Quantum singular-value transformation circuit

Inputs. A finite-dimensional complex-inner-product space $H$, a unitary $U:H\to H$, orthogonal projections ${\mathrm{\Pi }}_R,{\mathrm{\Pi }}_L$, an integer $d\ge 0$, and an exact real phase list $\mathrm{\Phi }=({\varphi }_0,\dots ,{\varphi }_d)$. The input access contract states separately whether $U$, $U^{\ast }$, controlled-$U$, and controlled-$U^{\ast }$ are supplied and gives their individual costs. The scalar signal sequence is $U_{\mathrm{\Phi }}(x)=Z({\varphi }_0){\prod }_{j=1}^d(W_R(x)Z({\varphi }_j))$ from QSP signal matrix and ordered phase sequence, with its admissible polynomial pair from Admissible polynomial pair for Hermitian-reflection QSP. A desired polynomial target is supplied with a certificate that $\mathrm{\Phi }$ realizes its scalar polynomial; generation of $\mathrm{\Phi }$ is a separate classical/circuit input.

Output. The procedure returns the finite unitary sequence below and uses its compression from $H_R=\mathrm{ran}({\mathrm{\Pi }}_R)$ to $H_L=\mathrm{ran}({\mathrm{\Pi }}_L)$ for odd $d$, or from $H_R$ to itself for even $d$. No measurement or state-preparation circuit is implicit in this block description.

Instructions. For $d\ge 0$, set the current side to $R$ and apply the projector phase $e^{i{\varphi }_d(2{\mathrm{\Pi }}_R-I)}$. For $j=d-1,d-2,\dots ,0$, apply $U$ and switch the current side from $R$ to $L$ if it is $R$; otherwise apply $U^{\ast }$ and switch from $L$ to $R$. Then apply $e^{i{\varphi }_j(2{\mathrm{\Pi }}_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}-I)}$. Thus there are $d$ alternating signal calls and $d+1$ projector-phase operations; $d=0$ has no signal call and consists only of the phase at ${\mathrm{\Pi }}_R$.

For a real target polynomial, run the two phase lists $\mathrm{\Phi }$ and $-\mathrm{\Phi }$ in coherent control branches, with the control initialized in $\mathrm{\mid }+⟩$, and take the $\mathrm{\mid }+⟩$-to-$\mathrm{\mid }+⟩$ block. The procedure uses the reflected signal convention $W_R$ throughout. If a circuit is built through the alternative SU(2) convention, its control-dependent scalar phase must be retained as a relative phase. The exact block action, the real-part extraction, and the implementation costs are established in the separate theorem items on this page. Every phase list, inverse call, controlled call, phase rotation, and finite-gate compilation is supplied or charged separately.

When the input is a bounded real target polynomial, its completion and phase-generation steps are supplied separately; finite bit-time and gate-synthesis bounds are not inferred from a bare phase list.