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Scrambling speed of random quantum circuits

Winton Brown, Omar Fawzi

TL;DR

The paper investigates how quickly random quantum circuits scramble information, framing scrambling in decoupling and quantum-error-correcting terms. By analyzing the second-moment operator and the induced Pauli-string Markov chain, it provides near-optimal depth bounds: strong scrambling at depth O(log^3 n) and decoupling-based coding results, including constant-rate stabilizer codes with linear distance. It resolves the fast scrambling conjecture for constant-size messages on complete graphs (depth O(log n)) and on d-dimensional lattices (depth ~ n^{1/d} log^2 n), while establishing fundamental lower bounds that match the scaling in respective geometries. These results illuminate how fast local quantum dynamics can distribute information globally, with implications for quantum coding, decoupling theory, and black hole information questions.

Abstract

Random transformations are typically good at "scrambling" information. Specifically, in the quantum setting, scrambling usually refers to the process of mapping most initial pure product states under a unitary transformation to states which are macroscopically entangled, in the sense of being close to completely mixed on most subsystems containing a fraction fn of all n particles for some constant f. While the term scrambling is used in the context of the black hole information paradox, scrambling is related to problems involving decoupling in general, and to the question of how large isolated many-body systems reach local thermal equilibrium under their own unitary dynamics. Here, we study the speed at which various notions of scrambling/decoupling occur in a simplified but natural model of random two-particle interactions: random quantum circuits. For a circuit representing the dynamics generated by a local Hamiltonian, the depth of the circuit corresponds to time. Thus, we consider the depth of these circuits and we are typically interested in what can be done in a depth that is sublinear or even logarithmic in the size of the system. We resolve an outstanding conjecture raised in the context of the black hole information paradox with respect to the depth at which a typical quantum circuit generates an entanglement assisted encoding against the erasure channel. In addition, we prove that typical quantum circuits of poly(log n) depth satisfy a stronger notion of scrambling and can be used to encode alpha n qubits into n qubits so that up to beta n errors can be corrected, for some constants alpha, beta > 0.

Scrambling speed of random quantum circuits

TL;DR

The paper investigates how quickly random quantum circuits scramble information, framing scrambling in decoupling and quantum-error-correcting terms. By analyzing the second-moment operator and the induced Pauli-string Markov chain, it provides near-optimal depth bounds: strong scrambling at depth O(log^3 n) and decoupling-based coding results, including constant-rate stabilizer codes with linear distance. It resolves the fast scrambling conjecture for constant-size messages on complete graphs (depth O(log n)) and on d-dimensional lattices (depth ~ n^{1/d} log^2 n), while establishing fundamental lower bounds that match the scaling in respective geometries. These results illuminate how fast local quantum dynamics can distribute information globally, with implications for quantum coding, decoupling theory, and black hole information questions.

Abstract

Random transformations are typically good at "scrambling" information. Specifically, in the quantum setting, scrambling usually refers to the process of mapping most initial pure product states under a unitary transformation to states which are macroscopically entangled, in the sense of being close to completely mixed on most subsystems containing a fraction fn of all n particles for some constant f. While the term scrambling is used in the context of the black hole information paradox, scrambling is related to problems involving decoupling in general, and to the question of how large isolated many-body systems reach local thermal equilibrium under their own unitary dynamics. Here, we study the speed at which various notions of scrambling/decoupling occur in a simplified but natural model of random two-particle interactions: random quantum circuits. For a circuit representing the dynamics generated by a local Hamiltonian, the depth of the circuit corresponds to time. Thus, we consider the depth of these circuits and we are typically interested in what can be done in a depth that is sublinear or even logarithmic in the size of the system. We resolve an outstanding conjecture raised in the context of the black hole information paradox with respect to the depth at which a typical quantum circuit generates an entanglement assisted encoding against the erasure channel. In addition, we prove that typical quantum circuits of poly(log n) depth satisfy a stronger notion of scrambling and can be used to encode alpha n qubits into n qubits so that up to beta n errors can be corrected, for some constants alpha, beta > 0.

Paper Structure

This paper contains 18 sections, 16 theorems, 92 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Strong scrambling, quantum error correction and decoupling
  3. Black holes and the fast scrambling conjecture
  4. Proof technique
  5. Preliminaries
  6. Generalities
  7. Random quantum circuits
  8. Strong scrambling, decoupling and quantum error correction
  9. Sequential random circuit
  10. Parallelizing the circuit
  11. Decoupling and quantum error correcting codes
  12. Scrambling and the black hole information paradox
  13. Parallel circuit model on the complete graph
  14. Random circuit on a $d$-dimensional lattice
  15. Lower bound on the scrambling time
  16. ...and 3 more sections

Key Result

Theorem 3.1

Let $\rho(0)$ be an initial arbitrary mixed state on $n$ qubits and $\rho(t)$ be the corresponding state after the application of $t$ random two-qubit gates (the sequential circuit model). Then provided $f$ is such that $f \log 3 + h(f) - \frac{\log 3}{2} < 0$ and $t > c n \log^2 n$ (for some large where the expectation is taken over the random circuit. This implies that

Figures (2)

  • Figure 1: Illustration of decoupling for random quantum circuits
  • Figure 2: A set of non-intersecting 1D-paths on a 2D 4x4 square lattice

Theorems & Definitions (16)

  • Theorem 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Corollary 3.4
  • Theorem 3.5
  • Theorem 3.6: Good codes from low-depth circuits
  • Theorem 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Theorem 4.4
  • ...and 6 more