Gate-free state preparation for fast variational quantum eigensolver simulations

Molecular modeling is crucial in various fields, including energy storage, materials design, and drug discovery. Traditional methods like density functional theory (DFT) and coupled cluster struggle with strongly correlated systems where electron correlation effects dominate. Quantum computing offers a potential solution by mapping the molecular wavefunction directly onto a quantum processing unit (QPU). The variational quantum eigensolver (VQE) is a prominent algorithm for this, using parameterized quantum circuits to prepare a target state and measure the molecular energy. However, limitations of noisy intermediate-scale quantum (NISQ) devices, including short coherence times and gate errors, restrict the depth of these circuits. This paper explores an alternative approach, where instead of using parameterized quantum circuits, the state preparation is achieved by directly optimizing the device-level pulse shapes. This 'gate-free' approach aims to bypass the limitations imposed by gate-based methods on NISQ devices, allowing for faster state preparation and potentially enabling the simulation of larger, more complex molecules.

The paper reviews existing classical methods for molecular modeling, highlighting the limitations of approximate methods like DFT and coupled cluster for strongly correlated systems. It then discusses existing quantum algorithms, focusing on the VQE algorithm and its limitations in the context of NISQ devices. The authors mention various approaches to improve gate-based VQE, such as hardware-efficient ansätze and physically motivated fixed ansätze, all aimed at minimizing circuit depth. The authors note that even the best gate-based approaches may still exceed the coherence times of NISQ devices, motivating the exploration of alternative strategies, including improved error mitigation, increasing the coherence-to-gate-time ratio, or abandoning gates altogether in favor of direct pulse optimization.

The proposed method, ctrl-VQE, replaces the parameterized quantum circuit in standard VQE with a direct optimization of laboratory-frame analog control settings. The authors argue that quantum control techniques offer advantages over circuit-based approaches for state preparation on NISQ devices. The study focuses on the optimization of pulse shapes, specifically square pulses and Gaussian pulses, using techniques like I-BFGS-b. The molecular Hamiltonian is transformed into a qubit representation (using methods like Jordan-Wigner or parity transformations). The initial state is typically the Hartree-Fock state. The expectation value of the molecular Hamiltonian is calculated, and the pulse parameters are iteratively optimized using a classical optimization routine to minimize the energy. The authors describe an adaptive scheme for increasing the number of pulse parameters (time segments) to efficiently find the minimal number needed to achieve target accuracy, preventing over-parameterization. To assess the efficiency of ctrl-VQE, the unitary operator represented by the optimized pulse is decompiled into a quantum circuit using techniques like KAK decomposition and then transpiled to estimate the execution time for a comparison with gate-based approaches. The simulations use a transmon platform model, considering the device Hamiltonian, control Hamiltonian, and the time evolution under the total Hamiltonian. The calculations involve the use of molecular integrals generated using PySCF with an STO-3G basis set.

Ctrl-VQE accurately reproduces Full Configuration Interaction (FCI) bond dissociation energy curves for H₂ and HeH⁺, with maximum energy differences from FCI less than 0.03 mHa. The average error is 0.002 mH. The state overlaps with the exact FCI ground states are around 99%. The pulse durations reflect the different dynamics of bond dissociation in the two molecules, increasing for H₂ (homolytic dissociation with increasing multiconfigurational character) and decreasing for HeH⁺ (heterolytic dissociation approaching a single-determinant description). The study examines the relationship between pulse duration and electron correlation, observing that strongly correlated molecules require longer pulses. For LiH, an adaptive scheme for pulse parameterization was employed. Chemical accuracy is achieved with single-segment square pulses for H₂ and HeH⁺, and three segments for LiH. Decompilation of the optimized ctrl-VQE pulses into equivalent quantum circuits shows that the gate-based state preparation circuits can be significantly deeper than necessary. A noise analysis demonstrates the robustness of ctrl-VQE to imprecise control parameters, achieving accuracies below 10⁻⁴ energy error even with substantial noise (σ ≤ 0.01). A comparison with gate-based ansätze (RY and UCCSD) reveals that ctrl-VQE achieves state preparation times that are three orders of magnitude shorter, effectively mitigating the impact of decoherence and dephasing.

The results demonstrate that ctrl-VQE offers a significant advantage over gate-based VQE methods for molecular simulations on NISQ devices. The gate-free approach dramatically reduces the state preparation time, making it more resistant to noise and decoherence. The observed correlation between pulse duration and the degree of electron correlation in the molecule underscores the method’s ability to capture essential physical effects. The success in applying ctrl-VQE to a larger molecule like LiH, while requiring an adaptive parameterization scheme, illustrates its potential scalability. The relatively modest number of parameters needed to achieve convergence even for LiH further emphasizes the efficiency of this approach.

Ctrl-VQE offers a novel, hardware-level variational quantum algorithm for molecular simulations, surpassing traditional gate-based approaches in speed and noise resilience. Its application to H₂, HeH⁺, and LiH showcases its accuracy and potential for simulating larger, strongly correlated molecules. The adaptive parameterization scheme aids in practical implementation. Future work will focus on scaling the method to larger systems and incorporating more sophisticated constraints on pulse shapes to further enhance performance and explore the effect of device limitations.

The classical computational cost of simulating ctrl-VQE is significantly higher than for standard VQE, limiting the size of the systems that can be studied currently. The simulations assumed ideal device control and didn't explicitly include effects like T1 and T2 times beyond confirming the short pulse durations made these effects insignificant. Future work will address these issues by improving the software and simulation methodology.