Variational Quantum Eigensolver Framework

• The VQE framework is a hybrid quantum–classical algorithm that approximates ground and excited state energies by optimizing parametrized quantum circuits.
• Its methodology involves mapping fermionic Hamiltonians to qubits and measuring weighted Pauli operators to efficiently minimize energy on NISQ devices.
• Practical applications include quantum chemistry and materials science, with advances in excited state simulation and qubit-efficient circuit designs.

The Variational Quantum Eigensolver (VQE) framework comprises a class of hybrid quantum–classical algorithms designed to approximate ground and excited state energies of quantum systems defined by a Hamiltonian $H$. VQE and its extensions are fundamentally important approaches for harnessing near-term noisy quantum hardware for quantum chemistry, materials science, and many-body physics. By encoding parametrized quantum states as output of circuits (ansätze) and minimizing their energy expectation values through classical optimization of parameters, VQE circumvents the exponential bottlenecks of classical algorithms while remaining implementable on shallow quantum devices.

1. Fundamental Framework and Theoretical Principles

The VQE framework is based on the quantum variational principle

$$E(\boldsymbol{\theta}) = \langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle \geq E_0,$$

where $|\psi(\boldsymbol{\theta})\rangle$ is a normalized variational state, controlled by a vector of real parameters $\boldsymbol{\theta}$, and $E_0$ is the ground state energy of $H$. The trial state is prepared on a quantum computer by a parametrized circuit $U(\boldsymbol{\theta})$, typically acting on an initial computational basis state. The standard VQE workflow involves:

• Mapping a fermionic many-body Hamiltonian to a qubit form, e.g., via the Jordan–Wigner or Bravyi–Kitaev transformation; for instance, Jordan–Wigner maps

$$\hat{a}_j \rightarrow Z_0 \otimes \dots \otimes Z_{j-1} \otimes \frac{X_j + iY_j}{2}$$

• Decomposing $H$ as a weighted sum of Pauli operators, $H = \sum_i c_i P_i$.
• Measuring each term’s expectation value on quantum hardware and reconstructing $E(\boldsymbol{\theta})$ as

$$E(\boldsymbol{\theta}) = \sum_i c_i \langle\psi(\boldsymbol{\theta})|P_i|\psi(\boldsymbol{\theta})\rangle.$$

• Running a classical optimizer to adjust $\boldsymbol{\theta}$ in order to minimize $E(\boldsymbol{\theta})$.

This quantum–classical feedback loop is efficient in circuit depth and resilient to quantum noise, making VQE particularly suitable for NISQ devices (Fonseca et al., 7 May 2025).

2. Algorithmic Advancements: Excited States, Symmetries, and Efficient Representations

Subspace and Symmetry-based Extensions

While standard VQE targets the ground state, obtaining excited states is essential for simulating chemical reactions, spectroscopy, and material properties. The Subspace-search VQE (SSVQE) generalizes the framework to excited states by variationally mapping $k+1$ mutually orthogonal input states $\{|\phi_j\rangle\}_{j=0}^k$ through a common ansatz $U(\theta)$, ensuring orthogonality of outputs via unitarity (Nakanishi et al., 2018). The k-th excited state is identified as the highest-energy state within the low-energy subspace found by minimizing

$$L_1(\theta) = \sum_{j=0}^k \langle\phi_j|U^\dagger(\theta)H U(\theta)|\phi_j\rangle,$$

and then maximizing within the subspace,

$$L_2(\phi) = \langle\phi_s|V^\dagger(\phi)U^\dagger(\theta^*) H U(\theta^*)V(\phi)|\phi_s\rangle,$$

with $s=k$. Weighted versions using a single optimization enable simultaneous extraction of all states up to $k$.

Symmetry-adapted VQE (Seki et al., 2019, Mondal et al., 2023) enforces irreducible representations and spin multiplicity directly by constructing reference states and ansätze that respect point group and SU(2) symmetries. Non-unitary Hermitian projector operators select components with targeted symmetry quantum numbers, handled in classical postprocessing.

Qubit Efficiency and Tensor Network Variational Forms

Qubit-efficient VQE strategies leverage tensor network representations to reduce circuit width. For instance, a sequential circuit can construct a matrix product state (MPS) or projected entangled pair state (PEPS) by reusing a smaller number of qubits, sweeping through a lattice by measurement and reset (Liu et al., 2019). For a Q-MPS of bond dimension $D=2^V$ (with $V$ virtual qubits), the ansatz incorporates both local and long-range entanglement with polynomial resource scaling for practical system sizes.

3. Optimization Schemes and Resource Considerations

Optimization in VQE encompasses both parameter tuning and circuit structure. Classical optimizers such as BFGS, Adam, or gradient-free routines adapt joint quantum-classical workflows. Innovations include:

• Quasi-dynamical recursive evolution (Jattana et al., 2022): sequentially stacking parameterized unitaries, each initialized on the previously optimized state, avoids local minima and barren plateaus, supporting energy convergence across larger system sizes.
• Multiobjective and evolutionary strategies (Chivilikhin et al., 2020, Li et al., 2022): circuit topology, gate count, and gradient norm are optimized via Pareto fronts (e.g., NSGA-II) and genetic algorithms. Covariance-matrix adaptation evolution strategy (CMA-ES) complements topology optimization with robust parameter searches.
• Gradient-sensitive ansatz selection ensures trainable circuits; optimization avoids circuits with vanishing normalized gradients, mitigating the barren plateau problem (Li et al., 2022).

These advances aim to balance expressibility, trainability, and NISQ resource constraints (number of qubits, circuit depth, two-qubit gate count).

4. Practical Applications: Quantum Chemistry and Beyond

VQE’s primary application lies in quantum chemistry, where electronic Hamiltonians are mapped to qubit forms and solved for ground and excited state energies. The unitary coupled-cluster (UCC) and adaptive ansatzes (e.g., ADAPT-VQE) have been benchmarked for small molecules and larger strongly correlated systems (Hu et al., 2022). For molecular benchmarks (e.g., H$_2$, H$_4$, N$_2$), innovations such as orbital expansion VQE (OE-VQE) (Wu et al., 2022) implement systematic, chemist-guided active space expansions, improving convergence in strongly correlated regimes with shallow circuits.

Further, contextual subspace VQE (Kirby et al., 2020) splits the Hamiltonian into efficiently simulable and quantum-corrected sectors, reducing required qubits and measurements by an order of magnitude, and is particularly useful for electronic structure calculations on NISQ hardware.

Extensions to open quantum systems are realized in frameworks such as the Variational Open Quantum Eigensolver (VOQE), which operates in a doubled Hilbert space with a "Hermitian-preserving ansatz" and post-selection measurements to obtain steady states described by Lindblad or non-Hermitian equations (Shang, 6 Mar 2024).

Resource-aware optimization (e.g., CVQE (Gunlycke et al., 2023)) achieves high throughput by decoupling quantum sampling from classical optimization, enabling extensive reuse of measurement data and flexible ansatz design.

5. Performance, Scaling, and Measurement Complexity

The circuit width (number of qubits) is dictated by the size of the mapped spin–orbital basis; typically, $N_f = 2N_e$ for $N_e$ electrons. The circuit depth and runtime scale with both the number of terms in the ansatz and the operator composition in the mapped Hamiltonian. For UCC (see (Xu et al., 2023)), the naïve scaling for applying high-rank excitation terms is exponential, but a linear combination of unitaries (LCU) approach exploiting hidden SU(2) symmetry compresses complexity to cubic with excitation rank ($O(n^3)$ for rank $n$).

The measurement overhead is governed by the number of Pauli strings and the target energy precision $\epsilon$: typically, $O(N^4/\epsilon^2)$ for chemical systems. For practical quantum devices, reducing circuit depth (e.g., by symmetry, tensor structure, or adaptive ansatz growth) and optimizing measurement grouping are critical. Benchmark studies (Hu et al., 2022, Fonseca et al., 7 May 2025) show that adaptive and chemically motivated ansätze reach higher accuracy for small systems, but challenges persist in attaining chemical precision for stretched or strongly correlated molecules.

6. Limitations, Challenges, and Ongoing Research

Several major challenges remain:

• Barren plateaus: Regions of exponentially vanishing gradients impede parameter optimization in deep or highly entangling circuits. Symmetry-aware circuit construction (Liu et al., 2019), gradient-sensitive ansatz selection (Li et al., 2022), and shallow adaptive circuit design are promising countermeasures.
• Fermion-to-qubit mappings and locality: The Jordan–Wigner transformation introduces nonlocal Pauli strings, exacerbating optimization problems for larger $n$. Bravyi–Kitaev mappings or problem-inspired transformations improve locality (Uvarov et al., 2020).
• Measurement complexity: Strategies for operator grouping, shot allocation (e.g., knowledge distillation inspired annealing (Li, 6 May 2025)), and post-processing (e.g., symmetry projection (Seki et al., 2019)) are under active development.
• Integration of environmental effects: The VQE–self-consistent field method in a polarizable embedding setting (PE–VQE–SCF) (Kjellgren et al., 2023) enables inclusion of solvent and environment response at only modest additional quantum resource cost, facilitating realistic chemical modeling.

7. Implications, Benchmarking, and Outlook

The VQE framework and its derivatives have demonstrated capabilities for simulating quantum chemistry, correlated materials, and open quantum dynamics on near-term devices. Systematic benchmarking across a range of ansätze and molecules reveals that no single approach is universally optimal, and trade-offs between accuracy, circuit depth, and convergence cost are domain dependent (Hu et al., 2022). Chemically motivated adaptive strategies (e.g., orbital expansion) outperform generic hardware efficient circuits in strongly correlated problems.

Research continues into ansatz expressibility, symmetry exploitation, noise resilience, and measurement economy, along with embedding frameworks (quantum/classical hybrid environments) and open-system generalizations. Ongoing algorithmic advances promise to reduce the critical bottlenecks—both in quantum resources and classical optimization—necessary for practical quantum advantage in computational chemistry, condensed matter, and beyond.