Gradient-based ADAPT-VQE: Adaptive Quantum Eigensolver

• Gradient-based ADAPT-VQE is an adaptive quantum eigensolver that constructs ansätze by iteratively selecting operators with the largest energy gradient.
• It builds compact, problem-specific circuits from a curated pool of generators, dynamically tailoring the ansatz during each iteration.
• While its parameter recycling and gradient evaluation enhance convergence, the full-pool scanning process can lead to significant measurement overhead.

Searching arXiv for papers on gradient-based ADAPT-VQE and related variants. Gradient-based ADAPT-VQE is an adaptive variational quantum eigensolver in which the ansatz is not fixed in advance, but is assembled iteratively from a pool of candidate generators by selecting, at each step, the operator with the largest energy-gradient magnitude on the current state. In quantum chemistry this pool typically consists of singlet-adapted fermionic single and double excitations or related qubit-space operators; in spin models it may consist of two-qubit or graph-structured generators. The method therefore combines a greedy operator-selection rule with repeated variational re-optimization, aiming to produce compact, problem-tailored circuits while avoiding the oversized parameter spaces of static ansätze (Claudino et al., 2020, Vahedi et al., 7 Jun 2026).

1. Formal definition and gradient criterion

Let $H$ denote the Hamiltonian, $\{A_i\}_{i=1}^{|\mathcal E|}$ a pool of anti-Hermitian generators, and $$\ket{\Psi(\boldsymbol\theta)} = U(\boldsymbol\theta)\ket{\psi_{\rm ref}}$$ the current variational state, where $\ket{\psi_{\rm ref}}$ is typically the Hartree–Fock state in chemistry and may be $\ket{0}$ in more abstract formulations. The variational energy is

$$E(\boldsymbol\theta)=\langle\Psi(\boldsymbol\theta)|H|\Psi(\boldsymbol\theta)\rangle.$$

The defining signal of gradient-based ADAPT-VQE is the energy derivative associated with appending a new infinitesimal rotation generated by $A_i$. A standard expression is

$$g_i(\boldsymbol\theta)=\frac{\partial E(\boldsymbol\theta)}{\partial \theta_i} =\langle\Psi(\boldsymbol\theta)|[H,A_i]|\Psi(\boldsymbol\theta)\rangle,$$

with operator selection based on $|g_i|$ (Vahedi et al., 7 Jun 2026). Closely related formulations write the same criterion for a pool generator $\tau_k$ as

$\{A_i\}_{i=1}^{|\mathcal E|}$0

or, under an alternative convention for the generators, as $\{A_i\}_{i=1}^{|\mathcal E|}$1 (Claudino et al., 2020, Ikhtiarudin et al., 22 Jul 2025). The common content is that the commutator expectation quantifies the first-order energy response of the current state to the candidate operator.

The ansatz assembled after $\{A_i\}_{i=1}^{|\mathcal E|}$2 iterations is pseudo-Trotterized. A common notation is

$\{A_i\}_{i=1}^{|\mathcal E|}$3

with $\{A_i\}_{i=1}^{|\mathcal E|}$4 chosen from the pool by the largest-gradient rule (Claudino et al., 2020). This construction is “greedy” in the specific sense that it adds the operator predicted to yield the steepest instantaneous energy decrease among the currently available directions (Vaquero-Sabater et al., 7 Apr 2025).

2. Canonical iterative procedure

A standard run initializes the circuit as the identity, or equivalently starts from the reference state with an empty parameter list. At iteration $\{A_i\}_{i=1}^{|\mathcal E|}$5, all pool gradients are evaluated,

$\{A_i\}_{i=1}^{|\mathcal E|}$6

and the next operator is chosen as

$\{A_i\}_{i=1}^{|\mathcal E|}$7

If the largest gradient falls below a convergence threshold, the algorithm terminates; otherwise the new unitary generated by $\{A_i\}_{i=1}^{|\mathcal E|}$8 is appended and all variational parameters are re-optimized against the full energy objective (Vahedi et al., 7 Jun 2026).

Two stopping conventions are both used. One is a largest-gradient criterion, $\{A_i\}_{i=1}^{|\mathcal E|}$9, which appears in several formulations (Grimsley et al., 2022). Another computes the full pool-gradient vector $$\ket{\Psi(\boldsymbol\theta)} = U(\boldsymbol\theta)\ket{\psi_{\rm ref}}$$0 and stops when $$\ket{\Psi(\boldsymbol\theta)} = U(\boldsymbol\theta)\ket{\psi_{\rm ref}}$$1; benchmarks have used $$\ket{\Psi(\boldsymbol\theta)} = U(\boldsymbol\theta)\ket{\psi_{\rm ref}}$$2 (Claudino et al., 2020). These criteria are not identical, but both implement the same principle: ansatz growth stops when no operator in the pool appears able to reduce the energy appreciably at first order.

A practically important detail is parameter recycling. Existing optimal parameters are reused from the previous ADAPT iteration, while the newly appended parameter is initialized at zero. This warm-start strategy is central to the algorithm’s behavior in rough landscapes, because each enlarged VQE subproblem begins close to the previous local optimum rather than from a random point (Grimsley et al., 2022). Later benchmarking work also distinguishes between “full” re-optimization, in which all accumulated ansatz parameters are updated, and “last” re-optimization, in which only the newly added parameter is varied. In weakly correlated cases such as equilibrium LiH, “last” can suffice, whereas for $$\ket{\Psi(\boldsymbol\theta)} = U(\boldsymbol\theta)\ket{\psi_{\rm ref}}$$3, $$\ket{\Psi(\boldsymbol\theta)} = U(\boldsymbol\theta)\ket{\psi_{\rm ref}}$$4, and stretched geometries, “full” re-optimization is required for convergence to $$\ket{\Psi(\boldsymbol\theta)} = U(\boldsymbol\theta)\ket{\psi_{\rm ref}}$$5 accuracy (Rossi et al., 3 Jun 2026).

3. Measurement model and cost scaling

The principal drawback of gradient-based ADAPT-VQE is the cost of repeatedly scanning the full operator pool. At the algorithmic level, each outer iteration requires one gradient evaluation for each pool element, so the selection step scales as $$\ket{\Psi(\boldsymbol\theta)} = U(\boldsymbol\theta)\ket{\psi_{\rm ref}}$$6 expensive commutator-expectation measurements or state-vector evaluations, and a depth-$$\ket{\Psi(\boldsymbol\theta)} = U(\boldsymbol\theta)\ket{\psi_{\rm ref}}$$7 run incurs $$\ket{\Psi(\boldsymbol\theta)} = U(\boldsymbol\theta)\ket{\psi_{\rm ref}}$$8 gradient calls (Vahedi et al., 7 Jun 2026). On quantum hardware, every $$\ket{\Psi(\boldsymbol\theta)} = U(\boldsymbol\theta)\ket{\psi_{\rm ref}}$$9 requires preparation of the current state and measurement of the Pauli strings appearing in $\ket{\psi_{\rm ref}}$0, so the overhead is linear in pool size even before considering Pauli decomposition (Vahedi et al., 7 Jun 2026).

In chemistry, the pool size is typically $\ket{\psi_{\rm ref}}$1 for fermionic singles-plus-doubles, whereas in long-range spin models it is $\ket{\psi_{\rm ref}}$2. One analysis states that this overhead can dominate runtime once $\ket{\psi_{\rm ref}}$3 (Vahedi et al., 7 Jun 2026). A more fine-grained measurement analysis shows why. If $\ket{\psi_{\rm ref}}$4 and $\ket{\psi_{\rm ref}}$5, then

$\ket{\psi_{\rm ref}}$6

so each pool gradient is a weighted sum of Pauli-string expectation values (Anastasiou et al., 2023). For hardware-efficient or qubit-ADAPT pools, the naive counting can reach $\ket{\psi_{\rm ref}}$7 observables, since there are $\ket{\psi_{\rm ref}}$8 pool operators and each commutator expansion can involve $\ket{\psi_{\rm ref}}$9 nonzero contributions (Anastasiou et al., 2023, Nykänen et al., 2022).

This measurement burden is not only a shot problem but also a basis-setting problem. The paper “How to really measure operator gradients in ADAPT-VQE” develops a commuting-observable grouping strategy for qubit-ADAPT pools that reduces the naive $\ket{0}$0 basis count to $\ket{0}$1, with a total shot bound

$\ket{0}$2

so that full-pool gradient measurement is only $\ket{0}$3 times as expensive as a standard VQE iteration (Anastasiou et al., 2023). Other work emphasizes that exact measurement of nested commutators would require ancilla-enabled circuits, whereas the operator-selection step can instead use direct measurement of $\ket{0}$4 on the current ansatz state (Claudino et al., 2020).

4. Optimization landscape, initialization, and failure modes

A central claim in the modern ADAPT-VQE literature is that the one-operator-at-a-time construction substantially alters the optimization landscape seen by the classical optimizer. The analysis in “ADAPT-VQE is insensitive to rough parameter landscapes and barren plateaus” does not claim that local minima disappear; rather, it argues that ADAPT-VQE provides an initialization strategy that is dramatically better than random initialization, and that even if one iteration converges to a local trap, later operator additions can “burrow” deeper into that trap until the ansatz reaches the exact solution (Grimsley et al., 2022).

The mechanism is tied to parameter recycling. At a new growth step, previously optimized parameters already lie at a stationary point, while the newly inserted parameter is chosen specifically because its pool gradient is the largest and exceeds the stopping threshold. The same work therefore argues that ADAPT-VQE should not suffer optimization problems due to barren plateaus, because the recycling point always retains at least one large-magnitude descent direction that can be estimated with a polynomial number of shots (Grimsley et al., 2022). In numerical studies on linear $\ket{0}$5, $\ket{0}$6, LiH, and $\ket{0}$7, recycled ADAPT initialization reached FCI or chemical accuracy with markedly fewer operators than random or Hartree–Fock starts: approximately 12 operators for $\ket{0}$8, approximately 30 for $\ket{0}$9, and fewer than 20 for LiH and $$E(\boldsymbol\theta)=\langle\Psi(\boldsymbol\theta)|H|\Psi(\boldsymbol\theta)\rangle.$$0 (Grimsley et al., 2022).

At the same time, gradient-based ADAPT-VQE has distinct failure modes. One is the emergence of “dead weight” operators: generators that were selected by the gradient heuristic but later optimize to near-zero amplitudes. Pruned-ADAPT-VQE attributes these to poor operator selection, operator reordering, and fading operators (Vaquero-Sabater et al., 7 Apr 2025). Another is the “gradient trough,” an interval in which $$E(\boldsymbol\theta)=\langle\Psi(\boldsymbol\theta)|H|\Psi(\boldsymbol\theta)\rangle.$$1 drops by many orders of magnitude while the energy remains above the ground-state value. In that regime, resolving the largest gradient requires $$E(\boldsymbol\theta)=\langle\Psi(\boldsymbol\theta)|H|\Psi(\boldsymbol\theta)\rangle.$$2 shots, and naive stopping can cause false convergence (Stadelmann et al., 31 Dec 2025). A mathematically explicit arbitrary-position insertion formula,

$$E(\boldsymbol\theta)=\langle\Psi(\boldsymbol\theta)|H|\Psi(\boldsymbol\theta)\rangle.$$3

shows that the gradient can depend strongly on where a new operator is inserted in the existing noncommutative ansatz, motivating non-appending insertion protocols as a trough-mitigation strategy (Stadelmann et al., 31 Dec 2025).

5. Resource-efficient modifications built around the gradient oracle

Because the full-pool gradient scan is the dominant cost center, many extensions preserve the gradient-based logic while modifying how gradients are measured, amortized, or approximated. TETRIS-ADAPT-VQE keeps the same gradient evaluation step but relaxes the one-operator-at-a-time growth rule: after sorting pool elements by $$E(\boldsymbol\theta)=\langle\Psi(\boldsymbol\theta)|H|\Psi(\boldsymbol\theta)\rangle.$$4, it adds multiple operators with disjoint qubit supports in the same iteration. Since those unitaries commute, they can be executed in parallel within one circuit layer. On $$E(\boldsymbol\theta)=\langle\Psi(\boldsymbol\theta)|H|\Psi(\boldsymbol\theta)\rangle.$$5, LiH, $$E(\boldsymbol\theta)=\langle\Psi(\boldsymbol\theta)|H|\Psi(\boldsymbol\theta)\rangle.$$6, and $$E(\boldsymbol\theta)=\langle\Psi(\boldsymbol\theta)|H|\Psi(\boldsymbol\theta)\rangle.$$7, reported ADAPT/TETRIS depth ratios at convergence are 1.6, 1.8, 2.3, and 2.6, respectively, and the number of gradient-measurement rounds is reduced by about 50–70% for $$E(\boldsymbol\theta)=\langle\Psi(\boldsymbol\theta)|H|\Psi(\boldsymbol\theta)\rangle.$$8–14 qubits (Anastasiou et al., 2022).

A second line of work attacks shot complexity directly. “Shot-Efficient ADAPT-VQE via Reused Pauli Measurements and Variance-Based Shot Allocation” reuses Pauli outcomes from the final VQE subroutine at iteration $$E(\boldsymbol\theta)=\langle\Psi(\boldsymbol\theta)|H|\Psi(\boldsymbol\theta)\rangle.$$9 for gradient estimation at iteration $A_i$0, because the state being measured is identical. It further allocates shots across commuting cliques according to estimated variances. In the reported H$A_i$1 benchmarks, total shots to chemical accuracy decrease from 5120 under uniform allocation to 4774 with VMSA and 2908 with VPSR; for LiH the corresponding figures are 9216, 8687, and 4498. Combined reuse plus VPSR cuts total shots by more than 65% relative to naive measurement, while keeping the same ansatz depth in the studied systems (Ikhtiarudin et al., 22 Jul 2025).

A third approach replaces explicit commutator measurement by informationally complete data reuse. AIM-ADAPT-VQE employs adaptive informationally complete generalized measurements so that the same measurement record used for energy estimation can also be classically post-processed to estimate all pool commutators. For the studied $A_i$2 and $A_i$3 Hamiltonians, this yields ADAPT-VQE with no additional measurement overhead beyond energy estimation, and when the energy is measured within chemical precision, the final CNOT counts are close to those obtained from exact-gradient ADAPT-VQE (Nykänen et al., 2022).

A fourth approach learns the gradient oracle statistically. “Graph Neural Networks for Fast Operator Selection in Adaptive VQE” reformulates operator selection as a graph-based decision problem and trains a GNN policy on exact simulations of disordered long-range spin chains using gradient magnitudes as supervision. In a VQE workflow, the resulting GNN-VQE achieves energy errors close to standard ADAPT-VQE while drastically reducing full-pool gradient evaluations. On LiH and $A_i$4, the GNN is reported to be highly effective as a shortlist generator: exact rescoring over a few GNN-proposed candidates recovers near-oracle rollout behavior while searching only a small fraction of the pool (Vahedi et al., 7 Jun 2026).

6. Benchmarks, applications, and relation to alternative selection rules

Gradient-based ADAPT-VQE has been benchmarked most extensively in molecular ground-state calculations. Numerical simulations on $A_i$5, NaH, and KH found that both fixed-ansatz VQE and ADAPT-VQE provide good energy estimates, but ADAPT-VQE is more robust to optimizer choice. The same study reported that gradient-based L-BFGS is more economical than COBYLA, that ADAPT circuits optimized with L-BFGS are shallower and use fewer gates than their COBYLA counterparts, and that L-BFGS typically yields ansätze with 10–20% fewer operators. Reported energy errors relative to FCI are below $A_i$6 for $A_i$7, below $A_i$8 across $A_i$9–$$g_i(\boldsymbol\theta)=\frac{\partial E(\boldsymbol\theta)}{\partial \theta_i} =\langle\Psi(\boldsymbol\theta)|[H,A_i]|\Psi(\boldsymbol\theta)\rangle,$$0 for NaH, and below $$g_i(\boldsymbol\theta)=\frac{\partial E(\boldsymbol\theta)}{\partial \theta_i} =\langle\Psi(\boldsymbol\theta)|[H,A_i]|\Psi(\boldsymbol\theta)\rangle,$$1 for KH except near $$g_i(\boldsymbol\theta)=\frac{\partial E(\boldsymbol\theta)}{\partial \theta_i} =\langle\Psi(\boldsymbol\theta)|[H,A_i]|\Psi(\boldsymbol\theta)\rangle,$$2; infidelities grow from $$g_i(\boldsymbol\theta)=\frac{\partial E(\boldsymbol\theta)}{\partial \theta_i} =\langle\Psi(\boldsymbol\theta)|[H,A_i]|\Psi(\boldsymbol\theta)\rangle,$$3 in NaH to approximately $$g_i(\boldsymbol\theta)=\frac{\partial E(\boldsymbol\theta)}{\partial \theta_i} =\langle\Psi(\boldsymbol\theta)|[H,A_i]|\Psi(\boldsymbol\theta)\rangle,$$4 in KH as system size increases (Claudino et al., 2020).

The same benchmark record also clarifies a common misconception. Gradient-based ADAPT-VQE is often described as if it always produced the minimal physically meaningful ansatz. The pruning literature shows that this is not generally true: the greedy commutator rule can admit redundant operators whose optimized amplitudes are essentially zero, and post-optimization pruning can reduce ansatz size without changing the energy. Reported examples include $$g_i(\boldsymbol\theta)=\frac{\partial E(\boldsymbol\theta)}{\partial \theta_i} =\langle\Psi(\boldsymbol\theta)|[H,A_i]|\Psi(\boldsymbol\theta)\rangle,$$5 from 14 to 11 operators, LiH from 34 to 27, and $$g_i(\boldsymbol\theta)=\frac{\partial E(\boldsymbol\theta)}{\partial \theta_i} =\langle\Psi(\boldsymbol\theta)|[H,A_i]|\Psi(\boldsymbol\theta)\rangle,$$6 from 46 to 35, all at the same final error or with no loss in accuracy (Vaquero-Sabater et al., 7 Apr 2025).

Recent work also frames gradient-based ADAPT-VQE as a baseline against which alternative operator-scoring rules are compared. Geo-ADAPT-VQE replaces the Euclidean gradient score with a natural-gradient rule based on the quantum information metric, and reports faster, more stable convergence, up to 4$$g_i(\boldsymbol\theta)=\frac{\partial E(\boldsymbol\theta)}{\partial \theta_i} =\langle\Psi(\boldsymbol\theta)|[H,A_i]|\Psi(\boldsymbol\theta)\rangle,$$7 fewer parameters than standard ADAPT-VQE, and up to 100-fold reduction in energy error in challenging geometries (Sohail et al., 11 Mar 2026). Param-ADAPT-VQE replaces the gradient score by a parameter-based importance criterion $$g_i(\boldsymbol\theta)=\frac{\partial E(\boldsymbol\theta)}{\partial \theta_i} =\langle\Psi(\boldsymbol\theta)|[H,A_i]|\Psi(\boldsymbol\theta)\rangle,$$8 obtained from one-parameter local VQEs on sub-Hamiltonians, and reports approximately 20–60% fewer variational parameters together with approximately 30–50% lower total measurement cost across LiH, $$g_i(\boldsymbol\theta)=\frac{\partial E(\boldsymbol\theta)}{\partial \theta_i} =\langle\Psi(\boldsymbol\theta)|[H,A_i]|\Psi(\boldsymbol\theta)\rangle,$$9, and $|g_i|$0 benchmarks (He et al., 4 Feb 2026). Efficient energy-based fermionic Rotoselect, via exact Hamiltonian transformation, reduces the cost of energy-based selection to within about 10% of gradient-based selection and clarifies that, in strongly correlated regimes, full ansatz re-optimization and orbital optimization dominate convergence behavior more than the choice between gradient-based and energy-based scoring alone (Rossi et al., 3 Jun 2026).

Taken together, these results place gradient-based ADAPT-VQE in a precise methodological position. It is the canonical first-order adaptive ansatz-construction procedure: compact, problem-tailored, and generally robust under recycled initialization, but classically and metrologically bottlenecked by repeated full-pool gradient scans. Much of the subsequent ADAPT-VQE literature therefore preserves its commutator-based selection principle while attempting to reduce, reuse, restructure, or learn the gradient oracle itself (Vahedi et al., 7 Jun 2026).