---
title: 'Pulse-Based VQE: Direct Control Optimization'
url: https://www.emergentmind.com/topics/pulse-based-variational-quantum-eigensolver-vqe
type: topic
---

# Pulse-Based VQE: Direct Control Optimization

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 [2008.04302]. 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 [2212.12911], [2303.02410], [2202.08908], [2311.17423], [2309.16756], [2606.17357].

## 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 [2008.04302]. 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 [2008.04302].

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 [2212.12911]. 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 \(E(\boldsymbol{\theta})=\langle\psi(\boldsymbol{\theta})|H|\psi(\boldsymbol{\theta})\rangle\) [2303.02410]. In the pulse-based variational quantum optimal control formulation for neutral atoms, the same principle is expressed through the Schrödinger propagator \(U(t)\) under \(H[z(t)]=H_d+H_c[z(t)]\), with a terminal cost
\[
J(U,z)=\langle\psi(T)|H_{\rm mol}|\psi(T)\rangle+\frac{\lambda}{2}\sum_{l=1}^L\int_0^T|z_l(t)|^2\,dt
\]
and often simply
\[
E(z)=\langle\psi(T;z)|H_{\rm mol}|\psi(T;z)\rangle
\]
[2202.08908].

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
\[
H_{\rm sys}
=\sum_{k=1}^N\Bigl(\omega_k\,a_k^\dagger a_k-\tfrac{\delta_k}{2}\,a_k^\dagger a_k^\dagger a_k a_k\Bigr)
+\sum_{\langle k\ell\rangle}g\,(a_k^\dagger a_\ell+a_\ell^\dagger a_k),
\]
and control terms
\[
H_k=c_k(t)\,(\mathrm e^{i\nu_k t}a_k+\mathrm e^{-i\nu_k t}a_k^\dagger),
\]
where \(\omega_k,\delta_k\) are the \(k\)th transmon’s frequency and anharmonicity, \(g\) the nearest-neighbor coupling, and \(\nu_k\) the drive frequency [2008.04302]. The control pulses can be piecewise-constant square pulses divided into \(n\) segments, with fixed total time \(T\) and variational amplitudes \(\{c_{k,j}\}\), optionally together with switching times \(\Delta t_j\). In that formulation, amplitudes satisfy \(\lvert c_{k,j}\rvert\le20\text{ MHz}\), and frequencies lie in \(\omega_k\pm2\pi\text{ GHz}\) [2008.04302].

A more systematic study of parameterization appears in the analysis of pulse-level VQE optimizability. There, the drive on qubit \(q\) is written in the lab frame as
\[
D_q(t)=A_q(t)\cos[\nu_q(t)t+\phi_q(t)],
\]
which under the rotating-wave approximation yields
\[
V_q^{(R)}(t)=\Omega_q(t)\,e^{i\Delta_q(t)t}a_q+\text{h.c.},
\]
with \(\Delta_q(t)=\nu_q(t)-\omega_q\) and
\[
\Omega_q(t)=A_q(t)e^{i\phi_q(t)}
\quad\text{or}\quad
\Omega_q(t)=\alpha_q(t)+i\beta_q(t)
\]
[2405.15166]. The paper investigates five finite-dimensional strategies: \(\{\alpha\beta\}\), \(\{A\phi\}\), \(\{\alpha\}\), \(\{\alpha\Delta\}\), and \(\{\alpha\beta\Delta\}\), all implemented as piecewise-constant windowed pulses of total duration \(T\) divided into \(W\) windows [2405.15166].

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 \(\tau\) and in-plane phases \(\phi\) as the variational parameters [2212.12911]. 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
\[
A(\theta)=\sin(\theta),\qquad
\tau(\theta)=\tfrac{16}{\pi}\bigl\lfloor\tfrac{\pi}{16}\sin(\theta)\bigr\rfloor
\]
to ensure amplitude in \([ -1,1]\) and \(\tau\) a multiple of \(16\,dt\) [2303.02410]. In neutral-atom optical tweezer arrays, pulse-based VQE uses the native Rydberg Hamiltonian with global detuning \(\Delta(t)\), global Rabi frequency \(\Omega(t)\), and geometry \(G\), with piecewise-linear interpolation between control values \(\{\Omega_i,\Delta_i\}_{i=0}^M\) [2507.19153].

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 \(\partial E/\partial c_{k,j}\) are computed via adjoint methods in \(\mathcal O(1)\) propagator calls, at roughly \(2.5\times\) the cost of an energy evaluation, and the full parameter vector \(\boldsymbol\theta=\{c_{k,j},\nu_k\}\) is updated with L-BFGS-B until the energy change falls below a threshold such as \(\Delta E<10^{-6}\) Ha [2008.04302].

The parameterization study for LiH adopts BFGS with analytic gradients of GRAPE type, with convergence declared when \(\max|\partial E/\partial\theta_k|<10^{-6}\) [2405.15166]. It explicitly recommends fixing all drive frequencies on resonance, optimizing only the two real parameters per window \(\{\alpha_{qj},\beta_{qj}\}\) in Cartesian form, using piecewise-constant windows of length \(s\approx3\) ns, initializing all amplitudes to zero, targeting total pulse duration \(T\approx1.2\times T_0\), enforcing amplitude bounds through a smooth penalty term, and using a quasi-Newton optimizer rather than gradient-free methods [2405.15166].

PANSATZ emphasizes hardware-executable optimization loops. Each iteration generates an OpenPulse schedule of total duration \(T_{\rm tot}(\theta)=\sum\tau\)’s, runs the schedule for \(10\,000\) shots, measures Pauli strings for the Hamiltonian, constructs \(E(\theta)\) with readout-error mitigation, and updates \(\theta\) with hill-climbing or SPSA [2212.12911]. 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 \(0.222\) ns unit [2212.12911].

An explicit analytic gradient framework for parametrized pulse programs is given by ODEgen. Starting from
\[
H(\theta,t)=H_{\rm drift}+\sum_{j=1}^{N_g}f_j(\theta,t)H_j,
\]
the method differentiates the time-dependent Schrödinger equation via a differentiable ODE solver to obtain \(U(\theta)\) and \(V_j=\partial_{\theta_j}U(\theta)\), then defines the effective generator
\[
G_j(\theta)\equiv i\,U(\theta)^\dagger\partial_{\theta_j}U(\theta)
\]
[2309.16756]. Decomposing \(G_j\) into Pauli words yields an analytic parameter-shift rule for \(\partial_{\theta_j}E(\theta)\), with worst-case quantum resource count \(2d_{\rm DLA}\) shifted circuits per parameter, where \(d_{\rm DLA}\) is the dimension of the dynamical Lie algebra [2309.16756]. 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 [2309.16756].

In the neutral-atom VQOC formulation, the classical update is derived from an adjoint equation with operator-valued Lagrange multiplier \(P(t)\), yielding a gradient expression for the control fields \(z_l(t)\) and a piecewise update rule for discretized pulse amplitudes [2202.08908]. 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 \(\mathrm H_2\) and \(\mathrm{HeH}^+\), each two-orbital problem was mapped to \(2\) active qubits via the parity transform. Using square pulses with \(n=2\) segments per qubit and total time \(T\approx9\) ns, ctrl-VQE reproduced the Full CI curve to better than \(0.03\) mHa at every bond length, with average error \(\sim0.002\) mHa, and state overlap with the exact ground state exceeding \(99\%\) [2008.04302]. For \(\mathrm H_2\), pulse durations grew from \(\sim9\) ns at equilibrium (\(R=0.75\) Å) to \(\sim25\) ns at \(R=2.5\) Å), whereas for \(\mathrm{HeH}^+\) they shrank from \(\sim12\) ns at equilibrium to \(\sim1\) ns at large \(R\) [2008.04302]. For LiH at \(R=1.5\) Å in a four-transmon simulation, an adaptive scheme increased the number of segments until chemical accuracy was reached; with \(n=5\) segments per qubit and \(T=40\) ns, the energy was within \(0.4\) mHa of FCI, leakage to higher transmon levels was below \(1\%\), and the overlap with the exact state was \(99.4\%\) [2008.04302].

PANSATZ reported both simulation and hardware results. On IBM’s parameterized-transmon model with \(T_1=T_2=100\,\mu\)s, \(3\)-level truncation, and shot noise, a one-layer PANSATZ achieved chemical accuracy \((\pm1.6\) mHartree) for \(\mathrm H_2\), \(\mathrm{HeH}^+\), and LiH with only \(5\)–\(11\) parameters and \(\sim50\) classical iterations [2212.12911]. On IBM hardware, raw energies for \(\mathrm H_2\) 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 \(400\) ns even at dissociation [2212.12911].

On cross-resonance hardware, pulse-level VQE was applied to \(\mathrm H_2\), \(\mathrm H_3\), and \(\mathrm H_4\). For \(\mathrm H_2\), both CNOT and pulse ansätze reached the full-CI curve within sampling error [2303.02410]. For \(\mathrm H_3\), with \(62\) Pauli terms in \(21\) groups, the pulse ansatz located the minimum at \(30.1^\circ\), closer to the ideal CI value \(29.3^\circ\) than the CNOT ansatz at \(36.4^\circ\); with readout-error mitigation, the error in the minimum angle dropped to \(27.7^\circ\) versus \(38.2^\circ\) for CNOT, and pulse energies were on average \(33\%\) closer to the CI minimum than CNOT, improving to \(52\%\) with readout-error mitigation [2303.02410]. For \(\mathrm H_4\), a depth-1 pulse ansatz reached energy \(\simeq-4.39\) Hartree versus \(\simeq-4.26\) Hartree for a depth-2 CNOT ansatz, and optimizing CR phases pushed the pulse result to \(-4.44\) Hartree [2303.02410].

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 \(289\) ns with achieved average gate fidelities \(\ge99.99\%\), while double-excitation pulses have durations \(\lesssim927\) ns with fidelities \(\ge99.9\%\) [2606.17357]. Compared with gate-based implementations at \(\Delta/h=8\) MHz, the same work gives \(T_{SQ}^{\rm gate}=942\) ns and \(T_{DQ}^{\rm gate}=4692\) ns versus pulse-optimized values \(T_{SQ}^{\rm pulse}\approx259\) ns and \(T_{DQ}^{\rm pulse}\approx1946\) ns, and states that in full UCCSD-VQE circuits the total runtime is reduced by up to a factor of \(15.3\) [2606.17357].

## 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 \(\sim500\)–\(800\) ns of analog pulses, whereas the optimized ctrl-VQE state preparation is \(\mathcal O(100)\times\) faster; for LiH, the RY circuit takes \(\sim3.5\,\mu\)s and UCCSD \(\sim82\,\mu\)s versus ctrl-VQE’s \(40\) ns [2008.04302]. Because typical transmon \(T_{1,2}^*\sim50\)–\(100\,\mu\)s, the paper concludes that ctrl-VQE comfortably fits under decoherence while gate-based VQE circuits approach or exceed it [2008.04302].

PANSATZ reports schedule durations up to \(7\times\) shorter than a gate-based ansatz. For \(\mathrm H_2\), a transpiled Real Amplitudes layer on ibm_lagos has \(T\approx800\) ns, whereas an optimally tuned one-layer PANSATZ gives \(T\approx120\) ns, corresponding to \(\sim6.7\times\) reduction [2212.12911]. The cross-resonance pulse-VQE study gives similar reductions across several molecules: for \(\mathrm H_2\), \(3360\) dt versus \(928\pm114\) dt; for \(\mathrm H_3\), \(9184\) dt versus \(\sim2700\) dt; and for \(\mathrm H_4\), \(\sim48\,700\) dt versus \(10\,000\) dt, corresponding to speed-ups of approximately \(3.6\times\), \(3.4\times\), and \(4.8\times\), respectively [2303.02410]. That work explicitly relates the improvement to transmon coherence times on the order of \(50\)–\(100\,\mu\)s, noting that the multi-microsecond savings in \(\mathrm H_3\) and \(\mathrm H_4\) translate into lower state-preparation noise [2303.02410].

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 \(10\%\) in some cases, and suggests mitigation by unnormalized cost functions or penalty terms [2008.04302]. The same work reports that imperfect parameter setting with Gaussian noise up to \(\sigma\sim10^{-2}\) still yields sub-\(10^{-4}\) Ha energy errors [2008.04302]. The parameterization study finds a large peak in iteration count near the effective minimal evolution time \(T_0\), attributing this to amplitude-bound saturation, and shows that including detuning greatly increases iterations without improving \(T_0\) [2405.15166]. 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 [2303.02410].

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 [2212.12911], [2303.02410], [2405.15166].

## 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 \(\tau_{\ell,(i,j)}^{CR}\) is a free variable, setting \(\tau^{CR}\to0\) removes the corresponding entangler or even an entire layer [2212.12911]. The work interprets this as adaptivity akin to ADAPT-VQE but without operator selection overhead, and reports that plots of \(\tau^{CR}\) versus bond distance rise smoothly from \(0\) to their optimal values as more entanglement is required [2212.12911].

A more explicit adaptive segmentation strategy is used for Rydberg-atom VQE. There, time is discretized into \(M\) segments, and an iterative loop randomly selects a segment, splits it at a random time \(t_s\), interpolates the new \((\Omega_s,\Delta_s)\), reoptimizes, and repeats until the relative energy error \(\eta_{\rm err}\) falls below a threshold \(\epsilon\) [2507.19153]. 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 [2507.19153]. For example, in the Heisenberg model, \(N=4\) reached best \(\eta_{\rm err}<0.01\%\) in \(3\) segments; \(N=8\) reached best \(\eta\approx0.1\%\) after \(50\) segments; and \(N=10\) with \(q=\pi\) initialization reached best \(\eta\sim0.16\%\) after \(50\) segments [2507.19153]. For the mixed-field Ising model at \(N=10\), mean relative errors were \(\lesssim0.02\%\) across \(h_x\in\{0.8,1.0,1.2,1.8\}\), with best runs converging in \(4\)–\(10\) segments [2507.19153].

Pulse-based VQE has also been combined with contextual-subspace methods in SpacePulse. There, the molecular Hamiltonian is partitioned into a noncontextual component \(H_{nc}\) that is solved classically and a contextual remainder \(H_c\) projected into a smaller subspace \(W\), after which only the contextual correction
\[
E_{\rm corr}(\theta)=
\frac{\langle0|U_{\rm pulse}(\theta)^\dagger H_c^W U_{\rm pulse}(\theta)|0\rangle}
{\langle0|U_{\rm pulse}(\theta)^\dagger U_{\rm pulse}(\theta)|0\rangle}
\]
is variationally minimized [2311.17423]. SpacePulse couples this with Pauli grouping based on a commutation graph and reports roughly a factor-of-three reduction in measurement settings [2311.17423]. It also reports strong qubit-count compression: for NH, the original parity-mapped Hamiltonian requires \(12\) qubits, tapering reduces this to \(4\), and contextual subspace with one-qubit threshold reduces the quantum workload to \(1\) qubit; for BeH\(^+\), \(12\to5\to1\); for F\(_2\), \(20\to6\to1\) [2311.17423].

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 [2405.15166]. That study further concludes that Cartesian \(\{\alpha,\beta\}\) controls and polar \(\{A,\phi\}\) controls achieve identical effective minimal evolution times, but Cartesian optimization is \(2\)–\(3\times\) faster in iteration count, while varying detunings adds optimizer difficulty without improving minimal time [2405.15166]. 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 [2212.12911].

Limitations are also consistent across platforms. PANSATZ states that the ideal unitary \(U(\theta)\) is not known analytically, reducing interpretability of ansatz states, and notes hardware-specific hyperparameter choices and potentially rugged pulse landscapes [2212.12911]. 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 \(\theta\)-dependent [2303.02410]. 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 [2507.19153].

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 [2212.12911]; pulse-level ADAPT-VQE, symmetry rotations such as WAHTOR, and chemistry-inspired pulse ansätze [2303.02410]; analytic derivative methods such as ODEgen for hardware-executable gradient estimation [2309.16756]; and modular offline optimization of pulse-compiled single- and double-excitation elements for scalable UCC-style chemistry [2606.17357]. 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 [2606.17357].

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 [2008.04302], [2212.12911], [2303.02410]. 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.

Source: https://www.emergentmind.com/topics/pulse-based-variational-quantum-eigensolver-vqe