Hybrid Quantum-Classical Eigensolver

• Hybrid quantum-classical eigensolvers are frameworks that integrate quantum state manipulation with classical optimization to iteratively compute eigenvalues and eigenstates of many-body systems.
• They employ diverse methodologies such as probabilistic cooling, variational optimization, and subspace expansion to reduce circuit depth and mitigate errors on near-term quantum hardware.
• These eigensolvers have broad applications in quantum chemistry, strongly correlated materials, and quantum simulations, leveraging the strengths of both quantum and classical computing.

A hybrid quantum-classical eigensolver is a computational framework that integrates quantum and classical resources in the iterative preparation of eigenstates and estimation of eigenvalues for many-body Hamiltonians. This approach exploits the complementary capabilities of quantum devices (for nontrivial wavefunction manipulation, state projection, or measurement) and classical optimization or postprocessing (for parameter updates, subspace diagonalization, or error mitigation), thereby overcoming the limitations inherent to both classical algorithms and near-term quantum hardware. Hybrid eigensolver protocols encompass a spectrum of methodologies, including probabilistic cooling, quantum subspace diagonalization, Hamiltonian and wavefunction co-optimization, and measurement-driven subspace reduction, with demonstrated applications ranging from ground and excited state preparation in trapped ions to simulations of correlated electron materials and quantum chemistry.

1. Algorithmic Principles and Types

The core architectural principle of hybrid quantum-classical eigensolvers is the partitioning of quantum and classical subtasks within an iterative workflow. Quantum resources are utilized either to (i) perform nonunitary projections through measurement (as in probabilistic eigenstate cooling), (ii) prepare and measure quantum states encoded by parameterized circuits (variational algorithms), or (iii) implement low-depth circuits to efficiently evaluate matrix elements or overlaps in subspaces of interest. The classical component may undertake variational parameter optimization, subspace selection via sampling, diagonalization of reduced effective Hamiltonians, or the orchestration of feedback loops.

The following typology summarizes several principal algorithmic families:

Method Family	Quantum Task	Classical Task
Probabilistic Eigensolving	Unitary evolution + projective measurement	Parameter (τ) optimization
VQE / Variational	State preparation, energy measurement	Variational optimization, updating θ
Subspace / Krylov Expansion	Overlap measurement, moment state prep	QCQP or generalized eigenproblem
Quantum Sampling Selection	State sampling in computational basis	Diagonalization, subspace selection
Tensor-Network Bridged	Local circuit sampling, symmetry projection	MPS contraction, symmetry enforcement

2. Probabilistic Quantum-Classical Protocols

Probabilistic eigensolving methods utilize an ancillary system and projective measurement to drive the quantum state toward the eigenstate of interest. In the trapped-ion probabilistic eigensolver protocol (Zhang et al., 2018), the central unitary is

$$W_{(\gamma)}(\tau) = \exp\left( -i (H_s + \gamma) \sigma_a^x \tau \right)$$

where $H_s$ is the system Hamiltonian, $\gamma$ shifts the spectrum, and $\tau$ is an evolution parameter. Post-selection on the ancilla in state $|0\rangle$ enacts a nonunitary operator $C_{(\gamma)}(\tau) = \cos[(E_j + \gamma)\tau]$, leading to selective amplitude amplification of low-energy eigenstates. The protocol is iterative; repeated successively, the state is probabilistically cooled towards the ground state. A fixed-step or a variational approach (the latter incorporating classical optimization of $\tau$ to minimize the system energy) may be employed.

Excited states can be accessed by applying an additional pre-evolution operator that cancels overlap with lower-lying states, such as $U_s = \exp[-i \frac{\pi}{2E_s} H_s \sigma_a^x]$. The overall process is especially suited to analog-digital trapped-ion platforms, using spin–spin and spin–boson couplings (e.g. Mølmer–Sørensen gates and sideband transitions), single-qubit rotations, and Trotterized decomposition for non-commuting Hamiltonians.

Key expressions:

• Conditional state update: $$|c_j^{(k)}|^2 = |c_j^{(k-1)}|^2 \cos^2[(E_j+\gamma)\tau] / P_k^0$$.
• Energy monotonicity (post-selecting $|0\rangle_a$): $$\langle H_s \rangle^{(0)}_{k} \leq \langle H_s \rangle^{(0)}_{k-1} < \langle H_s \rangle^{(1)}_{k}$$.

This class of protocols offers depth reduction and flexibility but inherits probabilistic run time and variable success rate depending on initial state overlap.

3. Variational and Subspace Expansion Strategies

Variational quantum eigensolvers (VQE) employ a quantum circuit parameterized by variables $\theta$, preparing trial states $|\psi(\theta)\rangle = U(\theta)|\Psi_0\rangle$ for the minimization of $$E(\theta) = \langle \psi(\theta)| H |\psi(\theta)\rangle$$ (Willsch et al., 2022, Rasmussen et al., 2022, Sobhani et al., 2 Oct 2024). The classical optimizer steers the parameters via iterative feedback based on measured expectation values, exploiting the variational principle as an energy upper bound. Variants such as VAQC (Harwood et al., 2021) traverse a family of interpolative Hamiltonians $H(t)$, bootstrapping circuit parameters along a homotopy to avoid local minima.

Hybrid eigensolvers without direct variational feedback include Krylov- or subspace-based methods. These construct an Ansatz as a linear combination of basis states formed by sequential action of Hamiltonian terms or “moments” on an easily prepared initial state (Bharti et al., 2020):

$$|\xi(\alpha)\rangle^{(K)} = \alpha_0 |\psi\rangle + \alpha_1 H |\psi\rangle + \dots + \alpha_K H^K|\psi\rangle$$

Quantum hardware is used to efficiently estimate all required overlaps, and ground-state finding is recast as a classical generalized eigenproblem or a quadratically constrained quadratic program (QCQP). This paradigm eliminates quantum–classical parameter loops and is robust against barren plateaus.

Quantum-selected configuration interaction (QSCI) algorithms (Kanno et al., 2023) and projective eigensolvers (Stair et al., 2021) use hybrid strategies where quantum sampling (in the computational basis) identifies important determinants; classical exact diagonalization within this subspace yields variational energies for both ground and excited states while maintaining robustness against quantum noise.

4. Measurement, Sampling, and Optimization Techniques

Hybrid eigensolvers leverage several measurement and sampling strategies:

• Ancilla-Assisted Overlap Measurement: Circuits prepare interference between basis states, enabling efficient evaluation of diagonal and off-diagonal matrix elements with a single ancilla (Jouzdani et al., 2020). For example, a controlled preparation of $|0\rangle|n'\rangle + |1\rangle|n\rangle$ followed by measurement discriminates real or imaginary parts of $\langle n'| H | n \rangle$ via ancilla observables.
• Symmetric Subspace Projection and Real-Space Sampling: Partitioning the system into blocks, local quantum circuits process configurations which are then “bridged” by tensor networks (MPS) to restore global entanglement (Xu et al., 22 Oct 2025). Symmetry mappings consolidate equivalent real-space configurations, substantially compressing the Hilbert space and enabling efficient Monte Carlo-like optimization.
• Tensor-Network Bridged Circuits: Wavefunctions are constructed from linear combinations of matrix product states (MPS) in multiple rotated orbital bases, with quantum circuits evaluating classically intractable off-diagonal matrix elements. Gradient-free sweep algorithms optimize expansion coefficients by quantum subspace diagonalization, achieving chemical accuracy with linear-depth circuits and high resilience to shot noise (Leimkuhler et al., 16 Apr 2024).
• Generative and Transfer-Learning Models: Generative Quantum Eigensolver (GQE) frameworks utilize transformer-based deep generative models to sequentially generate quantum operator strings (often as SMILES-inspired text), learning transferable operator representations between molecular systems to accelerate configuration search and circuit construction (Yin et al., 24 Sep 2025, Minami et al., 28 Jan 2025).

Classical optimization across these methods ranges from standard gradient descent (mutual gradient descent in Hamiltonian–wavefunction space (Yuan et al., 2020)), quasi-Newton residual updates (projective eigensolvers (Stair et al., 2021)), to convex relaxations and semidefinite programming in overlap-based techniques (Bharti, 2020).

5. Applications to Quantum Chemistry, Strongly Correlated Systems, and Quantum Simulation

Hybrid quantum-classical eigensolvers demonstrate versatility across domains:

• Quantum Chemistry: Application to molecular ground-state and excited-state energy estimation (e.g., LiH, BeH₂, H₂O, ammonia, N₂) (Kanno et al., 2023, Pellow-Jarman et al., 8 Mar 2025), geometry optimization using combined wavefunction and Hamiltonian space gradients (Yuan et al., 2020), and transfer learning of quantum operators between molecules (Yin et al., 24 Sep 2025, Minami et al., 28 Jan 2025).
• Strongly Correlated Materials: Gutzwiller embedding frameworks decompose the infinite lattice Hubbard- or Anderson-type Hamiltonians into impurity (embedding) problems tractable by VQE or UCC ansätze (Yao et al., 2020), yielding phase diagrams for heavy fermion and Mott states.
• Spin Models and Lattice Systems: Frustrated Heisenberg models in up to 2D torus geometries are addressed using hybrid tensor-network-bridged quantum circuits with symmetry enhancement and bond-dimension extrapolation to achieve energy errors as low as $10^{-5}$ (Xu et al., 22 Oct 2025).
• Electron–Phonon Systems: Hybrid approaches that combine VQE for electronic states with a variational non-Gaussian solver for lattice vibrations enable the paper of coupled charge density wave and antiferromagnetic phases without expanding the quantum register (Denner et al., 2023).
• Quantum Optimization, Factorization, and Control: Generation of custom eigensolver circuits via generative models for combinatorial optimization (e.g., Ising Hamiltonians) (Minami et al., 28 Jan 2025); prime factorization mapped as a ground state search within the VQE framework (Sobhani et al., 2 Oct 2024).

6. Advantages, Resource Scaling, and Current Limitations

Hybrid eigensolvers are designed to align with current and near-term quantum hardware capabilities, offering several operational advantages:

• Reduced Circuit Depth and Qubit Overhead: Many approaches rely on shallow circuits, short-depth operator compositions, or restricted subspaces, focusing quantum resources on evaluation (not parameter feedback), thus maximizing error resilience (Jouzdani et al., 2020, Leimkuhler et al., 16 Apr 2024).
• Flexible Integration with Classical Postprocessing: Tasks such as subspace diagonalization, mutual gradient updates, or tensor network contraction are well-suited for classical computation and allow the hybrid protocol to adapt as classical and quantum hardware evolves (Stenger et al., 2021, Xu et al., 22 Oct 2025).
• Natural Error Robustness and Mitigation: Quantum sampling for subspace selection (QSCI) and symmetry-projected measurement frameworks inherently provide robustness to quantum noise, as all final eigenvalue estimation and observable evaluation are performed classically within the optimized subspace (Kanno et al., 2023, Xu et al., 22 Oct 2025).
• Practical Scalability: For many-body correlated systems, the effective reduction of configuration space (e.g., using only a small fraction of the full Slater determinant basis in HiVQE (Pellow-Jarman et al., 8 Mar 2025)) and tensor-network compression enable simulations beyond the regime accessible by classical diagonalization or tensor contraction.

Key limitations include (i) probabilistic convergence and protocol “restart” in measurement-driven methods; (ii) the potential for classical feedback (barren plateaus or local minima in VQE-type optimization); (iii) overheads from noise in near-term quantum measurement; and (iv) challenges in mapping classical redundancy and symmetry into efficient quantum subspaces for large lattice or molecular systems.

7. Emerging Trends and Integration with Multi-Hybrid Architectures

Recent research extends hybrid eigensolver workflows by integrating multiple quantum and classical resources. For example, a triple-hybrid protocol incorporates a quantum annealer to optimize measurement grouping (via graph coloring of commuting Pauli terms expressed as a QUBO problem), a classical optimizer, and a gate-based quantum processor to minimize VQE circuit executions (Jattana, 16 Jul 2024). This modularization further reduces resource requirements, efficiently partitions pre-processing, and suggests a generalizable multi-hybrid paradigm. At the same time, transfer learning, generative modeling of circuit structures, and symmetric compression point to a growing intersection between quantum physics, machine learning, and graph-theoretic computation.

Hybrid quantum-classical eigensolvers thus provide a unifying theoretical and practical framework with the flexibility to adapt quantum resources, classical optimization, and domain symmetries for scalable, accurate simulation of quantum many-body systems across chemistry, materials science, and optimization.