Algorithms · Variational quantum eigensolver
Variance-minimizing VQE
Hamiltonian variance supplements or replaces energy to target eigenstates and diagnose convergence.
Attested & literatureUpdated
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A small worked example, computed in your browser. It is not a result from this record's papers.
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.
Step 1 of 5 · RY(π/2)
⟨H⟩ after this step0.500
H = Z₀Z₁ + 0.5 X₁ + 0.5 X₀
One of the four angles an optimiser would be free to turn. π/2 is not a textbook value: it came out of a deterministic coarse grid search over this file's own simulator (see the comment on VQE_ANGLES), which is how these four constants were fixed. A real VQE reaches angles like these by measuring the energy and adjusting; this example does not run that loop.
Probability spreads from 1 outcome to 2. All 2 are equally likely.
The same freedom on qubit 1, and a different answer: 5π/4 rather than π/2. The two qubits are not symmetric in this Hamiltonian, so the ansatz must be able to treat them differently.
Probability spreads from 2 outcomes to 4.
A CX entangles the two qubits, letting the ansatz reach states a product of single-qubit rotations alone cannot.
Probability moves between the same 4 outcomes. Up: 01 7.3% → 42.7%. Down: 11 42.7% → 7.3%. This is the step that entangles the qubits: from here on no single qubit has a state of its own, only the register as a whole does.
The second layer's rotation on qubit 0. The search settled on zero here, so it does nothing at all — the honest outcome, not a mistake: the layer has to be in the circuit for the search to have had the choice.
Nothing about the state changes here — the same outcomes with the same probabilities and the same phases as the step before.
And the last of the four, also zero. A two-layer ansatz turned out to need only its first layer for this Hamiltonian — something the search found, and not something you can see by inspecting the circuit.
Nothing about the state changes here — the same outcomes with the same probabilities and the same phases as the step before.
Readout 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 in Studio- Record typeLiterature method
- ExecutionRequires a concrete ansatz and Hamiltonian
- Papers1
Hamiltonian variance supplements or replaces energy to target eigenstates and diagnose convergence. This record separates the reusable method idea from any one molecule, Hamiltonian, optimizer, or device.
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 →
Hamiltonian variance supplements or replaces energy to target eigenstates and diagnose convergence. In a complete experiment, the method must be paired with a defined qubit Hamiltonian, reference state, parameterized circuit, measurement grouping, classical optimizer, stopping rule, and error analysis. The catalog therefore treats it as a literature-backed algorithm record rather than pretending that one generic snippet is the paper's implementation. Use the cited source to recover assumptions and compare energy error, variance, circuit resources, measurement cost, optimizer evaluations, and robustness under the same instance and budget.
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 method
- Execution
- Requires a concrete ansatz and Hamiltonian
- Evidence
- Literature curation
- Conversion
- Not applicable until a circuit is supplied
This record is not a concrete circuit.
Quantum vs classical
Classical baseline
Compare Variational quantum eigensolver 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
Report input loading, circuit depth, repetitions, classical preprocessing, post-processing, and wall-clock time together.
Nobody has reviewed this record for gaps yet.
METHOD: Variance-minimizing VQE
SCOPE: Hamiltonian variance supplements or replaces energy to target eigenstates and diagnose convergence.
This is a literature method record, not a fixed circuit.
Supply: Hamiltonian, reference state, ansatz, optimizer, measurement plan, and stopping rule.A reference record, not runnable source. Leona cannot execute it, so it cannot be saved to your Library as a circuit.
Where this sits
This record is named by the layer graph at:
- Minimise the energy variance Method
Takes A parameterised circuit family; an objective function of its parameters, evaluated only through estimates bought with a finite shot budget; a starting point; and a stopping rule — a tolerance, an iteration cap, or an exhausted budget. Returns A preparation routine for the state at the parameters the search stopped at, and the total number of objective evaluations it consumed. The routine is returned whether or not the search found a minimum; that it stopped is not evidence that it converged.
Related entries
References
- Variational quantum eigensolvers by variance minimization
Dan-Bo Zhang, Zhan-Hao Yuan, Tao Yin · 2020
Primary or survey context for Variance-minimizing VQE; consult the paper for assumptions and implementation details.