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Three Lectures on Complexity and Black Holes

Leonard Susskind

TL;DR

This work frames quantum complexity as a thermodynamic-like quantity whose growth governs the interior geometry of black holes. It draws a precise correspondence between circuit complexity and gravitational quantities—notably wormhole volume via the CV conjecture and the Wheeler–DeWitt patch via CA—and elucidates how a second-law of complexity shapes interior evolution, firewalls, and the fate of perturbations. By introducing negentropy and uncomplexity as resources, it explains how minimal resources, like one clean qubit, can enable substantial computational tasks and influence spacetime structure. The combined narrative connects quantum information, chaos, and holography to offer a coherent picture of how complexity underlies the emergence and dynamics of spacetime behind horizons.

Abstract

Given at PiTP 2018 summer program entitled "From Qubits to Spacetime." The first lecture describes the meaning of quantum complexity, the analogy between entropy and complexity, and the second law of complexity. Lecture two reviews the connection between the second law of complexity and the interior of black holes. I discuss how firewalls are related to periods of non-increasing complexity which typically only occur after an exponentially long time. The final lecture is about the thermodynamics of complexity, and "uncomplexity" as a resource for doing computational work. I explain the remarkable power of "one clean qubit," in both computational terms and in space-time terms. The lectures can also be found online at \url{https://static.ias.edu/pitp/2018/node/1796.html} .

Three Lectures on Complexity and Black Holes

TL;DR

This work frames quantum complexity as a thermodynamic-like quantity whose growth governs the interior geometry of black holes. It draws a precise correspondence between circuit complexity and gravitational quantities—notably wormhole volume via the CV conjecture and the Wheeler–DeWitt patch via CA—and elucidates how a second-law of complexity shapes interior evolution, firewalls, and the fate of perturbations. By introducing negentropy and uncomplexity as resources, it explains how minimal resources, like one clean qubit, can enable substantial computational tasks and influence spacetime structure. The combined narrative connects quantum information, chaos, and holography to offer a coherent picture of how complexity underlies the emergence and dynamics of spacetime behind horizons.

Abstract

Given at PiTP 2018 summer program entitled "From Qubits to Spacetime." The first lecture describes the meaning of quantum complexity, the analogy between entropy and complexity, and the second law of complexity. Lecture two reviews the connection between the second law of complexity and the interior of black holes. I discuss how firewalls are related to periods of non-increasing complexity which typically only occur after an exponentially long time. The final lecture is about the thermodynamics of complexity, and "uncomplexity" as a resource for doing computational work. I explain the remarkable power of "one clean qubit," in both computational terms and in space-time terms. The lectures can also be found online at \url{https://static.ias.edu/pitp/2018/node/1796.html} .

Paper Structure

This paper contains 50 sections, 109 equations, 42 figures.

Table of Contents

  1. Lecture I: Hilbert Space is Huge
  2. Preface
  3. How Huge?
  4. Volume of CP(N)
  5. Relative Complexity
  6. Dual Role of Unitaries
  7. Volume of $SU(2^K)$
  8. Exploring $SU(2^K)$
  9. Relative Complexity of Unitaries
  10. Complexity is Discontinuous
  11. Graph Theory Perspective
  12. Collisions and Loops
  13. The Second Law of Quantum Complexity
  14. Hamiltonian Evolution
  15. Lecture II: Black Holes and the Second Law of Complexity
  16. ...and 35 more sections

Figures (42)

  • Figure 1: Time flows from left to right. A gate acts on an incoming state of two-qubits to give an outgoing state.
  • Figure 2: k-local all-to-all circuit
  • Figure 3: Spatially local circuit
  • Figure 4: Standard circuit architecture. Note that in each step the gates all commute because they act on non-overlapping qubits.
  • Figure 5: The auxiliary system: The evolution of the unitary operator $U(t)$ can be thought of as the motion of a classical particle moving through the group space $SU(2^K)$.
  • ...and 37 more figures