Operators · VQE Hamiltonians and observables
Coulomb interaction operator
Represents two-electron repulsion in an orbital basis. Representative form: V = 1/2 Σ hₚqrs a†ₚa†_q a_r a_s.
Attested & literatureUpdated
- Paper about this operatorNone found
- OutcomeDissolved — nothing to cite beyond a definition
- Near missesTwo, both about adjacent objects
- Papers1
This record, used inside a worked example
This page says To use this record in VQE
— here is a concrete, runnable instance of it.
VQE on a 2-qubit transverse-field Ising Hamiltonian
2 qubits, H = Z⊗Z + 0.5*(X⊗I + I⊗X), a stated toy Hamiltonian (not a molecule), minimised with a hardware-efficient ansatz.
Where it ends up
- 1043%
- 0143%
- 007%
- 117%
At these angles the ansatz's energy ⟨H⟩ reaches E0 = −√2, the Hamiltonian's exact ground energy; a real VQE run reaches these angles by repeating measure-the-energy then update-the-angles, not by a closed-form solution.
Open that example in StudioCoulomb interaction operator is cataloged as an operator rather than a circuit. Represents two-electron repulsion in an orbital basis.
TakesNothingWhat joins here
No input port, deliberately. You measure a state with this entry; you do not apply it and pass a register on.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port, deliberately. An observable is measured with, never applied, so there is no register to hand on.
Nothing in the Atlas meets this end.
Not a stage. You measure a state with this; you do not apply it and pass a register on. It has a width and deliberately no ports. See all 60 →
This record was searched and did not deepen. That is the finding, not a gap in the search.
There is no primary research paper whose subject is . It is the standard second-quantized form of an interaction that was already known, written in the formalism of second quantization that was already established — textbook composition rather than a result. Nothing beyond the definition and the formula could be honestly added, so nothing was.
Two near misses were checked and rejected, and the reasons are the useful part.
The integral literature is about the coefficients, not the operator. There is a substantial and genuinely primary literature on evaluating the numbers — Gaussian basis functions and the recurrence schemes built on them. Those papers are about computing matrix elements. Citing one here would claim this record documents an evaluation algorithm, which it does not.
The analytic literature is about a different representation of the same physics. The Coulomb cusp condition — the constraint the singularity imposes on exact wavefunctions — is a real theorem and it is not about this object. It concerns the first-quantized operator in real space; what this record holds is the already-integrated orbital-basis form, in which the singularity has been integrated away. The connection is real but indirect: the cusp is why finite orbital-basis expansions converge slowly. Citing it as though it were about this operator is precisely the slippage the catalog's sourcing rule forbids.
So the record keeps its shared citation and its two authored strings, and adds this note about why. A search that returns nothing is worth recording once so the next person does not repeat it, and an object that genuinely cannot carry more than a formula should not be padded until it looks like one that can.
TakesNothingWhat joins here
No input port, deliberately. You measure a state with this entry; you do not apply it and pass a register on.
Nothing in the Atlas meets this end.
ReturnsNothingWhat joins here
No output port, deliberately. An observable is measured with, never applied, so there is no register to hand on.
Nothing in the Atlas meets this end.
Not a stage. You measure a state with this; you do not apply it and pass a register on. It has a width and deliberately no ports. See all 60 →
- Paper about this operator
- None found
- Outcome
- Dissolved — nothing to cite beyond a definition
- Near misses
- Two, both about adjacent objects
- What it is
- Standard second quantization of a known interaction
- Instance data
- Required for execution
- Read depth
- Not applicable — no source claimed
This record is not a concrete circuit.
Quantum vs classical
Classical baseline
Use a classical state-vector or matrix simulation at the same width, precision, and measurement objective.
Quantum claim
The quantum record demonstrates a state or operator behavior; it does not make classical simulation or communication costs disappear.
How to compare
Compare fidelity, samples, gate depth, noise, memory, and the cost of preparing and reading the state.
Nobody has reviewed this record for gaps yet.
OPERATOR: Coulomb interaction operator
REPRESENTATIVE FORM: V = 1/2 Σ hₚqrs a†ₚa†_q a_r a_s
ROLE: Represents two-electron repulsion in an orbital basis.
This is a mathematical operator record, not an executable circuit.A reference record, not runnable source. Leona cannot execute it, so it cannot be saved to your Library as a circuit.
Related entries
- Variational quantum eigensolver for ground-state energy
Algorithms
- Pauli-X operator
Operators
References
- OpenFermion: The Electronic Structure Package for Quantum Computers
Jarrod R. McClean, Kevin J. Sung, Ian D. Kivlichan, Yudong Cao, Chengyu Dai, E. Schuyler Fried, Craig Gidney, Brendan Gimby, Pranav Gokhale, Thomas Häner, Tarini Hardikar, Vojtěch Havlíček, Oscar Higgott, Cupjin Huang, Josh Izaac, Zhang Jiang, Xinle Liu, Sam McArdle, Matthew Neeley, Thomas O'Brien, Bryan O'Gorman, Isil Ozfidan, Maxwell D. Radin, Jhonathan Romero, Nicholas Rubin, Nicolas P. D. Sawaya, Kanav Setia, Sukin Sim, Damian S. Steiger, Mark Steudtner, Qiming Sun, Wei Sun, Daochen Wang, Fang Zhang, Ryan Babbush · 2017
Provides open-source representations and transformations for fermionic and qubit operators used in quantum simulation.