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arXiv:2211.07983 · Quantum

Differentiable matrix product states for simulating variational quantum computational chemistry

Chu Guo, Yi Fan, Zhiqian Xu, Honghui Shang

Published Dec 4, 2023• 4 citations
arXivHTMLPDF

Abstract

Quantum Computing is believed to be the ultimate solution for quantum chemistry problems. Before the advent of large-scale, fully fault-tolerant quantum computers, the variational quantum eigensolver (VQE) is a promising heuristic quantum algorithm to solve real world quantum chemistry problems on near-term noisy quantum computers. Here we propose a highly parallelizable classical simulator for VQE based on the matrix product state representation of quantum state, which significantly extend the simulation range of the existing simulators. Our simulator seamlessly integrates the quantum circuit evolution into the classical auto-differentiation framework, thus the gradients could be computed efficiently similar to the classical deep neural network, with a scaling that is independent of the number of variational parameters. As applications, we use our simulator to study commonly used small molecules such as HF, HCl, LiH and H 2 O, as well as larger molecules CO 2 , BeH 2 and H 4 with up to 40 qubits. The favorable scaling of our simulator against the number of qubits and the number of parameters could make it an ideal testing ground for near-term quantum algorithms and a perfect benchmarking baseline for oncoming large scale VQE experiments on noisy quantum computers.

DOI
10.22331/q-2023-12-04-1192

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Quantum Computing is believed to be the ultimate solution for quantum chemistry problems. Before the advent of large-scale, fully fault-tolerant quantum computers, the variational quantum eigensolver (VQE) is a promising heuristic quantum algorithm to solve real world quantum chemistry problems on near-term noisy quantum computers. Here we propose a highly parallelizable classical simulator for VQE based on the matrix product state representation of quantum state, which significantly extend the simulation range of the existing simulators. Our simulator seamlessly integrates the quantum circuit evolution into the classical auto-differentiation framework, thus the gradients could be computed efficiently similar to the classical deep neural network, with a scaling that is independent of the number of variational parameters. As applications, we use our simulator to study commonly used small molecules such as HF, HCl, LiH and H 2 O, as well as larger molecules CO 2 , BeH 2 and H 4 with up to 40 qubits. The favorable scaling of our simulator against the number of qubits and the number of parameters could make it an ideal testing ground for near-term quantum algorithms and a perfect benchmarking baseline for oncoming large scale VQE experiments on noisy quantum computers.
We develop a high-performance simulator for variational quantum eigensolver (VQE), the major innovations include: (1) A differentiable matrix product state (MPS) based VQE simulator that seamlessly integrates MPS into the automatic differentiation framework, which overcomes the exponential memory growth of state-vector simulator and can efficiently calculate gradients with a cost independent of the number of parameters; (2) A dynamic scheme to distribute the gradient calculations to achieve good load balance; (3) A parallel adaptive VQE which integrates our differentiable MPS simulator to further enhance the simulation performance; (4) Study of real chemical systems with convergence to chemical accuracy using our simulator, achieving nearly linearly strong and weak scaling for chemical systems with up to 100 qubits. Our simulator provides an ideal test ground for VQE and paves the way of benchmarking large-scale VQE experiments on near-term quantum computers.
Quantum computational chemistry (QCC) is the use of quantum computers to solve problems in computational quantum chemistry. We develop a high performance variational quantum eigensolver (VQE) simulator for simulating quantum computational chemistry problems on a new Sunway supercomputer. The major innovations include: (1) a Matrix Product State (MPS) based VQE simulator to reduce the amount of memory needed and increase the simulation efficiency; (2) a combination of the Density Matrix Embedding Theory with the MPS-based VQE simulator to further extend the simulation range; (3) A three-level parallelization scheme to scale up to 20 million cores; (4) Usage of the Julia script language as the main programming language, which both makes the programming easier and enables cutting edge performance as native C or Fortran; (5) Study of real chemistry systems based on the VQE simulator, achieving nearly linearly strong and weak scaling. Our simulation demonstrates the power of VQE for large quantum chemistry systems, thus paves the way for large-scale VQE experiments on near-term quantum computers.
Quantum chemistry is one of the most promising applications of quantum computers in the near future. For noisy intermediate-scale quantum devices, the quantum-classical hybrid framework based on the variational quantum eigensolver (VQE) has become the method of choice. In the literature, there are many different variants of VQE, but it is not known which one is optimal for a given molecule. For this purpose, we perform a thorough benchmarking on more than ten different kinds of VQE ansatzes (in systems up to 30 qubits), based on their performance on the energy accuracy, runtime until convergence, and number of parameters. Our results show that the ADAPT ansatz can be used to obtain more accurate energy for small systems (below 14 qubits), but it costs much more computational resources. For larger molecules, UCCSD0 has better performance. However, all the tested ansatzes can hardly reach chemical accuracy at stretched bond lengths. Our results were obtained using MindSpore Quantum, where the codes and the benchmarking toolkit are publicly available at Gitee.
As a hybrid quantum-classical algorithm, the variational quantum eigensolver is widely applied in quantum chemistry simulations, especially in computing the electronic structure of complex molecular systems. However, on existing noisy intermediate-scale quantum devices, some factors such as quantum decoherence, measurement errors, and gate operation imprecisions are unavoidable. To overcome these challenges, this study proposes an efficient noise-mitigating variational quantum eigensolver for accurate computation of molecular ground state energies in noisy environments. We design the quantum circuit with reference to the structure of matrix product states and utilize it to pre-train the circuit parameters, which ensures circuit stability and mitigates fluctuations caused by initialization. We also employ zero-noise extrapolation to mitigate quantum noise and combine it with neural networks to improve the accuracy of the noise-fitting function, which significantly eliminates noise interference. Furthermore, we implement an intelligent grouping strategy for measuring Hamiltonian Pauli strings, which not only reduces measurement errors but also improves sampling efficiency. We perform numerical simulations to solve the ground state energy of the H4molecule by using MindSpore Quantum framework, and the results demonstrate that our algorithm can constrain noise errors within the range of O(10−2)∼O(10−1)\mathcal{O}\left( {{{10}^{ - 2}}} \right)\sim \mathcal{O}\left( {{{10}^{ - 1}}} \right)O(10−2)∼O(10−1), outperforming mainstream variational quantum eigensolvers. This work provides a new strategy for high-precision quantum chemistry calculations on near-term noisy quantum hardware.
Quantum computing is moving beyond its early stage and seeking for commercial applications in chemical and biomedical sciences. In the current noisy intermediate-scale quantum computing era, the quantum resource is too scarce to support these explorations. Therefore, it is valuable to emulate quantum computing on classical computers for developing quantum algorithms and validating quantum hardware. However, existing simulators mostly suffer from the memory bottleneck so developing the approaches for large-scale quantum chemistry calculations remains challenging. Here we demonstrate a high-performance and massively parallel variational quantum eigensolver (VQE) simulator based on matrix product states, combined with embedding theory for solving large-scale quantum computing emulation for quantum chemistry on HPC platforms. We apply this method to study the torsional barrier of ethane and the quantification of the protein-ligand interactions. Our largest simulation reaches 1000 qubits, and a performance of 216.9 PFLOP/s is achieved on a new Sunway supercomputer, which sets the state-of-the-art for quantum computing emulation for quantum chemistry.
A programmable quantum device that has a large number of qubits without fault-tolerance has emerged recently. Variational quantum eigensolver (VQE) is one of the most promising ways to utilize the computational power of such devices to solve problems in condensed matter physics and quantum chemistry. As the size of the current quantum devices is still not large for rivaling classical computers at solving practical problems, Fujii et al. proposed a method called ``Deep VQE'', which can provide the ground state of a given quantum system with the smaller number of qubits by combining the VQE and the technique of coarse graining [K. Fujii, K. Mitarai, W. Mizukami, and Y. O. Nakagawa, arXiv:2007.10917]. In this paper, we extend the original proposal of Deep VQE to obtain the excited states and apply it to quantum chemistry calculation of a periodic material, which is one of the most impactful applications of the VQE. We first propose a modified scheme to construct quantum states for coarse graining in Deep VQE to obtain the excited states. We also present a method to avoid a problem of meaningless eigenvalues in the original Deep VQE without restricting variational quantum states. Finally, we classically simulate our modified Deep VQE for quantum chemistry calculation of a periodic hydrogen chain as a typical periodic material. Our method reproduces the ground-state energy and the first-excited-state energy with the errors up to O(1)O(1)%O(1) despite the decrease in the number of qubits required for the calculation by two or four compared with the naive VQE. Our result will serve as a beacon for tackling quantum chemistry problems with classically-intractable sizes by smaller quantum devices in the near future.