We present a deterministic investigation of Variational Quantum Monte Carlo optimization over discrete, exponentially large state spaces. By evaluating variational objectives and analytic gradients exactly through explicit summation, statistical sampling effects are avoided, enabling controlled study of smooth but highly nonconvex optimization landscapes. The framework is instantiated on small structured quantum systems with known exact optima, allowing direct validation of gradient implementations and optimization behavior. Using variational parameterization, gradient-based methods converge to the exact solution for systems of up to eight qubits, with energy deviations on the order of 10− 5 for small systems and 10− 9 for larger ones. The resulting setup provides a compact, stochasticity-free reference for investigating variational objectives and optimization dynamics in discrete high-dimensional settings.
We present a deterministic investigation of Variational Quantum Monte Carlo optimization over discrete, exponentially large state spaces. By evaluating variational objectives and analytic gradients exactly through explicit summation, statistical sampling effects are avoided, enabling controlled study of smooth but highly nonconvex optimization landscapes. The framework is instantiated on small structured quantum systems with known exact optima, allowing direct validation of gradient implementations and optimization behavior. Using variational parameterization, gradient-based methods converge to the exact solution for systems of up to eight qubits, with energy deviations on the order of 10− 5 for small systems and 10− 9 for larger ones. The resulting setup provides a compact, stochasticity-free reference for investigating variational objectives and optimization dynamics in discrete high-dimensional settings.
Abstract Variational quantum algorithms have the potential for significant impact on high-dimensional optimization, with applications in classical combinatorics, quantum chemistry, and condensed matter. Nevertheless, the optimization landscape of these algorithms is generally nonconvex, leading the algorithms to converge to local, rather than global, minima and the production of suboptimal solutions. In this work, we introduce a variational quantum algorithm that couples classical Markov chain Monte Carlo techniques with variational quantum algorithms, allowing the former to provably converge to global minima and thus assure solution quality. Due to the generality of our approach, it is suitable for a myriad of quantum minimization problems, including optimization and quantum state preparation. Specifically, we devise a Metropolis–Hastings method that is suitable for variational quantum devices and use it, in conjunction with quantum optimization, to construct quantum ensembles that converge to Gibbs states. These performance guarantees are derived from the ergodicity of our algorithm’s state space and enable us to place analytic bounds on its time-complexity. We demonstrate both the effectiveness of our technique and the validity of our analysis through quantum circuit simulations for MaxCut instances, solving these problems deterministically and with perfect accuracy, as well as large-scale quantum Ising and transverse field spin models of up to 50 qubits. Our technique stands to broadly enrich the field of variational quantum algorithms, improving and guaranteeing the performance of these promising, yet often heuristic, methods.
We present a technique for optimizing hundreds of thousands of variational parameters in variational quantum Monte Carlo. By introducing iterative Krylov subspace solvers and by multiplying by the Hamiltonian and overlap matrices as they are sampled, we remove the need to construct and store these matrices and thus bypass the most expensive steps of the stochastic reconfiguration and linear method optimization techniques. We demonstrate the effectiveness of this approach by using stochastic reconfiguration to optimize a correlator product state wave function with a Pfaffian reference for four example systems. In two examples on the two dimensional Fermionic Hubbard model, we study 16 and 64 site lattices, recovering energies accurate to 1 in the smaller lattice and predicting particle-hole phase separation in the larger. In two examples involving an ab initio Hamiltonian, we investigate the potential energy curve of a symmetrically dissociated $4\ifmmode\times\else\texttimes\fi{}4$ hydrogen lattice as well as the singlet-triplet gap in free base porphin. In the hydrogen system we recover 98 or more of the correlation energy at all geometries, while for porphin we compute the gap in a 24 orbital active space to within 0.02 eV of the exact result. The number of variational parameters in these examples ranges from $4\ifmmode\times\else\texttimes\fi{}{10}^{3}$ to $5\ifmmode\times\else\texttimes\fi{}{10}^{5}$.
The variational quantum Monte Carlo (VQMC) method received significant attention in the recent past because of its ability to overcome the curse of dimensionality inherent in many-body quantum systems. Close parallels exist between VQMC and the emerging hybrid quantum-classical computational paradigm of variational quantum algorithms. VQMC overcomes the curse of dimensionality by performing alternating steps of Monte Carlo sampling from a parametrized quantum state followed by gradient-based optimization. While VQMC has been applied to solve high-dimensional problems, it is known to be difficult to parallelize, primarily owing to the Markov Chain Monte Carlo (MCMC) sampling step. In this work, we explore the scalability of VQMC when autoregressive models, with exact sampling, are used in place of MCMC. This approach can exploit distributed-memory, shared-memory and/or GPU parallelism in the sampling task without any bottlenecks. In particular, we demonstrate the GPU-scalability of VQMC for solving up to ten-thousand dimensional combinatorial optimization problems.
The variational quantum Monte Carlo (VQMC) method received significant attention in the recent past because of its ability to overcome the curse of dimensionality inherent in many-body quantum systems. Close parallels exist between VQMC and the emerging hybrid quantum-classical computational paradigm of variational quantum algorithms. VQMC overcomes the curse of dimensionality by performing alternating steps of Monte Carlo sampling from a parametrized quantum state followed by gradient-based optimization. While VQMC has been applied to solve high-dimensional problems, it is known to be difficult to parallelize, primarily owing to the Markov Chain Monte Carlo (MCMC) sampling step. In this work, we explore the scalability of VQMC when autoregressive models, with exact sampling, are used in place of MCMC. This approach can exploit distributed-memory, shared-memory and/or GPU parallelism in the sampling task without any bottlenecks. In particular, we demonstrate GPU-scalability of VQMC for solving up to ten-thousand dimensional combinatorial optimization problems.
TitleVariational Monte Carlo for Ab Initio Electronic Structure with Neural Quantum States and Selective State-Space Models Overview This repository provides a research implementation of Variational Monte Carlo (VMC) for ab initio electronic-structure calculations using Neural Quantum States (NQS). The work focuses on stable natural-gradient optimization, physically motivated wavefunction constraints, and the exploration of alternative architectural inductive biases for modeling electronic correlation. The project is released as an exploratory research artifact. All results, limitations, and open questions are stated explicitly. The work is motivated by a central question in modern neural quantum chemistry:can alternative sequence-based architectures reduce the quadratic scaling bottleneck of electron–electron interaction modeling while preserving physical fidelity? Scientific Scope The framework targets: Atomic systems (H through Ne) Small molecules (e.g., H₂, LiH, H₂O) Exploratory extensions, including: Periodic boundary conditions Excited-state considerations Berry-phase estimation Entanglement diagnostics Time-dependent VMC via McLachlan’s variational principle At present, numerical validation is limited to a single-electron benchmark (Hydrogen) and is explicitly presented as a sanity check of the physics engine and optimization loop, not as evidence of improved expressivity for correlated fermions. Core Methodology The implementation follows the standard Variational Monte Carlo formalism, in which a parameterized wavefunctionψ is optimized by minimizing the energy expectation value Optimization is performed using Stochastic Reconfiguration (SR), a natural-gradient method that accounts for the geometry of the variational manifold and significantly improves stability compared to first-order optimizers. Key numerical features include: Log-domain arithmetic for wavefunction evaluation Stable determinant computation using slogdet Hutchinson trace estimation for Laplacian evaluation Explicit enforcement of Kato cusp conditions via analytic Jastrow factors Multi-determinant antisymmetric ansatz to satisfy fermionic exchange symmetry SSM-Backflow Hypothesis The primary research contribution explored in this repository is the SSM-Backflow hypothesis. Most contemporary NQS models (e.g., FermiNet, PauliNet) model electron–electron interactions using explicit quadratic aggregation, leading to O(Ne2)O(N_e^2)O(Ne2) computational complexity. This work investigates an alternative inductive bias: For each electron, neighboring electrons are sorted by inter-electron distance The resulting sequence is processed using a selective state-space model (SSM) inspired by Mamba architectures Correlation information is accumulated sequentially through a learned recurrence Aggregated messages are used to parameterize backflow transformations or orbital functions This approach yields a theoretical complexity of O(NelogNe)O(N_e \log N_e)O(NelogNe) per layer (dominated by sorting) and is motivated by the physical observation that electronic correlation typically decays approximately exponentially with distance. Importantly, this is treated strictly as a hypothesis. The manuscript explicitly notes that: SSMs are not permutation-invariant by construction Sequential ordering may introduce inductive biases misaligned with fermionic symmetry Empirical validation on multi-electron systems is required before drawing conclusions Results and Current Validation Status At present, the repository reports results for atomic hydrogen (Ne = 1): Exact energy: −0.500000 Ha Achieved energy: −0.5001 Ha Error: 0.12 mHa (well below chemical accuracy) Local-energy variance: 0.0454 Optimizer: Stochastic Reconfiguration Convergence: ~100 optimization steps Because hydrogen has no electron–electron correlation, this result is interpreted solely as a sanity check of: Hamiltonian implementation Local-energy estimator Natural-gradient optimization loop Numerical stability mechanisms The manuscript clearly states that multi-electron benchmarks (e.g., He, Li, or H₂) are required to meaningfully evaluate the SSM-Backflow hypothesis and are planned for future versions. Stability and Diagnostics The implementation includes multiple practical safeguards designed to stabilize VMC optimization: Dynamic energy-based divergence detection Variance-based walker reset mechanisms Controlled respawning of walkers near nuclear centers In addition, an exploratory commutator-based diagnostic (informally termed the “Noether Inverse”) is included as a monitoring signal during training. This quantity is explicitly not claimed to represent a new conserved symmetry and is treated purely as an empirical observation requiring further theoretical and numerical investigation. Intended Use and Limitations This repository is released as: A research implementation A transparent record of an ongoing investigation A foundation for future validation and benchmarking It is not presented as a production-ready electronic-structure solver, nor as evidence of state-of-the-art performance. All limitations are explicitly documented in the manuscript. Licensing and Reuse The work is shared to encourage: Reproducibility Methodological discussion Extension and validation by the community If you use or build upon this work, please cite the Zenodo DOI associated with this release.
Quantum Averages and Statistical Samplings In this chapter, we discuss the general framework in which the variational Monte Carlo methods are defined and few important implementations for interacting systems of bosons and fermions on the lattice. The main advantage of considering this approach relies on the variational principle that has been shown in section 1.4: the energy of a given quantum state is always bounded from below by the exact ground-state one, giving us the route to obtain the best possible solution to the problem. In most cases, a suitable parametrization of the variational wave function allows us to consider a wide range of different quantum phases (e.g., metals, superconductors, and insulators). By performing an optimization of the parameters, we can reach the lowest-energy state, which is expected to capture the correct ground-state behavior. Therefore, variational wave functions represent a flexible and valuable approach to get important insights into the low-energy properties of models that cannot be solved by exact methods. By contrast, the main limitation of this approach is the fact that it is based on a given Ansatz , which may contain a relevant bias that cannot be removed within the chosen parametrization. We also mention that variational wave functions can be easily defined and treated for a wide class of models, irrespective of the range of interactions and the dimension of the local (i.e., single-site) Hilbert space, which can be even infinite and does not need the use of an uncontrolled cutoff to work with a finite-dimensional space. In this respect, variational Monte Carlo is better than other methods, like densitymatrix renormalization group or tensor-network methods in which the complexity dramatically increases with both the range of the interaction and the dimension of the local Hilbert space (White, 1992; Schollwöck, 2005, 2011). Let us start by describing the general framework in which variational Monte Carlo methods are defined. First of all, we fix a complete basis set in the Hilbert space, in which (for simplicity) the states are taken to be orthogonal and normalized such that: Then, any quantum state can be written as: In turn, the expectation value of an operator over a given variational wave function takes the following form: The main problem in evaluating the expectation value is that the number of configurations in the sum is exponentially large with the number of particles.