Algorithms · Eigenvalue estimation
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.
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Phase estimation of P(3*pi/4)
3 counting qubits estimating the phase of P(3π/4), which is 2π·3/8 exactly: a phase 3-bit QPE can represent exactly.
Step 1 of 4 · X
Probability of each outcome after this step
- 1000100%
Puts the target qubit into |1⟩, which is an eigenstate of P(3*pi/4): applying the gate to it multiplies it by a phase and changes nothing else. That is the whole requirement QPE places on its input.
The whole state moves from 0000 to 1000 — one certain outcome before, one certain outcome after, no superposition either side.
Spreads the 3 counting qubits over all 8 values they can hold at once. Each one will later ask about a different power of the phase.
Probability spreads from 1 outcome to 8. All 8 are equally likely.
Counting qubit k applies P(3*pi/4) to the target 2^k times, controlled on itself. The target never changes — it is an eigenstate — so the phase it would have picked up is kicked back onto the control instead. This is the step that writes the answer into the register.
The outcome probabilities do not move at all. What this step changes is phase — 8 distinct relative phases across the state now — which no measurement can see until a later step turns it back into probability.
The inverse Fourier transform reads that phase pattern back out as a binary number. Phase in, bits out.
Probability gathers from 8 outcomes onto 1. 1011 goes from 12.5% to 100.0%.
Readout The counting register reads 011 = 3 with probability 1: angle/(2π) = 3/8, recovered exactly.
Open in Studio- Record typeLiterature reference
- Reference implementationqpe_for_molecules
- Reported costPolynomial 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.
- Papers1
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.
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 →
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..
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 →
- Record type
- Literature reference
- Reference implementation
- qpe_for_molecules
- 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.
- Complexity basis
- abstract of arXiv:quant-ph/0604193, quoted as written (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.
- Circuit
- Not supplied
This record is not a concrete circuit.
Quantum vs classical
Classical baseline
Compare Eigenvalue estimation with the strongest classical method for the same instance, input budget, and output metric.
Quantum claim
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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 as written (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.
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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Where this sits
This record is named by the layer graph at:
- Ground-state energy by phase estimation Method
Takes A Hermitian 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 and a confidence . 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.
Related entries
- Quantum phase estimation
Algorithms
- Iterative phase estimation
Algorithms
- Variational quantum eigensolver for ground-state energy
Algorithms
- Quantum adiabatic evolution
Algorithms
References
- Simulated Quantum Computation of Molecular Energies
Alán Aspuru-Guzik, Anthony D. Dutoi, Peter J. Love, Martin Head-Gordon · 2006
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.