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arXiv:2605.00807 · Physical Review A

Probability distribution analysis of the cascaded variational quantum eigensolver

Yi-Hua Lai, John P. T. Stenger, Gloria Bazargan, Igor V. Schweigert, Daniel Gunlycke

quant-phSubmitted May 1, 2026• Announced May 4, 2026• 0 citations• Revised Oct 16, 2026 (v2)
v1 · May 1, 2026v2 · Oct 16, 2026
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Abstract

The cascaded variational quantum eigensolver (CVQE) circumvents the need for iterative communication between the quantum and classical processing units that is necessary in the conventional VQE algorithm. While CVQE offers complete freedom to choose the guiding state as input, not all guiding states provide both accurate solutions and efficient resource usage. Our work presents a process based on trapezoidal-state preparation for selecting guiding states that yield accurate many-electron ground-state solutions with minimal resource consumption. By analyzing state probability distributions at different stages of CVQE calculations for the collision complex mediating the $\mathrm{H}_2 + \mathrm{H}_2^+ \rightarrow \mathrm{H}_3^+ + \mathrm{H}$ bimolecular reaction, we identify stage-specific errors, trace their impact on final classical optimization, and determine the optimal guiding-state parameters for given resource constraints. Our findings uncover a critical trade-off on noisy intermediate-scale quantum (NISQ) devices, demonstrating that shallower diabatic circuits provide the best compromise among optimization efficiency, state quality, sampling cost, and signal-to-noise ratio in our system.

Journal reference
Phys. Rev. A 114, 042436 (2026)
DOI
10.1103/h17r-4ldq
Comments
15 pages, 11 figures

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The cascaded variational quantum eigensolver (CVQE) circumvents the need for iterative communication between the quantum and classical processing units that is necessary in the conventional VQE algorithm. While CVQE offers complete freedom to choose the guiding state as input, not all guiding states provide both accurate solutions and efficient resource usage. Our work presents a process based on trapezoidal-state preparation for selecting guiding states that yield accurate many-electron ground-state solutions with minimal resource consumption. By analyzing state probability distributions at different stages of CVQE calculations for the collision complex mediating the H2+H2+→H3++H\mathrm{H}_2 + \mathrm{H}_2^+ \rightarrow \mathrm{H}_3^+ + \mathrm{H}H2​+H2+​→H3+​+H bimolecular reaction, we identify stage-specific errors, trace their impact on final classical optimization, and determine the optimal guiding-state parameters for given resource constraints. Our findings uncover a critical trade-off on noisy intermediate-scale quantum (NISQ) devices, demonstrating that shallower diabatic circuits provide the best compromise among optimization efficiency, state quality, sampling cost, and signal-to-noise ratio in our system.
We propose a hybrid variational quantum algorithm that has variational parameters used by both the quantum circuit and the subsequent classical optimization. Similar to the Variational Quantum Eigensolver (VQE), this algorithm applies a parameterized unitary operator to the qubit register. We generate this operator using diabatic state preparation. The quantum measurement results then inform the classical optimization procedure used by the Cascaded Variational Quantum Eigensolver (CVQE). We demonstrate the algorithm on a system of interacting electrons and show how it can be used on long-term error-corrected as well as short-term intermediate-scale quantum computers. Our simulations performed on IBM Brisbane produced energies well within chemical accuracy.
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Abstract The variational quantum eigensolver (VQE) is one of the most promising algorithms to find eigenstates of a given Hamiltonian on noisy intermediate-scale quantum devices (NISQ). The practical realization is limited by the complexity of quantum circuits. Here we present an approach to reduce quantum circuit complexity in VQE for electronic structure calculations. Our ClusterVQE algorithm splits the initial qubit space into clusters which are further distributed on individual (shallower) quantum circuits. The clusters are obtained based on mutual information reflecting maximal entanglement between qubits, whereas inter-cluster correlation is taken into account via a new “dressed” Hamiltonian. ClusterVQE therefore allows exact simulation of the problem by using fewer qubits and shallower circuit depth at the cost of additional classical resources, making it a potential leader for quantum chemistry simulations on NISQ devices. Proof-of-principle demonstrations are presented for several molecular systems based on quantum simulators as well as IBM quantum devices.
A variational quantum eigensolver (VQE) optimizes parametrized eigenstates of a Hamiltonian on a quantum processor by updating parameters with a classical computer. Such a hybrid quantum-classical optimization serves as a practical way to leverage classical algorithms in exploiting the power of near-term quantum devices. Here, aiming to solve a group of Hamiltonians more efficiently, we develop an extension of the conventional VQE. A snake algorithm is incorporated to couple optimizing processes for VQEs of different Hamiltonians by gradient descent. Such a so-called collective VQE (CVQE) is applied to simulate molecules with varied bond lengths for demonstration. Numeral simulations show that the CVQE exhibits clear collective behavior in the optimization process of updating parameters. Remarkably, the CVQE tends to avoid a single VQE task to be trapped in the local minimum. The collective optimization utilizes intrinsic relations between related tasks and may inspire advanced hybrid quantum-classical algorithms for solving practical problems with current quantum technologies.
The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm designed for noisy intermediate-scale quantum (NISQ) hardware to estimate eigenvalues of many-body Hamiltonians. Unlike fully quantum approaches such as quantum phase estimation (QPE), VQE trades deep coherent circuits for repeated state preparation, measurement, and classical optimization, making it more compatible with limited qubit counts and finite coherence times. Recent developments have focused on reducing quantum resource requirements while retaining chemically meaningful wavefunction structure. In this paper, we examine recent progress in VQE methods for molecular electronic structure with an emphasis on three themes: (i) strategies for circuit and ansatz complexity reduction, including adaptive and selectively screened approaches, (ii) chemically motivated workflows that combine VQE with orbital optimization, fragmentation, and localized active-space ideas to better address strong correlation, and (iii) extensions of VQE to excited-state calculations. Throughout, we emphasize the tradeoffs among parameter count, gate depth, symmetry preservation, measurement overhead, and classical preprocessing, and discuss where these approaches may become most useful for chemically challenging active spaces. We also highlight benchmarking considerations for assessing both accuracy and resource requirements, and conclude with a perspective on regimes in which VQE may offer the greatest long-term value, particularly multireference active spaces and low-lying excited-state manifolds.
The variational quantum eigensolver (VQE) is one of the most promising algorithms for low-lying eigenstates calculation on Noisy Intermediate-Scale Quantum (NISQ) computers. Specifically, VQE has achieved great success for ground state calculations of a Hamiltonian. However, excited state calculations arising in quantum chemistry and condensed matter often requires solving more challenging problems than the ground state as these states are generally further away from a mean-field description, and involve less straightforward optimization to avoid the variational collapse to the ground state. Maintaining orthogonality between low-lying eigenstates is a key algorithmic hurdle. In this work, we analyze three VQE models that embed orthogonality constraints through specially designed cost functions, avoiding the need for external enforcement of orthogonality between states. Notably, these formulations possess the desirable property that any local minimum is also a global minimum, helping address optimization difficulties. We conduct rigorous landscape analyses of the models' stationary points and local minimizers, theoretically guaranteeing their favorable properties and providing analytical tools applicable to broader VQE methods. A comprehensive comparison between the three models is also provided, considering their quantum resource requirements and classical optimization complexity.