive and potentially faster-to-evaluate quantum algorithms.

Stefan Legeza and colleagues investigate the statistical behaviour of measurable quantities, known as observables, within random quantum circuits constructed using linear combinations of unitaries (LCU). These circuits combine multiple quantum operations, represented mathematically by unitary transformations, in a random fashion. The research analytically derives a formula for the variance, a measure of spread, of expectation values – the average result of many measurements – when LCU is applied to parametrised quantum circuits. Crucially, this formula accounts for post-selection probability, the likelihood of obtaining a specific outcome after a measurement, and demonstrates that applying LCU to trainable circuits maintains ‘trainability’.

Trainability refers to the ability of a quantum circuit’s parameters to be effectively adjusted during a learning process, a critical requirement for variational quantum algorithms. Concerns have arisen regarding ‘barren plateaus’, regions in the parameter space where the gradient, used to guide the optimisation process, vanishes, hindering learning. This study addresses these concerns by showing that LCU circuits avoid these problematic regions. The mathematical framework leverages Weingarten calculus, a powerful tool for calculating expectation values involving random matrices, to understand how randomness within the unitary transformation impacts the variance of measured observables.

Researchers derive a precise expression for the variance, incorporating the trace of the observable and its square, alongside the density matrix, which represents the quantum system’s state, and the dimension of the Hilbert space, the mathematical space describing all possible states of the quantum system. The analysis demonstrates that the variance scales predictably with system size, providing insight into how the behaviour of these circuits changes as they become more complex. The study extends to incoherent superpositions, broadening the applicability of the findings, and numerical simulations, conducted on linear combinations of fermionic Gaussian unitaries – a specific type of matchgate circuit, validate the analytical results and confirm the trainability of LCU-based circuits.

The derived scaling laws and analytical expressions contribute to a deeper understanding of the statistical properties of quantum systems, informing the development of more expressive and efficient quantum algorithms. These findings have implications for benchmarking quantum devices, advancing quantum simulation techniques, and exploring novel approaches to quantum computation. The demonstrated trainability of LCU circuits provides a valuable tool for constructing more expressive and efficient quantum models, and opens up new avenues for designing more robust and scalable quantum circuits.

This research suggests a potential speed-up in evaluating these trainable circuits on quantum hardware, leveraging the structure of the LCU to reduce the number of required measurements and accelerate the optimisation process. This work provides a solid theoretical and numerical foundation for exploring LCU as a powerful tool for overcoming the challenges posed by barren plateaus and unlocking the full potential of variational quantum algorithms and models.

Future work will focus on exploring the limits of this approach and investigating its performance on more complex quantum circuits and architectures. Understanding the interplay between circuit structure, noise, and optimisation algorithms is crucial to realising the full potential of variational quantum algorithms and achieving practical quantum advantage.