Variational Quantum Computation of Excited States

Higgott, Oscar; Wang, Daochen; Brierley, Stephen

doi:10.22331/q-2019-07-01-156

Cited by 417 publications

(439 citation statements)

References 52 publications

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“…One advantage of this algorithm over the WAVES and OC-VQE algorithms is its ability to target a state whose energy is the closest to a preset ω, as in Eq. (16). Although this ability to variationally target specific excited states is very desirable, the algorithm inherently involves the evaluation of a quadratic term inĤ, which greatly increases the number of Hamiltonian terms.…”

Section: Previous Workmentioning

confidence: 99%

“…(2) We analyze options for variational optimization of excited states that are subject to orthogonalization constraints with a previously variationally optimized ground state. 16 We explore several distinct options and make an analysis of the possible errors encountered when using such a variational approach. We show that these excited state energies can be significantly improved by using a different reference state for the excited state variational calculation, specifically, by using single excitation reference states.…”

Section: Introductionmentioning

confidence: 99%

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Generalized Unitary Coupled Cluster Wave functions for Quantum Computation

Lee

Huggins

Head‐Gordon

et al. 2018

J. Chem. Theory Comput.

424

580

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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.

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Section: Previous Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Generalized Unitary Coupled Cluster Wave functions for Quantum Computation

Lee

Huggins

Head‐Gordon

et al. 2018

J. Chem. Theory Comput.

424

580

View full text Add to dashboard Cite

show abstract

“…Cost evaluation occurs on the quantum computer, with speedup over classical cost evaluation, and the classical computer uses this cost information to adjust the parameters of the gate sequence. Variational hybrid algorithms have been proposed for Hamiltonian ground state and excited state preparation [8,14,15], approximate optimization [7], error correction [16], quantum data compression [17,18], quantum simulation [19,20], and quantum compiling [21]. A key feature of such algorithms is their near-term relevance, since only the subroutine of cost evaluation occurs on the quantum computer, while the optimization procedure is entirely classical, and hence standard classical optimization tools can be employed.…”

Section: Introductionmentioning

confidence: 99%

Variational quantum state diagonalization

et al. 2019

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Variational hybrid quantum-classical algorithms are promising candidates for near-term implementation on quantum computers. In these algorithms, a quantum computer evaluates the cost of a gate sequence (with speedup over classical cost evaluation), and a classical computer uses this information to adjust the parameters of the gate sequence. Here we present such an algorithm for quantum state diagonalization. State diagonalization has applications in condensed matter physics (e.g., entanglement spectroscopy) as well as in machine learning (e.g., principal component analysis). For a quantum state ρ and gate sequence U , our cost function quantifies how far U ρU † is from being diagonal. We introduce novel short-depth quantum circuits to quantify our cost. Minimizing this cost returns a gate sequence that approximately diagonalizes ρ. One can then read out approximations of the largest eigenvalues, and the associated eigenvectors, of ρ. As a proof-of-principle, we implement our algorithm on Rigetti's quantum computer to diagonalize one-qubit states and on a simulator to find the entanglement spectrum of the Heisenberg model ground state.

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“…Finally, we note that the wavefunction overlap can be inferred using different approaches. One option corresponds to using the SWAP test [33,34], which represents a common digital strategy and requires system doubling. This can be conveniently realized in some near-term setups, being successfully demonstrated for cold atom lattices by manybody interferometry of two copies of a quantum state [35].…”

Section: Sequential Energy Estimationmentioning

confidence: 99%

Quantum inverse iteration algorithm for programmable quantum simulators

Kyriienko

2020

npj Quantum Inf

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We propose a quantum inverse iteration algorithm which can be used to estimate the ground state properties of a programmable quantum device. The method relies on the inverse power iteration technique, where the sequential application of the Hamiltonian inverse to an initial state prepares an approximate groundstate. To apply the inverse Hamiltonian operation, we write it as a sum of unitary evolution operators using the Fourier approximation approach. This allows to reformulate the protocol as separate measurements for the overlap of initial and propagated wavefunction. The algorithm thus crucially depends on the ability to run Hamiltonian dynamics with an available quantum device. We benchmark the performance using paradigmatic examples of quantum chemistry, corresponding to molecular hydrogen and beryllium hydride. Finally, we show its use for studying the ground state properties of relevant material science models which can be simulated with existing devices, considering an example of the Bose-Hubbard atomic simulator.

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Variational Quantum Computation of Excited States

Cited by 417 publications

References 52 publications

Generalized Unitary Coupled Cluster Wave functions for Quantum Computation

Generalized Unitary Coupled Cluster Wave functions for Quantum Computation

Variational quantum state diagonalization

Quantum inverse iteration algorithm for programmable quantum simulators

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