Researchers Develop Classical Algorithms for Adaptive Circuits, Simulating Systems

Quantum computers promise revolutionary computational power, but simulating their behaviour remains a significant challenge, particularly as systems grow more complex. Kemal Aziz, Haining Pan, and Michael J. Gullans, along with J. H. Pixley, have developed new classical simulation algorithms that tackle this problem by focusing on quantum circuits generating states with low “magic”, a measure of how far a quantum state is from being easily simulated by classical computers. These algorithms excel at modelling circuits with frequent measurements, common in quantum error correction, and allow researchers to simulate larger and deeper systems than previously possible. The team’s approach reveals how measurements limit the build-up of this “magic”, offering insights into entanglement, purification, and overall system behaviour, and provides a valuable complementary tool to existing simulation techniques like matrix-product states.

Adaptive quantum circuits generate states exhibiting low levels of “magic”, which represents non-stabilizerness. These algorithms prove particularly well-suited to circuits featuring high rates of Pauli measurements, commonly found in quantum error correction and monitored quantum circuits. Measurements effectively limit the accumulation of magic induced by non-Clifford operations, originating from both generic noise processes and unitary gates. The algorithms also facilitate a systematic truncation procedure, enabling approximate simulation of complex quantum systems.

Efficiently Simulating Low-Magic Quantum Circuits

Researchers have developed new classical simulation algorithms designed to efficiently model quantum circuits that produce states with limited “magic”, or non-stabilizerness, a property crucial for advanced quantum computation. These algorithms are particularly well-suited for circuits employing frequent Pauli measurements, common in error correction and monitored quantum systems, and allow for systematic approximations to reduce computational demands. The team successfully simulated circuits previously inaccessible due to their complexity, demonstrating the ability to analyze dynamics with high rates of measurements and a sub-extensive number of non-stabilizing gates per circuit depth. By focusing on circuits with controlled levels of magic, the researchers were able to explore dynamics complementary to those studied using matrix-product state approaches, offering a new avenue for understanding complex quantum systems.

The method involves carefully managing the growth of the computational space required to represent the quantum state, allowing for simulations of larger systems and longer timescales. Results demonstrate that the number of logicals, a measure of the complexity of the quantum state, remains largely unchanged as the truncation error is varied from extremely small values to approximately 0. 01, confirming the algorithm’s accuracy within a reasonable range. Further increasing the cutoff significantly reduces the number of logicals, decreasing computational cost at the expense of some accuracy. Importantly, the researchers found that the inclusion of measurements reduces the number of logicals required, enhancing the efficiency of the simulation.

Data confirms that in the area law magic phase, the number of logicals scales with system size as a power law, indicating a manageable growth in computational complexity. This allows for efficient simulation using the developed algorithms, even for relatively large systems. These advancements represent a significant step forward in the ability to classically simulate quantum systems, providing valuable insights into the behavior of complex quantum phenomena.

Magic, Entanglement and Phases in Quantum Circuits

This research introduces a new simulation method, based on a low-rank stabilizer decomposition, to model the behaviour of quantum circuits that produce states with limited ‘magic’, or non-stabilizerness. The method excels at simulating circuits with frequent measurements, such as those used in quantum error correction, and allows for systematic approximations to manage computational demands. By applying this approach, researchers have been able to simulate larger and deeper circuits than previously possible, characterising the build-up of magic, entanglement, and purification. The simulations reveal a distinct transition in magic alongside the known transition in entanglement, particularly in circuits measured solely in the Z-basis.

The team identified four distinct phases characterised by varying levels of entanglement and magic, demonstrating how the introduction of T-gates, essential for universal quantum computation, influences entanglement depending on the circuit’s structure. Importantly, the simulations confirm that entanglement and purification transitions coincide in the all-to-all circuit model, with critical exponents consistent with previous studies. A limitation of the current method is its efficiency is constrained to specific parameter ranges near critical measurement rates or within the area law magic phase. Future work will focus on extending the method to handle other types of non-Clifford operations and exploring the behaviour of these transitions in different circuit geometries, potentially revealing how the universality class of the entanglement transition might change.