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arXiv:2405.15166 · Physical Review Applied

Parametrization and optimizability of pulse-level variational quantum eigensolvers

Kyle Sherbert, Hisham Amer, Sophia E. Economou, Edwin Barnes, Nicholas J. Mayhall

Published Feb 14, 2025• 6 citations
arXivHTMLPDF

Abstract

In conventional variational quantum eigensolvers (VQEs), trial states are prepared by applying series of parametrized gates to a reference state, with the gate parameters being varied to minimize the energy of the target system. Recognizing that the gates are intermediates, which are ultimately compiled into a set of control pulses to be applied to each qubit in the lab, the recently proposed ctrl-VQE algorithm takes the amplitudes, frequencies, and phases of the pulse as the variational parameters used to minimize the molecular energy. In this work, we explore how all 3 degrees of freedom interrelate with one another. To this end, we consider several distinct strategies to parametrize the control pulses, assessing each one through numerical simulations of a transmonlike device. For each parametrization, we contrast the pulse duration required to prepare a good ansatz, and the difficulty to optimize that ansatz from a well-defined initial state. We deduce several guiding heuristics to implement practical ctrl-VQE in hardware, which we anticipate will generalize for generic device architectures.

DOI
10.1103/physrevapplied.23.024036

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In conventional variational quantum eigensolvers (VQEs), trial states are prepared by applying series of parametrized gates to a reference state, with the gate parameters being varied to minimize the energy of the target system. Recognizing that the gates are intermediates, which are ultimately compiled into a set of control pulses to be applied to each qubit in the lab, the recently proposed ctrl-VQE algorithm takes the amplitudes, frequencies, and phases of the pulse as the variational parameters used to minimize the molecular energy. In this work, we explore how all 3 degrees of freedom interrelate with one another. To this end, we consider several distinct strategies to parametrize the control pulses, assessing each one through numerical simulations of a transmonlike device. For each parametrization, we contrast the pulse duration required to prepare a good ansatz, and the difficulty to optimize that ansatz from a well-defined initial state. We deduce several guiding heuristics to implement practical ctrl-VQE in hardware, which we anticipate will generalize for generic device architectures.
The variational quantum eigensolver (VQE) is currently the flagship algorithm for solving electronic structure problems on near-term quantum computers. This hybrid quantum/classical algorithm involves implementing a sequence of parameterized gates on quantum hardware to generate a target quantum state, and then measuring the expectation value of the molecular Hamiltonian. Due to finite coherence times and frequent gate errors, the number of gates that can be implemented remains limited on current quantum devices, preventing accurate applications to systems with significant entanglement, such as strongly correlated molecules. In this work, we propose an alternative algorithm (which we refer to as ctrl-VQE) where the quantum circuit used for state preparation is removed entirely and replaced by a quantum control routine which variationally shapes a pulse to drive the initial Hartree-Fock state to the full CI target state. As with VQE, the objective function optimized is the expectation value of the qubit-mapped molecular Hamiltonian. However, by removing the quantum circuit, the coherence times required for state preparation can be drastically reduced by directly optimizing the pulses. We demonstrate the potential of this method numerically by directly optimizing pulse shapes which accurately model the dissociation curves of the hydrogen molecule (covalent bond) and helium hydride ion (ionic bond), and we compute the single point energy for LiH with four transmons.
Abstract The variational quantum eigensolver is currently the flagship algorithm for solving electronic structure problems on near-term quantum computers. The algorithm involves implementing a sequence of parameterized gates on quantum hardware to generate a target quantum state, and then measuring the molecular energy. Due to finite coherence times and gate errors, the number of gates that can be implemented remains limited. In this work, we propose an alternative algorithm where device-level pulse shapes are variationally optimized for the state preparation rather than using an abstract-level quantum circuit. In doing so, the coherence time required for the state preparation is drastically reduced. We numerically demonstrate this by directly optimizing pulse shapes which accurately model the dissociation of H2 and HeH+, and we compute the ground state energy for LiH with four transmons where we see reductions in state preparation times of roughly three orders of magnitude compared to gate-based strategies.
Quantum computers promise a great computational advantage over classical computers, which might help solve various computational challenges such as the simulation of complicated quantum systems, finding optimum in large optimization problems, and solving large-scale linear algebra problems. Current available quantum devices have only a limited amount of qubits and a high level of noise, limiting the size of problems that can be solved accurately with those devices. Variational quantum algorithms (VQAs) have emerged as a leading strategy to address these limitations by optimizing cost function based on measurement results of shallow depth circuits. Recently, various pulse engineering methods were suggested in order to improve VQA results, including optimizing pulse parameters instead of gate angles as part of the VQA optimization process. In this paper, we suggest a novel pulse-based ansatz, which is parameterized mainly by pulses’ duration of pre-defined pulse structures. This ansatz structure provides relatively low amounts of optimization parameters while maintaining high expressibility, allowing fast convergence. In addition, the ansatz has structured adaptivity to the entanglement level required by the problem, allowing low noise and accurate results. We tested this ansatz against quantum chemistry problems. Specifically, finding the ground-state energy associated with the electron configuration problem, using the variational quantum eigensolver (VQE) algorithm for several different molecules. We manage to achieve chemical accuracy both in simulation for several molecules and on one of IBM’s NISQ devices for the H2 molecule in the STO-3G basis, without the need for extensive error mitigation. Our results are compared to a common gate-based ansatz and show better accuracy and significant latency reduction—up to 7× shorter ansatz schedules.
Variational quantum algorithms (VQAs) have demonstrated great potentials in the Noisy Intermediate Scale Quantum (NISQ) era. In the workflow of VQA, the parameters of ansatz are iteratively updated to approximate the desired quantum states. We have seen various efforts to draft better ansatz with less gates. Some works consider the physical meaning of the underlying circuits, while others adopt the ideas of neural architecture search (NAS) for ansatz generator. However, these designs do not exploit the full advantages of VQAs. Because most techniques target gate ansatz, and the parameters are usually rotation angles of the gates. In quantum computers, the gate ansatz will eventually be transformed into control signals such as microwave pulses on superconducting qubits. These control pulses need elaborate calibrations to minimize the errors such as over-rotation and under-rotation. In the case of VQAs, this procedure will introduce redundancy, but the variational properties of VQAs can naturally handle problems of over-rotation and under-rotation by updating the amplitude and frequency parameters. Therefore, we propose NAPA, a native-pulse ansatz generator framework for VQAs. We generate native-pulse ansatz with trainable parameters for amplitudes and frequencies. In our proposed NAPA, we are tuning parametric pulses, which are natively supported on NISQ computers. Given the limited availability of gradient-based optimizers for pulse-level quantum programs, we choose to deploy non-gradient optimizers in our framework. To constrain the number of parameters sent to the optimizer, we adopt a progressive way to generate our native-pulse ansatz. Experiments are conducted on both simulators and quantum devices for Variational Quantum Eigensolver (VQE) tasks to evaluate our methods.
The Control Variational Quantum Eigensolver (ctrl-VQE) directly optimizes microwave pulses to enable faster and lower-error quantum-state preparation, but its continuous control landscape re- quires efficient search strategies. We demonstrate that a reinforcement-learning agent based on a deep Q learning network can autonomously discover high-performance pulse sequences using only system parameters and a reward function. The approach is fully general for superconducting qubit platforms, requires no ansatz, and operates at nanosecond resolution compatible with hardware con- straints. As a proof of concept, we apply the method to ground-state preparation of the Hydrogen molecule on a simulated superconducting device. The agent consistently identifies optimized control sequences that achieve high fidelity and outperform random-search baselines. These results highlight adaptive learning as a promising hardware-ready framework for pulse-level quantum control.
State-of-the-art noisy digital quantum computers can only execute short-depth quantum circuits. Variational algorithms are a promising route to unlock the potential of noisy quantum computers since the depth of the corresponding circuits can be kept well below hardware-imposed limits. Typically, the variational parameters correspond to virtual RZ{R}_{Z}RZ​ gate angles, implemented by phase changes of calibrated pulses. By encoding the variational parameters directly as hardware pulse amplitudes and durations, we succeed in further shortening the pulse schedule and overall circuit duration. This decreases the impact of qubit decoherence and gate noise. As a demonstration, we apply our pulse-based variational algorithm to the calculation of the ground state of different hydrogen-based systems (H2,H3{\mathrm{H}}_{2}, {\mathrm{H}}_{3}H2​,H3​, and H4{\mathrm{H}}_{4}H4​) using IBM cross-resonance-based hardware. We observe a reduction in schedule duration of up to $5\ifmmode\times\else\texttimes\fi{}$ compared to cnot-based Ans\"atze, while also reducing the measured energy. In particular, we observe a sizable improvement of the minimal energy configuration of H3{\mathrm{H}}_{3}H3​ compared to a cnot-based variational form. Finally, we discuss possible future developments including error-mitigation schemes and schedule optimizations, which will enable further improvements of our approach, paving the way towards the simulation of larger systems on noisy quantum devices.