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Projected-unitary encoding and singular vectors

Definition

Let $H$ be a finite-dimensional complex inner-product space, let $U : H \to H$ be unitary, and let $Π_{R}, Π_{L} : H \to H$ be orthogonal projections. Write $H_{R} = ran (Π_{R})$ and $H_{L} = ran (Π_{L})$ . The projected-unitary encoding (PUE) of this data is the linear map

$A : H_{R} \to H_{L}, A x = Π_{L} U x .$

Equivalently, its ambient-space compression is $Π_{L} U Π_{R}$ , restricted to $H_{R}$ and regarded as taking values in $H_{L}$ . The map $A$ is a contraction: $∥ A ∥_{o p} \leq 1$ .

Put $n = \dim H_{R}$ , $m = \dim H_{L}$ , and $r = rank (A)$ . A singular-value decomposition consists of orthonormal bases $v_{1}, \dots, v_{n}$ of $H_{R}$ and $u_{1}, \dots, u_{m}$ of $H_{L}$ , and positive singular values $1 \geq σ_{1} \geq \dots \geq σ_{r} > 0$ , such that

$A = \sum_{j = 1}^{r} σ_{j} ∣ u_{j} ⟩ ⟨ v_{j} ∣ .$

The remaining right basis vectors span $\ker A$ and the remaining left basis vectors span $\ker A^{*}$ . Empty bases and sums are allowed when a dimension or the rank is zero. The pairing in each rank-one map is the root-linear-first Dirac conversion of Dirac Notation Conversion.

An exact square block encoding is a special case: identify the system with the all-zero ancilla subspace and take $Π_{L} = Π_{R}$ equal to that subspace. Under the injection $J_{a}$ of Finite block encoding, the compression is $J_{a}^{*} U J_{a}$ ; an exact $(α, a, 0)$ block encoding has compressed block $A / α$ .

Facts & Assumptions

Given: A finite-dimensional complex inner-product space $H$ , a unitary $U : H \to H$ , orthogonal projections $Π_{R}, Π_{L}$ , and the definitions of operator norm and PUE above.

[F1]

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

[F2]

Pythagoras and finite orthogonal sums: the squared norm of a sum of orthogonal vectors is the sum of their squared norms.

[F3]

Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces: a unitary operator preserves the norm of every vector.

[F4]

The operator norm is zero on the zero domain and otherwise is max_{||v||=1} ||Tv||: the operator norm is the maximum of $∥ T x ∥$ over unit vectors, with norm zero on the zero domain.

[F5]

Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition: every linear map between finite-dimensional inner-product spaces has orthonormal singular bases and positive singular values such that the displayed SVD formula holds.

[F6]

Dirac Notation Conversion: $∣ u ⟩ ⟨ v ∣$ maps $x$ to $⟨ x, v ⟩_{r o o t} u$ .

[F7]

Finite block encoding: the all-zero ancilla compression of $U$ is $J_{a}^{*} U J_{a}$ , and an exact block encoding sets this compression equal to the normalized target matrix.

[F8]

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

[F9]

Every linear map between finite-dimensional inner product spaces has a unique adjoint: every linear map between finite-dimensional inner-product spaces has a unique adjoint.

[F10]

For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and $T^{*} T = I$ are equivalent: a finite-dimensional unitary satisfies $U^{*} = U^{- 1}$ and preserves inner products.

[F11]

Real and complex inner-product spaces and their induced length: the inner product is positive definite, so pairings with every vector distinguish vectors.

Proof

1.1givenF1F2F3F4algebra

If $H_{R} = 0$ , then $A$ is the zero map and its operator norm is zero by [F4]. Otherwise take any unit $x \in H_{R}$ . The decomposition $U x = Π_{L} U x + (I - Π_{L}) U x$ has orthogonal terms by [F1], so [F2] gives $∥ Π_{L} U x ∥^{2} \leq ∥ U x ∥^{2}$ . By [F3], $∥ U x ∥ = 1$ , hence $∥ A x ∥ \leq 1$ . Taking the maximum over unit $x$ proves $∥ A ∥_{o p} \leq 1$ .

1.2F7givenalgebra

In the square block-encoding case, $J_{a}$ is the isometric identification of the system with $H_{R} = H_{L} = ran (Π_{a})$ . The definition of the projected block in [F7] is exactly $J_{a}^{*} U J_{a}$ , and the exact case identifies it with the normalized target. Thus this is the stated special case.

2.1F4F5F6step 1.1algebra

Apply [F5] to the finite-dimensional map $A : H_{R} \to H_{L}$ . For each positive singular pair, $A v_{j} = σ_{j} u_{j}$ and $∥ v_{j} ∥ = ∥ u_{j} ∥ = 1$ , so [F4] and step 1.1 give $σ_{j} = ∥ A v_{j} ∥ \leq 1$ . The rank-one terms in the SVD sum have the coordinate action fixed by [F6]. When $r = 0$ , the positive singular list and the displayed sum are empty; the kernel bases still exist by [F5].

3.1F5F6F8F9F11step 2.1algebra

For each $k \leq r$ and every $x \in H_{R}$ , the SVD formula, root-linear-first rank-one action [F6], and orthonormality of the $u_{j}$ give $⟨ A x, u_{k} ⟩ = σ_{k} ⟨ x, v_{k} ⟩$ ; for $k > r$ the same pairing is zero. By [F8]–[F9], $⟨ A x, u_{k} ⟩ = ⟨ x, A^{*} u_{k} ⟩$ , so positive definiteness [F11] gives $A^{*} u_{k} = σ_{k} v_{k}$ for $k \leq r$ and $A^{*} u_{k} = 0$ for $k > r$ . The SVD formula also gives $A v_{k} = σ_{k} u_{k}$ for $k \leq r$ and $A v_{k} = 0$ for $k > r$ . Since each singular list is a basis, expanding an arbitrary vector in those bases shows that $\ker A = span {v_{r + 1}, \dots, v_{n}}$ and $\ker A^{*} = span {u_{r + 1}, \dots, u_{m}}$ : the positive-index coefficients must vanish because the corresponding singular values are positive and the other basis is orthonormal. The argument includes empty bases and sums.

4.1F1F8F9F10givenalgebra∎

By [F1], orthogonal decomposition gives $⟨ Π_{L} U x, y ⟩ = ⟨ U x, y ⟩$ and $⟨ x, Π_{R} U^{*} y ⟩ = ⟨ x, U^{*} y ⟩$ for $x \in H_{R}$ and $y \in H_{L}$ . The adjoint rule [F8] for $U$ therefore gives $⟨ A x, y ⟩ = ⟨ x, Π_{R} U^{*} y ⟩$ . Thus $y \mapsto Π_{R} U^{*} y$ is an adjoint of $A$ , and uniqueness [F9] yields $A^{*} = Π_{R} U^{*} ∣_{H_{L}}$ . Also, [F10] applies to the unitary $U$ and shows that $U^{*} = U^{- 1}$ is unitary.

Depends on

Used by

QSVT regularized pseudoinverse transformation Algorithm
Quantum singular-value transformation circuit Algorithm
Odd and even singular-value transforms Definition
Operator perturbation bound for QSVT Lemma
PUE singular-space invariant decomposition Lemma
Singular-value transform is independent of SVD choices Lemma
Projector phase operations from controlled reflections Theorem
QSVT action on singular subspaces Theorem
QSVT robustness, approximation error, and resources 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 11–12 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 29 §29.3 (standard reference, not scraped)