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Machine Learning

Generalized Unitary Coupled Cluster Wave functions for Quantum Computation

Joonho Lee(Lawrence Berkeley National Laboratory), William J. Huggins(Lawrence Berkeley National Laboratory), Martin Head‐Gordon(Lawrence Berkeley National Laboratory), K. Birgitta Whaley(Lawrence Berkeley National Laboratory)
November 28, 2018Journal of Chemical Theory and Computation475 citations
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Journal of Chemical Theory and Computation

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2018

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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 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 O(kN), compared with O(N3) for UCCGSD, and O((N−η)2η) 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 H4 (STO-3G, 6-31G), H2O (STO-3G), and N2 (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.

Analysis

Why This Paper Matters

This paper addresses a critical bottleneck in quantum chemistry on near-term quantum devices: the trade-off between ansatz expressivity and circuit depth. Unitary coupled-cluster (UCC) methods are promising for variational quantum eigensolvers (VQE), but standard UCCSD and UCCGSD require deep circuits that exceed current hardware capabilities. By introducing k-UpCCGSD, the authors provide a systematically improvable ansatz with dramatically lower circuit depth (O(kN) vs O(N^3) or O((N-η)^2 η)), making it feasible for noisy intermediate-scale quantum (NISQ) computers. This work is significant because it directly enables larger molecular simulations on near-term hardware, bridging the gap between theoretical quantum algorithms and experimental realization.

Technical Contributions

  • Sparse generalized doubles operators: The k-UpCCGSD ansatz uses a family of sparse operators that only include pair double excitations (pCCD) and generalized singles, reducing the number of parameters and circuit gates.
  • Circuit depth scaling: k-UpCCGSD requires O(kN) depth, where k is the number of product terms, compared to O(N^3) for UCCGSD and O((N-η)^2 η) for UCCSD. This is a polynomial reduction in gate count.
  • Systematic improvability: Increasing k improves accuracy, allowing users to trade off cost and precision.
  • Excited state treatment: Uses orthogonally constrained VQE and shows that multideterminantal references from classical linear response improve excited state energies.

Results

  • Ground state benchmarks on H4 (STO-3G, 6-31G), H2O (STO-3G), and N2 (STO-3G) show k-UpCCGSD achieves chemical accuracy (error < 1.6 mHa) with lower quantum resource requirements than UCCGSD and UCCSD.
  • For H4/STO-3G, k=2 UpCCGSD yields errors comparable to UCCGSD but with circuit depth O(N) instead of O(N^3).
  • Excited state energies from orthogonally constrained VQE are less accurate than ground states; using multideterminantal references reduces errors significantly.
  • The paper provides explicit scaling formulas: UCCSD depth ~ O((N-η)^2 η), UCCGSD ~ O(N^3), k-UpCCGSD ~ O(kN).

Significance

This work has broad implications for quantum computing and computational chemistry. By offering a practical ansatz with favorable scaling, it accelerates the timeline for quantum advantage in molecular simulations. The systematic improvability of k-UpCCGSD allows researchers to calibrate accuracy against hardware constraints, a key requirement for NISQ devices. Furthermore, the excited state methodology extends VQE beyond ground states, opening applications in photochemistry and spectroscopy. The paper's emphasis on classical pre-computation (multideterminantal references) also highlights hybrid quantum-classical workflows that are realistic for near-term hardware. Overall, this paper is a foundational contribution to the design of efficient quantum circuits for chemistry.

quantum-computingunitary-coupled-clustervariational-quantum-eigensolveransatz-designquantum-chemistrynear-term-quantumexcited-states
Scaling LawsBenchmarks

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