The unbearable hardness of deciding about magic

TL;DR

This paper proves that determining whether quantum states possess magic resources necessary for universal quantum computation is computationally intractable, requiring super-exponential time, thus revealing fundamental limits in quantifying and certifying quantum magic.

Contribution

The paper establishes the super-exponential complexity of deciding magic in quantum states, linking it to the exponential time hypothesis and demonstrating the computational hardness of related tasks.

Findings

Deciding membership in the stabilizer polytope is super-exponentially hard.

Quantifying and certifying magic are fundamentally intractable.

Even classically simulable regimes involve super-exponential complexity.

Abstract

Identifying the boundary between classical and quantum computation is a central challenge in quantum information. In multi-qubit systems, entanglement and magic are the key resources underlying genuinely quantum behaviour. While entanglement is well understood, magic - essential for universal quantum computation - remains relatively poorly characterised. Here we show that determining membership in the stabilizer polytope, which defines the free states of magic-state resource theory, requires super-exponential time $\class{exp} ( n^2)$ in the number of qubits $n$, even approximately. We reduce the problem to solving a $3$-\class{SAT} instance on $n^2$ variables and, by invoking the exponential time hypothesis, the result follows. As a consequence, both quantifying and certifying magic are fundamentally intractable: any magic monotone for general states must be super-exponentially hard to compute, and deciding whether an operator is a valid magic witness is equally difficult. As a corollary, we establish the robustness of magic as computationally optimal among monotones. This barrier extends even to classically simulable regimes: deciding whether a state lies in the convex hull of states generated by a logarithmic number of non-Clifford gates is also super-exponentially hard. Together, these results reveal intrinsic computational limits on assessing classical simulability, distilling pathological magic states, and ultimately probing and exploiting magic as a quantum resource.