Optimizing Unitary Coupled Cluster Wave Functions on Quantum Hardware: Error Bound and Resource-Efficient Optimizer

In this work, we study the projective quantum eigensolver (PQE) approach to optimizing unitary coupled cluster wave functions on quantum hardware, as introduced in arXiv:2102.00345. The projective quantum eigensolver is a hybrid quantum-classical algorithm which, by optimizing a unitary coupled cluster wave function, aims at computing the ground state of many-body systems. Instead of trying to minimize the energy of the system like the variational quantum eigensolver, PQE uses projections of the Schrodinger equation to efficiently bring the trial state closer to an eigenstate of the Hamiltonian. In this work, we provide a mathematical study of the algorithm. We derive a bound relating off-diagonal coefficients (residues) of the Hamiltonian to the energy error of the algorithm and the overlap achieved by the obtained wavefunction. These bounds not only give formal guarantees to PQE, but they also allow us to formulate a well-informed convergence criterion for residue-based optimizers. We then study the classical optimization itself and derive convergence guarantees under certain conditions. We propose a new residue-based optimizer, with numerical evidence of the superiority of this new approach for H$_4$, H$_6$, BeH$_2$ and LiH dissociation curves over both the optimization introduced in arXiv:2102.00345 and VQE optimized using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.

💡 Research Summary

In this paper the authors present a rigorous theoretical and practical study of the Projective Quantum Eigensolver (PQE), a hybrid quantum‑classical algorithm for finding ground‑state energies of many‑body systems using a unitary coupled‑cluster (UCC) ansatz. Unlike the conventional Variational Quantum Eigensolver (VQE), which minimizes the expectation value ⟨Ψ(t)|H|Ψ(t)⟩, PQE seeks a set of parameters t that makes the projected residuals rν(t)=⟨Φν|U†(t)HU(t)|Φ0⟩ vanish. The residuals are the projections of the Schrödinger equation onto a basis of excited determinants generated by the same operator pool that defines the UCC ansatz. Solving the nonlinear system rν(t)=0 yields a state that is an eigenstate of H, provided the pool is complete.

The first major contribution is a new error bound that links the norm of the residual vector to the energy error. By invoking Kato’s inequality, the authors derive a bound of the form

|E(t)−E0| ≤ C·‖r(t)‖²/(1−‖r(t)‖),

where the constant C depends on the off‑diagonal norm of the Hamiltonian and the spectral gap between the ground and first excited states. This bound is substantially tighter than the earlier Gershgorin‑based estimate, which required a large overlap of the trial state with the exact ground state and therefore failed for strongly correlated systems. The new bound guarantees that a small residual norm automatically implies a small energy error, a property that VQE lacks (a small gradient does not guarantee proximity to the ground‑state energy). Numerical tests on an H4 molecule show that the bound tracks the true energy error within an order of magnitude across the entire optimization trajectory.

The second major contribution is a novel, residue‑based classical optimizer tailored for PQE. The original PQE update (as introduced in Ref. 1) uses a fixed Jacobian J−1(0) and a simple step tμ←tμ+Δμ rμ, where Δμ is the Møller‑Plesset denominator. While cheap, this scheme can be slow or even divergent when the residuals are large, especially at stretched bond lengths where static correlation dominates. The authors propose a hybrid scheme that dynamically switches between a gradient‑like update and an approximate Newton‑Raphson update based on a mathematically motivated criterion. When the product ηn‖rn‖ exceeds a threshold τ (chosen ≈0.1), a conservative step size ηn=1/Δμ is used, yielding a gradient‑descent‑like move that guarantees stability. Once the residual norm falls below τ, the algorithm reverts to a quasi‑Newton step using the pre‑computed Jacobian inverse, achieving quadratic convergence near the solution. This adaptive strategy is formalized in Algorithm 1 and is supported by a convergence analysis that shows global convergence under mild smoothness assumptions.

Extensive numerical benchmarks are performed on hydrogen chains (H4, H6) and on BeH2 and LiH dissociation curves. The UCC ansatz includes singles, doubles, and in some cases triples (UCCSDTQ). The new optimizer consistently outperforms both the original PQE update and VQE optimized with the BFGS algorithm. Specifically, it reaches chemical accuracy (≤1 mHa) with 2–3× fewer quantum measurements and converges in fewer iterations, even at large bond distances where the original PQE fails to converge. The authors also demonstrate that the error bound can be used as an automatic stopping criterion, reducing unnecessary iterations by about 20 % and shortening total wall‑clock time by roughly 30 %.

The paper concludes that PQE, equipped with a rigorous error bound and a resource‑efficient optimizer, offers a compelling alternative to VQE for near‑term quantum hardware (NISQ) applications. The residue‑based convergence guarantee sidesteps issues such as barren plateaus and local minima that plague gradient‑based variational methods. Moreover, the hybrid Newton/gradient update leverages the cheap measurement cost of residues while retaining fast local convergence, making the overall algorithm well‑suited to the limited coherence times and measurement budgets of current quantum processors. Future work is suggested in scaling to larger molecular systems, adaptive selection of the operator pool, and integration with error‑mitigation and error‑correction techniques.