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QPE output law and conditional state for a spectral input

Statement

Let $V$ be a finite-dimensional complex inner-product space, let $H : V \to V$ be Hermitian with distinct eigenvalues $λ$ and spectral projections $P_{λ}$ , and let $ψ \in V$ be a unit vector. Run the ideal phase-estimation sampler on $ψ$ , with unitary $U$ whose declared phase class on $P_{λ} V$ is $θ (λ) \in R / Z$ . Write $\overset{ˉ}{θ} (λ) \in [0, 1)$ for its unique representative, so $U v = e^{2 π i \overset{ˉ}{θ} (λ)} v$ for $v \in P_{λ} V$ . Write $a_{y} (λ)$ for the phase-register amplitude at output $y$ when the input is in that eigenspace, so $∣ a_{y} (λ) ∣^{2} = P_{\overset{ˉ}{θ} (λ)} (y)$ as in Phase-estimation finite outcome law. Then the output law is $\Pr (Y = y) = \sum_{λ \in σ (H)} μ_{ψ} ({λ}) P_{\overset{ˉ}{θ} (λ)} (y) .$ For every outcome with positive probability, the normalized conditional system vector is $ψ_{y} = \frac{\sum_{λ} a_{y} (λ) P_{λ} ψ}{\sqrt{\Pr (Y = y)}} .$ Moreover, $μ_{ψ}$ is a probability distribution and, for every $g : σ (H) \to C$ , $⟨ ψ ∣ g (H) ∣ ψ ⟩ = \sum_{λ} μ_{ψ} ({λ}) g (λ) .$ The conditional system is generally a coherent sum across eigenspaces, not an exact eigenspace projection.

Facts & Assumptions

Given: A finite-dimensional Hermitian $H$ , its spectral resolution, a unit vector $ψ$ , the ideal sampler, and its phase encoding on every eigenspace.

[F1]

A normal endomorphism is a sum of its eigenvalues times pairwise orthogonal projections, and each spectral projection is a polynomial in the endomorphism: $I = \sum_{λ} P_{λ}$ , $P_{λ} P_{ν} = 0$ for distinct eigenvalues, and the projections are self-adjoint.

[F2]

State-dependent spectral distribution: $μ_{ψ} ({λ}) = ∥ P_{λ} ψ ∥^{2}$ .

[F3]

Quantum phase-estimation spectral sampler and Phase-estimation finite outcome law: on an eigenvector of phase representative $\overset{ˉ}{θ} (λ) \in [0, 1)$ , the phase register has amplitudes $a_{y} (λ)$ with squared magnitudes $P_{\overset{ˉ}{θ} (λ)} (y)$ .

[F4]

Finite projective measurements and Projective measurement outcome weights are probabilities give the computational-basis outcome probability as the squared norm of the projected vector. The readout in Quantum phase estimation specifies the Lüders branch $Z \mapsto P_{y} Z P_{y}$ ; on a pure input, its positive-probability branch is therefore the projected vector normalized.

[F5]

The spectral functional calculus f(T) for a normal endomorphism: $g (H) = \sum_{λ} g (λ) P_{λ}$ .

[F6]

Dirac Notation Conversion: $⟨ ψ ∣ P_{λ} ∣ ψ ⟩ = ∥ P_{λ} ψ ∥^{2}$ for an orthogonal projection.

Proof

1.1F1F2F5F6given

The orthogonality in [F1] gives $ψ = \sum_{λ} P_{λ} ψ$ and $∥ ψ ∥^{2} = \sum_{λ} ∥ P_{λ} ψ ∥^{2} = 1$ . Thus the weights in [F2] are nonnegative and sum to one. By [F5]–[F6], $⟨ ψ ∣ g (H) ∣ ψ ⟩ = \sum_{λ} g (λ) ⟨ ψ ∣ P_{λ} ∣ ψ ⟩ = \sum_{λ} μ_{ψ} ({λ}) g (λ)$ .

2.1F1F2F3F4step 1.1algebra

By linearity, the QPE unitary sends the initial phase state and $ψ$ to $\sum_{λ} \sum_{y} a_{y} (λ) ∣ y ⟩ \otimes P_{λ} ψ$ , using [F3] on each eigenspace. Projecting the phase register onto $∣ y ⟩$ gives the system vector $v_{y} = \sum_{λ} a_{y} (λ) P_{λ} ψ$ . Since the summands for distinct $λ$ are orthogonal by [F1], [F4] yields $\Pr (Y = y) = ∥ v_{y} ∥^{2} = \sum_{λ} ∣ a_{y} (λ) ∣^{2} ∥ P_{λ} ψ ∥^{2} = \sum_{λ} μ_{ψ} ({λ}) P_{\overset{ˉ}{θ} (λ)} (y)$ . Summing these probabilities over the finite output set gives one, as each eigenphase kernel is a measurement law.

3.1F1F3F4step 2.1algebra∎

When $\Pr (Y = y) > 0$ , [F4] normalizes the projected branch to $v_{y} / ∥ v_{y} ∥$ , which is the displayed formula. Its coefficients $a_{y} (λ)$ need not vanish outside one eigenvalue, so no exact eigenspace-collapse claim follows; the formula itself gives the precise conditional state.

Depends on

Used by

Cited to discharge well-definedness by State-dependent spectral distribution.

Dependency tree · two levels

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Sources

Andrew M. Childs, Lecture Notes on Quantum Algorithms, §4.3 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, version 6, §4.6 (standard reference, not scraped)