Qubit Loss and Errors Drive Transition to Nishimori Criticality in Circuits

Research demonstrates that introducing weak measurements beyond standard Clifford circuits destabilises percolation criticality in qubit systems, causing a transition to Nishimori universality. Simulations utilising over a million qubits reveal a rapid change at this point, with both resulting critical states exhibiting multifractal scaling dimensions, and the percolation spectrum precisely determined.

The fidelity of quantum computations relies heavily on minimising errors during the manipulation of qubits – the fundamental units of quantum information. Imperfections, such as qubit loss and coherent errors, inevitably arise in physical implementations, ultimately limiting the complexity of achievable quantum circuits. Recent research focuses on understanding the critical behaviour at the threshold where these errors overwhelm the system’s ability to maintain quantum coherence. A collaborative team – Malte Pütz, Romain Vasseur, Andreas W.W. Ludwig, Simon Trebst, and Guo-Yi Zhu – have investigated this threshold behaviour in weakly monitored quantum circuits, demonstrating a transition in critical behaviour when both qubit loss and coherent errors are present. Their work, entitled ‘Flow to Nishimori universality in weakly monitored quantum circuits with qubit loss’, details how the system’s critical point shifts from a behaviour characteristic of percolation to one governed by Nishimori criticality, even with minimal deviation from idealised conditions. The team employed extensive numerical simulations, utilising hybrid Gaussian fermion and tensor network techniques on systems exceeding one million qubits, to map the phase diagrams and characterise the resulting multifractal scaling behaviour.

Characterising Entanglement in Two-Dimensional Systems via Higher-Order Cumulants

This study investigates entanglement structure in two-dimensional systems subject to both qubit loss and coherent errors, with a particular focus on behaviour at the point where long-range entanglement breaks down. Researchers demonstrate a transition from percolation criticality to Nishimori universality as systems move away from purely projective stabilizer measurements – a crucial element of quantum error correction. The work employs extensive numerical simulations, utilising hybrid Gaussian fermion and tensor network techniques on systems exceeding one million qubits, to map phase diagrams and characterise critical behaviour.

The analysis centres on cumulants of entanglement entropy, offering a more detailed description of entanglement distribution than standard measures like entanglement entropy itself. Entanglement entropy quantifies the degree of quantum correlation between subsystems. Cumulants, in this context, provide statistical information about the distribution of entanglement entropy across the system, revealing subtleties missed by simply calculating the average value. Specifically, the research highlights that higher-order cumulants – beyond the second moment – reveal crucial information about entanglement structure, enabling a more nuanced understanding of quantum correlations.

Researchers meticulously address the influence of boundary conditions on these calculations, recognising their potential to distort results. Open boundary conditions introduce significant effects on higher-order cumulants, particularly the third and fourth, necessitating careful data analysis and potentially biasing outcomes. To mitigate these effects, the study focuses on bulk data, excluding boundary contributions and ensuring a more accurate representation of the system’s interior behaviour.

The study provides precise numerical results for the leading exponents characterising multifractality within both the percolation and Nishimori fixed points, revealing the complex scaling behaviour of entanglement. Multifractality describes systems exhibiting scaling properties that vary across different points, indicating a complex, fragmented structure. Researchers derive the exact multifractal spectrum of exponents for percolation, offering a complete characterisation of its entanglement structure and providing a benchmark for comparison with other systems.

Researchers establish a clear link between the order of Rényi entropy used and the observed behaviour of these cumulants, highlighting the importance of selecting an appropriate entropy measure for specific analyses. Rényi entropy is a generalisation of entanglement entropy, allowing for different sensitivities to the magnitude of entanglement.

Calculations reveal a sign change in the fifth cumulant when transitioning from the von Neumann entropy to higher-order Rényi entropies, indicating a reshaping of the entanglement distribution and signifying a shift in the underlying entanglement characteristics as system parameters vary. The von Neumann entropy is a specific case of Rényi entropy, often used as a standard measure of entanglement. This change signifies that the system’s entanglement characteristics are not fully described by the von Neumann entropy alone.

Researchers demonstrate a critical transition in the behaviour of the percolation model as it moves away from the Clifford regime of projective stabilizer measurements, revealing that even a small deviation from this regime causes a rapid and substantial change in the critical renormalization group flow. The renormalization group is a mathematical framework used to study how physical systems behave at different scales. This crossover is tracked through detailed phase diagrams, parameterised by the probability and strength of random weak measurements.

Researchers provide detailed analysis of the scaling dimensions at both the percolation and Nishimori fixed points, demonstrating multifractality in both cases and establishing a robust methodology for analysing entanglement using higher-order cumulants. Scaling dimensions characterise how physical quantities change under scale transformations, providing insights into the system’s critical behaviour.