Cyclic Variational Quantum Eigensolver (CVQE)

• CVQE is a variational quantum algorithm that simulates ground states on NISQ devices by adaptively expanding a multi-determinant reference state using measurement-driven sampling.
• The method employs a fixed, hardware-friendly entangler along with cyclic optimization (CAD and gradient descent) to reuse compiled circuits and ensure resource efficiency.
• Its distinctive staircase descent pattern, marked by abrupt energy drops after determinant expansion events, enables it to overcome barren plateaus and achieve chemical precision.

Cyclic Variational Quantum Eigensolver (CVQE) is a variational quantum algorithm for ground-state simulation on noisy intermediate-scale quantum (NISQ) devices that combines a fixed, hardware-friendly entangler with an adaptive, measurement-driven expansion of a multi-determinant reference state. In the formulation introduced in “Cyclic Variational Quantum Eigensolver: Escaping Barren Plateaus through Staircase Descent,” CVQE departs from conventional VQE by iteratively enlarging a reference superposition of Slater determinants according to measured sampling probabilities, while reusing a single entangling circuit across optimization cycles. The method is presented as fully automated on quantum hardware, parallel to multi-reference methods in quantum chemistry, and empirically characterized by a staircase-like descent pattern in which abrupt energy drops follow reference expansion events (Zhang et al., 16 Sep 2025).

1. Core algorithmic cycle

At the center of CVQE is a repeated four-step cycle indexed by $k=1,2,\dots$. The reference set $\mathcal S^{(k)}$ initially contains only the Hartree–Fock determinant, $\mathcal S^{(1)}=\{\lvert \mathrm{HF}\rangle\}$. At cycle $k$, the current determinants are assembled into a normalized reference superposition

$$\lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle = \sum_{i\in \mathcal S^{(k)}} c_i \lvert D_i\rangle .$$

This state is prepared on hardware using an ancilla-assisted state-preparation circuit whose cost scales linearly in $|\mathcal S|$ (Zhang et al., 16 Sep 2025).

A fixed entangling circuit is then applied. In the reported implementation this entangler is a single Trotter-step UCCSD operator, yielding

$$\lvert \psi_{\rm trial}(\mathbf c,\boldsymbol\theta)\rangle = \hat U_{\rm UCCSD}(\boldsymbol\theta)\, \lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle .$$

The energy

$$E(\mathbf c,\boldsymbol\theta) = \langle \psi_{\rm trial}\lvert \widehat H \rvert \psi_{\rm trial}\rangle$$

is minimized simultaneously over the determinant coefficients $\mathbf c$ and the entangler parameters $\boldsymbol\theta$, using Cyclic Adamax (“CAD”) with periodic resets for $\mathcal S^{(k)}$0 and standard gradient descent for $\mathcal S^{(k)}$1 (Zhang et al., 16 Sep 2025).

The cycle closes with sampling-based reference expansion. The trial state is measured in the computational basis over $\mathcal S^{(k)}$2, and any newly observed determinant $\mathcal S^{(k)}$3 whose empirical probability exceeds a dynamic threshold is admitted into the next reference set. If the expanded set exceeds the user-set determinant cap $\mathcal S^{(k)}$4, the smallest-$\mathcal S^{(k)}$5 configurations are pruned. Iteration continues until energy convergence or exhaustion of the determinant budget. This design keeps the entangler fixed while allowing the reference to grow in the directions suggested by measurement data, avoiding manual ansatz or operator-pool design and preserving compile-once circuits (Zhang et al., 16 Sep 2025).

2. Variational formulation and determinant representation

The variational state in CVQE has a two-level structure: an adaptive reference superposition and a fixed-structure entangler. The reference state

$\mathcal S^{(k)}$6

defines the multi-reference subspace, while the full trial state is obtained by the action of $\mathcal S^{(k)}$7 on that subspace. Each Slater determinant $\mathcal S^{(k)}$8 is represented as a computational-basis product state on $\mathcal S^{(k)}$9 qubits, with $\mathcal S^{(1)}=\{\lvert \mathrm{HF}\rangle\}$0 denoting occupied and $\mathcal S^{(1)}=\{\lvert \mathrm{HF}\rangle\}$1 denoting empty spin orbitals (Zhang et al., 16 Sep 2025).

The fixed entangler is specified under a first-order Trotter step as

$\mathcal S^{(1)}=\{\lvert \mathrm{HF}\rangle\}$2

where $\mathcal S^{(1)}=\{\lvert \mathrm{HF}\rangle\}$3 run over occupied orbitals and $\mathcal S^{(1)}=\{\lvert \mathrm{HF}\rangle\}$4 over virtual orbitals. The fermionic creation and annihilation operators are mapped to qubits באמצעות Jordan–Wigner, and the parameter pool $\mathcal S^{(1)}=\{\lvert \mathrm{HF}\rangle\}$5 remains fixed in size across cycles (Zhang et al., 16 Sep 2025).

This fixed-pool property is central to the distinction between CVQE and adaptive ansatz-growth procedures. The variational space expands through the reference coefficients and determinant support rather than through continual enlargement of the entangling operator set. A plausible implication is that CVQE relocates adaptivity from circuit synthesis to reference-state selection, while retaining a standard energy-minimization objective.

3. Staircase descent and escape from barren plateaus

The defining empirical signature of CVQE is its staircase-like descent trajectory. In the reported benchmarks, fixed UCCSD exhibits an error curve that quickly plateaus well above chemical precision in strongly correlated regimes, whereas CVQE shows long flat regions during which no new determinants are admitted, followed by abrupt energy drops immediately after reference expansion (Zhang et al., 16 Sep 2025).

The mechanism described for these drops couples two events. First, the CAD optimizer is periodically restarted; specifically, every $\mathcal S^{(1)}=\{\lvert \mathrm{HF}\rangle\}$6 iterations in the reported setting, its first- and second-moment estimates are zeroed, clearing stale gradient memory. Second, the admission of new determinants enlarges the accessible variational manifold by opening fresh descent directions. The coincidence of momentum reset and landscape enlargement is identified as the trigger for the sharp energy declines that allow CVQE to move beyond barren regions (Zhang et al., 16 Sep 2025).

The determinant-admission rule is tied to the norm of the entangler gradient: $\mathcal S^{(1)}=\{\lvert \mathrm{HF}\rangle\}$7 A new determinant event is flagged when the measured $\mathcal S^{(1)}=\{\lvert \mathrm{HF}\rangle\}$8 exceeds this threshold. The stated role of $\mathcal S^{(1)}=\{\lvert \mathrm{HF}\rangle\}$9 is to ensure that early optimization admits only dominant configurations, while later stages allow smaller-weight determinants to enter. This makes the growth criterion adaptive to the local optimization state rather than fixed a priori (Zhang et al., 16 Sep 2025).

4. Resource model and hardware characteristics

CVQE is formulated for NISQ-era quantum chemistry with explicit accounting of qubits, depth, and measurement overhead. The hardware requirement is $k$0 system qubits, one per spin orbital, together with $k$1 ancillas for superposition state preparation; in the reported examples, $k$2. Example system sizes are BeH$k$3 on $k$4 qubits, H$k$5 on $k$6 qubits, and N$k$7 on $k$8 qubits (Zhang et al., 16 Sep 2025).

The reference-superposition circuit has gate complexity

$k$9

which is described as nearly linear in $$\lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle = \sum_{i\in \mathcal S^{(k)}} c_i \lvert D_i\rangle .$$0. The UCCSD component consists of one Trotter layer whose depth is set by the number of single and double excitations, and this circuit is compiled once and reused across cycles. The compile-once feature follows directly from the fact that reference growth does not alter the fixed entangler itself (Zhang et al., 16 Sep 2025).

Measurement costs are split between state expansion and energy optimization. Each cycle uses $$\lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle = \sum_{i\in \mathcal S^{(k)}} c_i \lvert D_i\rangle .$$1 computational-basis measurements to identify promising determinants. Energies and gradients for $$\lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle = \sum_{i\in \mathcal S^{(k)}} c_i \lvert D_i\rangle .$$2 rely on standard Hamiltonian-term grouping, and coefficient updates use parameter-shift or finite-difference on the cost. This measurement model reflects a hybrid workflow in which the quantum device supplies both expectation information and the sampling statistics that drive reference growth (Zhang et al., 16 Sep 2025).

5. Benchmark behavior in quantum chemistry

The principal numerical claim is that CVQE maintains chemical precision across weakly and strongly correlated regimes for the dissociation curves of BeH$$\lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle = \sum_{i\in \mathcal S^{(k)}} c_i \lvert D_i\rangle .$$3, H$$\lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle = \sum_{i\in \mathcal S^{(k)}} c_i \lvert D_i\rangle .$$4, and N$$\lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle = \sum_{i\in \mathcal S^{(k)}} c_i \lvert D_i\rangle .$$5. In the benchmark summary, CVQE achieves

$$\lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle = \sum_{i\in \mathcal S^{(k)}} c_i \lvert D_i\rangle .$$6

identified as chemical accuracy, even in strongly correlated bond-stretch regimes where fixed UCCSD errors exceed $$\lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle = \sum_{i\in \mathcal S^{(k)}} c_i \lvert D_i\rangle .$$7 (Zhang et al., 16 Sep 2025).

A more granular comparison is given for H$$\lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle = \sum_{i\in \mathcal S^{(k)}} c_i \lvert D_i\rangle .$$8 at $$\lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle = \sum_{i\in \mathcal S^{(k)}} c_i \lvert D_i\rangle .$$9\,Å under a determinant budget $|\mathcal S|$0. In that setting, CVQE reaches errors of order $|\mathcal S|$1, one order of magnitude below fixed UCCSD and better than SHCI with a comparable determinant count $|\mathcal S|$2. The reported error table is:

$|\mathcal S|$3	$|\mathcal S|$4 (Ha)	$|\mathcal S|$5 (Ha)
50	$|\mathcal S|$6	$|\mathcal S|$7
100	$|\mathcal S|$8	$|\mathcal S|$9
150	$$\lvert \psi_{\rm trial}(\mathbf c,\boldsymbol\theta)\rangle = \hat U_{\rm UCCSD}(\boldsymbol\theta)\, \lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle .$$0	$$\lvert \psi_{\rm trial}(\mathbf c,\boldsymbol\theta)\rangle = \hat U_{\rm UCCSD}(\boldsymbol\theta)\, \lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle .$$1

These benchmarks are presented as evidence of high determinant efficiency and favorable accuracy-cost trade-offs relative to Selected Configuration Interaction. The abstract further states that CVQE outperforms fixed UCCSD by several orders of magnitude and achieves favorable accuracy-cost trade-offs compared to Selected Configuration Interaction (Zhang et al., 16 Sep 2025).

6. Relation to multi-reference theory, interpretability, and acronym usage

CVQE is explicitly connected to classical multi-reference quantum chemistry. The stated parallel is with methods such as CASSCF or MRPT2, which first construct an active multi-determinant reference to capture non-dynamical (static) correlation and then refine around that space. In CVQE, the reference is not selected manually; instead, measurement-driven reference growth automatically identifies important configurations. The method is therefore described as requiring no chemical intuition or manual pool construction (Zhang et al., 16 Sep 2025).

Interpretability is framed in determinant-level terms. Each admitted determinant corresponds to a chemically meaningful excitation, with bond-breaking configurations in N$$\lvert \psi_{\rm trial}(\mathbf c,\boldsymbol\theta)\rangle = \hat U_{\rm UCCSD}(\boldsymbol\theta)\, \lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle .$$2 given as an example. Post-iteration analysis of the final $$\lvert \psi_{\rm trial}(\mathbf c,\boldsymbol\theta)\rangle = \hat U_{\rm UCCSD}(\boldsymbol\theta)\, \lvert \psi_{\rm init}^{(k)}(\mathbf c)\rangle .$$3 can reveal which electronic configurations dominate in different regimes and may guide future ansatz or selection-rule refinement. The authors also suggest that the enriched reference could be combined with alternative entanglers, including qubit-ADAPT-style or hardware-native circuits, or with selective Trotterization guided by determinant structure (Zhang et al., 16 Sep 2025).

The acronym “CVQE” is not unique in the broader literature. Stenger et al. use “CVQE” to denote a “Cascaded Variational Quantum Eigensolver” in a hybrid VQE–CVQE algorithm based on diabatic state preparation and classical subspace diagonalization (Stenger et al., 4 Dec 2025). Separately, a VQE method for causal loop Feynman diagrams and directed acyclic graphs is also described as a cyclic VQE approach, where iterative runs and penalty projectors are used to sample multiple degenerate minima of a loop Hamiltonian (Clemente et al., 2022). This suggests that disambiguation by full expansion—“Cyclic” versus “Cascaded”—is important when comparing methods across subfields.

7. Conceptual significance and scope

Within the NISQ landscape, CVQE is presented as a framework that preserves a fixed entangler while allowing systematic enlargement of the variational space in promising directions. Its central claim is not that adaptivity is eliminated, but that adaptivity is shifted into measurement-driven reference growth. The resulting workflow combines hardware efficiency, automated determinant discovery, and an optimization trajectory marked by staircase descent rather than smooth monotone convergence (Zhang et al., 16 Sep 2025).

A recurring misconception in discussions of adaptive VQE methods is that adaptivity necessarily implies continual ansatz redesign or operator-pool searches. CVQE provides a contrasting template: the entangling circuit remains fixed, the parameter pool remains fixed in size, and the changing object is the reference superposition. Another possible misconception is that its multi-reference character makes it primarily a classical selected-CI procedure; the formalism instead keeps the reference-expansion rule on quantum hardware through computational-basis sampling of the entangled trial state. In that sense, CVQE occupies a hybrid position between conventional VQE and classical multi-reference selection strategies, with its distinctive staircase-descent dynamics serving as the operational signature of that design (Zhang et al., 16 Sep 2025).