Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets

Abstract: Quantum computers can be used to address molecular structure, materials science and condensed matter physics problems, which currently stretch the limits of existing high-performance computing resources. Finding exact numerical solutions to these interacting fermion problems has exponential cost, while Monte Carlo methods are plagued by the fermionic sign problem. These limitations of classical computational methods have made even few-atom molecular structures problems of practical interest for medium-sized quantum computers. Yet, thus far experimental implementations have been restricted to molecules involving only Period I elements. Here, we demonstrate the experimental optimization of up to six-qubit Hamiltonian problems with over a hundred Pauli terms, determining the ground state energy for molecules of increasing size, up to BeH2. This is enabled by a hardware-efficient variational quantum eigensolver with trial states specifically tailored to the available interactions in our quantum processor, combined with a compact encoding of fermionic Hamiltonians and a robust stochastic optimization routine. We further demonstrate the flexibility of our approach by applying the technique to a problem of quantum magnetism. Across all studied problems, we find agreement between experiment and numerical simulations with a noisy model of the device. These results help elucidate the requirements for scaling the method to larger systems, and aim at bridging the gap between problems at the forefront of high-performance computing and their implementation on quantum hardware.

• The paper introduces a hardware-efficient VQE approach that compactly encodes fermionic Hamiltonians using optimized trial states on superconducting qubits.
• It validates the method experimentally on small molecules and a Heisenberg antiferromagnet, showing consistency with numerical simulations despite noise.
• The study emphasizes that improved circuit depth and qubit coherence are crucial for advancing quantum computation in chemistry and magnetic systems.

Overview of the Paper: Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets

This paper presents the development and experimental implementation of a hardware-efficient variational quantum eigensolver (VQE) designed to address molecular structure and quantum magnetism problems on a superconducting quantum processor. The authors demonstrate the capacity of their approach to optimize quantum Hamiltonian problems involving up to six qubits with significant Pauli terms, advancing the potential of quantum computers in solving problems that are computationally intensive for classical systems.

The research focuses on determining the ground state energy of many-body interacting fermionic systems, relevant for quantum chemistry and condensed matter physics. It employs a VQE approach accompanied by trial states optimized for the hardware's specific entangling capabilities, offering advantages over classical methods that struggle with efficiency issues, such as Monte Carlo simulations afflicted by the fermionic sign problem.

Key Contributions

1. Efficient Hamiltonian Encoding and State Preparation:
   • The paper introduces a compact method for encoding fermionic Hamiltonians onto qubits. The authors leverage tapering techniques to reduce qubit requirements and use a "hardware-efficient" trial state preparation method that takes advantage of the chip's native interactions among superconducting qubits.
2. Experimental Execution and Validation:
   • Employing a superconducting quantum processor, the VQE was experimentally confirmed for small molecules like $\text{H}_2$, $\text{LiH}$, and $\text{BeH}_2$ and a Heisenberg antiferromagnet model. Results showed consistency with numerical simulations using a noise model of the device, validating the practical applicability of the quantum algorithm.
3. Quantum Magnetism Application:
   • The approach is extended to simulate quantum magnetism problems, demonstrating that increased circuit depths yield better approximations of the ground state energy when error rates are sufficiently low.

Numerical Results and Analysis

A significant achievement highlighted in the paper involves scaling the VQE to accommodate problems of practical interest by determining molecule energies like that of $\text{BeH}_2$. The optimization showed good agreement with exact diagonalization results under certain circuit depths, distinctly delineating the constraints imposed by coherence times and qubit connectivity.

The experimental results underscore differences when integrating more complex entanglers against the challenges due to decoherence and finite sampling on current quantum devices. Although deeper circuits theoretically provide better energy estimates, the benefits are mitigated by noise, emphasizing the need for noise-resilient strategies in future quantum computations.

Implications and Future Directions

The results indicate that increasing circuit depth and number of qubits, while controlling decoherent noise, are critical for extending the application of quantum hardware-efficient algorithms to larger systems and more complex molecules. These findings underscore the necessity for improved qubit coherence, faster gate times, and potentially more connected qubit architectures.

Speculation on Future Developments:

Future quantum systems may integrate advances in error mitigation techniques, as proposed by recent studies cited by the authors, to handle noise without substantially increasing hardware overhead. Consequently, practical quantum computing for chemical and physical applications could eventually make significant headway in high-performance computing landscapes.

This research highlights progress toward bridging classical and quantum computation capabilities, particularly for problems that are classically intractable. It lays foundational work for the continuous evolution of quantum algorithms that are sympathetic to existing quantum hardware's constraints while pushing the frontier of computable problems.