Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The complexity of perfect quantum state classification

Nathaniel Johnston, Benjamin Lovitz, Vincent Russo, Jamie Sikora

TL;DR

This work analyzes the zero-error quantum state classification task through the lens of $k$-learnability, where a state drawn from a known ensemble is identified by narrowing its identity to a $k$-subset via a POVM. It shows that $k$-learnability is equivalent to $k$-incoherence of the Gram matrix, enabling SDP-based decision procedures and revealing a sharp complexity boundary: poly-time solvable when either $k$ is fixed or the Hilbert-space dimension is fixed, but NP-hard when both $k$ and the dimension are part of the input. The authors connect learning width to factor width and prove NP-hardness via reductions from $k$-clique, while also providing succinct NP certificates and polynomial-time algorithms for constant-$k$ and fixed-rank regimes. The results delineate the tractability frontier for perfect (zero-error) quantum state classification and offer practical algorithms and software for computing $k$-learnability and related properties in quantum information contexts.

Abstract

The problem of quantum state classification asks how accurately one can identify an unknown quantum state that is promised to be drawn from a known set of pure states. In this work, we introduce the notion of $k$-learnability, which captures the ability to identify the correct state using at most $k$ guesses, with zero error. We show that deciding whether a given family of states is $k$-learnable can be solved via semidefinite programming. When there are $n$ states, we present polynomial-time (in $n$) algorithms for determining $k$-learnability for two cases: when $k$ is a fixed constant or the dimension of the states is a fixed constant. When both $k$ and the dimension of the states are part of the input, we prove that there exist succinct certificates placing the problem in NP, and we establish NP-hardness by a reduction from the classical $k$-clique problem. Together, our findings delineate the boundary between efficiently solvable and intractable instances of quantum state classification in the perfect (zero-error) regime.

The complexity of perfect quantum state classification

TL;DR

This work analyzes the zero-error quantum state classification task through the lens of -learnability, where a state drawn from a known ensemble is identified by narrowing its identity to a -subset via a POVM. It shows that -learnability is equivalent to -incoherence of the Gram matrix, enabling SDP-based decision procedures and revealing a sharp complexity boundary: poly-time solvable when either is fixed or the Hilbert-space dimension is fixed, but NP-hard when both and the dimension are part of the input. The authors connect learning width to factor width and prove NP-hardness via reductions from -clique, while also providing succinct NP certificates and polynomial-time algorithms for constant- and fixed-rank regimes. The results delineate the tractability frontier for perfect (zero-error) quantum state classification and offer practical algorithms and software for computing -learnability and related properties in quantum information contexts.

Abstract

The problem of quantum state classification asks how accurately one can identify an unknown quantum state that is promised to be drawn from a known set of pure states. In this work, we introduce the notion of -learnability, which captures the ability to identify the correct state using at most guesses, with zero error. We show that deciding whether a given family of states is -learnable can be solved via semidefinite programming. When there are states, we present polynomial-time (in ) algorithms for determining -learnability for two cases: when is a fixed constant or the dimension of the states is a fixed constant. When both and the dimension of the states are part of the input, we prove that there exist succinct certificates placing the problem in NP, and we establish NP-hardness by a reduction from the classical -clique problem. Together, our findings delineate the boundary between efficiently solvable and intractable instances of quantum state classification in the perfect (zero-error) regime.
Paper Structure (22 sections, 20 theorems, 68 equations)

This paper contains 22 sections, 20 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Perfectly distinguishability and learning width
  3. Our results
  4. Author note.
  5. Structure of the paper
  6. How to tell if a set of states is k-learnable?
  7. Gram matrices and factor width
  8. Weak membership and optimization problems
  9. Weak optimization problems
  10. NP-hardness of FactorWidth
  11. Succinct proofs of FactorWidth
  12. Polynomial-time algorithm for FactorWidth for constant k
  13. Polynomial-time algorithm for FactorWidth when the rank is constant
  14. Proof of Lemma \ref{['lem:SDP_same']}
  15. Proof that $\beta \leq \alpha$.
  16. ...and 7 more sections

Key Result

Theorem 3

Given an indexed list of (possibly sub-normalized) pure quantum states $\{ | \psi_1 \rangle, \ldots, | \psi_n \rangle \} \subset \mathbb{C}^n$, there exists a polynomial-time algorithm (in $n$) to test if the list is $k$-learnable when $k$ is a fixed constant.

Theorems & Definitions (38)

  • Definition 1
  • Remark 2
  • Theorem 3: Informal, see \ref{['sect:ellipsoid']}
  • Theorem 4: Informal, see \ref{['cor:learning_easy_small_dim']}
  • Theorem 5: Informal, see \ref{['sect:inNP']}
  • Theorem 6: Informal, see \ref{['thm:nphard']}
  • Lemma 7
  • Definition 8
  • Lemma 9
  • Definition 10
  • ...and 28 more