🔬 condensed matter

Non-adiabatic Optimisation via Localisation-Delocalisation Transitions

This paper introduces FRIO, a digitized non-adiabatic quantum optimization algorithm that leverages many-body localization transitions and Landau-Zener crossing density to achieve constant-time scaling for spin-glass ground state problems, demonstrating superior performance over QAOA in numerical simulations.

Original authors: Ryan Smith, Sabrina Yue Wang, Filippo Maria Gambetta, Raul A. Santos

Published 2026-10-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ryan Smith, Sabrina Yue Wang, Filippo Maria Gambetta, Raul A. Santos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a vast, rugged landscape of hills and valleys, where every point represents a possible solution to a complex problem. In the world of computing, finding the best solution is like finding the deepest valley in this terrain. For decades, scientists have hoped that quantum computers, which operate on the strange rules of the subatomic world, could navigate this landscape far faster than classical machines. The challenge is that these landscapes are often filled with countless small dips and traps that look like the bottom but are not. Traditional methods try to slide slowly down the slopes, hoping to avoid getting stuck, but this process can take an impossibly long time. A newer approach involves shaking the system to help it jump out of these traps, but doing so too quickly often causes the system to lose its way entirely. The question remains: is there a way to shake the system just enough to escape the traps without losing the path to the true bottom?

Researchers at Phasecraft and the University of Leeds have proposed a new method called Fast Reverse Iterative Optimisation, or FRIO, which attempts to solve this dilemma by carefully controlling how the system moves between different states of order and chaos. Their work focuses on a specific type of difficult problem known as a spin glass, which is a mathematical model used to describe systems with many conflicting interactions, similar to how a tangled knot of wires resists being straightened. In these systems, the energy landscape is so complex that it is easy for a computer to get stuck in a local low point, thinking it has found the best answer when a better one exists nearby. The team's strategy involves starting with a known, decent solution and then deliberately disturbing it. They apply a series of rapid changes, or "quenches," that push the system from a state where it is stuck in a specific configuration into a chaotic, fluid state where it can explore new possibilities, and then back again.

The core of their discovery lies in understanding exactly how much chaos is needed. Previous theories suggested that to find the best solution, the system must navigate a very specific, narrow boundary between a frozen, disordered state and a flowing, thermal state. The new research suggests that this boundary is not as strict as once thought. By analyzing the density of energy levels and how the system transitions between them, the authors found that they could accelerate the process significantly. They developed a mathematical description, similar to how one might track the spread of a drop of ink in water, to predict the probability of the system finding a lower energy state. This model indicates that even with very short, non-adiabatic processes—meaning the system is not given enough time to settle slowly—the algorithm can still succeed with high probability, provided the parameters are chosen correctly.

To test this, the team simulated the algorithm on digital quantum circuits, breaking the continuous process into small, manageable steps. They compared their method against the Quantum Approximate Optimisation Algorithm, or QAOA, which is currently one of the most popular tools for solving these types of problems on near-term quantum hardware. The results were striking. When the researchers tuned their parameters to ensure the system moved deep enough into the chaotic, thermal phase, their algorithm consistently outperformed the standard QAOA approach. This was true even when the QAOA settings were pre-optimized for the specific problem. The study showed that the new method could find the ground state, or the deepest valley, more frequently and with fewer computational resources, particularly for systems with a high number of connections between their parts.

A crucial part of their work was figuring out how to set these parameters without needing to know the answer in advance. They devised a testing protocol that uses short bursts of activity to see if the system is behaving in a "thermal" way, which indicates it is exploring the landscape effectively. By measuring how the system's internal states spread out over a small section of the computer, they could determine if the chosen settings were correct. This allows the algorithm to be self-correcting and adaptable, a vital feature for the noisy, imperfect quantum computers available today. The simulations covered systems ranging from a few dozen to nearly twenty nodes, and the success rate remained robust as the size increased, suggesting the method scales well.

The researchers emphasize that their findings are based on simulations and numerical evidence rather than physical experiments on a real quantum machine, though the methods they describe are designed to be directly implementable on current hardware. They found that the algorithm does not need to follow the slow, careful path of traditional methods, nor does it need to circle a specific critical point as previously believed. Instead, a more direct, non-adiabatic approach, when guided by the right understanding of how disorder and order interact, can be highly effective. This work offers a promising path forward for using quantum computers to solve real-world optimisation problems, from managing power grids to financial modelling, by turning the chaotic nature of quantum mechanics into a tool for finding the best solutions.

Technical Summary: Non-adiabatic Optimisation via Localisation–Delocalisation Transitions

Problem Statement
The paper addresses the challenge of efficiently solving classical combinatorial optimisation problems on quantum computers, specifically those mappable to finding the ground states of disordered spin-glass systems like the Sherrington-Kirkpatrick (SK) model. These problems feature complex energy landscapes with exponentially many local minima. While Variational Quantum Algorithms (VQAs) such as the Quantum Approximate Optimisation Algorithm (QAOA) are popular candidates for Near-term Intermediate-scale Quantum (NISQ) devices, they often require classical optimisation of parameters and may struggle with barren plateaus or local minima. Traditional Quantum Annealing (QA) faces the "adiabatic bottleneck," requiring exponentially long annealing times to avoid Landau-Zener transitions when the energy gap closes. Recent iterative approaches attempting to bypass this by circling the many-body localisation (MBL) tricritical point still require annealing times scaling polynomially with system size ( $O(\text{poly}(N))$ ) to strictly avoid diabatic transitions.

Methodology
The authors propose Fast Reverse Iterative Optimisation (FRIO), a digitised, non-adiabatic cyclic annealing protocol designed to traverse the MBL-to-thermal phase transition. The algorithm operates in four steps per cycle:

Initialization: A longitudinal magnetic field is applied to align the system with a reference bitstring $S_r$ (a known low-energy state).
Transverse Quench: A transverse field is ramped up, creating a Hamiltonian that mixes the reference state with nearby eigenstates.
Return Quench: The system is quenched back to the original SK Hamiltonian, crossing the phase boundary between the paramagnetic (thermal) and spin-glass (MBL) phases.
Measurement: The system is measured in the computational basis. If a lower energy state is found, it becomes the new reference for the next cycle.

The protocol is digitised using a first-order (and optionally second-order) Trotter decomposition. Crucially, the authors argue that strict adiabaticity is not required. Instead, they model the spectral weight propagation during the quench using a coarse-grained description based on the density of Landau-Zener crossings. By treating transitions as a Poisson process, they derive an effective Fokker-Planck equation for the success probability. This model suggests that even with short, fixed annealing times ( $O(1)$ ), the success probability of finding a lower energy state can exceed $1/2$ for reference states with non-vanishing energy density, provided the system traverses deep enough into the thermal phase.

To determine optimal parameters (specifically the ratio $\chi = B_x/B_z$ ) without prior knowledge of the phase diagram, the authors propose a parameter testing protocol based on short-time thermalization dynamics. They utilize the Inverse Participation Ratio (IPR) of a small subsystem (a 4-qubit marginal) as a metric. If the dynamics thermalize (converging to the Random Matrix Theory prediction for the IPR), the parameters are deemed to be in the thermal regime suitable for the algorithm.

Key Contributions

FRIO Algorithm: Introduction of a non-adiabatic, iterative optimisation algorithm that leverages the MBL-delocalisation transition to enhance tunnelling between local minima.
Theoretical Framework: Derivation of an effective Fokker-Planck equation describing the success probability, predicting that constant annealing times ( $O(1)$ ) can yield favorable success probabilities, contrasting with the $O(\text{poly}(N))$ requirements of previous cyclic protocols.
Parameter Selection Strategy: A method to identify suitable Hamiltonian parameters for NISQ devices by measuring short-time IPR dynamics, avoiding the need for exact diagonalization or complex classical optimisation of circuit angles.
Digitisation: Demonstration that the continuous protocol can be effectively Trotterised with a modest number of steps, making it feasible for near-term hardware.

Results
Numerical simulations using Exact Diagonalization (ED) on SK model instances (up to $N=18$ ) and weighted $k$ -regular graphs yielded the following:

Parameter Testing: The IPR-based metric successfully identified the transition from the MBL phase to the thermal phase. A ratio of $\chi = 1$ was found to be sufficient to ensure thermalization and successful optimisation, even though it is below the tricritical point ( $\chi_{tri} \approx 3.6$ ) identified in previous literature.
Performance vs. QAOA: The digitised FRIO algorithm (with $M=1, 2, 3$ $M = 1, 2, 3$ Trotter steps) was compared against QAOA circuits of equivalent depth ( $p=3, 6, 9$ $p = 3, 6, 9$ ).
- For $\chi = 1$ (deep in the thermal phase), FRIO consistently outperformed QAOA, even when QAOA angles were individually optimised for specific graph instances.
- For $\chi = 0.25$ (near the critical line), FRIO remained competitive with QAOA.
- The algorithm showed robustness across different system sizes, though performance degraded for large $N$ with very shallow circuits ( $M=1$ ) due to Trotter errors.
Low-Energy Dynamics: The algorithm demonstrated the ability to transition to lower energy states even when starting from low-lying excited eigenstates, though the success probability naturally decreases as the system approaches the ground state due to the changing density of states.

Significance and Claims
The paper claims that FRIO offers a pathway to solving spin-glass optimisation problems on NISQ devices without the need for variational parameter optimisation (which is resource-intensive and prone to local minima in classical optimisers). By exploiting non-adiabatic dynamics and the localisation-delocalisation transition, the authors suggest that the required annealing time can be reduced from polynomial to constant with respect to system size for reference states away from the ground state.

The authors note that while the algorithm outperforms optimised QAOA in the tested regime, the parameters chosen may not directly scale to larger, classically intractable systems without further tuning (e.g., longer quench durations or more Trotter steps). They explicitly state that the resilience to Trotterisation might be attributed to the introduction of counter-diabatic terms, though this requires further investigation. The work positions FRIO as a promising candidate for direct implementation on platforms with all-to-all connectivity (such as trapped ions or neutral atoms) where the lack of variational overhead is a distinct advantage.

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