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Chemical Physics Letters

Some comments on the coupled cluster with generalized singles and doubles (CCGSD) ansatz

Debashis Mukherjee, Werner Kutzelnigg

Published Sep 14, 2004• 25 citations
Publisher
DOI
10.1016/j.cplett.2004.08.100

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We investigate the structure of the exact wave function as a solution of the Schrödinger equation, aiming the singles and doubles description of the exact wave function. The basis is that the Hamiltonian involves only one and two body operators. We first present two theorems that indicate a possibility of the singles and doubles description of the exact wave function. We then examine the exponential ansatz, as this theorem implies it to be a possible structure of the exact wave function. Variational CCS (singles) wave function is shown to be certainly exact for one particle Hamiltonian. Thouless transformation plays an important role in the formulation. The conventional CCSD (singles and doubles) function is restrictive, even if it is solved variationally. A wider coupled cluster function with general singles and doubles substitution operators (CCGSD) is also not exact for the existence of noncommuting operators. We then analyze some formal properties of the full CI wave function, and finally, we propose an ansatz of the exact wave function and describe the method of solution. It involves successive solutions of the secular equations of the order of singles and doubles. It is variational and we can calculate both ground and excited states.
We introduce a unitary coupled-cluster (UCC) ansatz termed k -UpCCGSD that is based on a family of sparse generalized doubles operators, which provides an affordable and systematically improvable unitary coupled-cluster wave function suitable for implementation on a near-term quantum computer. k -UpCCGSD employs k products of the exponential of pair coupled-cluster double excitation operators (pCCD), together with generalized single excitation operators. We compare its performance in both efficiency of implementation and accuracy with that of the generalized UCC ansatz employing the full generalized single and double excitation operators (UCCGSD), as well as with the standard ansatz employing only single and double excitations (UCCSD). k -UpCCGSD is found to show the best scaling for quantum computing applications, requiring a circuit depth of, compared with for UCCGSD, and for UCCSD, where N is the number of spin orbitals and η is the number of electrons. We analyzed the accuracy of these three ansätze by making classical benchmark calculations on the ground state and the first excited state of H 4 (STO-3G, 6-31G), H 2 O (STO-3G), and N 2 (STO-3G), making additional comparisons to conventional coupled cluster methods. The results for ground states show that k -UpCCGSD offers a good trade-off between accuracy and cost, achieving chemical accuracy for lower cost of implementation on quantum computers than both UCCGSD and UCCSD. UCCGSD is also found to be more accurate than UCCSD but at a greater cost for implementation. Excited states are calculated with an orthogonally constrained variational quantum eigensolver approach. This is seen to generally yield less accurate energies than for the corresponding ground states. We demonstrate that using a specialized multideterminantal reference state constructed from classical linear response calculations allows these excited state energetics to be improved.
The coupled cluster (CC) ansatz is generally recognized as providing one of the best wave function-based descriptions of electronic correlation in small- and medium-sized molecules. The fact that the CC equations with double excitations (CCD) may be expressed as a handful of dense matrix-matrix multiplications makes it an ideal method to be ported to graphics processing units (GPUs). We present our implementation of the spin-free CCD equations in which the entire iterative procedure is evaluated on the GPU. The GPU-accelerated algorithm readily achieves a factor of 4-5 speedup relative to the multithreaded CPU algorithm on same-generation hardware. The GPU-accelerated algorithm is approximately 8-12 times faster than Molpro, 17-22 times faster than NWChem, and 21-29 times faster than GAMESS for each CC iteration. Single-precision GPU-accelerated computations are also performed, leading to an additional doubling of performance. Single-precision errors in the energy are typically on the order of 10(-6) hartrees and can be improved by about an order of magnitude by performing one additional iteration in double precision.
Distributed quantum computing (DQC) has been proposed as a way to scale quantum algorithms for practical applications beyond monolithic quantum processor architectures. Among these applications, quantum chemistry is widely regarded as one of the most promising use cases for quantum computing. In this work, we estimate the distributed-resource requirements of unitary coupled-cluster (UCC) ansätze for quantum chemistry, focusing on unitary coupled-cluster singles and doubles (UCCSD), unitary pair coupled-cluster doubles (UpCCD), and unitary pair coupled-cluster with generalized singles and doubles (UpCCGSD) circuits for hydrogen chains. We focus on a teleportation-based approach to DQC, quantitatively comparing a naive distribution method to the output of the TeleSABRE algorithm. For both approaches, we estimate the cost of handling nonlocal two-qubit gates across a fixed midpoint or quarter-point partition, reporting Bell-pair/classical-communication costs in teleportation. Across Jordan-Wigner and Bravyi-Kitaev, we find that UpCCD with spin-blocked Jordan-Wigner ordering gives the most favorable scaling, while UCCSD incurs substantially larger distributed-resource requirements.