Random Circuit Sampling Supremacy Conjecture

Determine whether any classical algorithm can efficiently perform random circuit sampling within polynomial time while achieving a small, reasonable error margin, i.e., sample from the output distribution generated by random quantum circuits to within a small error bound using polynomial computational resources.

Background

Random circuit sampling (RCS) is proposed as a leading task for demonstrating quantum advantage, where a quantum processor samples bit strings from the output distribution of randomly chosen quantum circuits. The authors note that the computational complexity of RCS has been investigated theoretically and highlight a central conjecture asserting the classical intractability of achieving efficient sampling within a small error bound.

This conjecture underpins experimental claims of quantum supremacy using superconducting qubit processors (e.g., Sycamore). Resolving the conjecture would clarify the hardness landscape of RCS and strengthen or refute the theoretical foundation of beyond-classical demonstrations in the noisy, intermediate-scale regime.

References

The RCS supremacy conjecture, in particular, suggests that no classical algorithm can efficiently perform RCS within polynomial time while maintaining a small, reasonable error margin.

— Google Quantum AI's Quest for Error-Corrected Quantum Computers  (2410.00917 - AbuGhanem, 2024) in Section 6.3 (Random circuit sampling)

We treat this as an optimistic baseline computed under assumptions that favor the classical side (Sec.~\ref{sec:classical}): unbounded memory, negligible communication, and nothing better than the contraction paths we found---slicing, approximate, multi-amplitude, and MPS methods all remain open.

— Quantum computational advantage in random-circuit sampling on IBM superconducting quantum computers  (2609.28657 - Sedrakyan et al., 23 Sep 2026) in Introduction, paragraph preceding Figure 1

Sampling from a distribution q that achieves χ(q,p) ≥ 1 against the ideal output distribution p of a random degree-3 IQP circuit with inverse polynomial failure probability requires time Ω(2{κ n}) for some constant κ > 1/2 or superpolynomial space.

— Verifiable quantum advantage based on polynomials with planted structures  (2609.39918 - Bläser et al., 30 Sep 2026) in Conjecture 2, Section 1, subsection “Verifiable advantage from planted independent spaces”

Extending our work to polynomial-time sampling of arbitrary-connectivity peaked shallow circuits is a natural open question.

— Polynomial-time additive-error estimation of output probabilities for shallow quantum circuits  (2610.02146 - Coudron et al., 1 Oct 2026) in Section 1, Introduction, paragraph following Table 1

There exist uniformly generated families of concordance-preserving circuits on $n$ qubits, with inputs of polynomial description given to the simulator, for which no classical algorithm running in time polynomial in $n$ samples the output of the individual runs to within total variation $1/\mathrm{poly}(n)$.

— Classical Verification of a Remote Quantum Processor via Degenerate Concordant Computations: An Application to Randomness Generation  (2610.08185 - Das et al., 6 Oct 2026) in Conjecture 1, Section 7, “Conjectures”