Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 324 / 4

First construct a controlled unitary gate from the uncontrolled oracle. Keep the supplied $|v_0⟩$ in a reference register. Controlled on an extra qubit, swap the data and reference registers, query $U$ on the register that contains $|v_0⟩$ in one branch, and swap back. Because $U|v_0⟩=|v_0⟩$, that branch is unchanged, while the other branch acquires $U$ on the data. Reversing the control convention with Pauli X gates gives controlled-$U$. The reference state is returned unchanged, and each use costs one query to $U$ and $O(n)$ controlled-SWAP gates.

Expand $|b⟩={\sum }_j{b}_j|v_j⟩$ in an eigenbasis of $U$. Exact quantum phase estimation with an $m$-qubit phase register producesApply the available phase gates, controlled by the corresponding bits of $c_j$, to multiply branch $j$ byThen reverse phase estimation. Although no $U^{†}$ oracle was supplied, every eigenvalue obeys ${\lambda }_j^{2^m}=1$, so $U^{-q}=U^{2^m-q}$. The inverse controlled powers can therefore be implemented with forward calls to $U$. Since $2^m=\mathrm{poly}(n)$, phase estimation and its inverse use only $\mathrm{poly}(n)$ queries. The phase register returns to $|0^m⟩$, while the data register isAs a direct check, the same spectral promise implies $U^{†}=U^{2^m-1}$.

The HHL algorithm requires coherent, efficient and repeatable preparation of the normalized state $|b⟩$, normally through a known preparation circuit and its inverse; possession of a single unknown physical specimen does not supply that access. The component of $b$ on any discarded or unresolved small-eigenvalue subspace must also be negligible. Here $U$ is a unitary operator, so it is invertible and all its singular values equal one, giving condition number $\kappa =1$.

Standard HHL is stated for a Hermitian matrix with an efficient sparse-access or block encoding oracle. A non-Hermitian $U$ can be embedded in the Hermitian block matrixpart (a) supplies efficient access to $U^{†}$. With inverse-polynomial target precision, $m=O(\log n)$ phase bits, and an efficient preparation oracle for $|b⟩$, the runtime is $\mathrm{poly}(n)$. The output is the normalized quantum state proportional to the solution $x$, rather than a classical list of all its amplitudes.

Implement the assumed efficient classical algorithm for $x\mapsto {\theta }_x$ as a reversible circuit. On input $|x⟩|0^s⟩$, it computes an $O(m)$-bit binary expansion of the angle in a work register using $\mathrm{poly}(m)$ Toffoli gates and elementary reversible gates. Write the computed angle as a sum of binary-weighted angles. For each angle bit, apply the corresponding controlled single-qubit $R_y$ rotation to the target. These rotations have the same axis, so their angles add and produceFinally apply uncomputation to erase the work register. Each Toffoli gate and controlled rotation has a constant-size decomposition into one- and two-qubit gates when arbitrary one-qubit rotations are available. Ignoring the stipulated precision costs, the resulting circuit has size $\mathrm{poly}(m)$ and returns every quantum ancilla to zero.