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Quantum Computing, Postselection, and Probabilistic Polynomial-Time

Scott Aaronson

TL;DR

The paper proves that the power of quantum computation with postselection, $PostBQP$, exactly equals the classical complexity class $PP$, providing a new bridge between quantum and classical computation. It presents a concise constructive proof, shows robust closure properties, and uses this equivalence to give new, simpler insights into classical results like the closure of $PP$ under intersection. The work also analyzes how hypothetical changes to quantum mechanics (e.g., alternative probability rules or nonunitary evolution) would dramatically increase computational power, thereby offering a fresh perspective on why quantum mechanics may be constrained to unitary evolution with the standard Born rule. Overall, it demonstrates that quantum concepts can yield streamlined proofs and deeper understanding of classical complexity theory.

Abstract

I study the class of problems efficiently solvable by a quantum computer, given the ability to "postselect" on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic Polynomial-Time. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PP-complete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.

Quantum Computing, Postselection, and Probabilistic Polynomial-Time

TL;DR

The paper proves that the power of quantum computation with postselection, , exactly equals the classical complexity class , providing a new bridge between quantum and classical computation. It presents a concise constructive proof, shows robust closure properties, and uses this equivalence to give new, simpler insights into classical results like the closure of under intersection. The work also analyzes how hypothetical changes to quantum mechanics (e.g., alternative probability rules or nonunitary evolution) would dramatically increase computational power, thereby offering a fresh perspective on why quantum mechanics may be constrained to unitary evolution with the standard Born rule. Overall, it demonstrates that quantum concepts can yield streamlined proofs and deeper understanding of classical complexity theory.

Abstract

I study the class of problems efficiently solvable by a quantum computer, given the ability to "postselect" on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic Polynomial-Time. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PP-complete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.

Paper Structure

This paper contains 6 sections, 5 theorems, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. The Class $\mathsf{PostBQP}$
  4. Fantasy Quantum Mechanics
  5. Open Problems
  6. Acknowledgments

Key Result

Proposition 2

$\mathsf{PostBQP}\subseteq\mathsf{PP}$.

Figures (1)

  • Figure 1: If $s$ and $2^{n}-2s$ are both positive, then as we vary the ratio of $\beta$ to $\alpha$, we eventually get close to $\left\vert +\right\rangle =\left( \left\vert 0\right\rangle +\left\vert 1\right\rangle \right) /\sqrt{2}$ (dashed lines). On the other hand, if $2^{n}-2s$ is not positive (dotted line), then we never even get into the first quadrant.

Theorems & Definitions (6)

  • Definition 1
  • Proposition 2
  • Proposition 3
  • Theorem 4
  • Proposition 5
  • Theorem 6