Low Depth Phase Oracle Using a Parallel Piecewise Circuit

• URL: http://arxiv.org/abs/2409.04587v2
• Date: Tue, 8 Oct 2024 17:53:38 GMT
• Title: Low Depth Phase Oracle Using a Parallel Piecewise Circuit
• Authors: Zhu Sun, Gregory Boyd, Zhenyu Cai, Hamza Jnane, Balint Koczor, Richard Meister, Romy Minko, Benjamin Pring, Simon C. Benjamin, Nikitas Stamatopoulos,
• Abstract summary: We explore the important task of applying a phase $exp(i f(x))$ to a computational basis state $left| x right>$. The closely related task of rotating a target qubit by an angle depending on $f(x)$ is also studied.
• Score: 3.629687485125086
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We explore the important task of applying a phase $exp(i f(x))$ to a computational basis state $\left| x \right>$. The closely related task of rotating a target qubit by an angle depending on $f(x)$ is also studied. Such operations are key in many quantum subroutines, and often the function $f$ can be well-approximated by a piecewise linear composition. Examples range from the application of diagonal Hamiltonian terms (such as the Coulomb interaction) in grid-based many-body simulation, to derivative pricing algorithms. Here we exploit a parallelisation of the piecewise approach so that all constituent elementary rotations are performed simultaneously, that is, we achieve a total rotation depth of one. Moreover, we explore the use of recursive catalyst 'towers' to implement these elementary rotations efficiently. Depending on the choice of implementation strategy, we find a depth as low as $O(log n + log S)$ for a register of $n$ qubits and a piecewise approximation of $S$ sections. In the limit of multiple repetitions of the oracle, we find that catalyst tower approaches have an $O(S \cdot n)$ T-count, whereas linear interpolation with QROM has an $O(n^{log_2(3)})$ T-count.

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