Operators · VQE Hamiltonians and observables
Commuting observable group
Measurement partition for shared basis estimation. Representative form: G_k={P_j : [P_i,P_j]_qw=0}.
Attested & literatureUpdated
- Primary sourceVerteletskyi, Yen & Izmaylov 2019, §II A
- RelationQubit-wise commutativity, Eq. (4)
- Not an equivalence relationTransitivity fails
- 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 StudioCommuting observable group is cataloged as an operator rather than a circuit. Measurement partition for shared basis estimation.
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 →
The relation is not the one this record used to state. Its form was — ordinary commutativity — and the cited paper shows that is the wrong condition. Two Pauli words that commute need not be simultaneously measurable by single-qubit projective measurements, and the paper's own counterexample is while the two are not qubit-wise commuting. Their common eigenstates are entangled superpositions rather than product states, so no set of single-qubit measurements resolves both. The relation that does the work is the qubit-wise commutator of Eq. (4), which vanishes only when every single-qubit factor of commutes with its counterpart in . Qubit-wise commuting implies commuting; the converse fails.
The property worth knowing about this object is that it is not a partition. Qubit-wise commutativity is reflexive and symmetric but not transitive — the paper's example is and while . So it is not an equivalence relation, there are no equivalence classes, and a Hamiltonian has no unique grouping of its terms.
That is why the object is a group in the loose sense and never a partition, and it is the reason the optimization problem is a minimum clique cover over a graph rather than a sort into buckets. Represent each Pauli word as a vertex and join qubit-wise commuting pairs by an edge; a set of mutually measurable terms is then a clique, and the best grouping is a minimum cover by cliques — which the paper notes is NP-hard, hence the heuristics.
Why this record exists beside the method record, which cites the same paper. It was written to test whether it should. The answer is that the two carry different halves and neither is the other's summary: the algebraic facts above — which relation, why it is not transitive, why no unique grouping exists — are properties of the object and hold whatever procedure you use. What belongs to vqe-measurement-grouping instead is the procedure and its measured result: which heuristics were benchmarked, and that grouping reduced the operator count roughly threefold against the total number of Hamiltonian terms. An object record that repeated the threefold figure would be restating the method from the object's side, and this one does not.
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
- Verteletskyi, Yen & Izmaylov 2019, §II A
- Relation
- Qubit-wise commutativity, Eq. (4)
- Not an equivalence relation
- Transitivity fails
- Counterexample
- [x₁x₂, y₁y₂] = 0 but not qubit-wise
- Consequence
- No unique partition — hence clique cover
- Read depth
- Full text
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: Commuting observable group
REPRESENTATIVE FORM: G_k={P_j : [P_i,P_j]_qw=0}
ROLE: Measurement partition for shared basis estimation.
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
- Measurement Optimization in the Variational Quantum Eigensolver Using a Minimum Clique Cover
Vladyslav Verteletskyi, Tzu-Ching Yen, Artur F. Izmaylov · 2019
Primary source for this record, read at the depth its deepened outcome states.