2025/10/21 by Jeongwan Haah, Douglas Stanford, Haah, Jeongwan +1 · 5 citations
Computer Science · Physics and Astronomy · #Computational complexity theory #Function (biology) #Limit (mathematics) #Product (mathematics) #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum many-body systems #Quantum state #State (computer science) #Time complexity #Unitary state
paper · pdf · open access · doi:10.21468/scipostphys.21.1.016
published in SciPost Physics 21(1) (SciPost.org)
openalex created_date 2025/10/24 · openalex publication_date 2026/07/22 · openalex updated_date 2026/08/05
For chaotic quantum dynamics modeled by random unitary circuits, we study the complexity of reduced density matrices of subsystems as a function of evolution time where the initial global state is a product pure state. The state complexity is defined as the minimum number of local quantum channels to generate a given state from a product state to a good approximation. In 1+1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> d, we prove that the complexity of subsystems of length ℓ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ℓ</mml:mi> </mml:math> smaller than half grows linearly in time T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>T</mml:mi> </mml:math> at least up to T = ℓ / 4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mi>/</mml:mi> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> but becomes zero after time T = ℓ /2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> in the limit of a large local dimension, while the complexity of the complementary subsystem of length larger than half grows linearly in time up to exponentially late times. Using holographic correspondence, we give some evidence that the state complexity of the smaller subsystem should actually grow linearly up to time T = ℓ/2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and then abruptly decay to zero.