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arXiv:1910.10329 · Journal of Chemical Theory and Computation

Is the Trotterized UCCSD Ansatz Chemically Well-Defined?

Harper R. Grimsley, Daniel Claudino, Sophia E. Economou, Edwin Barnes, Nicholas J. Mayhall

Published Dec 16, 2019• 157 citations
arXivHTMLPDF

Abstract

The variational quantum eigensolver (VQE) has emerged as one of the most promising near-term quantum algorithms that can be used to simulate many-body systems such as molecular electronic structures. Serving as an attractive ansatz in the VQE algorithm, unitary coupled cluster (UCC) theory has seen a renewed interest in recent literature. However, unlike the original classical UCC theory, implementation on a quantum computer requires a finite-order Suzuki-Trotter decomposition to separate the exponentials of the large sum of Pauli operators. While previous literature has recognized the nonuniqueness of different orderings of the operators in the Trotterized form of UCC methods, the question of whether or not different orderings matter at the chemical scale has not been addressed. In this Letter, we explore the effect of operator ordering on the Trotterized UCCSD ansatz, as well as the much more compact k -UpCCGSD ansatz recently proposed by Lee et al. [ J. Chem. Theory Comput., 2019, 15, 311. arXiv, 2019, quant-ph:1909.09114. https://arxiv.org/abs/1909.09114 ]. We observe a significant, system-dependent variation in the energies of Trotterizations with different operator orderings. The energy variations occur on a chemical scale, sometimes on the order of hundreds of kcal/mol. This Letter establishes the need to define not only the operators present in the ansatz but also the order in which they appear. This is necessary for adhering to the quantum chemical notion of a “model chemistry”, in addition to the general importance of scientific reproducibility. As a final note, we suggest a useful strategy to select out of the combinatorial number of possibilities, a single well-defined and effective ordering of the operators.

DOI
10.1021/acs.jctc.9b01083

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The Trotterized Unitary Coupled Cluster Single and Double (UCCSD) ansatz has recently attracted interest due to its use in Variation Quantum Eigensolver (VQE) molecular simulations on quantum computers. However, when the size of molecules increases, UCCSD becomes less interesting as it cannot achieve sufficient accuracy. Including higher-order excitations is therefore mandatory to recover the UCC's missing correlation effects. In this Letter, we extend the Trotterized UCC approach via the addition of (true) Triple T excitations introducing UCCSDT. We also include both spin and orbital symmetries. Indeed, in practice, these later help to reduce unnecessarily circuit excitations and thus accelerate the optimization process enabling to tackle larger molecules. Our initial numerical tests (12-14 qubits) show that UCCSDT improves the overall accuracy by at least two-orders of magnitudes with respect to standard UCCSD. Overall, the UCCSDT ansatz is shown to reach chemical accuracy and to be competitive with the CCSD(T) gold-standard classical method of quantum chemistry.
In quantum chemistry, the variational quantum eigensolver (VQE) is a promising algorithm for molecular simulations on near-term quantum computers. However, VQEs using hardware-efficient circuits face scaling challenges due to the barren plateau problem. This raises the question of whether chemically inspired circuits from unitary coupled cluster (UCC) methods can avoid this issue. Here we provide theoretical evidence indicating they may not. By examining alternated dUCC ansätzes and relaxed Trotterized UCC ansätzes, we find that in the infinite depth limit, a separation occurs between particle-hole one- and two-body unitary operators. While one-body terms yield a polynomially concentrated energy landscape, adding two-body terms leads to exponential concentration. Numerical simulations support these findings, suggesting that popular 1-step Trotterized unitary coupled-cluster with singles and doubles (UCCSD) ansätze may not scale. Our results emphasize the link between trainability and circuit expressiveness, raising doubts about VQEs’ ability to surpass classical methods. The variational quantum eigensolver (VQE) is a promising approach for molecular simulations on quantum computers but faces scaling issues due to the barren plateau problem. The authors’ findings indicate that unitary coupled cluster circuits may not overcome these challenges, raising doubts about VQE’s ability to outperform classical methods.
The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm for near-term quantum computers. It offers an approach to solving central problems in quantum chemistry, such as estimating molecular ground-state energies by creating a parametrized trial wavefunction or ansatz state and iteratively updating the parameters using classical optimization to minimize the energy expectation value. A critical challenge in VQE implementation arises from the non-commutativity of operators in the ansatz circuit, constructed using first-order Suzuki-Trotter expansions. The underlying operator order significantly impacts the performance of VQE calculations. In this thesis, we systematically investigate the effects of the operator order of the first-order Trotterization of the Unitary Coupled Cluster on VQE performance for different molecules. Using numerical simulations, we demonstrate that arbitrarily chosen orders can lead to substantial variations in convergence behavior and inaccurate ground-state energy estimation. Furthermore, we attempt to evaluate ordering strategies based on the Hamiltonian to test the possibility of using a predecided order strategy. Our studies show that no discernible ordering strategy emerges based on the Hamiltonian and specific random orders perform significantly better than others.
The Coupled Cluster (CC) method is used to compute the electronic correlation energy in atoms and molecules and often leads to highly accurate results. However, due to its single-reference nature, standard CC in its projected form fails to describe quantum states characterized by strong electronic correlations and multi-reference projective methods become necessary. On the other hand, quantum algorithms for the solution of many-electron problems have also emerged recently. The quantum unitary variant of CC (UCC) with singles and doubles (q-UCCSD) is a popular wavefunction Ansatz for the variational quantum eigensolver algorithm. The variational nature of this approach can lead to significant advantages compared to its classical equivalent in the projected form, in particular, for the description of strong electronic correlation. However, due to the large number of gate operations required in q-UCCSD, approximations need to be introduced in order to make this approach implementable in a state-of-the-art quantum computer. In this work, we evaluate several variants of the standard q-UCCSD Ansatz in which only a subset of excitations is included. In particular, we investigate the singlet and pair q-UCCD approaches combined with orbital optimization. We show that these approaches can capture the dissociation/distortion profiles of challenging systems, such as H4, H2O, and N2 molecules, as well as the one-dimensional periodic Fermi–Hubbard chain. These results promote the future use of q-UCC methods for the solution of challenging electronic structure problems in quantum chemistry.
Quantum computation represents a revolutionary approach to solving problems in quantum chemistry. However, due to the limited quantum resources in the current noisy intermediate-scale quantum (NISQ) devices, quantum algorithms for large chemical systems remain a major task. In this work, we demonstrate that the circuit depth of the unitary coupled cluster (UCC) and UCC-based ansatzes in the algorithm of the variational quantum eigensolver can be significantly reduced by an energy-sorting strategy. Specifically, subsets of excitation operators are first prescreened from the operator pool according to its contribution to the total energy. The quantum circuit ansatz is then iteratively constructed until convergence of the final energy to a typical accuracy. For demonstration, this method has been successfully applied to molecular and periodic systems. Particularly, a reduction of 50%-98% in the number of operators is observed while retaining the accuracy of the original UCCSD operator pools. This method can be straightforwardly extended to general parametric variational ansatzes.