Pulse-Based VQE: Direct Control Optimization

• Pulse-based VQE is a hybrid quantum–classical method where the variational ansatz is defined through hardware control pulses rather than gate sequences.
• It leverages continuous-time evolution and tailored pulse parameterizations to achieve schedule compression, reducing decoherence effects in quantum systems.
• The approach incorporates analytic gradient techniques and adaptive segmentation to optimize control parameters, enabling high-accuracy ground-state energy estimation.

Searching arXiv for recent and foundational work on pulse-based VQE, including ctrl-VQE, PANSATZ, cross-resonance implementations, and pulse-level parameterization. Pulse-Based Variational Quantum Eigensolver (VQE) denotes a family of hybrid quantum–classical eigensolvers in which the variational ansatz is specified directly at the level of hardware control pulses rather than as a sequence of parameterized logical gates. In this approach, a trial state is generated by time evolution under a drift-plus-control Hamiltonian whose amplitudes, phases, frequencies, durations, or detunings are themselves the variational parameters, and the energy is obtained from the expectation value of a qubit-mapped molecular or many-body Hamiltonian. The paradigm was articulated in gate-free form by ctrl-VQE, which replaces the state-preparation circuit by a variationally shaped pulse that drives a Hartree–Fock reference state toward the target full configuration interaction state, thereby reducing the coherence times required for state preparation (Meitei et al., 2020). Subsequent work extended the pulse-based formulation to cross-resonance superconducting hardware, neutral-atom platforms, contextual-subspace methods, analytic pulse-gradient evaluation, and modular pulse-optimized chemistry circuit elements (Meirom et al., 2022, Egger et al., 2023, Keijzer et al., 2022, Liang et al., 2023, Kottmann et al., 2023, Gothen et al., 15 Jun 2026).

1. Formal definition and variational structure

The defining mathematical feature of pulse-based VQE is that the ansatz state is generated by continuous-time evolution under a controlled device Hamiltonian rather than by a gate circuit. In ctrl-VQE this is written as

$$H(t)=H_{\rm sys}+\sum_k c_k(\boldsymbol\theta,t)\,H_k,$$

with propagator

$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$

and trial state

$$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$

where $|\psi_0\rangle$ is the Hartree–Fock reference (Meitei et al., 2020). The variational objective is the molecular energy

$$E(\boldsymbol\theta)=\langle\psi(\boldsymbol\theta)|H_{\rm mol}|\psi(\boldsymbol\theta)\rangle,$$

after mapping the second-quantized electronic Hamiltonian to qubits as a sum of Pauli strings (Meitei et al., 2020).

This continuous-time ansatz structure recurs across the literature. PANSATZ defines

$$U(\theta)=T\,\exp\biggl(-i\int_0^{T_{\rm tot}}H_{\rm ctrl}(t;\theta)\,dt\biggr),$$

and evaluates

$$E(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$

for a qubit-mapped chemistry Hamiltonian (Meirom et al., 2022). The cross-resonance pulse VQE likewise uses

$$U(\boldsymbol{\theta})=\mathcal{T}\exp\Bigl[-\,i\int_{0}^{T(\boldsymbol{\theta})} H_{\rm ctrl}\bigl(\boldsymbol{\theta},t\bigr)\,dt\Bigr]$$

with the cost function $$E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$$ (Egger et al., 2023). In the pulse-based variational quantum optimal control formulation for neutral atoms, the same principle is expressed through the Schrödinger propagator $U(t)$ under $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$0, with a terminal cost

$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$1

and often simply

$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$2

(Keijzer et al., 2022).

The common conceptual shift is therefore from a gate-parameter manifold to a control-function manifold. This suggests that pulse-based VQE should be understood less as a single ansatz family than as a control-theoretic reformulation of VQE in which the hardware Hamiltonian enters the variational layer explicitly.

2. Control Hamiltonians and pulse parameterizations

The pulse-level ansatz depends on how the control fields are parameterized. In superconducting transmon realizations of ctrl-VQE, the system Hamiltonian is decomposed into a fixed drift part

$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$3

and control terms

$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$4

where $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$5 are the $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$6th transmon’s frequency and anharmonicity, $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$7 the nearest-neighbor coupling, and $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$8 the drive frequency (Meitei et al., 2020). The control pulses can be piecewise-constant square pulses divided into $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$9 segments, with fixed total time $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$0 and variational amplitudes $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$1, optionally together with switching times $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$2. In that formulation, amplitudes satisfy $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$3, and frequencies lie in $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$4 (Meitei et al., 2020).

A more systematic study of parameterization appears in the analysis of pulse-level VQE optimizability. There, the drive on qubit $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$5 is written in the lab frame as

$$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$6

which under the rotating-wave approximation yields

$$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$7

with $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$8 and

$$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$9

(Sherbert et al., 2024). The paper investigates five finite-dimensional strategies: $|\psi_0\rangle$0, $|\psi_0\rangle$1, $|\psi_0\rangle$2, $|\psi_0\rangle$3, and $|\psi_0\rangle$4, all implemented as piecewise-constant windowed pulses of total duration $|\psi_0\rangle$5 divided into $|\psi_0\rangle$6 windows (Sherbert et al., 2024).

Other platform-specific parameterizations are more hardware-native. PANSATZ fixes the pulse envelope shapes—DRAG for single-qubit pulses and flat-top Gaussian for two-qubit cross-resonance pulses—and treats pulse durations $|\psi_0\rangle$7 and in-plane phases $|\psi_0\rangle$8 as the variational parameters (Meirom et al., 2022). The cross-resonance pulse VQE maps variational angles directly to the amplitude of a DRAG single-qubit pulse and the amplitude and duration of a Gaussian-square CR tone, using the wrapper functions

$|\psi_0\rangle$9

to ensure amplitude in $$E(\boldsymbol\theta)=\langle\psi(\boldsymbol\theta)|H_{\rm mol}|\psi(\boldsymbol\theta)\rangle,$$0 and $$E(\boldsymbol\theta)=\langle\psi(\boldsymbol\theta)|H_{\rm mol}|\psi(\boldsymbol\theta)\rangle,$$1 a multiple of $$E(\boldsymbol\theta)=\langle\psi(\boldsymbol\theta)|H_{\rm mol}|\psi(\boldsymbol\theta)\rangle,$$2 (Egger et al., 2023). In neutral-atom optical tweezer arrays, pulse-based VQE uses the native Rydberg Hamiltonian with global detuning $$E(\boldsymbol\theta)=\langle\psi(\boldsymbol\theta)|H_{\rm mol}|\psi(\boldsymbol\theta)\rangle,$$3, global Rabi frequency $$E(\boldsymbol\theta)=\langle\psi(\boldsymbol\theta)|H_{\rm mol}|\psi(\boldsymbol\theta)\rangle,$$4, and geometry $$E(\boldsymbol\theta)=\langle\psi(\boldsymbol\theta)|H_{\rm mol}|\psi(\boldsymbol\theta)\rangle,$$5, with piecewise-linear interpolation between control values $$E(\boldsymbol\theta)=\langle\psi(\boldsymbol\theta)|H_{\rm mol}|\psi(\boldsymbol\theta)\rangle,$$6 (Nagao et al., 25 Jul 2025).

These constructions show that pulse-based VQE does not require a unique control basis. The concrete choice of amplitude, phase, duration, or detuning variables is platform dependent, and the literature treats that choice as part of the ansatz design problem.

3. Optimization methods and gradient evaluation

Pulse-based VQE retains the hybrid variational loop of ordinary VQE but changes the structure of the classical subproblem. In ctrl-VQE simulations, analytic gradients $$E(\boldsymbol\theta)=\langle\psi(\boldsymbol\theta)|H_{\rm mol}|\psi(\boldsymbol\theta)\rangle,$$7 are computed via adjoint methods in $$E(\boldsymbol\theta)=\langle\psi(\boldsymbol\theta)|H_{\rm mol}|\psi(\boldsymbol\theta)\rangle,$$8 propagator calls, at roughly $$E(\boldsymbol\theta)=\langle\psi(\boldsymbol\theta)|H_{\rm mol}|\psi(\boldsymbol\theta)\rangle,$$9 the cost of an energy evaluation, and the full parameter vector $$U(\theta)=T\,\exp\biggl(-i\int_0^{T_{\rm tot}}H_{\rm ctrl}(t;\theta)\,dt\biggr),$$0 is updated with L-BFGS-B until the energy change falls below a threshold such as $$U(\theta)=T\,\exp\biggl(-i\int_0^{T_{\rm tot}}H_{\rm ctrl}(t;\theta)\,dt\biggr),$$1 Ha (Meitei et al., 2020).

The parameterization study for LiH adopts BFGS with analytic gradients of GRAPE type, with convergence declared when $$U(\theta)=T\,\exp\biggl(-i\int_0^{T_{\rm tot}}H_{\rm ctrl}(t;\theta)\,dt\biggr),$$2 (Sherbert et al., 2024). It explicitly recommends fixing all drive frequencies on resonance, optimizing only the two real parameters per window $$U(\theta)=T\,\exp\biggl(-i\int_0^{T_{\rm tot}}H_{\rm ctrl}(t;\theta)\,dt\biggr),$$3 in Cartesian form, using piecewise-constant windows of length $$U(\theta)=T\,\exp\biggl(-i\int_0^{T_{\rm tot}}H_{\rm ctrl}(t;\theta)\,dt\biggr),$$4 ns, initializing all amplitudes to zero, targeting total pulse duration $$U(\theta)=T\,\exp\biggl(-i\int_0^{T_{\rm tot}}H_{\rm ctrl}(t;\theta)\,dt\biggr),$$5, enforcing amplitude bounds through a smooth penalty term, and using a quasi-Newton optimizer rather than gradient-free methods (Sherbert et al., 2024).

PANSATZ emphasizes hardware-executable optimization loops. Each iteration generates an OpenPulse schedule of total duration $$U(\theta)=T\,\exp\biggl(-i\int_0^{T_{\rm tot}}H_{\rm ctrl}(t;\theta)\,dt\biggr),$$6’s, runs the schedule for $$U(\theta)=T\,\exp\biggl(-i\int_0^{T_{\rm tot}}H_{\rm ctrl}(t;\theta)\,dt\biggr),$$7 shots, measures Pauli strings for the Hamiltonian, constructs $$U(\theta)=T\,\exp\biggl(-i\int_0^{T_{\rm tot}}H_{\rm ctrl}(t;\theta)\,dt\biggr),$$8 with readout-error mitigation, and updates $$U(\theta)=T\,\exp\biggl(-i\int_0^{T_{\rm tot}}H_{\rm ctrl}(t;\theta)\,dt\biggr),$$9 with hill-climbing or SPSA (Meirom et al., 2022). The same work notes that gradients can in principle be evaluated either by parameter-shift rules or by stochastic gradient-free methods such as SPSA and discrete hill-climbing, with pulse durations discretized to the controller’s $$E(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$0 ns unit (Meirom et al., 2022).

An explicit analytic gradient framework for parametrized pulse programs is given by ODEgen. Starting from

$$E(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$1

the method differentiates the time-dependent Schrödinger equation via a differentiable ODE solver to obtain $$E(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$2 and $$E(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$3, then defines the effective generator

$$E(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$4

(Kottmann et al., 2023). Decomposing $$E(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$5 into Pauli words yields an analytic parameter-shift rule for $$E(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$6, with worst-case quantum resource count $$E(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$7 shifted circuits per parameter, where $$E(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$8 is the dimension of the dynamical Lie algebra (Kottmann et al., 2023). In simulated VQE examples for realistic superconducting transmons, ODEgen obtained lower energies with fewer quantum resources than SPS, and a pulse VQE run with gradients computed via ODEgen was demonstrated entirely on quantum hardware (Kottmann et al., 2023).

In the neutral-atom VQOC formulation, the classical update is derived from an adjoint equation with operator-valued Lagrange multiplier $$E(\theta)=\langle\psi(\theta)|H|\psi(\theta)\rangle$$9, yielding a gradient expression for the control fields $$U(\boldsymbol{\theta})=\mathcal{T}\exp\Bigl[-\,i\int_{0}^{T(\boldsymbol{\theta})} H_{\rm ctrl}\bigl(\boldsymbol{\theta},t\bigr)\,dt\Bigr]$$0 and a piecewise update rule for discretized pulse amplitudes (Keijzer et al., 2022). This places pulse-based VQE in direct continuity with quantum optimal control rather than only with VQA heuristics.

4. Demonstrated performance in molecular VQE

The earliest numerical demonstrations of ctrl-VQE focused on small molecules. For $$U(\boldsymbol{\theta})=\mathcal{T}\exp\Bigl[-\,i\int_{0}^{T(\boldsymbol{\theta})} H_{\rm ctrl}\bigl(\boldsymbol{\theta},t\bigr)\,dt\Bigr]$$1 and $$U(\boldsymbol{\theta})=\mathcal{T}\exp\Bigl[-\,i\int_{0}^{T(\boldsymbol{\theta})} H_{\rm ctrl}\bigl(\boldsymbol{\theta},t\bigr)\,dt\Bigr]$$2, each two-orbital problem was mapped to $$U(\boldsymbol{\theta})=\mathcal{T}\exp\Bigl[-\,i\int_{0}^{T(\boldsymbol{\theta})} H_{\rm ctrl}\bigl(\boldsymbol{\theta},t\bigr)\,dt\Bigr]$$3 active qubits via the parity transform. Using square pulses with $$U(\boldsymbol{\theta})=\mathcal{T}\exp\Bigl[-\,i\int_{0}^{T(\boldsymbol{\theta})} H_{\rm ctrl}\bigl(\boldsymbol{\theta},t\bigr)\,dt\Bigr]$$4 segments per qubit and total time $$U(\boldsymbol{\theta})=\mathcal{T}\exp\Bigl[-\,i\int_{0}^{T(\boldsymbol{\theta})} H_{\rm ctrl}\bigl(\boldsymbol{\theta},t\bigr)\,dt\Bigr]$$5 ns, ctrl-VQE reproduced the Full CI curve to better than $$U(\boldsymbol{\theta})=\mathcal{T}\exp\Bigl[-\,i\int_{0}^{T(\boldsymbol{\theta})} H_{\rm ctrl}\bigl(\boldsymbol{\theta},t\bigr)\,dt\Bigr]$$6 mHa at every bond length, with average error $$U(\boldsymbol{\theta})=\mathcal{T}\exp\Bigl[-\,i\int_{0}^{T(\boldsymbol{\theta})} H_{\rm ctrl}\bigl(\boldsymbol{\theta},t\bigr)\,dt\Bigr]$$7 mHa, and state overlap with the exact ground state exceeding $$U(\boldsymbol{\theta})=\mathcal{T}\exp\Bigl[-\,i\int_{0}^{T(\boldsymbol{\theta})} H_{\rm ctrl}\bigl(\boldsymbol{\theta},t\bigr)\,dt\Bigr]$$8 (Meitei et al., 2020). For $$U(\boldsymbol{\theta})=\mathcal{T}\exp\Bigl[-\,i\int_{0}^{T(\boldsymbol{\theta})} H_{\rm ctrl}\bigl(\boldsymbol{\theta},t\bigr)\,dt\Bigr]$$9, pulse durations grew from $$E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$$0 ns at equilibrium ($$E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$$1 Å) to $$E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$$2 ns at $$E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$$3 Å), whereas for $$E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$$4 they shrank from $$E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$$5 ns at equilibrium to $$E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$$6 ns at large $$E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$$7 (Meitei et al., 2020). For LiH at $$E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$$8 Å in a four-transmon simulation, an adaptive scheme increased the number of segments until chemical accuracy was reached; with $$E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle$$9 segments per qubit and $U(t)$0 ns, the energy was within $U(t)$1 mHa of FCI, leakage to higher transmon levels was below $U(t)$2, and the overlap with the exact state was $U(t)$3 (Meitei et al., 2020).

PANSATZ reported both simulation and hardware results. On IBM’s parameterized-transmon model with $U(t)$4s, $U(t)$5-level truncation, and shot noise, a one-layer PANSATZ achieved chemical accuracy $U(t)$6 mHartree) for $U(t)$7, $U(t)$8, and LiH with only $U(t)$9–$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$00 parameters and $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$01 classical iterations (Meirom et al., 2022). On IBM hardware, raw energies for $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$02 on ibm_lagos, with only readout-error mitigation, lay within chemical accuracy of FCI at multiple bond lengths, and the final schedules remained shorter than $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$03 ns even at dissociation (Meirom et al., 2022).

On cross-resonance hardware, pulse-level VQE was applied to $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$04, $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$05, and $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$06. For $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$07, both CNOT and pulse ansätze reached the full-CI curve within sampling error (Egger et al., 2023). For $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$08, with $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$09 Pauli terms in $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$10 groups, the pulse ansatz located the minimum at $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$11, closer to the ideal CI value $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$12 than the CNOT ansatz at $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$13; with readout-error mitigation, the error in the minimum angle dropped to $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$14 versus $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$15 for CNOT, and pulse energies were on average $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$16 closer to the CI minimum than CNOT, improving to $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$17 with readout-error mitigation (Egger et al., 2023). For $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$18, a depth-1 pulse ansatz reached energy $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$19 Hartree versus $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$20 Hartree for a depth-2 CNOT ansatz, and optimizing CR phases pushed the pulse result to $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$21 Hartree (Egger et al., 2023).

A different scaling strategy appears in pulse-optimized circuit elements for quantum chemistry on silicon spin qubits. Instead of optimizing an entire VQE pulse globally, that work optimizes modular single- and double-excitation circuit elements. It reports that single-excitation pulses can be implemented in less than $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$22 ns with achieved average gate fidelities $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$23, while double-excitation pulses have durations $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$24 ns with fidelities $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$25 (Gothen et al., 15 Jun 2026). Compared with gate-based implementations at $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$26 MHz, the same work gives $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$27 ns and $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$28 ns versus pulse-optimized values $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$29 ns and $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$30 ns, and states that in full UCCSD-VQE circuits the total runtime is reduced by up to a factor of $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$31 (Gothen et al., 15 Jun 2026).

5. Runtime reduction, decoherence, and noise trade-offs

A principal motivation for pulse-based VQE is schedule compression. Ctrl-VQE compares pulse durations against compiled gate schedules and reports that a two-qubit UCCSD or “RY” ansatz on a mock IBMQ device compiles to $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$32–$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$33 ns of analog pulses, whereas the optimized ctrl-VQE state preparation is $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$34 faster; for LiH, the RY circuit takes $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$35s and UCCSD $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$36s versus ctrl-VQE’s $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$37 ns (Meitei et al., 2020). Because typical transmon $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$38–$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$39s, the paper concludes that ctrl-VQE comfortably fits under decoherence while gate-based VQE circuits approach or exceed it (Meitei et al., 2020).

PANSATZ reports schedule durations up to $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$40 shorter than a gate-based ansatz. For $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$41, a transpiled Real Amplitudes layer on ibm_lagos has $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$42 ns, whereas an optimally tuned one-layer PANSATZ gives $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$43 ns, corresponding to $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$44 reduction (Meirom et al., 2022). The cross-resonance pulse-VQE study gives similar reductions across several molecules: for $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$45, $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$46 dt versus $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$47 dt; for $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$48, $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$49 dt versus $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$50 dt; and for $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$51, $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$52 dt versus $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$53 dt, corresponding to speed-ups of approximately $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$54, $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$55, and $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$56, respectively (Egger et al., 2023). That work explicitly relates the improvement to transmon coherence times on the order of $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$57–$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$58s, noting that the multi-microsecond savings in $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$59 and $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$60 translate into lower state-preparation noise (Egger et al., 2023).

The runtime advantage is accompanied by control-specific trade-offs. Ctrl-VQE observes that short, high-amplitude pulses can induce leakage out of the computational subspace, reaching up to $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$61 in some cases, and suggests mitigation by unnormalized cost functions or penalty terms (Meitei et al., 2020). The same work reports that imperfect parameter setting with Gaussian noise up to $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$62 still yields sub-$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$63 Ha energy errors (Meitei et al., 2020). The parameterization study finds a large peak in iteration count near the effective minimal evolution time $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$64, attributing this to amplitude-bound saturation, and shows that including detuning greatly increases iterations without improving $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$65 (Sherbert et al., 2024). The cross-resonance study notes that short, high-amplitude pulses may induce some leakage, but in its numerical controls and hardware runs leakage did not prevent convergence (Egger et al., 2023).

A common misconception is that pulse-level control necessarily removes all noise bottlenecks. The published results do not support that conclusion. They instead show a narrower claim: shortening the state-preparation schedule reduces decoherence exposure and can improve measured energies, but leakage, calibration dependence, and optimization ruggedness remain intrinsic constraints (Meirom et al., 2022, Egger et al., 2023, Sherbert et al., 2024).

6. Structured adaptivity, subspace methods, and hybrid formulations

Several pulse-based VQE variants use the control layer not only to shorten schedules but also to modify ansatz structure adaptively. PANSATZ emphasizes “structured adaptivity”: because each CR pulse duration $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$66 is a free variable, setting $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$67 removes the corresponding entangler or even an entire layer (Meirom et al., 2022). The work interprets this as adaptivity akin to ADAPT-VQE but without operator selection overhead, and reports that plots of $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$68 versus bond distance rise smoothly from $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$69 to their optimal values as more entanglement is required (Meirom et al., 2022).

A more explicit adaptive segmentation strategy is used for Rydberg-atom VQE. There, time is discretized into $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$70 segments, and an iterative loop randomly selects a segment, splits it at a random time $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$71, interpolates the new $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$72, reoptimizes, and repeats until the relative energy error $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$73 falls below a threshold $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$74 (Nagao et al., 25 Jul 2025). Using this ctrl-VQE-inspired procedure, the ground states of the one-dimensional antiferromagnetic Heisenberg model and mixed-field Ising model were accurately prepared for systems up to ten qubits (Nagao et al., 25 Jul 2025). For example, in the Heisenberg model, $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$75 reached best $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$76 in $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$77 segments; $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$78 reached best $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$79 after $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$80 segments; and $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$81 with $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$82 initialization reached best $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$83 after $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$84 segments (Nagao et al., 25 Jul 2025). For the mixed-field Ising model at $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$85, mean relative errors were $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$86 across $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$87, with best runs converging in $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$88–$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$89 segments (Nagao et al., 25 Jul 2025).

Pulse-based VQE has also been combined with contextual-subspace methods in SpacePulse. There, the molecular Hamiltonian is partitioned into a noncontextual component $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$90 that is solved classically and a contextual remainder $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$91 projected into a smaller subspace $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$92, after which only the contextual correction

$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$93

is variationally minimized (Liang et al., 2023). SpacePulse couples this with Pauli grouping based on a commutation graph and reports roughly a factor-of-three reduction in measurement settings (Liang et al., 2023). It also reports strong qubit-count compression: for NH, the original parity-mapped Hamiltonian requires $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$94 qubits, tapering reduces this to $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$95, and contextual subspace with one-qubit threshold reduces the quantum workload to $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$96 qubit; for BeH$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$97, $$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$98; for F$$U(\boldsymbol\theta)=\mathcal T\exp\Bigl(-i\!\int_0^T\!H(t)\,\mathrm dt\Bigr),$$99, $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$00 (Liang et al., 2023).

These developments indicate that pulse-based VQE is compatible with both ansatz-adaptive and problem-reduction techniques. A plausible implication is that pulse control and Hamiltonian-structure reduction address complementary resource bottlenecks: the former primarily compresses schedule duration, while the latter reduces qubit and measurement cost.

7. Practical heuristics, limitations, and research directions

The literature identifies several recurring practical heuristics. For transmon ctrl-VQE, the most explicit guidance is to use piecewise-constant windows of moderate granularity, initialize from the zero pulse so that the initial state remains the Hartree–Fock reference, and exploit analytic gradients with quasi-Newton optimization (Sherbert et al., 2024). That study further concludes that Cartesian $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$01 controls and polar $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$02 controls achieve identical effective minimal evolution times, but Cartesian optimization is $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$03–$$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$04 faster in iteration count, while varying detunings adds optimizer difficulty without improving minimal time (Sherbert et al., 2024). PANSATZ similarly fixes pulse-shape hyperparameters from device-native calibrations and optimizes only durations and virtual phases, thereby keeping the variational parameter count comparable to a gate-based ansatz (Meirom et al., 2022).

Limitations are also consistent across platforms. PANSATZ states that the ideal unitary $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$05 is not known analytically, reducing interpretability of ansatz states, and notes hardware-specific hyperparameter choices and potentially rugged pulse landscapes (Meirom et al., 2022). The cross-resonance pulse-VQE work remarks that pulse-level error-mitigation protocols such as probabilistic error cancellation and zero-noise extrapolation become more involved because the ideal CR operation and its noise model become $$|\psi(\boldsymbol\theta)\rangle=U(\boldsymbol\theta)|\psi_0\rangle,$$06-dependent (Egger et al., 2023). The Rydberg-array study finds that adaptive segmentation can generate sharp pulse features as the number of segments grows, potentially challenging experimental pulse-shaping bandwidth, and that classical optimization cost grows rapidly with the number of segments (Nagao et al., 25 Jul 2025).

Several forward directions are explicitly proposed. These include advanced error mitigation such as zero-noise extrapolation via pulse stretching and probabilistic error cancellation tailored to pulses (Meirom et al., 2022); pulse-level ADAPT-VQE, symmetry rotations such as WAHTOR, and chemistry-inspired pulse ansätze (Egger et al., 2023); analytic derivative methods such as ODEgen for hardware-executable gradient estimation (Kottmann et al., 2023); and modular offline optimization of pulse-compiled single- and double-excitation elements for scalable UCC-style chemistry (Gothen et al., 15 Jun 2026). The modular strategy is especially notable because it addresses a limitation already identified for whole-ansatz pulse optimization: as problem sizes increase, optimizing a single pulse that implements an entire VQE ansatz quickly becomes intractable, whereas optimized circuit elements can be reused as pulse modules (Gothen et al., 15 Jun 2026).

Taken together, pulse-based VQE represents a shift from circuit compilation to direct control synthesis. Across superconducting transmons, silicon spin qubits, neutral atoms, and contextual-subspace hybrids, the central empirical result is consistent: by embedding the variational layer into the pulse schedule, one can shorten state preparation substantially while retaining variational flexibility for accurate ground-state estimation (Meitei et al., 2020, Meirom et al., 2022, Egger et al., 2023). The exact form of the advantage, however, is architecture dependent and mediated by controllability, leakage, calibration overhead, and optimizer behavior rather than by schedule compression alone.