[Paper Review] Decoding Measurement-Prepared Quantum Phases and Transitions: from Ising model to gauge theory, and beyond

Jong Yeon Lee, Wenjie Ji+2 more|arXiv (Cornell University)|Aug 24, 2022

Quantum many-body systems29 citations

TL;DR

The paper shows that measurement-based preparation on cluster states in d≥2 can realize conformal quantum critical points (CQCP) and map post-measurement amplitudes to classical thermal partition functions; it introduces a decoding protocol to detect long-range order without post-selection.

ABSTRACT

Measurements allow efficient preparation of interesting quantum many-body states with long-range entanglement, conditioned on additional transformations based on measurement outcomes. Here, we demonstrate that the so-called conformal quantum critical points (CQCP) can be obtained by performing general single-site measurements in an appropriate basis on the cluster states in $d\geq2$. The equal-time correlators of the said states are described by correlation functions of certain $d$-dimensional classical models at finite temperatures and feature spatial conformal invariance. This establishes an exact correspondence between the measurement-prepared critical states and conformal field theories of a range of critical spin models, including familiar Ising models and gauge theories. Furthermore, by mapping the long-range entanglement structure of measured quantum states into the correlations of the corresponding thermal spin model, we rigorously establish the stability condition of the long-range entanglement in the measurement-prepared quantum states deviating from the ideal setting. Most importantly, we describe protocols to decode the resulting quantum phases and transitions without post-selection, thus transferring the exponential measurement complexity to a polynomial classical computation. Therefore, our findings suggest a novel mechanism in which a quantum critical wavefunction emerges, providing new practical ways to study quantum phases and conformal quantum critical points.

Motivation & Objective

• Motivate how measurements on cluster states can prepare quantum states with long-range entanglement and critical properties.
• Show that measurement-rotated bases yield amplitudes proportional to Boltzmann weights of d-dimensional classical spin models.
• Establish stability criteria for long-range entanglement under deviations from ideal measurements via corresponding classical phase transitions.
• Demonstrate a decoding protocol that extracts long-range order without post-selection, transferring measurement complexity to classical computation.
• Provide a framework linking measurement outcomes to conformal field theories and gauge theories across dimensions.

Proposed method

• Demonstrate that post-measurement wavefunction amplitudes equal Boltzmann weights of a d-dimensional classical model at inverse temperature β(θ)=artanh(cos θ).
• Derive a parent Hamiltonian for measured states via Witten-like conjugation and Kramers-Wannier duality.
• Compute correlation functions after general single-site measurements on cluster states to analyze stability of long-range entanglement.
• Map measurement outcomes to classical spin models (Ising and Ising gauge theory) and analyze phase transitions.
• Develop a classical decoding protocol that reveals hidden long-range order without post-selection, reducing experimental overhead.

Experimental results

Research questions

• RQ1 Can general single-site measurements on cluster states realize conformal quantum critical points (CQCP) in d≥2?
• RQ2 How do measurement-induced amplitudes relate to Boltzmann weights of corresponding classical spin models at finite temperature?
• RQ3 What is the stability of long-range entanglement under deviations from the ideal measurement basis, and how does it map to classical phase transitions?
• RQ4 Can a decoding protocol retrieve long-range order without post-selection, and what is the computational cost?
• RQ5 How do results extend to higher dimensions and to different symmetry structures (e.g., 1-form, 2-form) and gauge theories?

Key findings

• Measurement in rotated bases on cluster states yields post-measurement amplitudes proportional to Boltzmann weights of a classical spin model at β=artanh(cos θ).
• In d≥2, there exist CQCPs where spatial correlations of the measured state exhibit conformal invariance.
• A classical decoding protocol can extract long-range order without post-selection, linking quantum-state preparation to polynomial-classical computation.
• Long-range entanglement in 2D GHZ-like states remains robust to finite angle deviations when measuring edges, while toric-code-like entanglement is more fragile under deviation.
• In 3D, certain cluster-state constructions map to ordinary 3D Ising and 2-form Ising gauge theories, with stability dependent on the form of the symmetry (0-form/1-form).
• The stability of long-range entanglement corresponds to phase transitions of the associated classical models (Ising, gauge theories, Nishimori line analogues).