Shortcuts for Adiabatic and Variational Algorithms in Molecular Simulation

The gist: This study presents shortcuts-to-adiabaticity techniques integrated into adiabatic and variational algorithms to enhance molecular ground state calculation, achieving comparable accuracy while reducing circuit depth for near-term devices.

Introduction and Motivation

Quantum algorithms are prominent in the pursuit of achieving quantum advantage in various computational tasks <ref:2407.20957#pg2> Since the inception of quantum computing in the 1980s, simulation of quantum manybody systems has been considered a litmus test in the field <ref:2407.20957#pg2> Quantum chemistry focuses on simulating the properties and dynamics of molecules and materials by applying the principles of quantum mechanics <ref:2407.20957#pg2> Adiabatic algorithms for molecule simulation have received little attention, with relatively few studies conducted in this area <ref:2407.20957#pg2> The VQE algorithm has been extensively studied in quantum chemistry, demonstrating good performance in smaller molecule sizes <ref:2407.20957#pg2> Designing compact ansatzes is essential to overcome challenges such as decoherence, noise, and barren plateaus induced by circuit depth <ref:2407.20957#pg2>.

Shortcuts to Adiabaticity (STA) in Adiabatic Algorithms

The approach includes the counter-diabatic driving that accelerates adiabatic evolution by mitigating adiabatic errors <ref:2407.20957#pg2> In adiabatic algorithms, CD driving accelerates the adiabatic evolution, enabling faster convergence to the ground state of the molecule’s Hamiltonian with shallow circuits <ref:2407.20957#pg2>. The CD Hamiltonian introduces an additional term to the original time-dependent Hamiltonian Hλ(t), in order to mitigate non-adiabatic transitions induced by violations of the adiabatic theorem <ref:2407.20957#pg2> This method is particularly effective in reducing the computational resources required for large-scale simulations <ref:2407.20957#pg2>.

CD-Inspired Ansatz in Variational Quantum Eigensolver (VQE)

The paper introduces the adiabatic gauge ansatz (AGA), which leverages the evolution information encapsulated in the CD Hamiltonian to construct compact and expressive circuits for molecular simulations <ref:2407.20957#pg2>. This approach significantly enhances the efficiency and scalability of the algorithm <ref:2407.20957#pg2> In contrast to adiabatic algorithms with CD driving, by using AGA in the VQE algorithm, the complex optimization of the CD terms is intrinsically addressed within the classical optimization loop, thereby streamlining the overall computational process <ref:2407.20957#pg2>. The AGA is constructed by reparametrizing each Pauli operator: AGA(l)(θ) = XjθAj, where Aj are the individual Pauli string operators obtained from the nested commutators with the expansion order of l, and θ≡θj are the free parameters for optimization <ref:2407.20957#pg2>.

Performance Benchmarking and Results

The performance of different benchmarked ansatzes is compared with AGA(1) and AGAR(1) <ref:2407.20957#pg2>. Both AGA(1) and AGAR(1) consistently achieve convergence below the chemical accuracy threshold <ref:2407.20957#pg2>. The AGA(1) approach frequently outperforms other methods, achieving convergence several orders of magnitude bellow the chemical accuracy ∼ 1 kcal/mol <ref:2407.20957#pg2>. The AGAR(1) provides an interesting alternative, especially for hardware-limited quantum processors, due to its compact two-body term <ref:2407.20957#pg2>. For the BeH2 molecule, two distinct convergence regions are observed <ref:2407.20957#pg2>. The overall convergence shows significant improvement across all Trotter lengths and steps, particularly at short final times T = δtN when the adiabatic condition is not fulfilled <ref:2407.20957#pg2>.

Conclusion and Outlook

The incorporation of CD interaction into digitized AQC enhances the convergence to the ground state, reduces evolution time, and shortens circuit depth <ref:2407.20957#pg2>. This work establishes a critical starting point for leveraging CD-inspired ansatz in molecular simulations and paves the way for more advanced, resource-efficient quantum chemistry methods <ref:2407.20957#pg2>. With the advantages of CD driving, several promising avenues for further exploration arise <ref:2407.20957#pg2>. For instance, combining AGA with advanced techniques such as ADAPT-Givens VQE and STA Krylov space method could lead to the development of hybrid ansatzes <ref:2407.20957#pg2>. The simplicity of AQC and the reduced circuit complexity of AGAR are particularly promising for future NISQ devices <ref:2407.20957#pg2>. The work demonstrates that CD-assisted adiabatic evolution consistently produces results below the threshold for chemical accuracy (1 kcal/mol) <ref:2407.20957#pg2>.

Appendix A: Different ansatzes for VQE

The UCCSD ansatz is a unitary extension of the Coupled Cluster (CC) method, truncated at the second order of expansion <ref:2407.20957#pg2>. The k-Unitary Pair Coupled Cluster Generalized Singles and Doubles (k-UpCCGSD) ansatz is a variant within the CC family, designed to reduce the operator pool by focusing on a subset of excitation operators <ref:2407.20957#pg2>. ADAPT-VQE is not a specific ansatz but rather a variational protocol used to optimize the ansatz by selecting the most influential terms from a larger pool <ref:2407.20957#pg2>. The ADAPT-Givens VQE protocol is implemented using the Givens rotations (GR) protocol <ref:2407.20957#pg2>. The action of a GR on basis states is described by U01⟩ → a01⟩ + b10⟩, U10⟩ → c01⟩ + d10», where a, b, c, d are parameters defining the rotation <ref:2407.20957#pg2>.

References

[2] J. I. Cirac and P. Zoller, Quantum computations with cold trapped ions <ref:2407.20957#pg2> [3] N. Gisin and R. Thew, Quantum communication, Nature photonics 1, 165 (2007) <ref:2407.20957#pg2> [8] R. A. Friesner, Ab initio quantum chemistry: Methodology and applications, Proceedings of the National Academy of Sciences 102, 6648 (2005) <ref:2407.20957#pg2> [13] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, Quantum computation by adiabatic evolution, arXiv preprint quantph/0001106 (2000) <ref:2407.20957#pg2> [48] D. Sels and A. Polkovnikov, Minimizing irreversible losses in quantum systems by local counterdiabatic driving, Proceedings of the National Academy of Sciences 114 (2017) <ref:2407.20957#pg2> [63] Z. Zhan, C. Run, Z. Zong, L. Xiang, Y. Fei, Z. Sun, Y. Wu, Z Jia, P Duan, J Wu, Y Yin and G Guo Experimental determination of electronic states via digitized shortcut to adiabaticity and sequential digitized adiabaticity Phys Rev Appl 16 034050 (2021) <ref:2407.20957#pg2> [83] K. Takahashi and A del Campo Shortcuts to adiabaticity in krylov space Physical Review X 14 (2024) <ref:2407.20957#pg2> [85] V Bergholm, J Izaac, M Schuld, C Gogolin, S Ahmed, V Ajith, M S Alam, G Alonso-Linaje, B AkashNarayanan A Asadi et al Pennylane Automatic differentiation of hybrid quantum-classical computations arXiv preprint arXiv:1811.04968 (2018) <ref:2407.20957#pg2> [33] A Anand, P Schleich, S Alperin-Lea, P W Jensen, S Sim, M D´ıaz-Tinoco J S Kottmann M Degroote A F Izmaylov and A Aspuru-Guzik Quantum computing view on unitary coupled cluster theory Chemical Society Reviews 51 1659 (2022) <ref:2407.20957#pg2> [38] J Lee, W J Huggins, M Head-Gordon, and K B Whaley Generalized unitary coupled cluster wave functions for quantum computation Journal of chemical theory and computation 15 311 (2018) <ref:

1. The integration of counter-diabatic driving (CD driving) into adiabatic algorithms enhances convergence by mitigating non-adiabatic transitions, enabling faster convergence to the ground state of the molecule’s Hamiltonian with shallow circuits, which reduces computational resources required for large-scale simulations.

2. The introduction of the adiabatic gauge ansatz (AGA), derived from nested commutator expansions, allows for a compact and expressive circuits that significantly enhances efficiency and scalability in VQE by leveraging evolution information encapsulated in the CD Hamiltonian.

3. The Reduced Adiabatic Gauge Ansatz (AGAR) is proposed as a variant restricted to one- and two-body interactions, which significantly lowers the number of optimization terms, demonstrating robust performance at smaller inter-atomic distances for hardware-limited quantum processors.

4. The CD Hamiltonian, derived from the nested commutator expansion in Eq. (6), can be determined using classical optimization techniques like neural networks or genetic algorithms when action minimization is inefficient for complex many-body systems.

5. The resulting systems can achieve ground-state energy convergence below the chemical accuracy threshold (1 kcal/mol) for molecules like LiH and BeH2, improving the reliability of quantum simulations compared to standard methods.

Quantum algorithms offer a promising route toward computational advantage, but current implementations remain constrained by limited coherence and gate errors, making circuit complexity and noise sensitivity critical considerations. Here, we develop a shortcuts-to-adiabaticity strategy for molecular ground-state preparation and apply it to both digitized adiabatic quantum computing and the variational quantum eigensolver. In the adiabatic framework, we construct an approximate counterdiabatic (CD) Hamiltonian using a nested-commutator expansion. For the molecular systems considered, the CD correction accelerates state preparation and improves energy convergence in the fast-evolution regime, enabling a target accuracy to be reached with a lower implemented-operator count. In the variational setting, we introduce the adiabatic gauge ansatz (AGA), obtained by promoting the Pauli-string components generated by the approximate adiabatic gauge potential (AGP) to independent variational generators. We further propose a reduced variant, AGAR, that retains only Pauli generators with weight not larger than two, substantially reducing the ansatz size while maintaining competitive accuracy over the regimes considered. Benchmarks on LiH and BeH 2 show that the proposed ansätze achieve accuracies competitive with established approaches such as UCCSD. Experiments on the IBM_basquecountry superconducting quantum processor, together with simulations under depolarizing noise, show that the reduced CD-inspired ansatz yields smaller energy discrepancies than UCCSD under the noise conditions considered. Overall, our results demonstrate how AGP-inspired operator structures can be used to balance accuracy and implementation cost in molecular simulations on current noise-limited quantum hardware.