Robust subspace designs and the power of a unique small quantum witness

2026-10-05

AI Summary

This study addresses the quantum witness isolation problem for complex instances possessing multiple accepting witnesses in quantum computation. Methodologically, it introduces the novel concept of robust subspace design, integrating probabilistic and explicit constructions with linear algebraic subspace theory to achieve a space-bounded quantum variant of the Valiant–Vazirani theorem. This framework is subsequently applied to witness isolation and nullity testing reductions within quantum Merlin–Arthur (QMA) protocols. The primary contributions lie in effectively restricting NP-complete problems to at most one accepting witness while preserving their computational hardness, successfully isolating unique quantum witnesses, and streamlining the proof pathway for classical C_=L containment relations.

Abstract

The concept of subspace designs was introduced by Guruswami and Xing (STOC'13), and explicit constructions were given by Guruswami and Kopparty (FOCS'13). These are families of subspaces that have small intersection with any given subspace of a fixed dimension. We introduce \emph{robust subspace designs}. Informally, these are a quantitative extension in which we demand that not too many subspaces of the family contain directions that lie close to any given subspace of a fixed dimension. We give a probabilistic construction of such a robust subspace design of polynomial size, as well as a non-trivial explicit construction of superpolynomial size. Our main application of this new concept is a quantum space-bounded variant of the Valiant-Vazirani theorem (Theor.Comput.Sci.'86), which shows that restricting $\mathsf{NP}$-complete problems to instances with at most one accepting witness preserves hardness under randomized reductions. For quantum witnesses, the analogous quantity is the dimension of an accepting witness subspace. We use our probabilistic construction of robust subspace designs to isolate a unique witness for space-bounded quantum Merlin-Arthur protocols with perfect completeness and an acceptance gap outside their perfectly accepting subspace. As further applications, we give a randomized reduction of well-conditioned nullity testing to space-bounded quantum Merlin-Arthur protocols with perfect completeness. Using a similar idea, we find that ordinary subspace designs allow us to recover the classical $\mathsf{C_= L}$ containment of Allender, Beals, and Ogihara (STOC'96) for general nullity testing through a simpler proof.

Problem

• robust subspace designs
• Valiant-Vazirani theorem
• quantum Merlin-Arthur protocols
• nullity testing
• space-bounded quantum computation

Innovation

• Robust Subspace Designs
• Quantum Witness
• Valiant-Vazirani Theorem
• Space-bounded QMA
• Nullity Testing