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PUE singular-space invariant decomposition

Statement

Let $A : H_{R} \to H_{L}$ be the PUE map of Projected-unitary encoding and singular vectors, with positive singular pairs $(σ_{j}, v_{j}, u_{j})$ . For every $0 < σ_{j} < 1$ , set

$w_{j} = \frac{(I - Π_{R}) U^{*} u_{j}}{\sqrt{1 - σ_{j}^{2}}}, {\tilde{w}}_{j} = \frac{(I - Π_{L}) U v_{j}}{\sqrt{1 - σ_{j}^{2}}} .$

The vectors $w_{j}$ form an orthonormal family in $\ker Π_{R}$ and the vectors ${\tilde{w}}_{j}$ form an orthonormal family in $\ker Π_{L}$ . In the input basis $(v_{j}, w_{j})$ and output basis $(u_{j}, {\tilde{w}}_{j})$ ,

$U ∣_{span (v_{j}, w_{j})} = (\begin{array}{cc} σ_{j} & \sqrt{1 - σ_{j}^{2}} \\ \sqrt{1 - σ_{j}^{2}} & - σ_{j} \end{array}) .$

For $σ_{j} = 1$ , $U v_{j} = u_{j}$ and the complementary vector is absent. A right zero-singular vector $v \in \ker A \cap H_{R}$ satisfies $U v \in \ker Π_{L}$ ; a left zero-singular vector $u \in \ker A^{*} \cap H_{L}$ satisfies $U^{*} u \in \ker Π_{R}$ . Let $M_{R}$ be the orthogonal sum of all positive-singular input planes or lines, $\ker A$ , and $U^{*} (\ker A^{*})$ ; let $M_{L}$ be the corresponding sum of positive-singular output planes or lines, $U (\ker A)$ , and $\ker A^{*}$ . Then $U (M_{R}) = M_{L}$ . The residual ambient subspaces $M_{R}^{⊥}$ and $M_{L}^{⊥}$ lie in $\ker Π_{R}$ and $\ker Π_{L}$ , and $U (M_{R}^{⊥}) = M_{L}^{⊥}$ . Thus each named block is invariant under its corresponding projector, with no implicit pairing of the two kernels.

Facts & Assumptions

Given: A PUE $(U, Π_{R}, Π_{L})$ and the SVD data from Projected-unitary encoding and singular vectors.

[F1]

Projected-unitary encoding and singular vectors: $A = Π_{L} U ∣_{H_{R}}$ , $A^{*} = Π_{R} U^{*} ∣_{H_{L}}$ , the positive singular pairs satisfy $A v_{j} = σ_{j} u_{j}$ and $A^{*} u_{j} = σ_{j} v_{j}$ , the trailing bases span $\ker A$ and $\ker A^{*}$ , and $U^{*}$ is unitary.

[F2]

Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition: every finite-dimensional linear map has complete orthonormal singular bases and a positive-term SVD formula.

[F3]

The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ : $⟨ T x, y ⟩ = ⟨ x, T^{*} y ⟩$ .

[F4]

Adjoints satisfy $(S + T)^{*} = S^{*} + T^{*}$ , $(λ T)^{*} = \overline{λ} T^{*}$ , $(S T)^{*} = T^{*} S^{*}$ , and $T^{* *} = T$ : adjoints reverse products, $T^{* *} = T$ , and $I^{*} = I$ .

[F5]

The orthogonal projection $P_{W} v$ is the $W$ -component in $V = W \oplus W^{⊥}$ : a vector is the sum of its projection into a subspace and an orthogonal component.

[F6]

Pythagoras and finite orthogonal sums: orthogonal components have squared norms that add.

Proof

1.1givenF1F3F4F5F6algebra

For $0 < σ_{i}, σ_{j} < 1$ , [F1] gives $Π_{R} U^{*} u_{i} = σ_{i} v_{i}$ . Since $U^{*}$ preserves inner products by [F1], orthogonal projection and [F6] give $⟨ (I - Π_{R}) U^{*} u_{i}, (I - Π_{R}) U^{*} u_{j} ⟩ = ⟨ U^{*} u_{i}, U^{*} u_{j} ⟩ - ⟨ Π_{R} U^{*} u_{i}, Π_{R} U^{*} u_{j} ⟩ = δ_{i j} (1 - σ_{i}^{2})$ . Dividing by the positive square roots shows that the $w_{i}$ are orthonormal and lie in $\ker Π_{R}$ . The same calculation, using $Π_{L} U v_{i} = σ_{i} u_{i}$ , proves that the ${\tilde{w}}_{i}$ are orthonormal in $\ker Π_{L}$ .

1.2F1F3F5F6algebra

If $σ_{i} = 1$ , then $Π_{R} U^{*} u_{i} = v_{i}$ has norm one, as does $U^{*} u_{i}$ . By the orthogonal decomposition in [F5] and Pythagoras [F6], its component in $\ker Π_{R}$ is zero, so $U^{*} u_{i} = v_{i}$ and $U v_{i} = u_{i}$ .

2.1F1F3F4step 1.1algebra

Decompose $U v_{i} = Π_{L} U v_{i} + (I - Π_{L}) U v_{i} = σ_{i} u_{i} + \sqrt{1 - σ_{i}^{2}} {\tilde{w}}_{i}$ . Also $\sqrt{1 - σ_{i}^{2}} U w_{i} = U (I - Π_{R}) U^{*} u_{i} = u_{i} - U Π_{R} U^{*} u_{i} = u_{i} - σ_{i} U v_{i}$ , where the middle equality uses $U U^{*} = I$ and the last uses [F1]. Substitution of the first equation yields $U w_{i} = \sqrt{1 - σ_{i}^{2}} u_{i} - σ_{i} {\tilde{w}}_{i}$ , which is the displayed matrix.

3.1F1F2F3F4F5step 1.1step 1.2algebra∎

If $v \in \ker A \cap H_{R}$ , then $Π_{L} U v = A v = 0$ , so $U v \in \ker Π_{L}$ . If $u \in \ker A^{*} \cap H_{L}$ , then $Π_{R} U^{*} u = A^{*} u = 0$ , so $U^{*} u \in \ker Π_{R}$ . Every right singular vector lies in $H_{R}$ and is orthogonal to each $w_{i}$ and each $U^{*} u$ from the left kernel, since those vectors lie in $\ker Π_{R}$ . For a positive singular vector and a left-kernel vector, $⟨ (I - Π_{R}) U^{*} u_{i}, U^{*} u_{k} ⟩ = ⟨ U^{*} u_{i}, U^{*} u_{k} ⟩ - ⟨ Π_{R} U^{*} u_{i}, Π_{R} U^{*} u_{k} ⟩ = 0 - 0 = 0$ , because $u_{i} ⊥ u_{k}$ and $Π_{R} U^{*} u_{k} = 0$ . Thus the listed source blocks are mutually orthogonal. Their orthogonal complement is contained in $\ker Π_{R}$ , because the right singular vectors, including a basis of $\ker A$ , span $H_{R}$ . Applying $U$ preserves orthogonality. The listed output blocks are also mutually orthogonal: the left singular vectors, including a basis of $\ker A^{*}$ , span $H_{L}$ ; the vectors ${\tilde{w}}_{i}$ are orthonormal by step 1.1; and for $v \in \ker A$ , $⟨ {\tilde{w}}_{i}, U v ⟩ = ⟨ U v_{i}, U v ⟩ - ⟨ Π_{L} U v_{i}, Π_{L} U v ⟩ = 0$ , since $v_{i} ⊥ v$ and $A v = 0$ . The remaining named output vectors $U v$ lie in $\ker Π_{L}$ , so they are orthogonal to $H_{L}$ . The output complement is therefore contained in $\ker Π_{L}$ . Since $U$ maps each listed source block onto its corresponding output block, it maps the source complement onto the output complement. These are ambient-space decompositions; the residual blocks are not part of $H_{R}$ or $H_{L}$ . Each projector phase is scalar on the residual source and output blocks, so no residual component mixes into a projected singular-space block.

Depends on

Used by

Rank-one PUE singular-plane calculation Example
Hermitian block-encoding signal subspaces Lemma
QSVT action on singular subspaces Theorem

Dependency tree · two levels

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Sources

András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, Quantum singular value transformation and beyond, Definitions 11–14 and Lemma 14 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 29 §29.3 (standard reference, not scraped)