Editing Quantum algorithm (section)

==Hybrid quantum/classical algorithms==
Hybrid Quantum/Classical Algorithms combine quantum state preparation and measurement with classical optimization.<ref>{{cite journal|last1=Moll|first1=Nikolaj|last2=Barkoutsos|first2=Panagiotis|last3=Bishop|first3=Lev S.|last4=Chow|first4=Jerry M.|last5=Cross|first5=Andrew|last6=Egger|first6=Daniel J.|last7=Filipp|first7=Stefan|last8=Fuhrer|first8=Andreas|last9=Gambetta|first9=Jay M.|last10=Ganzhorn|first10=Marc|last11=Kandala|first11=Abhinav|last12=Mezzacapo|first12=Antonio|last13=Müller|first13=Peter|last14=Riess|first14=Walter|last15=Salis|first15=Gian|last16=Smolin|first16=John|last17=Tavernelli|first17=Ivano|last18=Temme|first18=Kristan|title=Quantum optimization using variational algorithms on near-term quantum devices|journal=Quantum Science and Technology|date=2018|volume=3|issue=3|pages= 030503|doi=10.1088/2058-9565/aab822|arxiv=1710.01022|bibcode=2018QS&T....3c0503M|s2cid=56376912}}</ref> These algorithms generally aim to determine the ground-state eigenvector and eigenvalue of a Hermitian operator.

=== QAOA ===
The [[quantum approximate optimization algorithm]] takes inspiration from quantum annealing, performing a discretized approximation of quantum annealing using a quantum circuit. It can be used to solve problems in graph theory.<ref>{{cite arXiv |last1=Farhi |first1=Edward |last2=Goldstone |first2=Jeffrey |last3=Gutmann |first3=Sam |date=2014-11-14 |title=A Quantum Approximate Optimization Algorithm |eprint=1411.4028 |class=quant-ph}}</ref> The algorithm makes use of classical optimization of quantum operations to maximize an "objective function."

=== Variational quantum eigensolver ===
The [[variational quantum eigensolver]] (VQE) algorithm applies classical optimization to minimize the energy expectation value of an [[Ansatz|ansatz state]] to find the ground state of a Hermitian operator, such as a molecule's Hamiltonian.<ref>{{Cite journal |last1=Peruzzo |first1=Alberto |last2=McClean |first2=Jarrod |last3=Shadbolt |first3=Peter |last4=Yung |first4=Man-Hong |last5=Zhou |first5=Xiao-Qi |last6=Love |first6=Peter J. |last7=Aspuru-Guzik |first7=Alán |last8=O’Brien |first8=Jeremy L. |date=2014-07-23 |title=A variational eigenvalue solver on a photonic quantum processor |journal=Nature Communications |language=En |volume=5 |issue=1 |pages=4213 |arxiv=1304.3061 |bibcode=2014NatCo...5.4213P |doi=10.1038/ncomms5213 |issn=2041-1723 |pmc=4124861 |pmid=25055053}}</ref> It can also be extended to find excited energies of molecular Hamiltonians.<ref>{{cite journal|last1=Higgott|first1=Oscar|last2=Wang|first2=Daochen|last3=Brierley|first3=Stephen|title=Variational Quantum Computation of Excited States|journal=Quantum|year=2019|volume=3|pages=156|doi=10.22331/q-2019-07-01-156|arxiv=1805.08138|bibcode=2019Quant...3..156H |s2cid=119185497}}</ref>

=== Contracted quantum eigensolver ===
The contracted quantum eigensolver (CQE) algorithm minimizes the residual of a contraction (or projection) of the Schrödinger equation onto the space of two (or more) electrons to find the ground- or excited-state energy and two-electron reduced density matrix of a molecule.<ref>{{Cite journal|last1=Smart|first1=Scott|last2=Mazziotti|first2=David|date=2021-02-18|title=Quantum Solver of Contracted Eigenvalue Equations for Scalable Molecular Simulations on Quantum Computing Devices|journal=Phys. Rev. Lett.|language=En|volume=125|issue=7|pages=070504|doi=10.1103/PhysRevLett.126.070504|pmid=33666467|arxiv=2004.11416|bibcode=2021PhRvL.126g0504S|s2cid=216144443}}</ref>  It is based on classical methods for solving energies and two-electron reduced density matrices directly from the anti-Hermitian contracted Schrödinger equation.<ref>{{Cite journal|last1=Mazziotti|first1=David|date=2006-10-06|title=Anti-Hermitian Contracted Schrödinger Equation: Direct Determination of the Two-Electron Reduced Density Matrices of Many-Electron Molecules|journal=Phys. Rev. Lett.|language=En|volume=97|issue=14|pages=143002|doi=10.1103/PhysRevLett.97.143002|pmid=17155245|bibcode=2006PhRvL..97n3002M}}</ref>