Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The unbearable hardness of deciding about magic

Lorenzo Leone, Jens Eisert, Salvatore F. E. Oliviero

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 $\exp( n^2)$ in the number of qubits $n$, even approximately. We reduce the problem to solving a $3$-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.

The unbearable hardness of deciding about magic

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 in the number of qubits , even approximately. We reduce the problem to solving a -SAT instance on 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.
Paper Structure (35 sections, 32 theorems, 106 equations, 1 figure)

This paper contains 35 sections, 32 theorems, 106 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. Quantifying magic
  4. Witnessing magic
  5. The classical-quantum boundary
  6. Summary of conceptual results
  7. Implications of our results
  8. Classical simulation of noisy quantum circuits
  9. Super-polynomial time magic-state distillation
  10. Discussion and open questions
  11. Acknowledgments
  12. Preliminaries
  13. Notation and basic notions
  14. Basics of complexity theory: $3$-SAT, quasi-polynomial classes and ETH
  15. Convex sets and related computational problems
  16. ...and 20 more sections

Key Result

Theorem 1

Deciding whether a state $\rho$ lies in the stabilizer polytope $\hat{\mathcal{S}}$, or is $\varepsilon$-far from every state in $\hat{\mathcal{S}}$, belongs to the complexity class $\class{QP}^2$ for any $\varepsilon = 1/\class{poly}(d)$.

Figures (1)

  • Figure 1: In this work, we show that any algorithm aimed at determining whether a $d \times d$ density matrix on $n$ qubits, $\rho$, belongs to the stabilizer polytope must necessarily run in super-exponential time. Consequently, any magic monotone also requires super-exponential time to compute. Concretely, we prove that the membership problem in the stabilizer polytope, formulated as a decision problem, lies in the complexity class $\class{QP}^2$. Also depicted in the upper picture is a schematic representation of a witness as a hyperplane separating a quantum state from the stabilizer polytope. At a higher level, it demonstrates that delineating the "classical" setting—where efficient classical simulation is possible, inside the polytope—from the "quantum" setting—where one can hope for computational quantum advantages—is itself a computationally difficult task.

Theorems & Definitions (75)

  • Theorem 1: Membership in the stabilizer polytope is hard. Informal of \ref{['thm:nphardrhowmem', 'wmemeforstabinqp2']}
  • Corollary 1: No efficient magic monotone
  • Theorem 2: Finding witnesses is hard. Informal of \ref{['thm:nphardwwd', 'wwdinqp2']}
  • Theorem 3: Tracing the classical-quantum boundary is hard. Informal of \ref{['thm:suptdop']}
  • Corollary 2: Deciding classical simulation of noisy circuits is hard. Informal of \ref{['cor1app', 'cor1appdoped']}
  • Corollary 3: Super-polynomial time magic-state distillation. Informal of \ref{['cor2app']}
  • Definition 1: Schatten $p$-norms
  • Definition 2: Decision problem
  • Definition 3: P and NP complexity classes
  • Definition 4: Quasipolynomial time and $\class{QP}^k$
  • ...and 65 more