Connecting phases of matter to the flatness of the loss landscape in analog variational quantum algorithms

TL;DR

The paper addresses barren plateaus in variational quantum algorithms by studying an analog VQA built from M quenches of a disordered Ising chain. It reveals that both thermalized and MBL phases can reach maximal expressivity as M grows, but BP appears much earlier in the thermalized regime, motivating an MBL-based initialization that preserves trainability while enabling later full expressivity. The authors formalize expressivity with frame potentials, derive an BP-related variance bound, and demonstrate the proposed initialization on ground-state and Max-Cut tasks, achieving superior or competitive performance depending on problem structure. This work provides practical, phase-aware guidelines for scaling analog VQAs and bridges digital-analog quantum computing approaches, with potential experimental validation in near-term quantum hardware.

Abstract

Variational quantum algorithms (VQAs) promise near-term quantum advantage, yet parametrized quantum states commonly built from the digital gate-based approach often suffer from scalability issues such as barren plateaus, where the loss landscape becomes flat. We study an analog VQA ansätze composed of $M$ quenches of a disordered Ising chain, whose dynamics is native to several quantum simulation platforms. By tuning the disorder strength we place each quench in either a thermalized phase or a many-body-localized (MBL) phase and analyse (i) the ansätze's expressivity and (ii) the scaling of loss variance. Numerics shows that both phases reach maximal expressivity at large $M$, but barren plateaus emerge at far smaller $M$ in the thermalized phase than in the MBL phase. Exploiting this gap, we propose an MBL initialisation strategy: initialise the ansätze in the MBL regime at intermediate quench $M$, enabling an initial trainability while retaining sufficient expressivity for subsequent optimization. The results link quantum phases of matter and VQA trainability, and provide practical guidelines for scaling analog-hardware VQAs.

Figures (16)

• Figure 1: Overview of our analog VQA framework (a) Schematic representation of a periodic Ising spin chain with nearest-neighbour interactions $J$, disordered on-site energy $h_i^{(m)}$ at the $m$-th quench, and a transverse field $B$. Each on-site disorder energy is uniformly drawn from the interval $[-W/2,W/2],$ with $W$ controlling the disorder strength. (b) Analog VQA modeled as a series of quench dynamics. Each block represents a unitary evolution under the quench Hamiltonian $H(\boldsymbol{\theta}_m)$ for a time $t_m$, parametrized by the disorder configuration $\boldsymbol{\theta}_m = \{h_i^{(m)}\}$ and measured by local observables. While the evolution times can in general be trainable parameters, we fix them here for our investigation on the connection with phases but they can be made trainable when solving the actual optimization problems (see the Appendix). (c) A schematic phase diagram of the system as a function of disorder strength $W$ (defined for a sufficiently long evolution time of a single quench dynamics) shows a transition between the many-body localized (MBL) phase and the thermalized phase. Bottom panels depict spin density profiles after long-time evolution: in the MBL phase ($W < W^*$), the system retains memory of the initial state, while in the thermalized phase ($W > W^*$), the system relaxes to a thermal state. This setup enables us to study the interplay between disorder-induced quantum phases and the performance and scalability of analog VQAs setup in (b).
• Figure 2: Summary of Results. A visualization of Hilbert space exploration and loss landscapes for our initialisation strategies using two different phases of matter: thermalized and MBL. The results are categorized by the number of quenches (analogous to the circuit depth in gate-based VQA approaches): shallow, intermediate, and deep. Shallow quenches -- Both initialisations exhibit non-flat loss landscapes. Thermalized dynamics lead to faster state evolution in Hilbert space compared to MBL dynamics, but the model expressivity in both initialisations is far from maximal. Intermediate quenches -- the thermalized initialisation begins to exhibit the barren plateaus and achieves maximal expressivity, while the MBL loss landscape remains non-flat and Hilbert space is not yet completely explored. This intermediate quench regime highlights our proposed initialisation strategy: using MBL initialisation strategy at intermediate quenches allows the model to attain high expressivity while retaining trainability. Deep quenches -- For both initialisations, barren plateaus appear and the Hilbert space is fully explored, indicating maximal expressivity.
• Figure 3: Level statistics of the disordered Ising model. The histograms display the level statistics of 500 instances of 9-qubit systems governed by the Hamiltonian in Eq. \ref{['eq:IsingModel']} for two different disorder strengths. (a) With $W=5J$, the histogram follows the GOE distribution, indicating that the system is in the thermalized phase characterized by level repulsion and correlated eigenvalues. (b) With $W=50J$, the histogram follows the Poisson distribution, indicating that the system is in the MBL phase where eigenvalues are uncorrelated and level crossings are allowed. These different disorder strengths are used to initialise the parameters in our ansätze for the thermalized and MBL initialisations.
• Figure 4: Second-order frame potentials (a) The difference between the second-order frame potential of the ansätze ensemble and of the Haar distribution is plotted against the number of quenches $M$ for the systems size ranging from $n=5$ to $n=9$, under the thermalized initialisation. The second-order frame potential is estimated by averaging the square of fidelity over $N(N-1)/2$ independent pairs with $N=30,000$. The saturated values of the second-order frame potential difference are plotted against the system size $n$ for (b) the thermalized and (c) MBL initialisations. (d) Same as panel (a), but under the MBL initialisation.
• Figure 5: Variance of the loss function in the thermalized and MBL initialisations. The top panels illustrate the comparison between the variance of the loss function for (a) thermalized and (b) MBL initialisations and its bounds against the number of quenches $M$. Specifically, the bounds are functions of the frame potential difference between the ansätze ensemble and the Haar distribution. The solid line represents the variance of the loss $\langle Z_1Z_2\rangle_{\boldsymbol{\theta}}$, the dotted line shows the empirical bound for the variance, and the dashed line indicates the theoretical bound for the variance as presented in Eq. \ref{['eq:variance-expressivity-bound']}. These panels display data for 5, 7, and 9 qubits from left to right. The middle panels show the variance of the loss $\langle Z_1Z_2\rangle_{\boldsymbol{\theta}}$ for (c) thermalized and (d) MBL initialisation as the number of quenches increases for systems with 5 to 13 qubits, averaged over 400 realizations. The bottom panels present the saturated variance of $\langle Z_1Z_2\rangle_{\boldsymbol{\theta}}$ plotted on a logarithmic scale against the number of qubits for (e) thermalized and (f) MBL phase. This provides evidence for the possible emergence of barren plateaus when the ansätze is initialised in both phases.