Molecular ground-state energies by phase estimation
Compute the ground-state energy of an atom or molecule, a calculation whose time the paper states scales exponentially with system size on a classical computer.
quantum chemistryphase estimationground state energystate preparationeigenvalue
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Compute the ground-state energy of an atom or molecule, a calculation whose time the paper states scales exponentially with system size on a classical computer. Phase estimation obtains an eigenvalue of the molecular Hamiltonian as a phase: an approximate ground state is prepared, the phase it acquires under evolution by the Hamiltonian is carried onto a separate readout register by controlled operations, and measuring that register returns the energy. The paper describes mappings of the molecular wave function to the quantum bits, and an adiabatic method for the preparation of a good approximate ground-state wave function, which it demonstrates for a stretched hydrogen molecule. Readout uses a recursive phase-estimation algorithm, which the paper reports reduces the number of quantum bits required for the readout register from about 20 to 4. The paper reports carrying out calculations of the water and lithium hydride molecular ground-state energies on a quantum computer simulator with this construction.
Circuit & simulation
problem
encodetransform
algorithm
transformmeasure
readout
measure
What this takes and returns
TakesNothingWhat joins here
No input port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port at this edge: the record publishes no gate sequence and no register, so there is nothing here to read one off — and unlike a declared hole, nothing has been recorded about what belongs here.
Nothing in the Atlas meets this end.
This record publishes no gate sequence and no register, so there is nothing here to read an interface off. Absent rather than empty. See all 152 →
TakesA Hermitian H reachable as a sum of terms, as sparse-access oracles or as a block-encoding; a way to prepare trial states, and — for the methods that need it — a trial state whose overlap with the ground state is not negligible; a target additive error ε and a confidence 1−δ.Returns A scalar estimate of the lowest eigenvalue with a stated additive-error guarantee, plus the run or query budget it consumed. Whether that estimate is also a rigorous upper bound is a property of the method and is not promised by the slot.
How it works
Phase estimation obtains an eigenvalue of the molecular Hamiltonian as a phase: an approximate ground state is prepared, the phase it acquires under evolution by the Hamiltonian is carried onto a separate readout register by controlled operations, and measuring that register returns the energy. The paper describes mappings of the molecular wave function to the quantum bits, and an adiabatic method for the preparation of a good approximate ground-state wave function, which it demonstrates for a stretched hydrogen molecule. Readout uses a recursive phase-estimation algorithm, which the paper reports reduces the number of quantum bits required for the readout register from about 20 to 4. The paper reports carrying out calculations of the water and lithium hydride molecular ground-state energies on a quantum computer simulator with this construction. The Classiq library carries this subject under applications · chemistry. Reported cost: Polynomial in place of exponential: the abstract states that the calculation time for the energy of atoms and molecules scales exponentially with system size on a classical computer but polynomially using quantum algorithms, that the number of quantum bits required scales linearly with the number of basis functions, and that the number of gates required grows polynomially with the number of quantum bits. No exponent, constant factor or target accuracy is attached to either polynomial. The one concrete count in the abstract is for readout: the recursive algorithm reduces the number of quantum bits required for the readout register from about 20 to 4..
Implementation
Unsupported
molecular-energy-phase-estimation.txt
ALGORITHM: Molecular ground-state energies by phase estimation
PROBLEM: Compute the ground-state energy of an atom or molecule, a calculation whose time the paper states scales exponentially with system size on a classical computer.
IDEA: Phase estimation obtains an eigenvalue of the molecular Hamiltonian as a phase: an approximate ground state is prepared, the phase it acquires under evolution by the Hamiltonian is carried onto a separate readout register by controlled operations, and measuring that register returns the energy. The paper describes mappings of the molecular wave function to the quantum bits, and an adiabatic method for the preparation of a good approximate ground-state wave function, which it demonstrates for a stretched hydrogen molecule. Readout uses a recursive phase-estimation algorithm, which the paper reports reduces the number of quantum bits required for the readout register from about 20 to 4. The paper reports carrying out calculations of the water and lithium hydride molecular ground-state energies on a quantum computer simulator with this construction.
REPORTED COST: Polynomial in place of exponential: the abstract states that the calculation time for the energy of atoms and molecules scales exponentially with system size on a classical computer but polynomially using quantum algorithms, that the number of quantum bits required scales linearly with the number of basis functions, and that the number of gates required grows polynomially with the number of quantum bits. No exponent, constant factor or target accuracy is attached to either polynomial. The one concrete count in the abstract is for readout: the recursive algorithm reduces the number of quantum bits required for the readout register from about 20 to 4.
BASIS: abstract of arXiv:quant-ph/0604193, quoted aswritten (the abstract contains no TeX): "The calculation time for the energy of atoms and molecules scales exponentially with system size on a classical computer but polynomially using quantum algorithms"; "The number of quantum bits required scales linearly with the number of basis functions, and the number of gates required grows polynomially with the number of quantum bits"; and "The recursive algorithm reduces the number of quantum bits required for the readout register from about 20 to 4". The abstract gives no exponent for either polynomial and no constant factor, so none is recorded here.
DEMONSTRATED BY: the Classiq library entry applications/chemistry/qpe_for_molecules
PRIMARY SOURCE: Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, Martin Head-Gordon (2006), Simulated Quantum Computation of Molecular Energies — https://arxiv.org/abs/quant-ph/0604193
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Quantum vs classical
Classical baseline
Compare Eigenvalue estimation with the strongest classical method for the same instance, input budget, and output metric.
Quantum claim
This reference exposes a quantum circuit pattern; it does not imply an application-level speedup without a matched benchmark.
How to compare
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Declared gaps
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Literature & references
Simulated Quantum Computation of Molecular Energies2006 · Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, Martin Head-Gordon
Primary source: it states the polynomial-versus-exponential scaling claim, the linear qubit count in the number of basis functions, the polynomial gate count, and the recursive phase-estimation readout that it reports cuts the readout register from about 20 qubits to 4. It also describes the mappings of the molecular wave function to the qubits and the adiabatic ground-state preparation demonstrated for a stretched hydrogen molecule, and reports the water and lithium hydride calculations on a quantum computer simulator. Consult it for the basis sets, the accuracy of those two energies and the cost of the adiabatic preparation, none of which the abstract states.