Day 1 (June 23)

Adiabatic quantum optimization is limited not only by small spectral gaps, but also by finite coherence times, environmental noise, and heating. Fast quantum optimization addresses these limitations by shortening the relevant adiabatic timescale through tailored schedules or by adding counterdiabatic Hamiltonians that suppress excitations during rapid quenches.
We first discuss nonadiabatic quantum optimization, where a single control parameter is tailored to shorten the effective adiabatic timescale across a quantum phase transition. [1] The framework minimizes the sum of simultaneous Landau--Zener transitions from the ground state to the relevant low-lying excited states near the critical region. The resulting variational principle yields nonlinear differential equations for the optimal schedule, which substantially outperforms local adiabatic driving, optimized power-law schedules, and quantum adiabatic brachistochrone, and can be extended to disordered nonintegrable spin chains with minimal numerical input.
As a complementary route, we study counterdiabatic driving across spin-glass bottlenecks with exponentially small gaps. [2] Local CD expansions can reduce excitations, but yield only negligible improvements in the final fidelity in realistic NP-hard instances such as Max-Cut and XORSAT. We therefore introduce quantum brachistochrone counterdiabatic driving (QBCD), which uses an approximate full CD Hamiltonian at a single parameter value near the critical point. Even after exponentially reducing its nonlocality, QBCD achieves finite fidelity on exponentially shorter timescales than local CD methods.
Finally, we discuss universal fast dynamics across quantum phase transitions, both without and with counterdiabatic driving. [3] We develop analytically tractable local CD expansion schemes and show that all defect cumulants follow universal power laws with the CD locality order, quantifying excitation suppression beyond the mean excitation number. [4] This theoretical picture is complemented by the experimental realization of digitized CD-assisted quantum simulations on superconducting processors with up to 156 qubits, where one- and two-dimensional transverse-field Ising models exhibit substantial excitation reduction. [5]

Quantum tunneling is expected to provide a computational speed-up in quantum computing, a phenomenon that Quantum Annealing (QA) aims to leverage. While the “Hamming weight with a spike” problem serves as an academic proof of concept, the analytical mechanisms underlying this effect during quantum evolution remain largely unexplored for more general potentials. In this work, we provide a discrete version of the result by Brascamp and Lieb [BL76], which in the continuous case states that the ground-state of a Schrödinger operator with a convex potential is log-concave. Log-concavity is a strong structural property that imposes a specific distribution shape, specifically unimodality with smooth variation. We prove that this property holds in the discrete setting for a large family of Hamming-weight-symmetric potentials (including convex ones) under a transverse field. We apply this general technical result to show that QA tunnels through spikes in quadratic potentials.

First, we formalize an implicit result from previous literature [JJ14] regarding the spectral gap of the transverse field with a Hamming-symmetric convex potential, explicitly demonstrating that the gap remains constant throughout the evolution. Second, we extend Reichardt's findings [Rei04] on the effect of a spike perturbation on the spectral gap, showing that the perturbed gap is proportional to the product of the spike's height and width divided by the variance of the unperturbed ground state. To derive a condition on the spike that guarantees an exponential speed-up over Simulated Annealing (the classical counterpart to QA), we bound the variance of the unperturbed ground state for a quadratic potential. Since the analytical shape of this ground state is unknown, we achieve this by proving that the ground state maintains a large, constant overlap with a binomial state throughout the evolution, a theoretical result we think of independent interest.

Ultimately, we show that the spectral gap of the perturbed Hamiltonian remains constant during the evolution if $h\frac{u-l}{\sqrt{l}}=\mathcal{O}(1)$, where $h$ represents the spike's height and the interval $[l, u]$ defines its width. This result naturally generalizes the “Hamming weight with a spike” problem to a wider class of potentials and strongly suggests its broader applicability to convex potentials.

[BL76] Herm Jan Brascamp and Elliott H Lieb. On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. Journal of functional analysis, 22(4):366–389, 1976.

[JJ14] Michael Jarret and Stephen P. Jordan. The fundamental gap for a class of Schrödinger operators on path and hypercube graphs. Journal of Mathematical Physics, 55(5), May 2014.

[Rei04] Ben W Reichardt. The quantum adiabatic optimization algorithm and local minima. In Proceedings of the thirty-sixth annual ACM symposium on Theory of computing, pages 502–510, 2004.

We introduce a simple framework for estimating lower bounds on the runtime of a broad class of adiabatic quantum algorithms. The central formula consists of calculating the variance of the final Hamiltonian with respect to the initial state. After examining adiabatic versions of certain keystone circuit-based quantum algorithms, this technique is applied to adiabatic quantum algorithms with undetermined speedup. In particular, we analytically obtain lower bounds on adiabatic algorithms for finding k-clique in random graphs. Additionally, for a particular class of Hamiltonian, it is straightforward to prove the equivalence between our framework and the conventional approach based on spectral gap analysis.
https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.033175

Quantum adiabatic optimization seeks to solve combinatorial problems using quantum dynamics, requiring the Hamiltonian of the system to align with the problem of interest. However, these Hamiltonians are often incompatible with the native constraints of quantum hardware, necessitating encoding strategies to map the original problem into a hardware-conformant form. While the classical overhead associated with such mappings is easily quantifiable and typically polynomial in problem size, it is much harder to quantify their overhead on the quantum algorithm, e.g., the transformation of the adiabatic timescale. 

In this work, we address this challenge on the concrete example of the encoding scheme proposed in [Nguyen et al., PRX Quantum 4, 010316 (2023)], which is designed to map optimization problems on arbitrarily connected graphs into Rydberg atom arrays. We consider the fundamental building blocks underlying this encoding scheme and determine the scaling of the minimum gap with system size along adiabatic protocols. Even when the original problem is trivially solvable, we find that the encoded problem can exhibit an exponentially closing minimum gap. We show that this originates from a quantum coherent effect, which gives rise to an unfavorable localization of the ground-state wave function. On the QuEra Aquila neutral atom machine, we observe such localization and its effect on the success probability of finding the correct solution to the encoded optimization problem. Finally, we propose quantum-aware modifications of the encoding scheme that avoid this quantum bottleneck and lead to an exponential improvement in the adiabatic performance. This highlights the crucial importance of accounting for quantum effects when designing strategies to encode classical problems onto quantum platforms.

I will present tensor network algorithms for optimizing quantum circuits for adiabatic quantum computing [PRXQuantum.5.020362]. To suppress diabatic transitions, counterdiabatic driving is incorporated into the optimization and variational matrix product operators are used to represent adiabatic gauge potentials. Traditionally, adiabatic time evolution is converted into quantum circuits using Trotter product formulas, and the inclusion of counterdiabatic driving further increases the circuit depth required at each time step. In this work, we classically optimize a fixed-depth parameterized quantum circuit that simultaneously captures adiabatic evolution and counterdiabatic driving over many time steps. These methods are applied to ground-state preparation in quantum Ising chains with transverse and longitudinal fields. We show that the resulting classically optimized circuits can significantly outperform Trotter product formulas. Finally I will discuss how this approach can be extended to combinatorial optimization.

N. Feinstein, S. Schulz, R. Banks, R. Ghosh, S. Bose, P. A. Warburton

A critical problem in quantum annealing is the appearance of small spectral gaps which limit the rate at which the algorithm can progress adiabatically. Of particular concern are so-called perturbative crossings which can form as a result of competing local optima and introduce gap minima that close exponentially with the problem size [1]. In some cases the introduction of XX-interactions, typically in the form of a catalyst Hamiltonian, have been shown to achieve gap enhancement in such settings. However, this success is highly dependent on the placements of the couplings within the problem graph as well as the problem itself [2,3,4,5] suggesting that it may be necessary to incorporate problem-specific information into catalyst design. In previous work, we motivated an approach to XX-term selection and demonstrated the effectiveness of the resulting catalysts at removing multiple perturbative crossings from example toy instances of the maximum weighted-independent-set (MWIS) problem [6,7]. Furthermore, we suggested that polynomial annealing runs could be used to obtain the information needed to construct these catalysts for more general settings. Here we report results implementing this idea as a recursive algorithm which builds a catalyst based on knowledge of the local optima which could be obtained from successive annealing runs. By simulating this process for MWIS problem instances, we find that we can significantly improve the probability of obtaining the optimal solution by comparison with performing the equivalent number of catalyst-free anneals. Finally, we discuss the possibility of implementing effective XX-terms on hardware that does not support such interactions natively.

[1] Amin and Choi, Physical Review A 80 (2009)

[2] Nishimori and Takada, Frontiers in ICT 4 (2017)

[3] Albash, Physical Review A 99 (2019)

[4] Takada et al., Journal of the Physical Society of Japan 89 (2020)

[5] Ghosh et al. Phys. Rev. Research 8 (2026)

[6] Feinstein et al. Physical Review A 110 (2024)

[7] Feinstein, "Targeted XX-catalysts for quantum annealing", University College London PhD thesis (2025)

We present a new framework to unravel the structure of the adiabatic algorithm based on the idea of Closest Accessible Symmetries (CAS). Our approach consists on recursively projecting onto reduced representations of the initial and final Hamiltonians such that the distance between this simplified representation of the computation and the original one is minimised. This method provides an analytical estimate of the energy landscape that can be used to make statements about the hardness of the process, both qualitative (e.g., the gap will remain large), as well as quantitative (e.g., bounds on the size of the minimum gap along the computation). The CAS framework also points to a new heuristic for classical optimisation problems when applied to the standard quantum annealing setting, and we believe it could have further applications in the design of analogue algorithms or the study of phase diagrams.

Recent results show that quantum annealing and optimal QAOA trajectories generate pseudo-Boltzmann states in diabatic regimes [DOI: https://doi.org/10.1103/hxv2-sbr7]. Motivated by this observation, we introduce a new approach to efficiently approximate catalysts that improve the performance of both QAOA and quantum annealing by controlling the non-Boltzmann contributions to the output probability distribution. These catalytic terms can be implemented on superconducting quantum hardware at the cost of a modest increase in the connectivity of the Hamiltonian during the evolution. For 3-regular MaxCut instances of up to 30 qubits, we show that these catalysts improve both the probability of sampling the ground state and the time to solution required to reach high-quality cuts, such as 85% or 90% of the total edges. Notably, no variational optimization is required, since the catalyst acts as a plug-and-play term that consistently enhances solution quality when incorporated via the usual s(1−s) schedule. Finally, we show that its performance can be further improved by optimizing the annealing schedule. We support this intuition by substantially improving performance on the frustrated Ising ring using a simple modified catalyst schedule that mirrors the usual schedule but vanishes at s>s', with s' between 0 and 1.