Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Black Holes and Complexity Classes

Leonard Susskind

TL;DR

The paper investigates whether the complexity of universal quantum circuits grows linearly in time up to an exponential maximum and how this relates to black hole interior growth in AdS/CFT. It builds a bridge between gravitational conjectures about ERB growth and quantum complexity class separations, formalizing this through the Weak Complexity Hypothesis (WCH) and the Stronger SerCH. The main results show that WCH is equivalent to PSPACE not being contained in BQP/poly, and SerCH is equivalent to PSPACE not in BQSUBEXP/subexp, thereby linking physical limits of gravity to fundamental limits on quantum computation. Although a complete proof remains elusive, the work highlights a profound interplay between high-energy physics and computational complexity with potential implications for both fields.

Abstract

It is not known what the limitations are on using quantum computation to speed up classical computation. An example would be the power to speed up PSPACE-complete computations. It is also not known what the limitations are on the duration of time over which classical general relativity can describe the interior geometry of black holes. What is known is that these two questions are closely connected: the longer GR can describe black holes, the more limited are quantum computers. This conclusion, formulated as a theorem, is a result of unpublished work done by Scott Aaronson and myself which I explain here.

Black Holes and Complexity Classes

TL;DR

The paper investigates whether the complexity of universal quantum circuits grows linearly in time up to an exponential maximum and how this relates to black hole interior growth in AdS/CFT. It builds a bridge between gravitational conjectures about ERB growth and quantum complexity class separations, formalizing this through the Weak Complexity Hypothesis (WCH) and the Stronger SerCH. The main results show that WCH is equivalent to PSPACE not being contained in BQP/poly, and SerCH is equivalent to PSPACE not in BQSUBEXP/subexp, thereby linking physical limits of gravity to fundamental limits on quantum computation. Although a complete proof remains elusive, the work highlights a profound interplay between high-energy physics and computational complexity with potential implications for both fields.

Abstract

It is not known what the limitations are on using quantum computation to speed up classical computation. An example would be the power to speed up PSPACE-complete computations. It is also not known what the limitations are on the duration of time over which classical general relativity can describe the interior geometry of black holes. What is known is that these two questions are closely connected: the longer GR can describe black holes, the more limited are quantum computers. This conclusion, formulated as a theorem, is a result of unpublished work done by Scott Aaronson and myself which I explain here.

Paper Structure

This paper contains 10 sections, 8 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Two Conjectures
  3. Qubits and Black Holes
  4. The Complexity Hypotheses
  5. Proof of Theorem 1
  6. The L Problem
  7. Stronger Hypothesis
  8. Conclusion
  9. Some Black Hole Terminology
  10. Some Complexity Terminology

Figures (3)

  • Figure 1: Penrose diagram for a "two-sided" black hole. The spacetime is sliced by maximal space-like slices anchored on the boundaries. It is obvious that the portions of the slices behind the horizon grow with time.
  • Figure 2: A circuit built by repeating a finite depth circuit $W$. $W^t$ is the unitary evolution operator after $t$ discrete time steps.
  • Figure 3: Polynomial size circuit described in the text that can compute $A|x\rangle$ under the stated hypothesis that $L_i$ can be solved by a polynomial size quantum circuit $C.$