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Unitarity, determinant, and polynomial form of a QSP sequence

Statement

For every $x \in [- 1, 1]$ and phase list $Φ$ of length $d + 1$ , $U_{Φ} (x)$ is unitary with determinant $(- 1)^{d}$ . There are polynomials $P, Q \in C [x]$ such that $U_{Φ} (x) = (\begin{array}{cc} P (x) & s (x) Q (x) \\ (- 1)^{d + 1} s (x) Q^{*} (x) & (- 1)^{d} P^{*} (x) \end{array}),$ where $P = 0$ or $\deg P \leq d$ with parity $d$ , $Q = 0$ or $\deg Q \leq d - 1$ with parity $d - 1$ , and $P P^{*} + (1 - x^{2}) Q Q^{*} = 1$ as a formal polynomial identity.

Facts & Assumptions

Given: $x \in [- 1, 1]$ , $s = \sqrt{1 - x^{2}} \geq 0$ , and real phases $ϕ_{0}, \dots, ϕ_{d}$ .

[F1]

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

[F2]

Euler's formula: $\exp (i θ) = \cos θ + i \sin θ$ for every real $θ$ : $e^{i ϕ} = \cos ϕ + i \sin ϕ$ for real $ϕ$ .

[F3]

$\exp (z + w) = \exp z \exp w$ , and the complex exponential extends the real exponential: $e^{z + w} = e^{z} e^{w}$ for complex $z, w$ .

[F4]

Real and imaginary parts, complex conjugation, and modulus: conjugation fixes real numbers, sends $i$ to $- i$ , and $z \overline{z} = ∣ z ∣^{2}$ .

[F5]

For the displayed $2 \times 2$ complex matrices, $M^{†} : = {\overline{M}}^{T}$ ; “unitary” means $M^{†} M = M M^{†} = I$ .

Proof

1.1F1F5algebra

Since $s^{2} = 1 - x^{2}$ , direct multiplication gives $W_{R}^{†} = W_{R}$ , $W_{R}^{2} = I$ , and $\det W_{R} = - 1$ . Thus $W_{R}^{†} W_{R} = W_{R} W_{R}^{†} = I$ , so $W_{R}$ is unitary in the matrix convention [F5].

1.2F1F2F3base

We prove the polynomial form by induction on $d$ . For $d = 0$ , $U_{Φ} = Z (ϕ_{0})$ , so $P = e^{i ϕ_{0}}$ and $Q = 0$ satisfy the degree and parity bounds and $P P^{*} = 1$ .

2.1F2F3F4F5step 1.1algebra

By [F2] and [F4], $\overline{e^{i ϕ}} = \cos ϕ - i \sin ϕ = e^{- i ϕ}$ ; by [F3], $e^{i ϕ} e^{- i ϕ} = 1$ . Thus $Z (ϕ)^{†} Z (ϕ) = Z (ϕ) Z (ϕ)^{†} = I$ and $\det Z (ϕ) = 1$ . Each factor $W_{R} Z (ϕ)$ is unitary with determinant $- 1$ , so the $d$ signal factors give $U_{Φ}^{†} U_{Φ} = I$ and $\det U_{Φ} = (- 1)^{d}$ .

3.1F5step 2.1algebra

Write a $2 \times 2$ unitary matrix as $M = (\begin{array}{cc} a & b \\ c & e \end{array})$ with determinant $ϵ \in {1, - 1}$ . The cofactor formula gives $M^{- 1} = ϵ (\begin{array}{cc} e & - b \\ - c & a \end{array})$ ; by [F5], $M^{- 1} = M^{†} = (\begin{array}{cc} \overline{a} & \overline{c} \\ \overline{b} & \overline{e} \end{array})$ . Equating entries yields $e = ϵ \overline{a}$ and $c = - ϵ \overline{b}$ . Applying this with $ϵ = (- 1)^{d}$ shows the displayed lower row is forced by the first row.

4.1F1step 3.1step 1.2IHinductionalgebra

Suppose the representation holds at degree $d$ . Multiplying on the right by $W_{R} (x) Z (ϕ_{d + 1})$ gives $P_{d + 1} = e^{i ϕ_{d + 1}} (x P_{d} + (1 - x^{2}) Q_{d})$ and $Q_{d + 1} = e^{- i ϕ_{d + 1}} (P_{d} - x Q_{d})$ . These are polynomials with degree bounds $d + 1$ and $d$ and parities $d + 1$ and $d$ . The lower row then has the required form by step 3.1.

5.1step 1.2step 4.1inductiondischarge-induction: step 4.1algebra∎

Put $s^{2} = 1 - x^{2}$ . The scalar phase in each recurrence factor cancels with its coefficient conjugate. Expanding the first norm gives $P_{d + 1} P_{d + 1}^{*} = x^{2} P_{d} P_{d}^{*} + x s^{2} (P_{d} Q_{d}^{*} + Q_{d} P_{d}^{*}) + s^{4} Q_{d} Q_{d}^{*} .$ Expanding the second gives $s^{2} Q_{d + 1} Q_{d + 1}^{*} = s^{2} P_{d} P_{d}^{*} - x s^{2} (P_{d} Q_{d}^{*} + Q_{d} P_{d}^{*}) + x^{2} s^{2} Q_{d} Q_{d}^{*} .$ The mixed terms cancel and the unmixed terms sum to $(x^{2} + s^{2}) P_{d} P_{d}^{*} + s^{2} (x^{2} + s^{2}) Q_{d} Q_{d}^{*} = P_{d} P_{d}^{*} + s^{2} Q_{d} Q_{d}^{*}$ . Thus the identity is preserved as a formal polynomial identity, and induction from step 1.2 proves it for every $d$ .

Depends on

Used by

Constructive factorization of an admissible QSP polynomial pair Theorem
QSVT action on singular subspaces Theorem

Cited to discharge well-definedness by QSP signal matrix and ordered phase sequence.

Dependency tree · two levels

24 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 (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 29 (standard reference, not scraped)