What Is BEAST-VQE?

Background: Challenges for Chemistry Algorithms on NISQ Devices

Current state-of-the-art quantum algorithms for chemical calculations typically encounter two primary limitations:

1. Qubit scarcity — Most quantum devices still have too few qubits to represent chemically significant problems in full fidelity.

2. High noise levels — Executing deep quantum circuits with high accuracy is still challenging, which limits our ability to extract chemically meaningful results from the quantum state.

Most conventional approaches (VQE, ADAPT-VQE, FAST-VQE) aim to recover the exact ground state of the Hamiltonian. On NISQ (Noisy Intermediate-Scale Quantum) devices, it may instead be advantageous to solve an approximate problem with significantly lower resource requirements.

This is the motivation for the Bosonic Encoding Adaptive Sampling Theory Variational Quantum Eigensolver (BEAST-VQE) algorithm.

BEAST-VQE: Pair-Correlated Bosonic Encoding

In the BEAST-VQE ansatz, all electrons are paired. This allows each electron pair to be treated as an effective (hard-core) bosonic particle.

Key consequences: - Qubit halving — Spatial orbitals map directly to qubits (instead of spin orbitals), halving the number of qubits. - Simplified Hamiltonian — The Hamiltonian can be expressed in terms of electron pair creation and annihilation operators, which obey bosonic commutation relations. - Efficient mapping — The mapping is simpler compared to Jordan–Wigner encoding, significantly reducing overhead for energy measurements. Only 2-3 measurements are required. - Efficient circuits — The gate pool is smaller, and the ansatz yields shallow circuits.

This bosonic formulation is the reason for the name: Bosonic Encoding Adaptive Sampling Theory (BEAST).

Adaptive BEAST-VQE

BEAST-VQE fits naturally into the adaptive VQE framework, where a circuit is constructed iteratively by selecting and adding operators from a predefined pool.

Differences from fermionic adaptive methods (ADAPT/FAST-VQE): - Reduced operator pool — Contains only single excitations in the bosonic picture. Pool size scales as \(\mathcal{O}(N^2)\) rather than \(\mathcal{O}(N^4)\) in the fermionic case. - Faster convergence — Empirically, energy convergence is often more rapid than for FAST-VQE. - Lower cost per iteration — The simpler measurement requirements make BEAST experiments relatively inexpensive.

Approximations and Limitations

Because BEAST-VQE assumes all electrons are paired, it is inherently approximate.

The main limitation of the BEAST model is that we are (by design) solving an approximate version of the actual chemical problem, and it thus can only be used to describe certain chemical properties - for example, dispersion effects are not well approximated, and open-shell systems are not directly accessible with the model.

One way of addressing these limitations is to leverage orbital optimization techniques, which effectively approximate single excitations and unpaired electrons within the paired-electron framework. For this purpose, we provide OO-BEAST-VQE — Orbital-Optimized BEAST-VQE in Qrunch.

See Also

• What Is an Adaptive-VQE and FAST-VQE — adaptive VQE algorithms in Kvantify Qrunch

• Gate Selection — gate selection in adaptive VQE algorithms

• Calculate the Ground State Energy Using the BEAST-VQE — using BEAST-VQE in Kvantify Qrunch

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References

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• Sokolov, I. O. et al. Quantum orbital-optimized unitary coupled cluster methods in the strongly correlated regime: can quantum algorithms outperform their classical equivalents? J. Chem. Phys. 152, 124107 (2020)

• Elfving, V. E., Millaruelo, M., Gámez, J. A., & Gogolin, C. Simulating quantum chemistry in the seniority-zero (paired-electron) space. Phys. Rev. A 103, 032605 (2021)

• O’Brien, T. E. et al. Purification-based quantum error mitigation of pair-correlated electron simulations. Nat. Phys. 19, 1787–1792 (2023)

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• Zhao, L., Goings, J., Wang, Q., Shin, K., Kyoung, W., Noh, S., Rhee, Y. M., & Kim, K. Enhancing the electron-pair approximation with measurements on trapped-ion quantum computers. . npj Quantum Inf 10, 76 (2024)