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Quantum Optimization, Semidefinite Programs, and Variational Algorithms — Examples

1 · Prerequisites

2 · Summary

The companion page works a scalar primal-dual SDP completely, exhibits a restricted ansatz that misses the ground state, and calculates a two-qubit VQE energy together with the full one-edge depth-one QAOA output distribution. Each computation uses the declared finite matrix, circuit and measurement conventions.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

A one-constraint scalar SDP and its dual

Example

Take $n = m = 1$ , $A_{1} = C = [1]$ , and $b_{1} = 1$ . The primal is maximize $x$ subject to $x \geq 0$ and $x \leq 1$ , so its feasible set is $[0, 1]$ and its attained optimum is $1$ . The dual is minimize $y$ subject to $y \geq 0$ and $y - 1 \geq 0$ , so its feasible set is $[1, \infty)$ and its attained optimum is $1$ . Every feasible pair obeys weak duality.

Facts & Assumptions

Given: Scalar positivity is the condition $x \geq 0$ ; the primal and dual signs are those of the finite SDP definition.

[F1]

Finite semidefinite primal-dual pair: the scalar primal constraint is $x \leq 1$ and the dual constraint is $y - 1 \geq 0$ .

[F2]

Weak duality for the finite SDP pair: every primal-feasible/dual-feasible pair satisfies objective $_{P} \leq$ objective $_{D}$ .

Verification

technique · direct

1.1givenF1algebra

The primal feasible interval is $[0, 1]$ , so $x \leq 1$ for every feasible $x$ , and $x = 1$ is feasible; hence the attained primal maximum is $1$ . The dual constraints are $y \geq 0$ and $y \geq 1$ , so every feasible $y$ is at least $1$ , with equality feasible; hence the attained dual minimum is $1$ .

2.1givenF2step 1.1algebra∎

For arbitrary feasible $x, y$ , step 1.1 gives $x \leq 1 \leq y$ , so the objectives satisfy weak duality, in agreement with [F2]. The equality of the two attained values is verified for this example only and is not inferred in general.

CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

A variational circuit family may miss the optimum

Statement refuted

A variational circuit family is guaranteed to contain a ground state of every finite Hermitian Hamiltonian.

Facts & Assumptions

Given: $H = diag (0, 1)$ on $C^{2}$ and the ansatz consisting of the fixed state $∣ 1 ⟩$ .

[F1]

Variational eigenvalue objective: the VQE objective is the Dirac expectation and $E_{0}$ is the least eigenvalue.

[F2]

VQE Rayleigh–Ritz upper bound and equality case: every ansatz energy is at least $E_{0}$ , with equality only for a ground-space state.

Counterexample

Given: $H = diag (0, 1)$ and the ansatz consisting of $∣ 1 ⟩$ .

Proof technique: direct.

1.1givenF1algebra

The eigenvalues of $H$ are $0, 1$ , so $E_{0} = 0$ and its ground eigenspace is $span {∣ 0 ⟩}$ . The only ansatz state $∣ 1 ⟩$ is orthogonal to that eigenspace.

2.1givenF2step 1.1algebra∎

Direct multiplication gives $H ∣ 1 ⟩ = ∣ 1 ⟩$ , hence the ansatz energy is $1 > 0 = E_{0}$ . It still satisfies [F2], so the universal ansatz-coverage claim is false.

ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

VQE and MaxCut-QAOA objective calculation

Example

For VQE, take $H = Z_{1} + 2 Z_{2}$ and the two-qubit ansatz $∣ ψ (θ, ϕ) ⟩ = R_{y} (θ) ∣ 0 ⟩ \otimes R_{y} (ϕ) ∣ 0 ⟩$ , where $R_{y} (t) ∣ 0 ⟩ = \cos (t / 2) ∣ 0 ⟩ + \sin (t / 2) ∣ 1 ⟩$ . Its energy is $E (θ, ϕ) = \cos θ + 2 \cos ϕ$ . At $(θ, ϕ) = (π / 2, 0)$ the energy is $2$ , while at $(π, π)$ the ansatz is $∣ 11 ⟩$ and reaches the ground energy $- 3$ .

For QAOA take the one-edge graph $V = {1, 2}$ , $E = {(1, 2)}$ , depth $p = 1$ , $γ = π / 2$ , $β = π / 8$ . Starting from $∣ + + ⟩$ , the cost phase is $diag (1, - i, - i, 1)$ in the order $00, 01, 10, 11$ ; the mixer $R_{x} (2 β) \otimes R_{x} (2 β)$ produces $- i (∣ 01 ⟩ + ∣ 10 ⟩) / \sqrt{2}$ . Thus the output distribution gives probability $1 / 2$ to each cut string $01, 10$ , zero to $00, 11$ , and expected cut value $1$ .

Facts & Assumptions

Given: The Pauli matrices, the declared QAOA product order, and the finite projective measurement law.

[F1]

VQE Rayleigh–Ritz upper bound and equality case: a normalized ansatz has energy at least the least eigenvalue.

[F2]

QAOA circuit and expected combinatorial objective: the one-edge cost operator is $(I - Z_{1} Z_{2}) / 2$ and the depth-one circuit order is $e^{- i β (X_{1} + X_{2})} e^{- i γ (I - Z_{1} Z_{2}) / 2}$ .

[F3]

QAOA objective is an expected feasible score only: measured bitstrings are feasible cuts and expected cut lies between feasible extrema.

[F4]

Projective measurement outcome weights are probabilities: finite projective measurement probabilities are squared basis amplitudes.

Verification

technique · direct

1.1givenF1algebra

The eigenvalues of $Z_{1} + 2 Z_{2}$ on $∣ 00 ⟩, ∣ 01 ⟩, ∣ 10 ⟩, ∣ 11 ⟩$ are $3, - 1, 1, - 3$ , so the ground energy is $- 3$ . The product ansatz has energy $\cos θ + 2 \cos ϕ$ : at $(π / 2, 0)$ this is $2$ , and at $(π, π)$ the state is $∣ 11 ⟩$ with energy $- 3$ . Thus both the displayed trial value and attainment claim are verified.

2.1givenF2F3F4algebra∎

For QAOA, the initial state is $(∣ 00 ⟩ + ∣ 01 ⟩ + ∣ 10 ⟩ + ∣ 11 ⟩) / 2$ . The cost unitary multiplies only $∣ 01 ⟩, ∣ 10 ⟩$ by $e^{- i π / 2} = - i$ . With $R_{x} (π / 4) = (\begin{array}{cc} \cos (π / 8) & - i \sin (π / 8) \\ - i \sin (π / 8) & \cos (π / 8) \end{array})$ on each wire, direct multiplication gives zero amplitudes for $00, 11$ and amplitude $- i / \sqrt{2}$ for each of $01, 10$ . By [F4], the probabilities are $0, 1 / 2, 1 / 2, 0$ in the stated order, so the expected cut value is $1$ , in agreement with [F2]–[F3].

Quantum Optimization, Semidefinite Programs, and Variational Algorithms — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, theorems and proofs

5 · Examples, counterexamples and false statements

A one-constraint scalar SDP and its dual

Example

Facts & Assumptions

Verification

A variational circuit family may miss the optimum

Statement refuted

Facts & Assumptions

Counterexample

VQE and MaxCut-QAOA objective calculation

Example

Facts & Assumptions

Verification

Sources