Operators · VQE Hamiltonians and observables
Fermi–Hubbard Hamiltonian
Correlated lattice-fermion benchmark Hamiltonian. Representative form: H = -tΣ⟨ij⟩σ(c†ᵢσcⱼσ+h.c.) + UΣᵢnᵢ↑nᵢ↓.
Attested & literatureUpdated
- Primary sourceHubbard 1963, Proc. R. Soc. A 276
- Read depthAbstract only — full text paywalled
- Free routeNone found; pre-arXiv
- 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 StudioFermi–Hubbard Hamiltonian is cataloged as an operator rather than a circuit. Correlated lattice-fermion benchmark Hamiltonian.
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 is limited by what could be read, and says so. Hubbard's 1963 paper is the right primary source and its full text is paywalled, with no free preprint, author copy or open mirror found — it predates arXiv by nearly thirty years. The abstract was read directly from the publisher's page; everything below comes from it, and nothing is claimed from the body.
What the abstract states: that a main effect of correlation in - and -bands is to produce behaviour characteristic of the atomic or Heitler–London picture; that the paper introduces a simple approximate model for the interaction of electrons in narrow energy bands to study this; that Hartree–Fock is applied to that model and its results examined; that a Green function technique then yields an approximate solution to the correlation problem; that this solution reduces to the exact atomic solution in one limit and to the ordinary uncorrelated band picture in the opposite one; and that the condition for ferromagnetism of the solution is discussed, with a two-electron example given to clarify the physical meaning.
The limiting behaviour is the part that matters for this catalog, and it is in the abstract rather than needing the body: a model that interpolates between the atomic limit and the uncorrelated band limit is exactly a model with a tunable correlation strength, which is why the ratio became the standard dial and why this Hamiltonian is a benchmark rather than a curiosity.
What is deliberately not here. No equation number, no section reference, no statement about the paper's derivation or its notation, because the body was not read. The catalog's paper register records this source with reportsBasis: "abstract", so the gap is a fact the corpus knows about itself rather than an absence a reader has to infer. Closing it needs the PDF, which is an owner decision — he offered precisely that on ai-ops#44, and this record is the first concrete instance of the offer being needed.
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 →
- Primary source
- Hubbard 1963, Proc. R. Soc. A 276
- Read depth
- Abstract only — full text paywalled
- Free route
- None found; pre-arXiv
- Model
- Hopping t, on-site repulsion U
- Paper's method
- Hartree–Fock, then a Green function treatment
- Access
- Owner decision — he offered exactly this on ai-ops#44
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: Fermi–Hubbard Hamiltonian
REPRESENTATIVE FORM: H = -tΣ⟨ij⟩σ(c†ᵢσcⱼσ+h.c.) + UΣᵢnᵢ↑nᵢ↓
ROLE: Correlated lattice-fermion benchmark Hamiltonian.
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.
Where the map uses this
This record is an instance of an object the map names, so these are the processes that consume or produce one. None of them is about this record in particular.
Hamiltonian you can query 6 of 34 processes
- Graph-Laplacian finite differences hands one back
- Recast a non-Hermitian generator as Hamiltonian evolution hands one back
- Block-encode a matrix takes one
- Simulate Hamiltonian evolution takes one
- Frame an operator as a ground-state question takes one
- Frame an operator as a ground-state question hands one back
Related entries
- Variational quantum eigensolver for ground-state energy
Algorithms
- Pauli-X operator
Operators
References
- Electron correlations in narrow energy bands
J. Hubbard · 1963
Primary source for this record, read at the depth its abstract-only outcome states.