Generalized Unitary Coupled Cluster Wave functions for Quantum Computation

Abstract

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 wavefunction 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 (UC-CCSD). k-UpCCGSD is found to show the best scaling for quantum computing applications, requiring a circuit depth of O(kN ), compared with O(N 3 ) for UCCGSD and O((N − η) 2 η) 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 thatk-UpCCGSD offers a good tradeoff 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 multi-determinantal reference state constructed from classical linear response calculations allows these excited state energetics to be improved.

Is the Trotterized UCCSD Ansatz Chemically Well-Defined?

“…The advantage due to a larger set of variational parameters is also reflected in the un-Trotterized version of the ansatz, 2-UpCCGSD, whose dissociation curve practically overlays with the FCI results. The significant improvement in the results with k = 2, accompanied by a virtually absent spread in the computed energies, is in agreement with the findings of Lee et al, 31 which implies that these ansätze are relatively insensitive to the ordering of the operators.…”

“…However, this comes at the cost of an increased number of measurements, and the introduction of a wavefunction ansatz that can limit the accuracy of the simulation (although our recent approach, ADAPT-VQE, can remove the ansatz error). 8 The initial demonstration of VQE 7 was followed by several theoretical studies [9][10][11][12][13][14][15] and demonstrations on other hardware such as superconducting qubits 10,14,16 and trapped ions. 17,18 A key ingredient in VQE is the ansatz, which is implemented as a quantum circuit which constructs trial wavefunctions that are measured and then updated in a classical optimization loop.…”

A Robust and Unified Solution for Choosing the Phases of Adiabatic States as a Function of Geometry: Extending Parallel Transport Concepts to the Cases of Trivial and Near-Trivial Crossings

“…Nevertheless, to implement the Meek-Levine approach 16,19 or other existing trivial crossing approaches, 13,14,17,18,20 the inevitable question remains: what are the phases of U? [22][23][24] And, for the skeptical reader who thinks that choosing +/-phases cannot be very important, consider this:…”

“…Note further that, in principle, the algorithm in Eqs. (20) and (23) could fail also by finding a local minimum (as opposed to a global minimum) of Re(Tr(3U 2 − 16U)), and so to assess our approach, we will also benchmark how well Jacobi sweeps find the global minimum of our target function Re(Tr(3U 2 − 16U)). As a side note, in all cases, our method is able to find a proper rotation matrix U with a real matrix logarithm, i.e.…”

“…This phase convention is used nowadays routinely for modern ab initio non-adiabatic dynamics calculations (Ehrenfest, FSSH, 10,11 To better understand this failure conceptually, consider how one would apply the concept of parallel transport in the extreme non-adiabatic limit of curve crossings, also called a trivial crossing. [13][14][15][16][17][18][19][20] Consider a two-level model Hamiltonian:…”