Topological obstructions to quantum computation with unitary oracles

TL;DR

This paper uses topological methods to prove fundamental limitations of quantum algorithms with unitary oracles, showing certain tasks like implementing the if clause are impossible regardless of query complexity.

Contribution

It introduces a topological framework to establish limitations on quantum algorithms with unitary oracles, including impossibility results for controlled operations and process tomography.

Findings

No number of queries to U and U† can implement the if clause.

Limitations are shown for process tomography and certain U algorithms.

Results challenge relaxed causality experiments and suggest new measurement strategies.

Abstract

Algorithms with unitary oracles can be nested, which makes them extremely versatile. An example is the phase estimation algorithm used in many candidate algorithms for quantum speed-up. The search for new quantum algorithms benefits from understanding their limitations: Some tasks are impossible in quantum circuits, although their classical versions are easy, for example, cloning. An example with a unitary oracle $U$ is the if clause, the task to implement controlled $U$ (up to the phase on $U$). In classical computation the conditional statement is easy and essential. In quantum circuits the if clause was shown impossible from one query to $U$. Is it possible from polynomially many queries? Here we unify algorithms with a unitary oracle and develop a topological method to prove their limitations: No number of queries to $U$ and $U^\dagger$ lets quantum circuits implement the if clause, even if admitting approximations, postselection and relaxed causality. We also show limitations of process tomography, oracle neutralization, and $\sqrt[\dim U]{U}$, $U^T$, and $U^\dagger$ algorithms. Our results strengthen an advantage of linear optics, challenge the experiments on relaxed causality, and motivate new algorithms with many-outcome measurements.