TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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LCU projected-block and postselection correctness

Statement

Let $M = \sum_{j < m} c_{j} V_{j}$ be a finite LCU representation with $α = \sum_{j} ∣ c_{j} ∣ > 0$ , let $W$ be the unitary from LCU block encoding, and let $J x = ∣ 0_{a n c} ⟩ \otimes x$ include every coefficient and PREPARE-work ancilla. Then $J^{†} W J = M / α .$ For every unit input vector $ψ$ , the clean-ancilla zero-branch vector is $M ψ / α$ and its postselection weight, under Clean-ancilla zero-branch postselection, is $∥ M ψ ∥^{2} / α^{2}$ . If this weight is positive, the conditional system vector is $M ψ / ∥ M ψ ∥$ ; if it is zero, the conditional vector is undefined.

Facts & Assumptions

Given: The finite LCU representation, complete PREPARE and SELECT access, explicitly supplied PREPARE $^{†}$ access with its declared cost, and a unit system vector $ψ$ .

[L1]

LCU block encoding: $W = P^{†} S_{s e l} P$ , where PREPARE sends the all-zero ancilla to the coefficient-amplitude state, up to one global phase, and SELECT applies each coefficient phase with its unitary term.

[F1]

Dirac Notation Conversion: the bra is the adjoint functional, and $∣ x ⟩ ⟨ y ∣$ acts by $z \mapsto ⟨ z, y ⟩_{r o o t} x$ .

[F2]

Finite registers and their ordered computational bases: the index register has the declared ordered computational basis, including its one-dimensional zero-qubit case.

[F3]

Clean-ancilla zero-branch postselection: the zero-branch vector of $ϕ$ is $J^{†} ϕ$ and its weight is the squared norm of that vector.

[F4]

Clean-ancilla postselection has normalized branch weights: the zero and complementary branch weights of a unit vector are in $[0, 1]$ and sum to one; a normalized conditional vector exists only when the corresponding weight is positive.

[F5]

The computational basis is orthonormal: the computational index basis, including the zero-qubit register's basis, is orthonormal.

Proof

Given: As in Facts & Assumptions.

1.1givenL1F1F2F5algebra

Write $β_{j} = \sqrt{∣ c_{j} ∣ / α}$ and $ω_{j} = c_{j} / ∣ c_{j} ∣$ for $c_{j} \neq 0$ (take $ω_{j} = 1$ when $c_{j} = 0$ ). PREPARE gives $e^{i θ} \sum_{j < m} β_{j} ∣ j, 0_{w o r k} ⟩$ from the all-zero ancilla. After SELECT, the $j$ th component is $e^{i θ} β_{j} ∣ j, 0_{w o r k} ⟩ \otimes ω_{j} V_{j} ψ$ . The all-zero matrix element of PREPARE $^{†}$ on $∣ j, 0_{w o r k} ⟩$ is $e^{- i θ} β_{j}$ by [F1], so the global phase cancels and $J^{†} W J ψ = \sum_{j} β_{j}^{2} ω_{j} V_{j} ψ = M ψ / α$ . Padded invalid indices have zero prepared amplitude. Since this holds for every $ψ$ , it proves the block identity; orthonormality [F5] validates these basis matrix elements.

2.1givenF3F4step 1.1algebra

Apply [F3] to $ϕ = W J ψ$ : its zero-branch vector is $J^{†} W J ψ = M ψ / α$ . Its branch weight is therefore $∥ M ψ ∥^{2} / α^{2}$ , and [F4] makes this a valid probability in $[0, 1]$ .

3.1F3F4step 2.1cases∎

If $∥ M ψ ∥ > 0$ , division of the branch vector by its norm gives $M ψ / ∥ M ψ ∥$ . If $M ψ = 0$ , the weight is zero and [F4] forbids assigning a normalized conditional vector to that branch.

Depends on

Used by

Segmented Taylor LCU simulator Algorithm
Two-term LCU postselection is input dependent Example
Segmented Taylor error and resources Theorem

Dependency tree · two levels

28 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 28 §28.3 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §9.3 (standard reference, not scraped)