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Projector phase operations from controlled reflections

Statement

Let $Π$ be an orthogonal projection on a finite-dimensional space $H$ , with reflection $R_{Π} = 2 Π - I$ and $S_{Π} = I - 2 Π = - R_{Π}$ . Suppose the control-first call $C (R_{Π})$ is explicitly supplied, together with an ancilla initialized in $∣ 0 ⟩$ and exact one-qubit $H$ and $R_{z}$ gates. Then the unitary $e^{i ϕ R_{Π}}$ is implemented exactly by two controlled-reflection calls and one variable phase rotation, with a constant number of additional one-qubit gates, and the ancilla is returned to $∣ 0 ⟩$ . If an external branch qubit is present and a branch-controlled $R_{z} (2 ϕ)$ is supplied, controlling only that rotation implements the controlled projector phase; the two controlled-reflection calls remain unconditional with respect to the external branch. In QSVT the construction is applied separately to $Π_{R}$ and $Π_{L}$ ; each projector-phase slot uses two controlled calls to the corresponding reflection. The access and gate costs are charged separately.

Facts & Assumptions

Given: The finite space $H$ , an orthogonal projection $Π$ , the specified controlled-reflection access, and a clean ancilla qubit.

[L1]

Orthogonal-Projector Reflection and Access Contract: $R_{Π} = 2 Π - I$ and $S_{Π} = I - 2 Π = - R_{Π}$ ; $R_{Π}$ is $+ I$ on $ran Π$ and $- I$ on $\ker Π$ .

[L2]

Controlled Unitary Access: $C (V) = ∣ 0 ⟩ ⟨ 0 ∣ \otimes I_{H} + ∣ 1 ⟩ ⟨ 1 ∣ \otimes V$ is the control-first unitary, and its availability is explicitly declared.

[L3]

Projector Reflections Are Hermitian Unitary Involutions: $C (S_{Π}) = Z_{a} C (R_{Π})$ , where $Z_{a} = diag (1, - 1)$ acts on the control ancilla.

[L4]

Exact one-qubit gates and rotations: $H = 2^{- 1 / 2} (\begin{array}{cc} 1 & 1 \\ 1 & - 1 \end{array})$ and $R_{z} (θ) = \exp (- i θ Z / 2)$ are exact one-qubit gates.

[L5]

Denotational semantics of a finite layered circuit: chronological circuit operations compose in reverse order in the operator product.

Proof

1.1L1L2L3L4algebra

Define the ancilla-controlled bit flip $X_{Π} : = H_{a} C (S_{Π}) H_{a}$ , where $H_{a} = H \otimes I_{H}$ . On $ran Π$ , $S_{Π} = - I$ , so [L2] makes $C (S_{Π})$ act as $Z_{a}$ on the ancilla; conjugating by $H_{a}$ gives $X_{a}$ . On $\ker Π$ , $S_{Π} = I$ , so $C (S_{Π})$ acts as $I$ and $X_{Π}$ is the identity. Thus $X_{Π}$ flips the ancilla exactly on the range of $Π$ . By [L3], each use of $C (S_{Π})$ is one supplied call to $C (R_{Π})$ plus one ancilla $Z_{a}$ gate.

2.1L1L4step 1.1algebra

Apply $X_{Π}$ , then $R_{z} (2 ϕ) = e^{- i ϕ Z_{a}}$ , then $X_{Π}$ again. On $ran Π$ , the ancilla operation is $X_{a} e^{- i ϕ Z_{a}} X_{a} = e^{i ϕ Z_{a}}$ ; on $\ker Π$ it is $e^{- i ϕ Z_{a}}$ . Acting on $∣ 0 ⟩$ , these give phases $e^{i ϕ}$ and $e^{- i ϕ}$ , respectively, and both branches return the ancilla to $∣ 0 ⟩$ . The induced system operation is therefore $e^{i ϕ} (Π) + e^{- i ϕ} (I - Π) = e^{i ϕ (2 Π - I)}$ .

3.1L2L3L4L5step 1.1step 2.1

The construction uses two occurrences of $X_{Π}$ , hence two controlled-reflection calls by step 1.1, and one variable $R_{z} (2 ϕ)$ rotation. Its Hadamard and ancilla- $Z$ gates have constant count per phase slot. Applying the same construction with $Π_{R}$ or $Π_{L}$ gives the two projector-phase operations used by the QSVT sequence; no direct or controlled reflection cost is inferred from the matrix description alone.

4.1L2L4step 1.1step 2.1algebra∎

Let $c$ be an external branch qubit. Replace the middle rotation by its controlled version, leaving both occurrences of $X_{Π}$ unchanged. On branch $c = 1$ , step 2.1 gives $e^{i ϕ R_{Π}}$ . On branch $c = 0$ , the middle operation is identity and the two adjacent $X_{Π}$ operations cancel because $X_{Π}^{2} = I$ ; the system operation is identity. Thus this is the controlled projector phase. It uses the same two controlled-reflection calls as the uncontrolled construction and one separately supplied and charged branch-controlled rotation.

Depends on

Used by

QSVT action on singular subspaces Theorem
QSVT signal and reflection query counts Theorem
Real bounded polynomial extraction in QSVT Theorem

Dependency tree · two levels

34 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

András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, Quantum singular value transformation and beyond, Lemma 19 and Figure 1 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 29 §29.3 (standard reference, not scraped)