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1. Test of the unitary coupled-cluster variational quantum eigensolver for a simple strongly correlated condensed-matter system

   https://doi.org/10.1142/S0217984920400497  

   Xu, Luogen; Lee, J. T.; Freericks, J. K. (July 2020, Modern Physics Letters B)

   null (Ed.)

   The variational quantum eigensolver has been proposed as a low-depth quantum circuit that can be employed to examine strongly correlated systems on today’s noisy intermediate-scale quantum computers. We examine details associated with the factorized form of the unitary coupled-cluster variant of this algorithm. We apply it to a simple strongly correlated condensed-matter system with nontrivial behavior — the four-site Hubbard model at half-filling. This work show some of the subtle issues one needs to take into account when applying this algorithm in practice, especially to condensed-matter systems. 

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2. An adaptive variational algorithm for exact molecular simulations on a quantum computer

   https://doi.org/10.1038/s41467-019-10988-2  

   Grimsley, Harper R.; Economou, Sophia E.; Barnes, Edwin; Mayhall, Nicholas J. (July 2019, Nature Communications)

   Abstract Quantum simulation of chemical systems is one of the most promising near-term applications of quantum computers. The variational quantum eigensolver, a leading algorithm for molecular simulations on quantum hardware, has a serious limitation in that it typically relies on a pre-selected wavefunction ansatz that results in approximate wavefunctions and energies. Here we present an arbitrarily accurate variational algorithm that, instead of fixing an ansatz upfront, grows it systematically one operator at a time in a way dictated by the molecule being simulated. This generates an ansatz with a small number of parameters, leading to shallow-depth circuits. We present numerical simulations, including for a prototypical strongly correlated molecule, which show that our algorithm performs much better than a unitary coupled cluster approach, in terms of both circuit depth and chemical accuracy. Our results highlight the potential of our adaptive algorithm for exact simulations with present-day and near-term quantum hardware. 

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3. Unleashed from constrained optimization: quantum computing for quantum chemistry employing generator coordinate inspired method

   https://doi.org/10.1038/s41534-024-00916-8  

   Zheng, Muqing; Peng, Bo; Li, Ang; Yang, Xiu; Kowalski, Karol (December 2024, npj Quantum Information)

   Abstract Hybrid quantum-classical approaches offer potential solutions to quantum chemistry problems, yet they often manifest as constrained optimization problems. Here, we explore the interconnection between constrained optimization and generalized eigenvalue problems through the Unitary Coupled Cluster (UCC) excitation generators. Inspired by the generator coordinate method, we employ these UCC excitation generators to construct non-orthogonal, overcomplete many-body bases, projecting the system Hamiltonian into an effective Hamiltonian, which bypasses issues such as barren plateaus that heuristic numerical minimizers often encountered in standard variational quantum eigensolver (VQE). Diverging from conventional quantum subspace expansion methods, we introduce an adaptive scheme that robustly constructs the many-body basis sets from a pool of the UCC excitation generators. This scheme supports the development of a hierarchical ADAPT quantum-classical strategy, enabling a balanced interplay between subspace expansion and ansatz optimization to address complex, strongly correlated quantum chemical systems cost-effectively, setting the stage for more advanced quantum simulations in chemistry. 

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4. Quantum simulation of electronic structure with a transcorrelated Hamiltonian: improved accuracy with a smaller footprint on the quantum computer

   https://doi.org/10.1039/D0CP04106H  

   Motta, Mario; Gujarati, Tanvi P.; Rice, Julia E.; Kumar, Ashutosh; Masteran, Conner; Latone, Joseph A.; Lee, Eunseok; Valeev, Edward F.; Takeshita, Tyler Y. (November 2020, Physical Chemistry Chemical Physics)

   Quantum simulations of electronic structure with a transformed Hamiltonian that includes some electron correlation effects are demonstrated. The transcorrelated Hamiltonian used in this work is efficiently constructed classically, at polynomial cost, by an approximate similarity transformation with an explicitly correlated two-body unitary operator. This Hamiltonian is Hermitian, includes no more than two-particle interactions, and is free of electron–electron singularities. We investigate the effect of such a transformed Hamiltonian on the accuracy and computational cost of quantum simulations by focusing on a widely used solver for the Schrödinger equation, namely the variational quantum eigensolver method, based on the unitary coupled cluster with singles and doubles (q-UCCSD) Ansatz. Nevertheless, the formalism presented here translates straightforwardly to other quantum algorithms for chemistry. Our results demonstrate that a transcorrelated Hamiltonian, paired with extremely compact bases, produces explicitly correlated energies comparable to those from much larger bases. For the chemical species studied here, explicitly correlated energies based on an underlying 6-31G basis had cc-pVTZ quality. The use of the very compact transcorrelated Hamiltonian reduces the number of CNOT gates required to achieve cc-pVTZ quality by up to two orders of magnitude, and the number of qubits by a factor of three. 

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5. Approximating large-basis coupled-cluster theory vibrational frequencies using focal-point approximations

   https://doi.org/10.1063/5.0168608  

   Nelson, Philip M; Glick, Zachary L; Sherrill, C David (September 2023, The Journal of Chemical Physics)

   The focal-point approximation can be used to estimate a high-accuracy, slow quantum chemistry computation by combining several lower-accuracy, faster computations. We examine the performance of focal-point methods by combining second-order Møller–Plesset perturbation theory (MP2) with coupled-cluster singles, doubles, and perturbative triples [CCSD(T)] for the calculation of harmonic frequencies and that of fundamental frequencies using second-order vibrational perturbation theory (VPT2). In contrast to standard CCSD(T), the focal-point CCSD(T) method approaches the complete basis set (CBS) limit with only triple-ζ basis sets for the coupled-cluster portion of the computation. The predicted harmonic and fundamental frequencies were compared with the experimental values for a set of 20 molecules containing up to six atoms. The focal-point method combining CCSD(T)/aug-cc-pV(T + d)Z with CBS-extrapolated MP2 has mean absolute errors vs experiment of only 7.3 cm−1 for the fundamental frequencies, which are essentially the same as the mean absolute error for CCSD(T) extrapolated to the CBS limit using the aug-cc-pV(Q + d)Z and aug-cc-pV(5 + d)Z basis sets. However, for H2O, the focal-point procedure requires only 3% of the computation time as the extrapolated CCSD(T) result, and the cost savings will grow for larger molecules. 

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