Variational quantum algorithms for discovering Hamiltonian spectra

Abstract

Calculating the energy spectrum of a quantum system is an important task, for example to analyse reaction rates in drug discovery and catalysis. There has been significant progress in developing algorithms to calculate the ground state energy of molecules on near-term quantum computers. However, calculating excited state energies has attracted comparatively less attention, and it is currently unclear what the optimal method is. We introduce a low depth, variational quantum algorithm to sequentially calculate the excited states of general Hamiltonians. Incorporating a recently proposed technique, we employ the low depth swap test to energetically penalise the ground state, and transform excited states into ground states of modified Hamiltonians. We use variational imaginary time evolution as a subroutine, which deterministically propagates towards the target eigenstate. We discuss how symmetry measurements can mitigate errors in the swap test step. We numerically test our algorithm on Hamiltonians which encode 3SAT optimisation problems of up to 18 qubits, and the electronic structure of the Lithium Hydride molecule. As our algorithm uses only low depth circuits and variational algorithms, it is suitable for use on near-term quantum hardware.

9 pages, 8 figures; changed title, fixed grammar

References in corpus (20)

• A variational eigenvalue solver on a quantum processor
• Hardware-efficient Variational Quantum Eigensolver for Small Molecules and Quantum Magnets
• The theory of variational hybrid quantum-classical algorithms
• Simulated Quantum Computation of Molecular Energies
• Error mitigation for short-depth quantum circuits
• Scalable Quantum Simulation of Molecular Energies
• Elucidating Reaction Mechanisms on Quantum Computers
• Towards Practical Quantum Variational Algorithms
• Efficient variational quantum simulator incorporating active error minimisation
• The Bravyi-Kitaev transformation for quantum computation of electronic structure
• Practical Quantum Error Mitigation for Near-Future Applications
• Hybrid Quantum-Classical Hierarchy for Mitigation of Decoherence and Determination of Excited States
• Robust determination of molecular spectra on a quantum processor
• Quantum chemistry calculations on a trapped-ion quantum simulator
• Learning the quantum algorithm for state overlap
• Witnessing eigenstates for quantum simulation of Hamiltonian spectra
• Tapering off qubits to simulate fermionic Hamiltonians
• The SWAP test and the Hong-Ou-Mandel effect are equivalent
• Quantum Simulation of Helium Hydride in a Solid-State Spin Register
• Low-depth circuit ansatz for preparing correlated fermionic states on a quantum computer

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