No Size-Preserving Amplification with Quantum Advice

Abstract:Marriott and Watrous showed that quantum Merlin--Arthur games admit generic error reduction without increasing witness size [Computational Complexity, 2005]. In this work, we show that this state-size-preserving amplification property does not hold for polynomial-time quantum computation with quantum advice. In particular, we present decision problems for which even a vanishing additive error reduction requires longer advice. More precisely, for every polynomially bounded advice length $m(n)\geq n^4$ and every error bound $\varepsilon(n)$ that stays below $1/2$ by at least an inverse polynomial, there is a positive function $\delta$ with $\delta(n)=O\bigl(\min\{(\log m/m)^{1/4},\ \sqrt{\log m/m}\,/(1/2-\varepsilon(n))\}\bigr)$ such that $\mathsf{BQP}_{\varepsilon}/\mathsf{q}m \subsetneq \mathsf{BQP}_{\varepsilon + \delta}/\mathsf{q}m$; for constant $\varepsilon$ the gap is $O(\sqrt{\log m/m})$. Here, $\mathsf{BQP}_\varepsilon/\mathsf{q}m$ is the class of languages recognizable with error at most $\varepsilon(n)$ by a polynomial-time quantum algorithm with an $m(n)$-qubit advice state that only depends on the input length $n$. We show this by proving a stronger separation $\mathsf{P}_{\varepsilon+\delta}/\mathsf{r} m \not\subset \mathsf{BQP}_{\varepsilon}/\mathsf{q} m$, where $\mathsf{P}_{\varepsilon}/\mathsf{r}m$ is the class of languages recognizable with error at most $\varepsilon(n)$ by a deterministic polynomial-time algorithm with an $m(n)$-bit advice string sampled from a distribution that depends only on $n$.

Quantum Physics (quant-ph); Computational Complexity (cs.CC)

arXiv:2610.03377 [quant-ph]

(or arXiv:2610.03377v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.03377 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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