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Qubitization lifts scalar QSP

Statement

Let $(U, J, α)$ be an exact Hermitian block encoding of $K$ , with $K = K^{*}$ and $∥ K ∥_{o p} \leq α$ . The qubiterate $W_{q}$ of Qubitization access model has, on each eigenvalue $x = λ / α$ , a two-dimensional invariant signal space on which it is a rotation with eigenvalues $e^{\pm i \arccos x}$ ; at $x = \pm 1$ the space collapses to the corresponding eigenline.

For a phase list $Φ = (ϕ_{0}, \dots, ϕ_{d})$ , alternating $U, U^{*}$ calls and projector phases $e^{i ϕ (2 Π - I)}$ acts on each paired signal space as the scalar Hermitian-reflection QSP sequence $U_{Φ} (x)$ of QSP signal matrix and ordered phase sequence. If $P_{Φ}$ is its upper-left polynomial, the clean compressed block is $P_{Φ} (K / α)$ on all eigenspaces simultaneously, with the endpoint and null directions handled by the same polynomial evaluation. Admissible phases are supplied by Constructive factorization of an admissible QSP polynomial pair; translating to the SU(2) convention carries the scalar $i^{d}$ , which remains a relative phase under control.

Facts & Assumptions

Given: An exact Hermitian block encoding, its declared controlled/inverse access, and a scalar QSP phase list.

[F1]

Qubitization access model: $J^{*} U J = K / α$ , and $W_{q} = R_{J^{'}} S \tilde{U}$ uses $\tilde{U} = ∣ 0 ⟩ ⟨ 0 ∣ \otimes U + ∣ 1 ⟩ ⟨ 1 ∣ \otimes U^{*}$ , $S = X \otimes I$ , and $R_{J^{'}} = 2 J^{'} J^{' *} - I$ .

[F2]

Hermitian block-encoding signal subspaces: on eigenvalue $x$ , the paired $U, U^{*}$ signal spaces have matrix $(\begin{array}{cc} x & \sqrt{1 - x^{2}} \\ \sqrt{1 - x^{2}} & - x \end{array})$ , with endpoint and residual cases specified.

[F3]

QSP signal matrix and ordered phase sequence: $U_{Φ} (x) = Z (ϕ_{0}) \prod_{j = 1}^{d} (W_{R} (x) Z (ϕ_{j}))$ , where $W_{R} (x) = (\begin{array}{cc} x & \sqrt{1 - x^{2}} \\ \sqrt{1 - x^{2}} & - x \end{array})$ .

[F4]

QSVT action on singular subspaces: the alternating $U, U^{*}$ sequence with the left/right projector phases acts on each signal plane by its scalar QSP matrix; its compressed block is the corresponding singular-space polynomial, including null and endpoint spaces.

[F5]

Constructive factorization of an admissible QSP polynomial pair: every admissible polynomial pair has an exact real phase list for the fixed Hermitian-reflection signal convention.

[F6]

Projector Reflections Are Hermitian Unitary Involutions: $R_{Π} = 2 Π - I$ is $+ I$ on $ran Π$ and $- I$ on $\ker Π$ .

[F7]

Hermitian-reflection and SU(2) QSP convention conversion: the SU(2) phase list conversion contributes the explicit scalar $i^{d}$ ; under a control this is a branch-relative phase.

Proof

Given: As in Facts & Assumptions.

1.1F1F6algebra

In control-first order, $F : = S \tilde{U} = (\begin{array}{cc} 0 & U^{*} \\ U & 0 \end{array}),$ so $F^{*} = F$ and $F^{2} = I$ . Also $J^{' *} F J^{'} = \frac{1}{2} (J^{*} U^{*} J + J^{*} U J) = K / α$ because $J^{*} U J = K / α$ is self-adjoint. For a unit eigenvector $K ψ_{λ} = λ ψ_{λ}$ , put $g_{λ} = J^{'} ψ_{λ}$ , $x = λ / α$ , and $s = \sqrt{1 - x^{2}}$ . The compression identity gives $F g_{λ} = x g_{λ} + s h_{λ}$ for a unit $h_{λ} ⊥ ran J^{'}$ when $∣ x ∣ < 1$ . Applying $F^{2} = I$ then gives $F h_{λ} = s g_{λ} - x h_{λ}$ . The reflection $R_{J^{'}}$ has matrix $diag (1, - 1)$ in this basis, so $W_{q} = R_{J^{'}} F$ has matrix $(\begin{array}{cc} x & s \\ - s & x \end{array})$ . Its characteristic polynomial is $z^{2} - 2 x z + 1$ , whose roots are $x \pm i \sqrt{1 - x^{2}} = e^{\pm i \arccos x}$ . If $∣ x ∣ = 1$ , the orthogonal component has norm zero and the clean eigenline has eigenvalue $x$ .

1.2F2F3F5F6algebra

On a paired signal plane from [F2], $2 Π - I$ is diagonal with entries $+ 1, - 1$ , so $e^{i ϕ (2 Π - I)}$ is $Z (ϕ)$ . The signal calls switch between the input and output bases of [F2]; therefore their chronological product, written in matrix order, is $Z (ϕ_{0}) W_{R} (x) Z (ϕ_{1}) \dots W_{R} (x) Z (ϕ_{d}) = U_{Φ} (x)$ by [F3]. The determinant and polynomial-entry conventions are those proved by the scalar QSP supplier in [F5].

2.1F2F4F5step 1.2algebra

Every clean system vector is a finite linear combination of eigenvectors of $K$ . Step 1.2 acts on each eigenvector by the same scalar polynomial $P_{Φ} (λ / α)$ , so linearity gives the compressed operator $P_{Φ} (K / α)$ . Repeated eigenvalues cause no ambiguity because the same scalar is applied to every vector in that eigenspace; zero signal and endpoint spaces have the actions specified in [F2] and [F4].

3.1F3F4F5F7step 1.2algebra∎

The reflected convention in step 1.2 is the one used by [F3], [F4], and [F5]. If the implementation instead uses the SU(2) signal, [F7] supplies the scalar $i^{d}$ ; applying it to only the active control branch changes a relative phase, so it must be implemented as stated rather than discarded as an uncontrolled global scalar.

Depends on

Used by

Qubitized Hamiltonian simulator Algorithm

Cited to discharge well-definedness by Qubitization access model.

Dependency tree · two levels

44 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

Guang Hao Low and Isaac L. Chuang, Hamiltonian Simulation by Qubitization (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 29 §29.3 (standard reference, not scraped)
András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, Quantum singular value transformation and beyond (standard reference, not scraped)