[Paper Review] The Measurement-induced Transition in Long-range Interacting Quantum Circuits

Block, Maxwell, Bao, Yimu+3 more|arXiv (Cornell University)|Apr 27, 2021

Quantum many-body systemsPhysics and Astronomy184 citations

TL;DR

This paper investigates the measurement-induced phase transition in one-dimensional long-range interacting quantum circuits with power-law interactions (1/r^α). Using numerical simulations and an exact mapping to a long-range quantum Ising model, it demonstrates that for α ≲ 3, the transition belongs to a continuum of non-conformal universality classes with continuously varying critical exponents, while for α ≳ 3, it reverts to conformal field theory behavior. The critical power-law α ≈ 3 marks a transition where long-range interactions become relevant, fundamentally altering the universality class.

ABSTRACT

The competition between scrambling unitary evolution and projective measurements leads to a phase transition in the dynamics of quantum entanglement. Here, we demonstrate that the nature of this transition is fundamentally altered by the presence of long-range, power-law interactions. For sufficiently weak power-laws, the measurement-induced transition is described by conformal field theory, analogous to short-range-interacting hybrid circuits. However, beyond a critical power-law, we demonstrate that long-range interactions give rise to a continuum of non-conformal universality classes, with continuously varying critical exponents. We numerically determine the phase diagram for a one-dimensional, long-range-interacting hybrid circuit model as a function of the power-law exponent and the measurement rate. Finally, by using an analytic mapping to a long-range quantum Ising model, we provide a theoretical understanding for the critical power-law.

Motivation & Objective

• To understand how long-range power-law interactions (1/r^α) modify the measurement-induced phase transition in hybrid quantum circuits.
• To determine the phase diagram of the transition as a function of the measurement rate p and the power-law exponent α.
• To establish an exact correspondence between long-range hybrid circuits and a long-range quantum Ising model to analytically explain the critical behavior at α ≈ 3.
• To resolve whether conformal field theory (CFT) remains valid for long-range interactions or if new universality classes emerge.

Proposed method

• Numerical simulation of one-dimensional hybrid quantum circuits with random two-qubit Clifford gates and projective measurements, where gate strength decays as P(r) ∼ 1/r^α.
• Use of four diagnostics: half-chain entanglement entropy (S_L/2), anti-podal mutual information (I_AB), global purification dynamics (S(t)), and single-qubit purification time (τ_p).
• Finite-size scaling analysis of τ_p to extract critical exponents ν (correlation length) and z (dynamical critical exponent).
• Analytic mapping of the hybrid circuit model to a long-range quantum Ising Hamiltonian to relate the transition to ground-state properties of spin chains.
• Comparison of operator spreading in the circuit model to chaotic Hamiltonian dynamics, showing equivalence when α is mapped to α/2.
• Supplemental analysis via effective Hamiltonian formalism to derive the critical behavior at α ≈ 3.

Experimental results

Research questions

• RQ1 How does the universality class of the measurement-induced transition change with the power-law exponent α in long-range interacting circuits?
• RQ2 Does conformal field theory (CFT) still describe the critical point for α < 3, or does it break down?
• RQ3 What is the nature of the phase transition for α < 2, where the area law is replaced by a sub-volume law?
• RQ4 Why does the transition exhibit continuously varying critical exponents for α ≲ 3, and what is the physical origin of this behavior?
• RQ5 Can the critical power-law α ≈ 3 be understood analytically through a mapping to a quantum Ising model?

Key findings

• For α ≳ 3, the measurement-induced transition is described by conformal field theory (CFT), with z ≈ 1 and ν ≈ 0.7, consistent with short-range models.
• For α ≲ 3, the transition belongs to a continuum of non-conformal universality classes, with both z and ν varying continuously as α decreases.
• The critical power-law α ≈ 3 marks a transition where long-range interactions become a relevant perturbation in the effective field theory, explaining the breakdown of CFT.
• For α < 2, the area-law phase is replaced by a sub-volume law phase, where half-chain entanglement entropy scales as S_L/2 ∼ L^{2−α}.
• The exact mapping to a long-range quantum Ising model provides an analytic explanation for the critical behavior at α ≈ 3, linking it to the relevance of long-range interactions in the effective Hamiltonian.
• Finite-size scaling of the purification time τ_p confirms the continuous variation of critical exponents, with ν and z deviating from CFT values below α ≈ 3.