Variational quantum algorithms and geometry

Variational Quantum Algorithms (VQAs), such as the Variational Quantum Eigensolver (VQE) and the Quantum Approximate Optimization Algorithm (QAOA), are widely studied as candidates for near-term quantum advantage. Recent work has shown that training VQAs is NP-hard in general. In this paper, we present a conditional result suggesting that the training of VQAs is undecidable, even in idealized, noiseless settings. We reduce the decision version of the digitized VQA training problem-where circuit parameters are drawn from a discrete set-to the question of whether a universal Diophantine equation (UDE) has a root. This reduction relies on encoding the UDE into the structure of a variational quantum circuit via the matrix exponentials. The central step involves establishing a correspondence between the objective function of the VQA and a known UDE of 58 variables and degree 4. Our main result is conditional on a natural conjecture: that a certain system of structured complex polynomial equations-arising from the inner product of a VQA circuit output and a fixed observable-has at least one solution. We argue this conjecture is plausible based on dimension-counting arguments (degrees of freedom in the Hamiltonians, state vector, and observable), and the generic solvability of such systems in algebraic geometry over the complex numbers. Under this assumption, we suggest that deciding whether a digitized VQA achieves a given energy threshold is undecidable. This links the limitations of variational quantum algorithms to foundational questions in mathematics and logic, extending the known landscape of quantum computational hardness to include uncomputability. Additionally, we establish an unconditional undecidability result for VQA convergence in open quantum systems.Variational Quantum Algorithms (VQAs), such as the Variational Quantum Eigensolver (VQE) and the Quantum Approximate Optimization Algorithm (QAOA), are widely studied as candidates for near-term quantum advantage. Recent work has shown that training VQAs is NP-hard in general. In this paper, we present a conditional result suggesting that the training of VQAs is undecidable, even in idealized, noiseless settings. We reduce the decision version of the digitized VQA training problem-where circuit parameters are drawn from a discrete set-to the question of whether a universal Diophantine equation (UDE) has a root. This reduction relies on encoding the UDE into the structure of a variational quantum circuit via the matrix exponentials. The central step involves establishing a correspondence between the objective function of the VQA and a known UDE of 58 variables and degree 4. Our main result is conditional on a natural conjecture: that a certain system of structured complex polynomial equations-arising from the inner product of a VQA circuit output and a fixed observable-has at least one solution. We argue this conjecture is plausible based on dimension-counting arguments (degrees of freedom in the Hamiltonians, state vector, and observable), and the generic solvability of such systems in algebraic geometry over the complex numbers. Under this assumption, we suggest that deciding whether a digitized VQA achieves a given energy threshold is undecidable. This links the limitations of variational quantum algorithms to foundational questions in mathematics and logic, extending the known landscape of quantum computational hardness to include uncomputability. Additionally, we establish an unconditional undecidability result for VQA convergence in open quantum systems.

arXiv:2503.23723Mar 31, 20250 citations

Variational quantum algorithms

Marco Cerezo, Andrew Arrasmith, Ryan Babbush +8 more

arXiv:2012.09265Nature Reviews PhysicsAug 12, 20213,385 citations

Variational quantum algorithm for molecular geometry optimization

Alain Delgado, Juan Miguel Arrazola, Soran Jahangiri +4 more

Classical algorithms for predicting the equilibrium geometry of strongly correlated molecules require expensive wave function methods that become impractical already for few-atom systems. In this work, we introduce a variational quantum algorithm for finding the most stable structure of a molecule by explicitly considering the parametric dependence of the electronic Hamiltonian on the nuclear coordinates. The equilibrium geometry of the molecule is obtained by minimizing a more general cost function that depends on both the quantum circuit and the Hamiltonian parameters, which are simultaneously optimized at each step. The algorithm is applied to find the equilibrium geometries of the ${\mathrm{H}}_{2},$ ${\mathrm{H}}_{3}{}^{+},$ ${\mathrm{BeH}}_{2},$ and ${\mathrm{H}}_{2}\mathrm{O}$ molecules. The quantum circuits used to prepare the electronic ground state for each molecule were designed using an adaptive algorithm where excitation gates in the form of Givens rotations are selected according to the norm of their gradient. All quantum simulations are performed using the PennyLane library for quantum differentiable programming. The optimized geometrical parameters for the simulated molecules show an excellent agreement with their counterparts computed using classical quantum chemistry methods.Classical algorithms for predicting the equilibrium geometry of strongly correlated molecules require expensive wave function methods that become impractical already for few-atom systems. In this work, we introduce a variational quantum algorithm for finding the most stable structure of a molecule by explicitly considering the parametric dependence of the electronic Hamiltonian on the nuclear coordinates. The equilibrium geometry of the molecule is obtained by minimizing a more general cost function that depends on both the quantum circuit and the Hamiltonian parameters, which are simultaneously optimized at each step. The algorithm is applied to find the equilibrium geometries of the ${\mathrm{H}}_{2},$ ${\mathrm{H}}_{3}{}^{+},$ ${\mathrm{BeH}}_{2},$ and ${\mathrm{H}}_{2}\mathrm{O}$ molecules. The quantum circuits used to prepare the electronic ground state for each molecule were designed using an adaptive algorithm where excitation gates in the form of Givens rotations are selected according to the norm of their gradient. All quantum simulations are performed using the PennyLane library for quantum differentiable programming. The optimized geometrical parameters for the simulated molecules show an excellent agreement with their counterparts computed using classical quantum chemistry methods.

arXiv:2106.13840Physical Review ANov 2, 202140 citations

Variational quantum algorithms: fundamental concepts, applications and challenges

Han Qi, Sihui Xiao, Zhuo Liu +2 more

Quantum Information ProcessingJun 3, 202437 citations

Title: Chaos Theory in High-Dimensional Spacetime - Predictive Modeling

Jincheng Zhang

Chaos Theory, a branch of mathematics dealing with complex systems exhibiting unpredictable behavior, presents a significant challenge to traditional modeling and prediction. This research delves into the application of variational quantum algorithms (VQAs) to enhance our understanding and predictive capabilities within high-dimensional spacetime. The core claim is to develop a novel framework for analyzing chaos theory within this context, generating complex dynamics and identifying critical points with unprecedented accuracy. This work proposes a quantum-enhanced approach to explore the space of possible spacetime geometries, leveraging variational quantum algorithms to create robust predictive models. The investigation focuses on the unique challenges posed by high dimensionality, and aims to overcome these limitations through a fundamentally new algorithmic strategy. This research represents a significant advancement in the field, potentially unlocking new avenues for predicting complex system behavior and advancing our comprehension of chaotic dynamics.Chaos Theory, a branch of mathematics dealing with complex systems exhibiting unpredictable behavior, presents a significant challenge to traditional modeling and prediction. This research delves into the application of variational quantum algorithms (VQAs) to enhance our understanding and predictive capabilities within high-dimensional spacetime. The core claim is to develop a novel framework for analyzing chaos theory within this context, generating complex dynamics and identifying critical points with unprecedented accuracy. This work proposes a quantum-enhanced approach to explore the space of possible spacetime geometries, leveraging variational quantum algorithms to create robust predictive models. The investigation focuses on the unique challenges posed by high dimensionality, and aims to overcome these limitations through a fundamentally new algorithmic strategy. This research represents a significant advancement in the field, potentially unlocking new avenues for predicting complex system behavior and advancing our comprehension of chaotic dynamics.

Sep 4, 20260 citations

Variational Quantum Algorithms for A System of Linear Equations

D. (Dildora) Muhamediyeva

Dec 1, 20230 citations

{\mathrm{H}}_{2},

{\mathrm{H}}_{3}{}^{+},

{\mathrm{BeH}}_{2},

and

{\mathrm{H}}_{2}\mathrm{O}

molecules. The quantum circuits used to prepare the electronic ground state for each molecule were designed using an adaptive algorithm where excitation gates in the form of Givens rotations are selected according to the norm of their gradient. All quantum simulations are performed using the PennyLane library for quantum differentiable programming. The optimized geometrical parameters for the simulated molecules show an excellent agreement with their counterparts computed using classical quantum chemistry methods.