Non-adiabatic Optimisation via Localisation-Delocalisation Transitions

arXiv:2610.30301 · cond-mat.dis-nn, cond-mat.stat-mech, quant-ph · Submitted 2026-10-21 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Non-adiabatic Optimisation via Localisation-Delocalisation Transitions".

Mira: Here's the gist of the paper. The gist The continuous dynamics can be Trotterised with a number of steps that remains modest as system size increases, making it appealing for near-term devices FRIO Algorithm Overview The paper introduces Fast Reverse Iterative Optimisation (FRIO), a novel algorithm combining elements from several types of Quantum Annealing protocols to find low energy states of spin-glasses that is more resilient to non-adiabaticity, Trotterisation, and amenable to implementation on NISQ devices The efficacy of this method is tested on the problem of finding the ground state of disordered spin-glasses, specifically the Sherrington-Kirkpatrick (SK) model and weighted k-local graphs FRIO's resilience comes from exploring the localisationdelocalisation transition which controls the tunneling between local minima when a transverse field is turned on Continuous-Time Annealing Mechanism The protocol consists of four steps designed to traverse the phase diagram across the many-body localized to thermal phase transition and back 1.

Kai: Step 1 involves applying a longitudinal magnetic field that has S r as its minimum energy state, which is purely classical and no Landau-Zener transitions can occur 2.

Mira: Step 2 performs a second quench by adding a transverse magnetic field to the system Hamiltonian, where the annealing time is t ∈

0, χBz: , with χ = Bx/Bz, so that at the end of the ramp Bx(χBz) = Bx 3.

Kai: Step 3 performs a third quench back to the SK model using the Hamiltonian H 3(t), where this step crosses the phase boundary between the paramagnetic and spin glass phases 4.

Paper summary: Kai: So this paper, "Non-adiabatic Optimisation via Localisation-Delocalisation Transitions," it's looking at how we can speed up solving classical optimisation problems on quantum computers, like finding spin-glass ground states. It suggests that using intuition from quantum statistical mechanics could guide us in building new algorithms.

Mira: Exactly, and the core idea they present is how continuous reverse quantum annealing across a many-body localisation transition can be accelerated by looking at the density of Landau-Zener crossings as the system gets quenched. It's about understanding that transition better than just running a slow anneal.

Lev: So what does this mean for us in terms of actual hardware? It suggests we might not need ridiculously long annealing times if we consider these crossings and model them with an effective Fokker-Planck equation for the success probability of the algorithm.

Kai: Right, so they suggest that this interpretation points toward an improvement in the required annealing time as a function of system size N, going from something like poly(N) down to constant time for reference states with non-vanishing energy density. That's a big difference if you're dealing with large problems.

Mira: And they show that the continuous dynamics can be Trotterised using a number of steps that stays modest even as the system size N increases, which makes this approach appealing for near-term devices like NISQ hardware.

Lev: That’s good to hear regarding the Trotterisation; if we keep the number of steps small, it keeps the overhead manageable for real hardware constraints. But what about parameter selection? How do you actually set up these protocols without needing huge amounts of time or resources?

Kai: Well, they provide a quantitative strategy for determining the parameters of the time-dependent Hamiltonian through inspecting short-time thermalization dynamics. They suggest measuring the IPR on a subsystem A to see if it thermalizes, and when it does, that tells you something about how far you've quenched.

Mira: That makes sense, they're using those initial thermalization checks to pick parameters for the protocol itself, which is smart because manually picking them is usually a nightmare. They test this by observing excellent scaling in the number of nodes at later times when chi is set to zero point five, showing that the dynamics have successfully thermalized for all system sizes considered <ref:2610.30301#pg2>.

Lev: So they're suggesting that setting chi = zero point five opens up a gap that really hinders those Landau-Zener transitions, which is a concrete parameter choice based on their observation of thermalization behavior <ref:2610.30301#pg2>. I wonder how robust this is when we move to more complex graphs or larger N?

Kai: They do test the robustness by looking at the success ratio analysis, and they find that at Bz = one and chi = one for all N greater than twelve we are highly likely to achieve a successful energy transition for that choice of quench sequence <ref:2610.30301#pg2>. Plus, the finite-size scaling shows that the success probability actually increases as you increase system size.

Paper summary: Mira: That's interesting because the paper notes that the dependence on the initial reference state S r is independent of which reference state you choose, and they found that this X-basis measurement isn't a good basis for their parameter selection algorithm because it remains fully delocalized for all values of Bx.

Lev: So they’re saying that the choice of the initial state doesn't really matter much once you get past those large system sizes, which is helpful when we think about implementing this on real quantum annealers where state preparation can be tricky. What about the comparison to other methods?

Kai: They compare their digitised FRIO algorithm directly against QAOA for finding the ground state of spin-glass problems, specifically the Sherrington-Kirkpatrick SK model and weighted k-local graphs. The numerical results suggest that with suitably selected circuit parameters which can be found efficiently using thermalizing dynamics, FRIO almost always outperforms the conventional QAOA ansatz with pre-optimized angles.

Mira: That comparison is telling because they find that choosing a chi so that the system crosses the many-body localized to thermal phase transition almost guarantees that FRIO will beat QAOA, even when the QAOA circuits are individually optimized for a specific graph instance.

Lev: So it's not just about finding *a* good setting; it’s about finding a setting that puts you right in the right place on the phase diagram, crossing that transition boundary. But they also looked at digitised versions, like FRIO2 with second-order Trotterisation, and found that it adds no notable benefit over a first-order one except for graphs with N greater than sixteen <ref:2610.30301#pg2>.

Kai: So the main point is that this cyclic optimisation algorithm doesn't need variational optimisation at all; it’s well suited for direct implementation because it works by traversing the phase diagram, not by iterating on an ansatz. It seems much more direct for platforms with limited shot budgets.

Mira: And they used level statistics analysis, specifically the ratio of adjacent spacings r n = (s n, s n-one) / (s n, s n-one), to detect chaotic phases <ref:2610.30301#pg1>. Localized systems show a Poisson distribution for P POI(r) with an average gap ratio of r POI about zero point three eight six, while chaotic phases follow a GOE distribution with an average gap ratio of r GOE about zero point five three six.

Lev: That level statistics check is useful for diagnosing the phase, but it’s not a direct measurement of the algorithm's success itself, which is what we really care about when you're building error-corrected circuits. What does this mean for running on hardware that has noise?

Paper summary: Kai: The paper also discussed how to use these statistics to detect chaotic phases, which means you can potentially tune your quench parameters based on the structure of the Hamiltonian itself, not just by brute force trial and error. It's a way to use the physics of the system to guide the algorithm's path.

Mira: And they mentioned that their approach retains a significant success probability over a wide range of graph sizes once you scale beyond classically tractable graph instances, which suggests this method has some promise beyond small examples. They also noted that while SKatan isn't a good choice for parameter setting on k-regular graphs, optimizing the angles is generally still required.

Lev: So they're showing that even with those necessary angle optimizations and the limitations of the phase diagram crossing, this method is still performing better than standard approaches like QAOA when you tune it right for these spin-glass problems. The paper itself has a few limitations, though, like the fact that the X-basis measurement isn't ideal for their parameter selection algorithm.

Kai: So to wrap up on "Non-adiabatic Optimisation via Localisation-Delocalisation Transitions," this paper shows how we can leverage the physics of localisation and delocalisation transitions to design an optimisation algorithm that is more resilient to non-adiabaticity. It gives us a concrete way, through the Fokker-Planck equation interpretation, to improve performance based on how fast you quench.

Mira: Essentially, it suggests that for these many-body localisation problems, if you can find the right quench sequence parameters using short-time thermalization data, you can get better results than standard QAOA by steering the system across that critical phase boundary.

Lev: For anyone building hardware for this, the implication is that you need a way to quickly probe those early dynamics to set your time evolution parameters correctly, rather than running long simulations hoping for the best.

Kai: So it's about making these quantum optimisation algorithms more efficient and less dependent on extremely precise timing during the anneal process.

Kai: The paper "Non-adiabatic Optimisation via Localisation-Delocalisation Transitions" is about using the transition between many-body localised and thermal phases to accelerate solving spin-glass problems on quantum computers. They propose a cyclic optimisation algorithm that traverses this phase diagram by carefully choosing quench parameters based on how fast the system thermalizes in its early stages.

Mira: The main claim is that by considering the density of Landau-Zener crossings, they can improve the success probability interpretation using an effective Fokker-Planck equation, which suggests you can achieve a constant annealing time that yields favorable success probability.

Lev: What this means for us practically is that we need to use thermalization dynamics to determine the protocol's parameters instead of relying on pre-optimized angles, because they found that FRIO almost always outperforms QAOA for the SK model when tuned correctly.

Kai: In short, it’s about building an algorithm that leverages quantum statistical mechanics intuition to make continuous reverse annealing more efficient for finding ground states in complex spin-glass models.

Conclusion: Kai: So we're wrapping up this look at "Non-adiabatic Optimisation via Localisation-Delocalisation Transitions." Essentially, this paper is about using the physics of how quantum systems localize and delocalize to make finding low energy states on a quantum computer faster.

Mira: Exactly, it's proposing a new way to think about continuous quantum annealing by focusing on those transition points between localized and thermal phases.

Lev: From my side, what I see is that they’re trying to avoid the long waits you usually have when the system gets stuck in a bad configuration during the process.

Kai: Right, and they use this concept of Landau-Zener crossings—those sudden changes—to figure out how fast you need to run the anneal. It suggests that if you look at those crossings, specifically their density as the system is quenched, it can speed things up significantly.

Mira: They interpret this behavior through an effective Fokker-Planck equation for the success probability of the algorithm, which is a way of modeling how likely your optimization routine is to land in a good spot.

Lev: That's interesting because if you can model that success probability like that, it changes what we think about the required annealing time as you scale up to bigger systems.

Kai: They suggest that for certain reference states, this required time doesn't have to be polynomial in the system size N; they claim it can become constant time.

Mira: That's a big leap from what we usually see in these kinds of simulations, and they show that you can still Trotterise the continuous dynamics with a number of steps that stays small even when N gets quite large.

Lev: Modest Trotterisation steps sound promising for real hardware because it keeps the computational cost manageable when you're running these kinds of quantum circuits.

Kai: And for implementing the actual circuit, they give a quantitative strategy to pick those time-dependent Hamiltonian parameters by looking at short-time thermalization dynamics.

Mira: They test this by observing thermalization scaling on subsystems, and they found that setting a specific quench parameter like chi equals zero point five opens up a gap that really helps avoid those problematic Landau-Zener transitions.

Lev: That's a concrete setting to try for hardware; if you can find the right chi value based on how fast your system thermalizes, it might save you from waiting too long.

Kai: So in the end, it seems like they're showing that by tuning the algorithm based on these physical transition points rather than just brute force optimization, we can get better results compared to existing methods.

Mira: It’s about moving beyond just running a standard anneal and instead using the underlying physics of localization to guide the entire process.

Lev: This opens up a path for us to design more resilient algorithms that don't break down as the system gets larger or noisier.

Kai: But we still have to figure out how robust this is when you move beyond these specific spin-glass models and onto more complex Hamiltonians.

Mira: That's where the real work lies, exploring if this idea of using localization transitions applies to other types of many-body problems.

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

Phasecraft Ltd. · University of Leeds

cond-mat.dis-nn, cond-mat.stat-mech, quant-ph

Submitted: 2026-10-21

Updated: 2026-10-21

Comments: 25 pages, 5 figures + supplementary information

License: http://creativecommons.org/licenses/by/4.0/

The gist: The gist The continuous dynamics can be Trotterised with a number of steps that remains modest as system size increases, making it appealing for near-term devices FRIO Algorithm Overview The paper

Key concepts

Many-Body Localisation (MBL): A phenomenon in many-body systems where particles become trapped in localized regions due to disorder. In this paper, the transition between a localized phase and a thermal phase is key. Crossing this boundary controls the tunneling between different local energy minima, which is crucial for the algorithm's success.
Landau-Zener Crossings: These are specific points in time during quantum evolution where two energy levels approach each other closely enough to cause a transition. The paper analyzes the density of these crossings as the system is quenched, using this information to predict and optimize the algorithm's success probability.
FRIO Algorithm: Fast Reverse Iterative Optimisation is a novel quantum annealing protocol designed for spin-glass problems. It involves a four-step cyclic process that traverses the phase diagram between localized and thermal phases, aiming to find low-energy states more robustly than standard methods.
Trotterisation: This is a mathematical technique used to approximate continuous time evolution using discrete steps. The paper shows that FRIO dynamics can be Trotterised with a small, modest number of steps even for large systems, which is beneficial for implementation on current noisy quantum computers (NISQ).

Summary

The gist The continuous dynamics can be Trotterised with a number of steps that remains modest as system size increases, making it appealing for near-term devices

FRIO Algorithm Overview

The paper introduces Fast Reverse Iterative Optimisation (FRIO), a novel algorithm combining elements from several types of Quantum Annealing protocols to find low energy states of spin-glasses that is more resilient to non-adiabaticity, Trotterisation, and amenable to implementation on NISQ devices The efficacy of this method is tested on the problem of finding the ground state of disordered spin-glasses, specifically the Sherrington-Kirkpatrick (SK) model and weighted k-local graphs FRIO's resilience comes from exploring the localisationdelocalisation transition which controls the tunneling between local minima when a transverse field is turned on

Continuous-Time Annealing Mechanism

The protocol consists of four steps designed to traverse the phase diagram across the many-body localized to thermal phase transition and back

Step 1 involves applying a longitudinal magnetic field that has S r as its minimum energy state, which is purely classical and no Landau-Zener transitions can occur
Step 2 performs a second quench by adding a transverse magnetic field to the system Hamiltonian, where the annealing time is t ∈ [0, χBz], with χ = Bx/Bz, so that at the end of the ramp Bx(χBz) = Bx
Step 3 performs a third quench back to the SK model using the Hamiltonian H 3(t), where this step crosses the phase boundary between the paramagnetic and spin glass phases
Step 4 involves measuring in the computational basis, where if E meas is less than Er, then the cycle can be repeated with S meas being the new reference state

Non-Adiabatic Dynamics and Success Probability

The success probability is defined as P(E(tf) < Er) = X n:En(t)<Er ⟨bn(tf)S r⟩2, even in the presence of a shorter annealing time The analysis models the energy level transitions as a Poisson process, where the average Landau-Zener probability is calculated using eq. (8) The success probability is obtained by integrating eq. (12) for t = tf over the energy, P(E(tf) < Er) = 1/2 erfc(-x), with x given by x = p/tf A(1)/√2A(2) The main insight is that a constant annealing time can produce a favourable success probability

Parameter Selection via Thermalization

The paper proposes a method to determine the protocol’s parameters by inspecting short-time thermalization dynamics, choosing to measure the IPR on a subsystem A In fig. 3(b), for χ = 0.5, we observe excellent scaling in the number of nodes at later times, showing that the dynamics have successfully thermalized for all the system sizes considered This indicates that χ = 0.5 would open a reasonably large gap that would hinder Landau-Zener transitions

Comparison with QAOA

The digitised FRIO can directly outperform QAOA when solving the max-cut problem on graph instances corresponding to the quantum SK model For all considered values of χ, our algorithm either remains competitive or outperforms QAOA for all considered graph sizes In fig. 5(b)-(c), we find that choosing a χ so that the system crosses the MBL to thermal phase transition almost guarantees that FRIO will beat QAOA, even when the QAOA circuits are individually optimized for a specific graph instance

Trotterisation Effects

The analysis is extended to digitised versions of FRIO, including a second-order Trotterisation dubbed FRIO2 The use of a second-order Trotter decomposition adds no notable benefit over using a first-order one, except when we consider graphs with N ≥ 16 nodes The M = 1 decomposition identifies another breakdown in the dynamics but not as severe as the first-order decomposition with there still being a small but not negligible chance of a successful energy transition occurring

Phase Diagram Diagnostics

Level statistics analysis, using the ratio of adjacent spacings rn = min(sn, sn−1) / max(sn, sn−1), can be used as a detector of chaotic phases The level-spacing statistics of localized systems should follow a Poisson distribution P POI(r) = 2/(1 + r) squared, with an average gap ratio of ⟨r⟩ POI ≈ 0.386 For chaotic phases, it is described by the GOE distribution P GOE(r) = (27/4)r + r 2/(5/2)(1 + r + r 2), with a corresponding mean gap ratio of ⟨r⟩ GOE ≈ 0.536

Conclusion on Performance

For the chosen quenches, which are far from adiabatic, the Trotterisation remains accurate enough despite the small number of steps used In principle, our algorithm could be implemented on current quantum annealing platforms but will be limited to certain quench sequences The cyclic optimisation algorithm does not require variational optimisation, it is well suited for direct implementation for solving optimisation problems on platforms with a limited shot budget

Supplemental Material Insights

The success ratio analysis shows that at Bz = 1 and χ = 1 for all N ≳ 10 we are highly likely to achieve a successful energy transition for this choice of quench sequence The finite-size scaling in fig. S7(a) shows excellent finite-size scaling where the success probability actually increases as we increase system size The dependence on the initial reference state S r is shown to be independent of the chosen reference state The X-basis measurement is not a good measurement basis for our parameter selection algorithm, as it remains fully delocalized for all values of Bx

Parameter Testing Robustness

The efficacy of our parameter testing algorithm becomes significantly better as we increase the number of graph edges k, indicating that the speed at which these types of Hamiltonians thermalize is dependent on the connectivity of the graph The SKatan method is a poor choice as a parameter setting technique for k-regular graphs and that optimisation of the angles is generally required The paper concludes that our algorithm retains a significant success probability over a wide range of graph sizes once we scale beyond classically tractable graph instances

AI Use Disclosure

Section II B was partially developed through discussions with Claude, and all AI-assisted outputs were critically reviewed by the authors The cyclic optimisation algorithm was implemented in Qiskit [44] and the exact diagonalization simulations were performed in QuSpin [45]

Acknowledgments

We would like to thank Ieva Cepait˙e, Elizabeth ˇCrosson, Niam Vaishnav and Andrew Hallam for helpful discussions The cyclic optimisation algorithm was implemented in Qiskit [44] and the exact diagonalization simulations were performed in QuSpin [45] R.S. acknowledges support by the Leverhulme Trust Research Leadership Award RL2019-015 and EPSRC Grants No. EP/Z533634/1 and No.

Improvements for AI systems

Optimized quantum annealing protocols can be accelerated by considering the density of Landau-Zener crossings as the system is quenched, which allows for a faster exploration of local minima in spin-glass problems. This leads to a required annealing time that goes from O(poly(N)) to O(1) for reference states with non-vanishing energy density.
The Fast Reverse Iterative Optimisation (FRIO) algorithm can be implemented on near-term devices with a number of Trotter steps that remains modest as system size increases, making it appealing for NISQ hardware. This is achieved by using a digitised version of the protocol where the evolution unitary is approximated by a small number of Trotter steps.
The algorithm can be used to find ground states for all-to-all connected graphs described by the quantum SK model with better performance than optimized QAOA when appropriate quench parameters are correctly identified. This is particularly true when deep into the thermal phase, where our algorithm readily outperforms optimized QAOA for all system sizes considered.
A method for determining protocol parameters can be implemented on quantum hardware by inspecting short-time thermalization dynamics, which provides a way to test whether Hamiltonian parameters are located in the thermal phase. This is done by measuring metrics like the inverse participation ratio (IPR) on a subsystem A.
The AI system can perform parameter selection for cyclic quenching using IPR dynamics measured in the Z-basis, as this basis is an excellent basis to choose, and this method is easily implementable on quantum hardware compared to traditional thermalization measures with favourable resolution scaling.

Abstract

The promise of efficiently solving classical optimisation problems on quantum computers is tantalizing. As many such problems can be mapped to approximating the ground state of spin-glasses, paradigmatic models for studying many-body localisation, it is natural to ask if some of the intuition coming from quantum statistical mechanics can help guide the construction of novel quantum optimisation algorithms. In this work, we discuss how continuous reverse quantum annealing crossing a many-body localisation transition can be accelerated by considering the density of Landau-Zener crossings as the system is quenched. We provide an interpretation of this behaviour in terms of an effective Fokker-Planck equation for the success probability of the algorithm. This interpretation suggests an improvement in the required annealing time as a function of the system size N, which goes from O(poly (N)) to O(1) for reference states with non-vanishing energy density. We show that the continuous dynamics can be Trotterised with a number of steps that remains modest as system size increases, making it appealing for near-term devices. For the resulting circuit implementation, we provide a quantitative strategy for determining the parameters of the time-dependent Hamiltonian. We compare the performance of our digitised algorithm, which we call FRIO, against that of the QAOA for finding the ground state of spin-glass problems, in particular the SK model. The numerical results suggest that with suitably selected circuit parameters which can be efficiently found using thermalizing dynamics, FRIO almost always outperforms the conventional QAOA ansatz with pre-optimized angles.