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arXiv:2609.27488

On Relationship Between Circuit Depth and Trainability of VQAs

Jan Michálek, Martin Friák, Petr Vašík

quant-phSubmitted Sep 23, 2026• Announced Sep 24, 2026• 0 citations
arXivHTMLPDF

Abstract

The training efficiency of Variational Quantum Algorithms (VQAs) is dictated by the geometry of their loss landscapes, which can be formally analysed by mapping these functions to random fields on manifolds. Instead of VQAs, we can then directly study the corresponding random fields, in our case, the Wishart Hypertoroidal Random Fields (WHRFs). We are mostly interested in the distribution of critical points (especially local minima) of WHRFs. For this purpose, the Kac-Rice formula is presented, reformulated, and simulated. The findings identify a phase transition in the distribution of local minima. Beyond a specific threshold, local minima concentrate near the global minimum in function value, meaning that even local minima are good approximators of the global one. The threshold is governed by the ratio between the problem Hamiltonian degrees of freedom and by the number of independent parameters in the VQA. Since the degrees of freedom parameter scales exponentially, we propose symmetry reduction operations to lower the degrees of freedom. This mathematically reduces the dimension, scaling down the required parameter threshold and enabling to solve bigger problems.

AI Summary

The paper asks how ansatz parameter count affects the distribution of local minima in variational quantum algorithms, and whether symmetry restrictions can lower the parameter count associated with favourable landscapes. It uses an existing mapping from shifted VQA energies to Wishart hypertoroidal random fields, with spectral parameter $m=\|H-E_0I\|_*^2/\|H-\bar E I\|_F^2$ and overparameterisation ratio $\gamma=p/(2m)$. A pointwise critical-value density is obtained by conditioning the Kac–Rice formula on field value and vanishing gradient. The resulting Wishart-plus-GOE Hessian expectation is evaluated by Monte Carlo sampling, using batched linear algebra and, for larger cases, chunking and logarithmic arithmetic. Simulations report migration of local-minimum density toward zero shifted energy near $p=2m-2$, compared with the previously established asymptotic boundary $p\approx2m$. Examples cover a toy model with $m=8$ and spectral parameters from two- to four-site Hubbard Hamiltonians, reaching $m=749$ and $p\approx1496$. Separate calculations apply particle-number, spin-flip and spatial-parity restrictions. For seven Hubbard sites at $U=2,t=1$, the Hilbert-space dimension falls from $16{,}384$ to $326$, and $m$ from approximately $130{,}000$ to $2{,}540$. The stated limitations include rare-event sampling noise, numerical and memory constraints, and continued exponential scaling after symmetry reduction. Local-minimum statistics do not directly establish optimisation performance or exclude barren plateaus. The random-field mapping may deviate for symmetry-informed ansatzes, and hardware validation remains future work; preliminary compilation observations highlight connectivity and routing overheads.

Likely largely AI-generatedOverclaimsIncrementalStrong assumptionsPossible error

Central claim may not hold: The real Wishart field defined in the paper has a chi-square_m/m field-value distribution and conditional gradient variance 4x/m, whereas the central calculation uses Gamma(m, rate m) and variance 2x/m. The asserted singular concentration at p=2m-2 contradicts the paper's own finite, positive limiting density there, and the energy-dependent Hessian determinant cannot generally be discarded when locating a transition. The simulations establish properties only of a random-matrix surrogate parameterized by Hamiltonian spectra, not the claimed optimization or circuit-depth consequences for actual VQAs. The reduced spectral parameter m is not shown to provide an upper bound on trainability thresholds for symmetry-preserving ansatzes, whose random-field equivalence the paper expressly leaves uncertain. For odd lattice sizes, the purported half-filled symmetry-reduction examples actually use L-1 particles, and the statement that odd-L half-filling is impossible is incorrect.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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The training efficiency of Variational Quantum Algorithms (VQAs) is dictated by the geometry of their loss landscapes, which can be formally analysed by mapping these functions to random fields on manifolds. Instead of VQAs, we can then directly study the corresponding random fields, in our case, the Wishart Hypertoroidal Random Fields (WHRFs). We are mostly interested in the distribution of critical points (especially local minima) of WHRFs. For this purpose, the Kac-Rice formula is presented, reformulated, and simulated. The findings identify a phase transition in the distribution of local minima. Beyond a specific threshold, local minima concentrate near the global minimum in function value, meaning that even local minima are good approximators of the global one. The threshold is governed by the ratio between the problem Hamiltonian degrees of freedom and by the number of independent parameters in the VQA. Since the degrees of freedom parameter scales exponentially, we propose symmetry reduction operations to lower the degrees of freedom. This mathematically reduces the dimension, scaling down the required parameter threshold and enabling to solve bigger problems.
The paper asks how ansatz parameter count affects the distribution of local minima in variational quantum algorithms, and whether symmetry restrictions can lower the parameter count associated with favourable landscapes. It uses an existing mapping from shifted VQA energies to Wishart hypertoroidal random fields, with spectral parameter m=∥H−E0I∥∗2/∥H−EˉI∥F2m=\|H-E_0I\|_*^2/\|H-\bar E I\|_F^2m=∥H−E0​I∥∗2​/∥H−EˉI∥F2​ and overparameterisation ratio γ=p/(2m)\gamma=p/(2m)γ=p/(2m).
A pointwise critical-value density is obtained by conditioning the Kac–Rice formula on field value and vanishing gradient. The resulting Wishart-plus-GOE Hessian expectation is evaluated by Monte Carlo sampling, using batched linear algebra and, for larger cases, chunking and logarithmic arithmetic. Simulations report migration of local-minimum density toward zero shifted energy near p=2m−2p=2m-2p=2m−2, compared with the previously established asymptotic boundary p≈2mp\approx2mp≈2m. Examples cover a toy model with m=8m=8m=8 and spectral parameters from two- to four-site Hubbard Hamiltonians, reaching m=749m=749m=749 and p≈1496p\approx1496p≈1496.
Separate calculations apply particle-number, spin-flip and spatial-parity restrictions. For seven Hubbard sites at U=2,t=1U=2,t=1U=2,t=1, the Hilbert-space dimension falls from 16,38416{,}38416,384 to 326326326, and mmm from approximately 130,000130{,}000130,000 to 2,5402{,}5402,540.
The stated limitations include rare-event sampling noise, numerical and memory constraints, and continued exponential scaling after symmetry reduction. Local-minimum statistics do not directly establish optimisation performance or exclude barren plateaus. The random-field mapping may deviate for symmetry-informed ansatzes, and hardware validation remains future work; preliminary compilation observations highlight connectivity and routing overheads.
One of the most important properties of classical neural networks is the clustering of local minima of the network near the global minimum, enabling efficient training. This has been observed not only numerically, but also has begun to be analytically understood through the lens of random matrix theory. Inspired by these results in classical machine learning, we show that a certain randomized class of variational quantum algorithms can be mapped to Wishart random fields on the hypertorus. Then, using the statistical properties of such random processes, we analytically find the expected distribution of critical points. Unlike the case for deep neural networks, we show the existence of a transition in the quality of local minima at a number of parameters exponentially large in the problem size. Below this transition, all local minima are concentrated far from the global minimum; above, all local minima are concentrated near the global minimum. This is consistent with previously observed numerical results on the landscape behavior of Hamiltonian agnostic variational quantum algorithms. We give a heuristic explanation as to why ansatzes that depend on the problem Hamiltonian might not suffer from these scaling issues. We also verify that our analytic results hold experimentally even at modest system sizes.
Variational quantum algorithms (VQAs) are a leading approach to near-term quantum computation, but their utility is limited by barren plateaus and other pathologies in their loss landscapes. Existing landscape theories based on dynamical Lie algebras, Jordan-algebraic Wishart systems, approximate t-designs, and Haar-random circuits are foundational, but they often neglect the finite-depth geometry of realistic ansätze and are therefore ill-suited to the shallow-depth regime, where VQAs are poor approximators of 2-designs and trainability is most feasible. This work introduces Krylov algebras, algebraic structures induced by the Krylov span of a finite generator set acting on one or more seed vectors, as a framework for VQA landscape theory. We show that VQA reachable manifolds can be approximated in a numerically robust, geometrically faithful fashion by Krylov-Lie algebras and groups, and that these structures induce canonical invariant measures for computing expectation values and variances under general sampling measures. In particular, we derive weighted non-Haar variance formulas that recover the usual Lie-algebraic Haar formulas as a special case while isolating non-Haar effects into explicit correction terms. We also show that the common heuristic that sufficiently deep circuit ensembles must converge to Haar fails in general without additional hypotheses, identify concrete obstructions to naive Haar convergence, and recover convergence under natural necessary and sufficient ergodic conditions. Lastly, our formulas further imply that non-Haar contributions to landscape statistics may mitigate barren plateaus by reweighting the visible sectors of the loss landscape, suggesting that VQAs may be more trainable than recent literature has posited.
One of the most important properties of neural networks is the clustering of local minima of the loss function near the global minimum, enabling efficient training. Though generative models implemented on quantum computers are known to be more expressive than their traditional counterparts, it has empirically been observed that these models experience a transition in the quality of their local minima. Namely, below some critical number of parameters, all local minima are far from the global minimum in function value; above this critical parameter count, all local minima are good approximators of the global minimum. Furthermore, for a certain class of quantum generative models, this transition has empirically been observed to occur at parameter counts exponentially large in the problem size, meaning practical training of these models is out of reach. Here, we give the first proof of this transition in trainability, specializing to this latter class of quantum generative model. We use techniques inspired by those used to study the loss landscapes of classical neural networks. We also verify that our analytic results hold experimentally even at modest model sizes.
Variational quantum algorithms (VQAs) are among the leading approaches for near-term quantum computing, yet their performance can degrade in barren plateau regimes characterized by vanishing gradients. A widely held intuition is that increasing circuit expressivity, often associated with random-state behavior, leads to a loss of trainability. Existing results show that sufficiently random circuits can lead to barren plateaus. Here we show that standard statistical signatures of randomness can emerge well before this regime, without degrading trainability. We demonstrate this behavior in structured variational circuits applied to the one-dimensional cluster-Ising model and a generalized toric code Hamiltonian. To characterize state complexity, we analyze Porter-Thomas statistics, entanglement-spectrum level statistics, and inverse participation ratios. Across both models, increasing circuit depth drives these diagnostics toward random-state-like or random-matrix-like behavior, while variational optimization remains effective, with no evidence of exponential gradient suppression in the regime studied. We interpret this behavior in terms of locality. Spectral correlations develop at relatively shallow depth through locally generated structure, while global state randomization and the associated concentration-of-measure effects are not yet realized. These results show that commonly used statistical diagnostics of complexity do not by themselves determine trainability. Instead, they point to a separation between different aspects of complexity in finite-depth variational circuits.
Exploring quantum applications of near-term quantum devices is a rapidly growing field of quantum information science with both theoretical and practical interests. A leading paradigm to establish such near-term quantum applications is variational quantum algorithms (VQAs). These algorithms use a classical optimizer to train a parameterized quantum circuit to accomplish certain tasks, where the circuits are usually randomly initialized. In this work, we prove that for a broad class of such random circuits, the variation range of the cost function via adjusting any local quantum gate within the circuit vanishes exponentially in the number of qubits with a high probability. This result can unify the restrictions on gradient-based and gradient-free optimizations in a natural manner and reveal extra harsh constraints on the training landscapes of VQAs. Hence a fundamental limitation on the trainability of VQAs is unraveled, indicating the essential mechanism of the optimization hardness in the Hilbert space with exponential dimension. We further showcase the validity of our results with numerical simulations of representative VQAs. We believe that these results would deepen our understanding of the scalability of VQAs and shed light on the search for near-term quantum applications with advantages.
Quantum circuits suffer from severe trainability issues: even shallow circuits are swamped with poor local minima. Yet matrix product states (MPS), which can be prepared by sequential circuits, are remarkably trainable in practice -- as demonstrated by decades of successful density matrix renormalization group calculations. In this work, we resolve this apparent paradox by proving that the energy landscapes of MPS are free from poor local minima, under the same setting where brickwork circuits are not. The key insight is that the gauge freedom of MPS creates an effective local overparametrization that causes local minima to concentrate near the global minimum, analogous to overparametrized classical neural networks. We rigorously prove that the local minimum distribution is invariant under moves of the orthogonality center of MPS representations. Numerical experiments further confirm that the optimization of sequential circuits converges to near-optimal solutions even for random Hamiltonians, in stark contrast to brickwork circuits. Our findings establish a theoretical understanding of the trainability of MPS, providing a valuable guide for designing variational quantum circuits and algorithms with better trainability in the future.