Gate-free state preparation for fast variational quantum eigensolver simulations

The paper addresses the key challenge in applying the variational quantum eigensolver (VQE) to molecular electronic structure on NISQ devices: circuit depth is limited by short coherence times, noise, and gate errors. Traditional VQE uses parameterized quantum circuits (ansätze), where deeper circuits provide greater expressibility but exceed hardware limits. The authors propose replacing the circuit with direct optimization of hardware-level analog control pulses (ctrl-VQE) to prepare target quantum states more rapidly, reducing the required coherence time. The study is motivated by the exponential classical cost of storing wavefunctions and by the potential of quantum devices to represent states natively. The purpose is to achieve chemically accurate energies with much shorter state-preparation times, enabling larger or more strongly correlated problems on NISQ hardware.

The authors situate their work within: (1) classical electronic structure methods, which struggle with strong correlation; (2) quantum VQE approaches with various ansätze (hardware-efficient, physically motivated, adaptive) that still require deep circuits; (3) circuit compilation techniques, including GRAPE-based optimal control compilers and partial compilation strategies, which can reduce pulse time but suffer from compilation latency and scalability issues; and (4) mappings from fermions to qubits (Jordan-Wigner, Bravyi-Kitaev, parity) and qubit tapering. Prior benchmarks on small molecules (e.g., H2, HeH+) have been done with gate-based methods. The literature highlights a tradeoff between expressibility and hardware limits and motivates direct pulse-level control as a more feasible near-term strategy.

The proposed control variational quantum eigensolver (ctrl-VQE) replaces the parameterized quantum circuit with a parameterized laboratory-frame pulse representation, optimized variationally to minimize the molecular Hamiltonian’s energy. Key steps: (1) Compute one- and two-electron integrals and map the second-quantized Hamiltonian to qubits (e.g., Jordan-Wigner, parity, or Bravyi-Kitaev). (2) Choose a pulse representation (e.g., piecewise-constant/square pulses or sums of Gaussians), parameterize amplitudes and drive frequencies, and initialize parameters. (3) Initialize the device in a reference state (Hartree–Fock). (4) Measure the expectation value of the mapped molecular Hamiltonian on the device. (5) Use a classical optimizer to update pulse parameters. (6) Iterate until convergence. The total pulse duration T is treated as a hyper-parameter (fixed during each optimization but varied across experiments). The transmon device model: a 1D chain with always-on couplings is described by H0 = Σk ωk a†k ak − (δk/2) a†k ak a†k ak + Σ(kl) gkl (a†k al + ak a†l). Control drives are applied to each transmon with Hc = Σk Ωk(t)(a†k e^{iνk t} + ak e^{-iνk t}). The total Hamiltonian H = H0 + Hc yields an entangling interaction in the interaction frame due to the static couplings. The ansatz state is |ψtrial(Ω(t),ν)⟩ = T exp[−i ∫0^T dt′ Hc(t′,Ω(t′),ν)] |ψ0⟩. Although Hc comprises single-qubit terms, inter-qubit couplings generate entangling dynamics. The objective is E(Ω(t),ν) = ⟨ψtrial|Hmolecule|ψtrial⟩ (normalized; an unnormalized objective was also tested). Pulse parameterizations: piecewise-constant (square) pulses with n time segments per qubit, Ω(t) = ci on segment i with fixed switching times; and Gaussian sums (results similar; Gaussian data in SI). Typical constraints reported: amplitudes within ±20 MHz and drive frequencies within ω ± 2 GHz (in Methods); and in adaptive studies, amplitudes within ±40 MHz and frequencies within ω ± 3 GHz. For N transmons and n segments per transmon, there are 2Nn parameters (amplitude and frequency per segment per qubit). Optimization uses L-BFGS-B; analytical gradients of energy with respect to pulse amplitudes are implemented, costing ~2.5× an energy evaluation. Molecular mappings: For H2 and HeH+, parity mapping with qubit tapering removes two qubits (two-transmon simulations). For LiH, a four-transmon model is used. Classical simulation and code: CtrlQ (open-source), with checks via Qiskit and QuTiP; integrals from PySCF (STO-3G). Device parameters typical of superconducting transmons are provided (Table 2). An adaptive pulse-parameter growth scheme is introduced to avoid over-parameterization: start with a single time segment, iteratively split segments (increasing n) and reoptimize until target accuracy is reached; drive frequencies are optimized; switching times are fixed after selection. Leakage (population outside computational |0⟩,|1⟩ states) is monitored, and mitigation via post-selection or unnormalized objectives is discussed. Noise robustness is assessed by sampling Gaussian noise on optimal parameters and averaging energies.

- Accuracy on bond dissociation curves: For H2 and HeH+ using simple square pulses with two time segments per qubit, ctrl-VQE reproduces FCI dissociation curves with maximum deviation 0.03 mHa and average error 0.002 mHa for both molecules. State overlaps with FCI exceed 99% along the curves. For H2 at large bond distances (singlet–triplet degeneracy), adding a spin penalty can enforce the singlet ground state. - Pulse duration vs correlation: Stronger correlation requires longer pulses. As bond length increases, H2 requires longer pulse durations, while HeH+ requires shorter pulses; at large separations for HeH+, total pulse duration can be as short as ~1.0 ns. Example optimized total pulse durations at selected geometries for H2/HeH+ include minima around 9 ns near equilibrium. - Leakage dynamics: Energy trajectory during evolution is non-monotonic; different optimized pulses (for a fixed T) vary in leakage. Some solutions show up to ~10% leakage; this can be handled by increased shot counts or minimized by using an unnormalized energy objective. Example evolutions are shown for H2 at R=0.75 Å with T in {9, 15, 21, 29} ns and for HeH+ at R=0.90 Å with T in {9, 14, 21, 24, 30} ns. - Robustness to control noise: Adding Gaussian noise to each optimal parameter with σ ∈ {1e−4, 1e−3, 1e−2, 1e−1} shows that accuracies better than 1e−4 Hartree are maintained up to σ ≈ 1e−2; significant degradation appears only for σ ≥ 0.01 (errors ~10 MHz or ~1 ns), comparable to parameter magnitudes. Lindblad simulations including T1/T2 show negligible impact due to the short pulse times. - Decompiled circuits vs pulses: For H2 at 0.75 Å, the ctrl-VQE pulse duration is 9 ns. The equivalent circuit obtained via: (a) KAK unitary decomposition yields 1202 ns; (b) arbitrary circuit for the state vector yields 1294 ns; (c) transpiled version yields 825 ns. This highlights substantial overhead in gate-based execution compared to direct pulses. - Adaptive pulse growth and LiH: Using the adaptive scheme, total pulse durations achieving chemical accuracy: H2 (1 segment suffices; converges by 2 segments) and HeH+ (1 segment; converges by 2). For LiH at R=1.5 Å (4 transmons), chemical accuracy requires 3 segments and converges by 5 segments with total pulse duration 40 ns. LiH performance (5 segments): ctrl-VQE energy −7.8806399 Ha vs FCI −7.8810157 Ha (error 0.0004 Ha), leakage 0.59%, overlap 99.38%. Multiple random restarts indicate the 40 ns pulse approaches the achievable accuracy under these constraints. - Comparison to gate-based ansätze (based on IBMQ mock Johannesburg calibrations): For H2 and HeH+, RY ansatz (4 params, 1 CNOT) takes 519 ns; UCCSD (3 params, 2 CNOTs) takes 825 ns; ctrl-VQE takes 9 ns. For LiH, RY (16 params, 9 CNOTs) takes 3485 ns; UCCSD (8 params, 245 CNOTs) takes 82,169 ns; ctrl-VQE takes 40 ns. Device T1 averages ~71,262 ns and T2 ~74,490 ns; UCCSD LiH durations approach decoherence limits, whereas ctrl-VQE is far below.

The results demonstrate that direct pulse-level optimization (ctrl-VQE) can prepare high-quality molecular ground states with orders-of-magnitude shorter state-preparation times than gate-based VQE, directly addressing the primary bottleneck of circuit depth and coherence time in NISQ devices. By exploiting the natural dynamics of coupled transmons and optimizing analog controls, ctrl-VQE achieves chemical accuracy for small molecules while minimizing exposure to decoherence and gate errors. The observed correlation between electron correlation strength and minimal pulse duration (longer for strongly correlated H2 at stretched bonds, shorter for HeH+ as it dissociates to closed shells) supports the intuition that entanglement generation requires finite time. Leakage, while present for some pulses, can be mitigated via post-selection or objective design and need not be a fundamental limitation. Robustness analyses show the approach tolerates realistic imperfections in control parameters. Decompilation studies quantify the substantial overhead of gate execution relative to direct pulses, reinforcing the efficiency gains. The adaptive pulse-parameterization scheme avoids over-parameterization, improves convergence, and scales modestly in parameter count while maintaining short total durations, as shown for LiH. Overall, ctrl-VQE provides a practical route to more accurate and larger-scale VQE simulations on NISQ hardware by shifting from circuits to hardware-native control.

The paper introduces ctrl-VQE, a gate-free variational algorithm that directly optimizes hardware-level control pulses to prepare molecular ground states rapidly and accurately on NISQ devices. Numerical demonstrations on H2 and HeH+ reproduce FCI dissociation curves within chemical accuracy with pulse durations around 9 ns, and on LiH achieve 0.0004 Ha error with a 40 ns pulse and high overlap while keeping leakage below 1%. Compared to gate-based ansätze and compiled circuits, ctrl-VQE achieves state preparation that is up to three orders of magnitude faster, greatly alleviating coherence-time constraints. The adaptive pulse-parameterization strategy efficiently balances expressibility and practicality. Future work will improve software implementations for larger systems, incorporate more sophisticated pulse-shape constraints, and study how device controllability impacts the description of electron correlation, moving toward scalable, hardware-native variational quantum simulations.

- Current demonstrations are limited to small molecules (H2, HeH+, LiH) in minimal STO-3G basis due to the high classical cost of simulating pulse-level dynamics and optimization. - Optimization landscapes can be challenging; over-parameterization impedes convergence, necessitating adaptive parameter growth strategies. - Reported device parameters and comparisons use simulated/mock hardware; exact device-specific parameters (anharmonicities, couplings) may vary across platforms. - Pulse leakage into non-computational states can increase shot requirements; while mitigable, it adds overhead. - Noise modeling focused on control parameter imprecision; explicit decoherence/dephasing was not a major factor due to short pulses, but broader noise sources and calibration drifts on real hardware may affect performance. - Some constraints (amplitude/frequency bounds, fixed switching times) may limit achievable accuracy for larger systems; LiH results suggest reaching an accuracy limit with 40 ns pulses under given constraints.