Quantum State’s Perspective: How Basis Choice Impacts Optimization

Author: Denis Avetisyan

A new study reveals that even without changing a quantum system’s energy, altering its mathematical representation can dramatically hinder the efficiency of variational optimization algorithms.

The study demonstrates that infidelity minimization within a Noisy Quantum System (NQS) - specifically for a five-qubit system with $h=0.5$ and varying $J$ values of $-1$ and $+1$ - yields a quantifiable relationship between the Fubini Study distance $\gamma\gamma$ and the associated loss $\mathcal{L}$ as a function of rotation angle $\phi$, indicating the system’s sensitivity to state transformations. — The study demonstrates that infidelity minimization within a Noisy Quantum System (NQS) – specifically for a five-qubit system with $h=0.5$ and varying $J$ values of $-1$ and $+1$ – yields a quantifiable relationship between the Fubini Study distance $\gamma\gamma$ and the associated loss $\mathcal{L}$ as a function of rotation angle $\phi$, indicating the system’s sensitivity to state transformations.

Research demonstrates that basis rotations significantly affect the curvature of the loss landscape in Neural Quantum States, impacting optimization performance despite a constant underlying energy.

Variational quantum algorithms rely on efficiently representing quantum states, yet the impact of representational choices remains poorly understood. In ‘Exploring the Effect of Basis Rotation on NQS Performance’, we investigate how changes in the basis used to represent quantum states affect the optimization landscape encountered by Neural Quantum States (NQS). Our analysis of a solvable Ising model reveals that basis rotations, while leaving the underlying energy landscape unchanged, effectively relocate the target wavefunction, increasing its distance from typical initializations and exposing information-geometric barriers like saddle points. This suggests that optimization challenges in shallow NQS architectures aren’t solely determined by landscape complexity, but also by the accessibility of the target state within that landscape – prompting the question of how to design NQS models that are intrinsically robust to basis-induced optimization impediments.

The Quantum Challenge: Representing the Irreducible

Describing the behavior of quantum many-body systems – those comprised of numerous interacting particles – presents a formidable challenge to classical computation. The difficulty arises because the information needed to fully define a quantum state grows exponentially with the number of particles. For instance, a system of $N$ spins requires $2^N$ complex numbers to specify its wavefunction, quickly exceeding the capacity of even the most powerful supercomputers. This exponential scaling means that simulating systems beyond a relatively small size – a few dozen particles – becomes practically impossible using traditional methods like direct wavefunction expansion or perturbation theory. Consequently, significant progress in understanding complex quantum phenomena, from high-temperature superconductivity to materials science, is often hampered by limitations in computational power and algorithmic efficiency.

Neural Quantum States (NQS) represent a significant departure from traditional methods of simulating quantum systems by embracing the flexibility of machine learning. Rather than attempting to directly solve the Schrödinger equation for many interacting particles-a task that quickly becomes computationally intractable-NQS utilizes neural networks as a variational ansatz for the quantum wavefunction. This approach encodes the complex relationships between quantum particles within the weights and biases of a trainable network. By adjusting these parameters, the network learns to approximate the ground state, or other relevant states, of the quantum system. The power of NQS lies in the network’s ability to efficiently represent highly entangled states, offering a potentially scalable pathway to study materials and phenomena currently beyond the reach of conventional computational methods. This variational principle, combined with the expressive capacity of deep learning, allows researchers to circumvent the exponential scaling issues inherent in describing many-body quantum systems, promising advancements in fields like materials science and quantum chemistry.

Neural Quantum States (NQS) represent a significant departure from traditional methods for simulating quantum systems, offering a potential solution to the ‘curse of dimensionality’ that plagues conventional approaches. Instead of explicitly calculating and storing the exponentially growing wavefunction – a task quickly becoming impossible even for modest system sizes – NQS cleverly encodes the quantum state within the weights and biases of a neural network. This allows the complex relationships between quantum particles to be learned and represented by the network’s trainable parameters. Through optimization techniques, the network adjusts these parameters to minimize the system’s energy, effectively ‘discovering’ the ground state wavefunction. Consequently, NQS opens the door to studying quantum systems previously out of reach, promising insights into materials science, high-energy physics, and potentially revolutionizing the design of novel quantum technologies by enabling the simulation of more complex and realistic scenarios.

The Shannon entropy of quantum coherence in a logRBM, calculated over 5000 iterations with h=0.5 and N=5, 7, 9, reveals distinct rotational dependencies for J=-1 and J=+1 couplings.

Navigating the Energy Landscape: A Descent Towards Solution

Neural Quantum State (NQS) training fundamentally relies on energy minimization to determine optimal network parameters. This process involves iteratively adjusting the weights and biases of the neural network to minimize the expectation value of the system’s Hamiltonian operator, $H$. The goal is to find the parameter set that corresponds to the ground state, or lowest energy state, of the quantum system being modeled. This minimization is typically achieved through gradient-based optimization algorithms, where the gradient of the energy with respect to the network parameters guides the parameter update. Successful energy minimization indicates the NQS has effectively learned to represent the target quantum state, enabling accurate predictions of system properties.

The optimization of Neural Quantum States (NQS) is critically dependent on the characteristics of the loss landscape. This landscape, representing the energy of the system as a function of network parameters, frequently contains saddle points – points where the gradient is zero but do not represent local minima – and extensive flat regions. Saddle points can cause optimization algorithms to stall or move in unproductive directions, while flat regions result in slow convergence as the gradient provides minimal guidance for parameter updates. The density and distribution of these features directly impact training efficiency; landscapes with a high density of saddle points or large flat regions necessitate more sophisticated optimization strategies or increased computational resources to locate the global or low-energy minima, as standard gradient descent methods may become trapped or progress very slowly.

The Quantum Natural Gradient (QNG) represents an optimization algorithm that improves upon standard gradient descent by incorporating the Quantum Fisher Information (QFI) matrix. Unlike traditional methods that apply a uniform learning rate across all parameters, QNG pre-conditions the gradient with the inverse of the QFI, effectively rescaling each parameter update based on its sensitivity to changes in the output. The QFI, calculated as the expected value of the second derivative of the output with respect to the parameters, quantifies this sensitivity. This adaptation allows for more efficient navigation of the loss landscape, particularly in regions characterized by high curvature or ill-conditioning, and demonstrably accelerates convergence towards optimal parameters compared to methods employing fixed step sizes. Computation of the QFI typically involves additional quantum circuits and post-processing, representing a computational overhead balanced by the reduction in the number of optimization steps required.

Analysis of Lanczos, DMRG, and RBM algorithms on a ferromagnetic Ising model (N=5, h=-0.5, φ=π/3) reveals they converge to a common saddle point, as visualized by energy evolution and state trajectory projections using UMAP.

Refining the Search: Curvature and Basis as Guides

Stochastic Reconfiguration (SR) builds upon the Quantum Natural Gradient (QNG) optimization method by introducing an iterative parameter update scheme driven by local curvature information. Unlike traditional gradient descent which uses first-order derivatives, QNG and subsequently SR leverage the Fisher Information Matrix to approximate the local geometry of the loss landscape. SR specifically refines this approach by repeatedly sampling and applying updates based on this curvature, allowing the optimization process to navigate complex, high-dimensional spaces more efficiently. This iterative refinement, informed by local curvature, aims to accelerate convergence and potentially escape local minima that might trap standard gradient-based methods. The technique effectively adapts the optimization trajectory based on the sensitivity of the loss function to parameter changes, as quantified by the curvature.

The optimization of quantum neural networks is demonstrably affected by the chosen basis representation. Basis Rotation is a technique used to explore alternative representations of the network’s parameter space, potentially facilitating improved convergence during training. This involves applying a unitary transformation to the basis vectors, effectively changing the coordinate system in which the optimization problem is expressed. While the Hamiltonian spectrum – and thus the underlying physics – remains invariant under such transformations, the curvature of the optimization landscape changes, influencing the efficiency with which gradient-based methods can locate optimal solutions. Different basis rotations can therefore lead to varying optimization trajectories and final results, highlighting the importance of basis selection or exploration during network training.

The study demonstrates that basis rotations, while maintaining the Hamiltonian spectrum, demonstrably impact the convergence rate of optimization algorithms. Analysis of rotated configurations revealed a relative energy error of ≤ 0.5 in certain instances, suggesting a potential degradation in solution accuracy despite the small magnitude of the error. This indicates that the optimization landscape’s curvature influences performance, and that even minor alterations to the basis can affect the algorithm’s ability to locate optimal parameters. These findings highlight a sensitivity to landscape features during optimization, even when the underlying Hamiltonian remains unchanged.

Averaged relative energy error, plotted on a log-log scale for N=5, 7, and 9 using a logRBM architecture (α=4) and stochastic reconfiguration optimization (τ=10⁵, η=1e-2, δ=1e-6), converges to the expected error level indicating accurate saddle point localization.

Measuring Fidelity: Assessing the Accuracy of Representation

Infidelity, quantified as the distance between the neural network’s reconstructed quantum state and the actual state, provides a rigorous measure of the neural quantum simulator’s (NQS) performance. A lower infidelity score indicates a more accurate representation, signaling the NQS’s ability to effectively capture the nuances of the quantum system. This metric isn’t merely a technical detail; it directly reflects the simulator’s reliability in predicting experimental outcomes and exploring complex quantum phenomena. Researchers utilize infidelity to benchmark different NQS architectures and training strategies, striving to minimize errors and unlock the potential of machine learning in quantum simulation. Essentially, infidelity acts as a crucial diagnostic tool, pinpointing areas where the neural network struggles and guiding improvements to enhance its fidelity to the underlying quantum reality, even as system complexity increases.

Entanglement entropy serves as a powerful diagnostic for evaluating how effectively a neural network quantum state (NQS) representation captures the complex relationships within a quantum system. This measure quantifies the degree of quantum correlation – the interconnectedness of particles beyond what classical physics allows – present in the NQS. Higher entanglement entropy values suggest the network has learned to represent states with significant non-local correlations, indicative of a richer and more accurate quantum depiction. Conversely, a low entanglement entropy implies the NQS primarily captures simple, separable states, potentially missing crucial features of the original quantum system and hindering its ability to accurately model complex quantum phenomena. Essentially, the level of entanglement captured by the NQS directly reflects the model’s capacity to represent the system’s intrinsic complexity and is therefore a vital indicator of its performance.

Research indicates a direct correlation between quantum state rotation and the difficulty of accurate representation using neural quantum states (NQS). As the rotation angle increases, so too does the Shannon Entropy, a measure of quantum coherence, signifying more complex quantum correlations within the system. This heightened coherence presents a significant challenge for shallow NQS models, which struggle to converge on an accurate representation of these rotated states. Notably, the study demonstrates that for antiferromagnetic systems exceeding five quantum particles ($N>5$), NQS consistently fails to converge, highlighting a fundamental limitation in representing highly coherent, rotated states with current NQS architectures and suggesting the need for more expressive models or training strategies to capture the intricacies of quantum entanglement.

A UMAP projection reveals distinct trajectory landscapes corresponding to varying levels of infidelity with respect to the ground state of a ferromagnetic Ising model (N=5, h=0.5).

Towards Scalability: Expanding the Horizon of Simulation

Neural Quantum States (NQS) rely heavily on the chosen neural network architecture, with significant consequences for both the expressivity of the quantum state it can represent and the computational scalability of the approach. Restricted Boltzmann Machines (RBMs), for example, offer a probabilistic framework well-suited to capturing correlations within quantum many-body systems, but may struggle with representing highly entangled states efficiently. Conversely, feedforward neural networks, while potentially more expressive, can require exponentially increasing numbers of parameters to accurately represent the same quantum state, hindering their ability to scale to larger systems. Researchers are actively investigating hybrid architectures and novel network designs – including those inspired by tensor networks – to strike an optimal balance between representational power and computational feasibility, ultimately aiming to unlock the full potential of NQS for simulating complex quantum phenomena. The selection of an appropriate architecture is therefore paramount to advancing the field and tackling increasingly challenging problems in quantum physics and chemistry.

A significant hurdle in advancing neural quantum simulations (NQS) lies in effectively characterizing and enhancing the representation of quantum coherence within the neural network. Quantum coherence, the superposition of quantum states, is crucial for capturing the full potential of quantum systems, yet its accurate depiction using classical machine learning models proves difficult. Researchers are actively investigating information-theoretic measures, such as Shannon Entropy – a metric quantifying uncertainty or randomness – to assess how well these networks preserve coherence during computations. Improving this representation isn’t simply about achieving higher numerical accuracy; it’s about ensuring the neural network can genuinely capture the quantumness of the system being simulated, enabling the exploration of phenomena inaccessible to classical methods. Future investigations will likely focus on developing novel network architectures and training strategies specifically designed to maximize the preservation and faithful representation of $quantum$ coherence, ultimately unlocking the potential of NQS for tackling increasingly complex quantum challenges.

Advancing the field of Neural Quantum Simulation (NQS) necessitates a concentrated effort on refining optimization algorithms to overcome the computational bottlenecks inherent in simulating larger, more intricate quantum systems. Current methodologies often struggle with the exponentially growing Hilbert space, demanding substantial computational resources and time. Future research will therefore prioritize the development of algorithms that can efficiently navigate this complex landscape, potentially leveraging techniques like adaptive optimization or variational quantum eigensolvers to reduce computational cost. Simultaneously, investigations into the fundamental limits of NQS are crucial; determining the types of quantum systems and the level of accuracy achievable with neural networks will guide the development of hybrid quantum-classical approaches and illuminate the potential for simulating phenomena currently inaccessible to classical computers. This pursuit not only expands the scope of simulatable systems, but also establishes a clearer understanding of where NQS excels and where alternative methods remain superior.

The study meticulously demonstrates how seemingly innocuous transformations-basis rotations-can drastically alter optimization trajectories within the loss landscape of Neural Quantum States. This sensitivity to representation echoes a fundamental principle articulated by Paul Dirac: “I have not the slightest idea what sort of wave function a photon represents, but I know it must be a function of the coordinates.” Dirac’s statement, though concerning photons, reveals an appreciation for the profound impact of the chosen mathematical framework. Similarly, this work highlights that performance isn’t solely dictated by the underlying energy landscape, but critically by how that landscape is expressed, demonstrating that a change in perspective can obscure the path to an optimal solution.

The Road Ahead

The observation that optimization falters not from changes to the underlying problem, but from alterations in its presentation, is less a revelation than a restatement of fundamental principles. A system that needs a particular coordinate system to function correctly has already failed to achieve a truly general solution. The demonstrated sensitivity of Neural Quantum States to basis rotation highlights a persistent reliance on representational crutches. The entanglement entropy measures, while informative, remain descriptive; the focus must shift toward invariants – quantities demonstrably unaffected by such arbitrary transformations.

Future work should not concern itself with mitigating the effects of basis dependency, but with eliminating the dependency itself. The search for robust variational algorithms requires a move beyond state-vector parametrization, toward methods grounded in observable properties. To map the loss landscape is merely to chart the symptoms; the cure lies in constructing states inherently independent of representational choices.

Ultimately, the field must ask not how to optimize within a given basis, but how to transcend the need for a basis altogether. Clarity is courtesy, and a truly elegant solution will require none.

Original article: https://arxiv.org/pdf/2512.17893.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/