Separating quantum and classical efficient computation

Establish whether the class of problems solvable efficiently by quantum algorithms with bounded error (BQP) is strictly different from the class of problems solvable efficiently by classical algorithms; for example, show the existence (or nonexistence) of a decision problem in BQP that is not efficiently solvable by any classical randomized algorithm.

Background

After introducing BQP, the author notes that it is not currently known whether quantum computers are computationally different from classical computers in terms of efficient solvability. This emphasizes a foundational uncertainty about the comparative power of quantum versus classical computation.

Resolving this separation would underpin claims of inherent quantum advantage beyond specific algorithms and provide a formal basis for the distinct capabilities of quantum processing units relative to classical architectures.

References

We don't really know if quantum computers are any different from classical computers, but we highly suspect they are.

— What You Shouldn't Know About Quantum Computers  (2405.15838 - Ferrie, 2024) in Myth 5: Quantum Computers Will Replace Digital Computers — BQP subsection

Shortly after, Fortnow and Rogers asked if the same is true relative to a \emph{random} oracle, a problem that remains open today.

— Quantum Speedups Require Structure or Depth  (2608.19158 - Blanc et al., 19 Aug 2026) in Section 5, subsection “Implications for random-oracle separations”

Quantum computers challenge the Strong Church-Turing thesis, but it remains unproven whether they can be efficiently simulated or not.

— On quantum channels: extreme points, topology, and categorical properties  (2608.20689 - Silva, 21 Aug 2026) in Chapter 1, Section "CPTP maps in Quantum Computation and Quantum Information Theory"

There is currently no general theory of speed-ups. While algorithms such as Shor's are generally believed to be robust examples of exponential speedups, there is no classification of speedups in which to place the SQU test.

— For Whom Does Bell Hold?  (2609.10535 - Green et al., 9 Sep 2026) in Section 5.3.3, “Summary and Future Directions”