QubitLogic
arXiv

Quantum Speedups Require Structure or Depth

One of the most basic conjectures in quantum complexity theory states that every $t$-query quantum algorithm can be simulated on most inputs by a $\mathrm{poly}(t)$-query classical algorithm.

We settle this conjecture for parallel quantum algorithms, showing that every t-query d-round quantum algorithm can be simulated on most inputs with t^{O(d^2)} classical queries.

Why this matters

Understanding the conditions for quantum speedups helps delineate the advantages that quantum computing may have over classical computing. This can guide future research and investments in quantum technology, especially for unstructured problems.

What they actually achieved

The researchers showed that every t-query d-round quantum algorithm can be simulated on most inputs with t^{O(d^2)} classical queries. This finding suggests that unstructured problems require quantum circuits of superconstant depth for superpolynomial speedups.

What they did not achieve

They did not achieve a solution to quantum speedups in all scenarios, particularly for structured problems. The results are specific to parallel quantum algorithms, and broader implications for all quantum algorithms remain unproven.

Sources