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On the hardness of cloning and connections to representation theory

Vojtěch Havlíček, Chinmay Nirkhe

Abstract

The states accepted by a quantum circuit are known as the witnesses for the quantum circuit's satisfiability. The assumption BQP does not equal QMA implies that no efficient algorithm exists for constructing a witness for a quantum circuit from the circuit's classical description. However, a similar complexity-theoretic lower bound on the computational hardness of cloning a witness is not known. In this note, we derive a conjecture about cloning algorithms for maximally entangled states over hidden subspaces which would imply that no efficient algorithm exists for cloning witnesses (assuming BQP does not contain NP). The conjecture and result follow from connections between quantum computation and representation theory; specifically, the relationship between quantum state complexity and the complexity of computing Kronecker coefficients.

On the hardness of cloning and connections to representation theory

Abstract

The states accepted by a quantum circuit are known as the witnesses for the quantum circuit's satisfiability. The assumption BQP does not equal QMA implies that no efficient algorithm exists for constructing a witness for a quantum circuit from the circuit's classical description. However, a similar complexity-theoretic lower bound on the computational hardness of cloning a witness is not known. In this note, we derive a conjecture about cloning algorithms for maximally entangled states over hidden subspaces which would imply that no efficient algorithm exists for cloning witnesses (assuming BQP does not contain NP). The conjecture and result follow from connections between quantum computation and representation theory; specifically, the relationship between quantum state complexity and the complexity of computing Kronecker coefficients.

Paper Structure

This paper contains 25 sections, 7 theorems, 41 equations.

Table of Contents

  1. The computational problem of cloning
  2. The computational hardness of cloning
  3. Aaronson and Christiano's oracle separation
  4. Zhandry's quantum money
  5. Our results
  6. Definitions and problem statements
  7. Computational task
  8. Representation theory notations and definitions
  9. Representations of symmetric groups
  10. The computational complexity of Kronecker coefficients
  11. A construction of hard verification algorithms from representation theory
  12. The complexity of generating states in $\Xi_\lambda$
  13. An improved construction
  14. Hardness of cloning
  15. Hardness of state generation
  16. ...and 10 more sections

Key Result

Theorem 1

If the internal state test passes with probability $1-\epsilon$ for a state in $\Gamma_\lambda$, then the internal state $\in \mathbb{C}^{D \times D}$ is $\leq 2 \sqrt{2\epsilon}$-close to the subspace $M_\lambda$.

Theorems & Definitions (19)

  • Definition 1: Verification circuit
  • Definition 2: Cloning
  • Conjecture 1: The hardness of witness cloning
  • Theorem 1: Characterization of internal states
  • proof
  • Lemma 1
  • Corollary 2: Complete characterization
  • proof
  • Theorem 3
  • proof
  • ...and 9 more