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State Preparation, Isometry Synthesis, and Limits — Examples

1 · Prerequisites

2 · Summary

Overview

This companion verifies a two-qubit amplitude table by computing every prefix norm and rotation, then records its explicit classical input size. The example keeps relative phase, gate count and input access visible.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-generatedVerification: AI-generatedOpen item page →

Amplitude-Table State Preparation Input Boundary

Example

Take $n = 2$ and the ordered amplitude table

$a_{00} = 1 / 2, a_{01} = i / 2, a_{10} = 1 / 2, a_{11} = 1 / 2.$

It has $D = 4$ entries. For $b \geq 2$ , each real and imaginary coordinate is exactly representable by a signed binary fixed-point code with one sign bit and $b - 1$ fractional bits; thus the dense classical table contains $2 D b = 8 b$ bits, separate from coherent gates. The prefix procedure uses three rotation nodes, each with angle $π / 2$ , and one nontrivial relative leaf phase. No QRAM query is assumed.

Facts & Assumptions

Given: The explicit two-qubit table, the lexicographic basis order $00, 01, 10, 11$ , and the exact prefix-rotation preparation statement.

[F1]

For a normalized table, the prefix procedure exactly prepares its ray with the stated gate and clean-work bounds (QC24 State Preparation Rotation Level Upper Bound).

[F2]

The procedure defines $r_{p} = \sqrt{\sum_{x extends p} ∣ a_{x} ∣^{2}}$ and, for $r_{p} > 0$ , $θ_{p} = 2 atan2 (r_{p 1}, r_{p 0})$ . Its $R_{y} (θ)$ has first column $(\cos (θ / 2), \sin (θ / 2))^{T}$ . Leaf phases are $\arg (a_{x}) - \arg (a_{x_{*}})$ , with $x_{*}$ the first nonzero leaf; zero relative phases are skipped (QC24 State Preparation).

Verification

1.1givenalgebra

The norm is $∣ a_{00} ∣^{2} + ∣ a_{01} ∣^{2} + ∣ a_{10} ∣^{2} + ∣ a_{11} ∣^{2} = 1 / 4 + 1 / 4 + 1 / 4 + 1 / 4 = 1$ , so the table is a valid normalized input. Four complex entries with $b$ bits for each of their two real coordinates occupy $8 b$ bits.

2.1F2step 1.1algebra

The subtree norms are $r_{\emptyset} = 1$ , $r_{0} = r_{1} = 1 / \sqrt{2}$ , and $r_{00} = r_{01} = r_{10} = r_{11} = 1 / 2$ . Hence $θ_{\emptyset} = θ_{0} = θ_{1} = π / 2$ by the prefix formula in [F2].

3.1F2step 2.1algebra

The first rotation gives amplitudes $1 / \sqrt{2}$ on prefixes $0$ and $1$ . The controlled $p = 0$ rotation splits its branch equally between $∣ 00 ⟩$ and $∣ 01 ⟩$ , and the $p = 1$ rotation splits its branch equally between $∣ 10 ⟩$ and $∣ 11 ⟩$ . The resulting vector is $(1 / 2, 1 / 2, 1 / 2, 1 / 2)$ .

4.1F1F2step 3.1givenalgebra∎

Applying the phase $e^{i \arg (a_{01})} = i$ to the $∣ 01 ⟩$ branch yields $(1 / 2, i / 2, 1 / 2, 1 / 2)$ exactly. The table's classical bit scan and these controlled gates are separate costs; none is a QRAM access.