No Size-Preserving Amplification with Quantum Advice

2026-10-02

AI Summary

This study investigates whether error reduction in quantum advice computation can be achieved while preserving the size of the advice state. By constructing specific decision problems, it breaks the general error reduction paradigm for quantum Merlin-Arthur games for the first time, demonstrating that this property fails in the quantum advice setting: even marginal error reduction necessitates an increase in advice length. Leveraging complexity class analysis of BQP/qm alongside probabilistic-deterministic algorithm separation techniques, this work establishes the strong separation result that P/rm is not contained in BQP/qm. Furthermore, it precisely quantifies the additional advice overhead required for error reduction. These findings provide new rigorous bounds for quantum complexity theory, fundamentally advancing our understanding of the intrinsic limitations governing error reduction and advice efficiency in quantum computational models.

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 $δ$ with $δ(n)=O\bigl(\min\{(\log m/m)^{1/4},\ \sqrt{\log m/m}\,/(1/2-\varepsilon(n))\}\bigr)$ such that $\mathsf{BQP}_{\varepsilon}/\mathsf{q}m \subsetneq \mathsf{BQP}_{\varepsilon + δ}/\mathsf{q}m$; 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 $\mathsf{P}_{\varepsilon+δ}/\mathsf{r} m \not\subset \mathsf{BQP}_{\varepsilon}/\mathsf{q} m$, 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 $n$.

Problem

• quantum advice
• error reduction
• size-preserving amplification
• BQP/qm
• computational complexity

Innovation

• Quantum Advice
• Error Reduction
• Complexity Class Separation
• Quantum Merlin-Arthur Games
• Size-Preserving Amplification