Algorithms · Variational quantum eigensolver
Subspace-search VQE
One shared unitary transforms several orthogonal inputs while a weighted objective orders multiple eigenstates.
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
- Source reportsSimulation
- Papers1
One shared unitary transforms several orthogonal inputs while a weighted objective orders multiple eigenstates. 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.
Returns3-wire register, not stated in the sourceWhat joins here
The edge is real and its shape is not. Somebody read the source and recorded that it does not state what leaves here. The record still publishes a 3-wire register, so what is withheld is the rest: whether it carries qubits or classical bits, and what a next stage would have to assume.
17 entries line up on shape, composition unverified. Named below.
This record publishes no gate sequence, and unlike a bare literature entry it says so on purpose: somebody read the source and recorded which part is missing and why, with a citation — the gap is listed further down this page. The register width above is what the record does publish. What it withholds is the rest of that edge, so the entries below are candidates whose widths line up and nothing more; a block may ship with a hole, but never with a guess in the hole. See all 1 →
- Bernstein–Vazirani all-ones oracle · 3 qubits
- Clifford brickwork benchmark · 3 qubits
- GHZ chain benchmark · 3 qubits
- RY-CX hardware-efficient ansatz · 3 qubits
- RZ-RY-CZ hardware-efficient ansatz · 3 qubits
- Ising Trotter-step benchmark · 3 qubits
- and 11 more
Only the register widths line up. This entry's source does not state what crosses this edge, so whether it carries qubits or classical bits is unknown — along with everything a width does not carry. These are the entries a filled-in gap could connect to, not entries that connect.
One shared unitary transforms several orthogonal inputs while a weighted objective orders multiple eigenstates. 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.
Returns3-wire register, not stated in the sourceWhat joins here
The edge is real and its shape is not. Somebody read the source and recorded that it does not state what leaves here. The record still publishes a 3-wire register, so what is withheld is the rest: whether it carries qubits or classical bits, and what a next stage would have to assume.
17 entries line up on shape, composition unverified. Named below.
This record publishes no gate sequence, and unlike a bare literature entry it says so on purpose: somebody read the source and recorded which part is missing and why, with a citation — the gap is listed further down this page. The register width above is what the record does publish. What it withholds is the rest of that edge, so the entries below are candidates whose widths line up and nothing more; a block may ship with a hole, but never with a guess in the hole. See all 1 →
- Bernstein–Vazirani all-ones oracle · 3 qubits
- Clifford brickwork benchmark · 3 qubits
- GHZ chain benchmark · 3 qubits
- RY-CX hardware-efficient ansatz · 3 qubits
- RZ-RY-CZ hardware-efficient ansatz · 3 qubits
- Ising Trotter-step benchmark · 3 qubits
- and 11 more
Only the register widths line up. This entry's source does not state what crosses this edge, so whether it carries qubits or classical bits is unknown — along with everything a width does not carry. These are the entries a filled-in gap could connect to, not entries that connect.
- 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.
The paper reports no error or precision figures for its own 4-qubit simulations — there is no data table behind the simulated results, so the accuracy an implementer should expect is not recoverable from this source. Owner review of arXiv:1810.09434, G1 field B5-9, 2026-08-06.
- Subspace-search variational quantum eigensolver for excited states · Ken M Nakanishi, Kosuke Mitarai, Keisuke Fujii, 2018
METHOD: Subspace-search VQE
SCOPE: One shared unitary transforms several orthogonal inputs while a weighted objective orders multiple eigenstates.
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:
- Subspace-search variational eigensolver Method
Takes A Hermitian H reachable as a sum of terms that can be measured separately; a statement of which state is wanted — an index k, a symmetry sector, or a target energy to sit nearest; for most of the methods here, the ground state already solved, and for the deflation route every lower state as well; a target additive error and a shot budget. Returns A scalar estimate — of that eigenvalue, or of the gap between it and the ground state — together with the run budget it consumed. Some methods return a whole low-energy subspace at once and others return one state per run, and that difference is a cost, not a convenience.
Related entries
References
- Subspace-search variational quantum eigensolver for excited states
Ken M Nakanishi, Kosuke Mitarai, Keisuke Fujii · 2018
Primary or survey context for Subspace-search VQE; consult the paper for assumptions and implementation details.