Cascaded Variational Quantum Eigensolver (CVQE)

• CVQE is a hybrid algorithm that approximates ground-state energies by cascading classical optimization on fixed quantum measurement data.
• It decouples quantum circuit execution from parameter tuning, drastically reducing quantum resource requirements while enabling non-unitary operator incorporation.
• CVQE demonstrates scalability and chemical accuracy in simulations of strongly correlated molecules and lattice models on current NISQ devices.

The Cascaded Variational Quantum Eigensolver (CVQE) is a class of hybrid quantum-classical algorithms designed to efficiently approximate ground-state energies of quantum many-body Hamiltonians on noisy intermediate-scale quantum (NISQ) hardware. CVQE methods generalize the Variational Quantum Eigensolver (VQE) by decoupling quantum circuit execution from the classical parameter optimization loop, exploiting measurement re-use and subspace diagonalization to greatly reduce quantum resource requirements. Several instantiations exist, unified by the principle of "cascading" classical post-processing atop fixed quantum measurement data, and often allow extension to non-unitary or adaptive ansatz forms while maintaining hardware efficiency (Gunlycke et al., 2023, Stenger et al., 2023, Stenger et al., 4 Dec 2025).

1. Fundamental Algorithmic Design

CVQE separates the variational parameter search from quantum circuit executions. A typical CVQE workflow comprises the following staged protocol:

1. Preparation of a guiding state $\ket{\Psi_0}$ by a fixed quantum circuit (often parameterized only for initialization) on the quantum processor.
2. Quantum measurement phase: For each Hamiltonian Pauli term, apply a pre-determined basis change circuit and measure bitstrings, obtaining empirical probability distributions associated with each basis.
3. Classical cascade: All subsequent parameter optimization—including implementation of Hamiltonian symmetries, imposition of non-unitary or correlator operations, and calculation of Rayleigh quotients—is performed classically, with variational parameters affecting only post-processing of the fixed measurement record.
4. Optimization involves updating variational parameters (e.g., occupation-basis phase shifts, Jastrow factors, or classical correlator coefficients) to minimize a cost function, generally a Rayleigh quotient over energy and normalization, evaluated exclusively from the quantum-collected bitstrings.
5. Convergence and output: The process is repeated—typically without additional quantum measurements—until the optimized energy converges below a user-set threshold.

This division obviates the need to recompile or re-execute quantum circuits for each variational update, making CVQE particularly suited to hardware-limited NISQ devices (Gunlycke et al., 2023).

2. Mathematical Formulation and Ansatz Structure

The unifying mathematical construct in CVQE is the representation of the trial wavefunction as

$$\ket{\Psi(\boldsymbol\theta)} = e^{i\hat{\lambda}(\boldsymbol\theta)} \hat{U}\ket{0}$$

where $\hat{U}$ prepares the guiding state $\ket{\Psi_0}$ and $\hat{\lambda}(\boldsymbol\theta)$ is a generally diagonal operator in the Fock or occupation basis, parameterized to be surjective over possible basis labelings. This ansatz enables:

• Full Fock space coverage: By judicious choice of $\boldsymbol\theta$ and $\Psi_{0n}$, all possible occupation-number basis vectors are accessible.
• Symmetry enforcement: Non-physical sectors can be excluded via divergent shifts in $\hat{\lambda}(\boldsymbol\theta)$ or parameter tying.
• Non-unitary operator inclusion: For instance, the Jastrow–Gutzwiller factor

$$\hat{G}(\theta) = \exp\left( - \sum_{qq'} \theta_{qq'}\, \hat{n}_q \hat{n}_{q'} \right)$$

may be included since the resulting operator remains measurable after circuit preparation, with coefficients absorbed as classical weights (Stenger et al., 2023).

The evaluation of all cost function terms is achieved via classical sums (or expectation values) over bitstring-measured data, re-weighted analytically according to current parameter settings: $$E(\boldsymbol\theta) = \frac{\Upsilon(\boldsymbol\theta)}{\Lambda(\boldsymbol\theta)}$$ with $\Upsilon, \Lambda$ finite sums over empirical probabilities mixed with parameter-dependent exponentials, enabling closed-form gradients for classical optimization (Gunlycke et al., 2023).

3. Quantum-to-Classical Cascade and Measurement Strategy

A key property of CVQE is the one-time quantum measurement of all relevant bases, followed by unrestricted classical parameter optimization:

• Quantum hardware execution: For each Hamiltonian Pauli term or related block, the quantum device prepares $\hat{R}\ket{\Psi_0}$ (where $\hat{R}$ is a fixed basis rotation) and measures the computational basis, recording histograms of bitstrings.
• Measurement aggregation: All Pauli operator expectation values needed for energy calculation are estimated via these histograms, independent of variational parameter values.
• Classical cost function evaluation: Upon updating any variational parameter, the classical computer recalculates $E(\boldsymbol\theta)$, its gradients, and higher derivatives as analytic functions of the same fixed measurement data, enabling arbitrary exploration of the variational landscape without additional quantum runs.

This strategy yields an exponential reduction in total quantum circuit executions compared to conventional VQE, particularly for highly parameterized ansätze or resource-hungry optimizers (e.g., thousands of gradient evaluations) (Gunlycke et al., 2023).

4. Extensions: Non-Unitary and Adaptive Variants

CVQE readily adapts to generalized ansatz structures and hybrid workflows:

a) Non-unitary correlators

The inclusion of non-unitary Jastrow–Gutzwiller correlators and related diagonal operators is natural in CVQE. For example, the total state

$$|\Psi(\theta, \varphi)\rangle = \hat{G}(\theta)\, \hat{U}(\varphi)\,|0\rangle$$

permits completely classical handling of the non-unitary part: only the guiding state $\hat{U}(\varphi)\,|0\rangle$ is prepared and measured, and energy evaluations for any $\theta$ value are accomplished by re-weighting the previously collected bitstring outcomes. This approach allows parameter scans and optimization "offline" from the quantum backend, with improved resilience to hardware errors, as only short-depth circuits need physical realization (Stenger et al., 2023).

b) Adaptive and hybrid quantum–classical schemes

Recent advances integrate CVQE with adaptive or diabatic circuit generation. For example, a hybrid VQE–CVQE protocol uses variationally parameterized quantum circuits (e.g., via diabatic Trotterized state preparation) to generate a guiding state, then cascades a classical subspace extraction via measurement-induced basis expansion and classical diagonalization. The cost function is the minimum eigenvalue of the projected Hamiltonian $H_\mathcal{B}$ extracted in the measurement-induced subspace, enabling robust optimization with minimal quantum depth (Stenger et al., 4 Dec 2025).

5. Resource Scaling and Practical Performance

CVQE achieves substantial resource advantages over VQE, particularly with respect to quantum hardware constraints:

Aspect	CVQE	VQE
# Quantum Executions	$|\mathcal{L}| \times 2^{|\mathcal{Q}_l|} \times S$ (single batch)	$|\mathcal{L}| \times S \times M \times N$ (per iteration)
Parameter Space	Arbitrarily large (post-processing only)	Tightly coupled to quantum runs
Circuit Depth	Fixed by guiding state preparation	Increases with ansatz complexity
Error Propagation	Fixed noise per batch, filterable	Aggregate over all iterations

Here $|\mathcal{L}|$ is the number of Hamiltonian terms, $|\mathcal{Q}_l|$ the support size per term, $S$ the shots per circuit, $M$ evaluations per step, and $N$ optimization steps (Gunlycke et al., 2023).

Benchmarks demonstrate chemical accuracy (energy error $<1.6\times10^{-3}$ Ha) for realistic molecular and lattice models using determinant budgets an order of magnitude smaller than full configuration-interaction, and quantum resource use greatly reduced versus UCCSD-VQE (Zhang et al., 16 Sep 2025, Stenger et al., 4 Dec 2025).

6. Applications and Demonstrations

Several key applications and hardware demonstrations highlight the versatility of CVQE:

• Strongly correlated molecular dissociation: CVQE achieves chemical accuracy for challenging molecular systems (BeH$_2$, H$_6$, N$_2$), outperforming fixed-depth VQE and selected CI with comparable or fewer determinants (Zhang et al., 16 Sep 2025).
• Jastrow–Gutzwiller operators: Practical implementation on IBM Q Lagos demonstrates near-exact ground-state energies for four-site Hubbard models, requiring 24 CX gates total and leveraging classical relabeling to further reduce hardware error (Stenger et al., 2023).
• Large orbital-count simulations: A 50-orbital, 25-electron lattice model on IBM Brisbane maintains energy errors below chemical accuracy with minimal depth and subspace sizes of $\sim 2000$, showcasing NISQ applicability to otherwise classically intractable problems (Stenger et al., 4 Dec 2025).

7. Scalability, Limitations, and Prospects

While CVQE accelerates NISQ ground-state simulation, several caveats and directions remain:

• The guiding circuit depth can still limit achievable system size, especially as orbital counts increase; resource scaling is dominated by either state-preparation (e.g., Givens rotations) or measurement pool sizes.
• The efficiency of the classical cascade relies on the compressibility of the true ground state—systems with delocalized or highly entangled ground states may require large subspace expansions, challenging active-space reduction strategies.
• Scalability can be improved via Pauli-string grouping, hardware-efficient ansätze, or subspace expansion techniques using measurement-driven feedback.
• Potential improvements include enrichment of the measurement selection rule, integration of perturbative (MRPT2-type) energy corrections, or gradient-based optimizers for closed-form cost function landscapes (Zhang et al., 16 Sep 2025, Stenger et al., 2023).

These properties, together with resilience to barren-plateau gradients and robustness of error propagation, position CVQE as a leading resource-efficient paradigm for quantum simulation on both near- and long-term hardware.