[Paper Review] Robust determination of molecular spectra on a quantum processor

James Colless, Vinay Ramasesh+6 more|arXiv (Cornell University)|Jul 20, 2017

Quantum Computing Algorithms and ArchitectureComputer Science28 references94 citations

TL;DR

Demonstrates a two-qubit VQE with a novel Quantum Subspace Expansion (QSE) to compute ground and excited-state energies of H2, achieving near chemical accuracy and mitigating incoherent errors.

ABSTRACT

Harnessing the full power of nascent quantum processors requires the efficient management of a limited number of quantum bits with finite lifetime. Hybrid algorithms leveraging classical resources have demonstrated promising initial results in the efficient calculation of Hamiltonian ground states--an important eigenvalue problem in the physical sciences that is often classically intractable. In these protocols, a Hamiltonian is parsed and evaluated term-wise with a shallow quantum circuit, and the resulting energy minimized using classical resources. This reduces the number of consecutive logical operations that must be performed on the quantum hardware before the onset of decoherence. We demonstrate a complete implementation of the Variational Quantum Eigensolver (VQE), augmented with a novel Quantum Subspace Expansion, to calculate the complete energy spectrum of the H2 molecule with near chemical accuracy. The QSE also enables the mitigation of incoherent errors, potentially allowing the implementation of larger-scale algorithms without complex quantum error correction techniques.

Motivation & Objective

• Motivate hybrid quantum-classical strategies for solving electronic structure problems with near-term quantum hardware.
• Demonstrate a complete VQE implementation augmented by QSE to access excited states and mitigate stochastic incoherent errors.
• Show that the approach can yield near-chemical accuracy for H2 across a range of internuclear separations.

Proposed method

• Project the electronic structure Hamiltonian of H2 onto a minimal STO-3G basis and map to a two-qubit Hamiltonian H_Q(R).
• Prepare trial states with a parametrized quantum circuit U(theta) on two superconducting transmon qubits and estimate <H_Q> term-wise via Pauli measurements.
• Use a classical particle swarm optimizer to minimize the VQE energy and obtain theta_min.
• Apply Quantum Subspace Expansion by measuring an expanded set of Pauli/fermionic-like operators to form H and S matrices in the subspace and diagonalize classically to obtain excited-state energies.
• Employ a linear-response QSE expansion to mitigate incoherent errors and improve ground-state energy estimates.
• Analyze dependence on internuclear distance R for H2 and compare energies to chemical accuracy.

Experimental results

Research questions

• RQ1 Can a two-qubit VQE with a QSE framework reproduce ground and excited-state energies of a simple molecular system across varying bond lengths?
• RQ2 Does QSE mitigate incoherent errors sufficiently to prevent spurious states and improve accuracy without full quantum error correction?
• RQ3 What is the impact of different QSE operator choices on the extracted spectrum and robustness to noise?

Key findings

• Ground-state and excited-state energies of H2 are obtained with near chemical accuracy (1.6×10^−3 Ha) across a range of internuclear distances.
• Linear-response QSE expansion significantly reduces energy estimation errors for most bond lengths compared to bare VQE, by almost two orders of magnitude.
• A spurious state can appear at certain distances due to incomplete error correction, which can be mitigated by expanding the QSE operator set or by filtering based on continuity.
• The QSE enables extraction of excited states with a polynomial overhead in measurements, without requiring full state tomography.
• Coherent gate errors are controllable; the algorithm converges despite mis-calibrated gate phases, illustrating robustness of the approach.