VQE Suite / Physics

State Representations & Measurement

The same statevector that the Convergence tab optimizes also drives the Statevector table, the QSphere, and the Measure button — this page derives what each of those views actually shows.

From θ to amplitudes and probabilities

The ansatz statevector, derived on the Hamiltonian & Ansatz page by tracing the circuit gate-by-gate, is

|\psi(\theta)\rangle = \cos\tfrac{\theta}{2}\,|01\rangle + \sin\tfrac{\theta}{2}\,|10\rangle

using this site's basis-index convention (basis index $= 2q_1 + q_0$ , see src/lib/physics/statevector.ts) — the same convention the Statevector table's BASIS_LABELS and the QSphere's point layout both use. Both amplitudes are always real for this ansatz, so the Statevector table's Phase column only ever reads 0° or 180°, and the QSphere colors points by the sign of the real amplitude (cyan positive, violet negative) rather than a general complex phase.

Measurement probabilities are the squared amplitudes — exactly what probabilitiesOf() in src/lib/physics/measurement.ts computes:

P(01) = \cos^2\tfrac{\theta}{2}, \qquad P(10) = \sin^2\tfrac{\theta}{2}

At the VQE-converged $\theta^\ast \approx -0.22974$ : $P(01) \approx 98.69\%$ , $P(10) \approx 1.31\%$ — the near-certainty of the |01⟩ outcome you see reflected in both the Statevector table and the QSphere's point sizes (point radius $\propto \sqrt{P}$ , so area encodes probability).

The reduced state of qubit 0, and why it's mixed

reducedDensityMatrixQubit0() partial-traces qubit 1 out of the full 2-qubit state: $\rho_0[a][b] = \sum_{q_1} \langle 2q_1{+}a | \psi \rangle \langle \psi | 2q_1{+}b \rangle$ . For this ansatz only the $|01\rangle$ and $|10\rangle$ amplitudes are nonzero, so every off-diagonal cross-term vanishes and the sum collapses to a diagonal matrix:

\rho_0 = \begin{pmatrix} \sin^2\tfrac{\theta}{2} & 0 \\ 0 & \cos^2\tfrac{\theta}{2} \end{pmatrix}

For this particular ansatz,

\rho_0

's diagonal entries are literally the same numbers as the measurement probabilities above — a coincidence of this ansatz's structure (only two, mutually-exclusive-on-qubit-0 basis states are populated), not a general fact about reduced states.

purity() then computes $\operatorname{Tr}(\rho_0^2)$ , which for a diagonal matrix is just the sum of the squared diagonal entries:

\operatorname{Tr}(\rho_0^2) = \sin^4\tfrac{\theta}{2} + \cos^4\tfrac{\theta}{2}

At $\theta = 0$ (the untrained ansatz, exactly $|01\rangle$ ) this gives $1$ exactly — qubit 0 is a pure, unentangled product state. At $\theta^\ast \approx -0.22974$ it drops to $\approx 0.97407$ : the converged ground state is genuinely, if only slightly, entangled across the two qubits — a real property of the H₂ ground state this ansatz reaches, not an artifact of the visualization.

Measurement is a real single sample, not an animation

Clicking Measure ▸ on the Statevector tab calls sampleMeasurement(), which draws one Math.random() value and walks the cumulative distribution of $|\text{amplitude}|^2$ until it exceeds the draw (inverse-CDF sampling) — a genuine single projective measurement, not a scripted collapse effect. Run repeatedly, the outcome frequencies converge to the probabilities above; a 20,000-sample check against this exact implementation matched the true probabilities to within statistical error (max deviation ≈0.00074, consistent with the expected $\sqrt{p(1-p)/N}$ sampling noise).

See it running live on the VQE Suite playground — the Statevector, QSphere, and Step-by-Step tabs all read from the same runH2AnsatzStatevector(theta) call.

Source: src/lib/physics/statevector.ts, entanglement.ts, measurement.ts, vqe.ts