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arXiv:2406.14109 · Physical Review Letters

Protect Measurement-Induced Phase Transition from Noise

Dongheng Qian, Jing Wang

Published Jan 16, 2025• 8 citations
arXivHTMLPDF

Abstract

Scrambling dynamics induced by random unitary gates can protect information from low-rate measurements, which underpins the phenomenon known as the measurement-induced phase transition (MIPT). However, typical decoherence noise disrupts the volume-law phase, complicating the observation of the MIPT on noisy intermediate-scale quantum devices. Here, we demonstrate that incorporating quantum-enhanced operations can effectively protect the MIPT from environmental noise, thereby enabling its detection in experiment. The transition is characterized by conditional entanglement entropy, which is associated with a statistical mechanics model wherein noise and quantum-enhanced operations act as competing external random fields. When the net external field is zero, a ferromagnetic-paramagnetic phase transition is expected, resulting in the MIPT. This zero-field condition also ensures an average apparatus-environment symmetry, making conditional entanglement entropy a valid probe of entanglement and establishing the transition as a genuine entanglement phase transition. Additionally, we provide numerical results that demonstrate the MIPT in a (2+1)-dimensional quantum circuit under dephasing noise. We also propose a method to estimate the noise rate, enabling the zero-field condition to be achieved experimentally and ensuring the feasibility of our protocol. Our result serves as a concrete example of the power of quantum enhancement in combating noise.

DOI
10.1103/physrevlett.134.020403

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Scrambling dynamics induced by random unitary gates can protect information from low-rate measurements, which underpins the phenomenon known as the measurement-induced phase transition (MIPT). However, typical decoherence noise disrupts the volume-law phase, complicating the observation of the MIPT on noisy intermediate-scale quantum devices. Here, we demonstrate that incorporating quantum-enhanced operations can effectively protect the MIPT from environmental noise, thereby enabling its detection in experiment. The transition is characterized by conditional entanglement entropy, which is associated with a statistical mechanics model wherein noise and quantum-enhanced operations act as competing external random fields. When the net external field is zero, a ferromagnetic-paramagnetic phase transition is expected, resulting in the MIPT. This zero-field condition also ensures an average apparatus-environment symmetry, making conditional entanglement entropy a valid probe of entanglement and establishing the transition as a genuine entanglement phase transition. Additionally, we provide numerical results that demonstrate the MIPT in a (2+1)-dimensional quantum circuit under dephasing noise. We also propose a method to estimate the noise rate, enabling the zero-field condition to be achieved experimentally and ensuring the feasibility of our protocol. Our result serves as a concrete example of the power of quantum enhancement in combating noise.
Measurement-induced phase transitions (MIPTs) have attracted increasing attention due to the rich phenomenology of entanglement structures and their relation with quantum information processing. Since physical systems are unavoidably coupled to environment, quantum noise, which can qualitatively modify or even destroy certain entanglement structure, needs to be considered in analyzing a system with MIPT. In this Letter, we investigate the effect of quantum noise modeled by a reset quantum channel acting on each site with a probability qqq on MIPT. Based on the numerical results from Clifford circuits, we show that the quantum noise can qualitatively change the entanglement properties---the entanglement obeys ``area law'' instead of ``volume law'' with a measurement rate p<pcp<{p}_{c}p<pc​. In the noise-induced area law phase, the entanglement exhibits a novel ${q}^{\ensuremath{-}1/3}$ power-law scaling. Using an analytic mapping from the quantum model to a classical statistical model, we further show that the area law entanglement is the consequence of noise-driven symmetry-breaking field, and the ${q}^{\ensuremath{-}1/3}$ scaling can be understood as the result of Kardar-Parisi-Zhang fluctuations of directed polymer with an effective length scale ${L}_{\mathrm{eff}}\ensuremath{\sim}{q}^{\ensuremath{-}1}$ in a random environment.
Coherent information quantifies the transmittable quantum information through a channel and is directly linked to the channel's quantum capacity. In a monitored quantum circuit, regarded as a quantum channel, extensive and positive coherent information is sustained at low measurement rates, protected by the scrambling dynamics. However, noise suppresses coherent information, driving it to zero or negative values. Here, we show that incorporating quantum-enhanced operations facilitates reliable quantum information transmission even in the presence of noise, as evidenced by a phase transition in coherent information from a recoverable phase with positive values to an irrecoverable phase with negative values. We provide both analytical understanding and numerical evidence demonstrating this transition, which is modulated by the relative frequencies of noise and quantum-enhanced operations. Additionally, we propose a resource-efficient protocol to characterize this phase transition in experiments, effectively avoiding postselection by utilizing every run of the quantum circuit. This approach bridges the gap between theoretical insights and practical implementation, making the phase transition feasible to demonstrate on realistic noisy intermediate-scale quantum devices.
Measurement-induced phase transition (MIPT) describes the nonanalytical change of the entanglement entropy resulting from the interplay between measurement and unitary evolution. In this paper, we investigate the relaxation critical dynamics near the MIPT for different initial states in a one-dimensional quantum circuit. Specifically, when the initial state is in the volume-law phase with vanishing measurement probability, we find that the half-chain entanglement entropy SSS decays as S∝t−1S\propto t^{-1}S∝t−1 with the coefficients proportional to the size of the system in the short-time stage; In contrast, when the initial state is the product state, SSS increases with time as S∝ln⁡tS\propto \ln{t}S∝lnt, consistent with previous studies. Despite these contrasting behaviors, we develop a unified scaling form to describe these scaling behaviors for different initial states where the off-critical-point effects can also be incorporated. This framework offers significant advantages for experimental MIPT detection. Our novel scheme, leveraging relaxation dynamical scaling, drastically reduces post-selection overhead, and can eliminate it completely with trackable classical simulation.
This is a conference proceeding in the framework of workshop “OpenQMBP2023” * at Institute Pascal (Orsay, France) and associated with the lecture given by Prof. Ehud Altman. We provide a comprehensive analysis of recent results in the context of measurement-induced phase transitions (MIPT) in quantum systems, with a particular focus on hybrid quantum circuits as a model system in one-dimension. Recent resutls, demonstrate how varying the rate of projective measurements can induce phase transitions, resulting in abrupt changes in the properties of the entanglement. The interplay between unitary evolution and measurement processes can be investigated, through mappings to classical statistical models and the application of replica field theory techniques. Starting from a low-entangled state, there can be three regimes characterized by different dynamics of bipartite entanglement entropies for a portion of the system: high-rate measurements leading to rapid entanglement saturation (area law), low-rate measurements allowing linear entanglement growth (up to volume law), and a critical rate at which entanglement grows logarithmically. Finally, we present results on the non-local effects of local measurements by ex-amining the field theory of critical ground states in Tomonaga-Luttinger liquids.
FINDING: Measurement-induced phase transitions (MIPT) in quantum systems reveal a sharp entanglement phase boundary driven by the competition between unitary evolution and projective measurements, with critical points often characterized by universal exponents and effective central charges. | MATH: The MIPT is typically studied via the entanglement entropy S=−Tr(ρAln⁡ρA)S = -\text{Tr}(\rho_A \ln \rho_A)S=−Tr(ρA​lnρA​) scaling as S∼c6ln⁡LS \sim \frac{c}{6} \ln LS∼6c​lnL at criticality (for 1+1D conformal field theory), where ccc is the effective central charge. The critical measurement rate pcp_cpc​ separates volume-law (S∝LS \propto LS∝L) from area-law (S∝constS \propto \text{const}S∝const) phases. For Clifford circuits, the transition is in the percolation universality class with c=0c = 0c=0 (or c=ln⁡2c = \ln 2c=ln2 for Haar random circuits). The "1/(2π)" appears in the context of the quantum Zeno effect: for a system measured at rate γ\gammaγ, the effective Hamiltonian is \(H_{\text{eff}} = H - i \frac{\gamma}{2} \sum_m |m\rangle\langle Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
When subject to a nonlocal unitary evolution, qubits in a quantum circuit become increasingly entangled. Conversely, measurements applied to individual qubits lead to their disentanglement from the collective system. The extent of entanglement reduction depends on the frequency of local projective measurements. A delicate balance emerges between unitary evolution, which enhances entanglement, and measurements which diminish it. In the thermodynamic limit, there is a phase transition from volume law entanglement to area law entanglement at a critical value of measurement frequency. This phenomenon, occurring in hybrid quantum circuits with both unitary gates and measurements, is termed as measurement-induced phase transition (MIPT). We study the behavior of MIPT in circuits comprising of two-qubit unitary gates parameterized by Cartan decomposition. We show that the entangling power and gate typicality of the two-qubit local unitaries employed in the circuit can be used to explain the behavior of global bipartite entanglement the circuit can sustain. When the two-qubit gate throughout the circuit is the identity and measurements are the sole driver of the entanglement behavior, we obtain an analytical estimate for the entanglement entropy that shows remarkable agreement with numerical simulations. We also find that the entangling power and gate typicality enable the classification of the two-qubit unitaries by different universality classes of phase transitions that can occur in the hybrid circuit. For all unitaries in a particular universality class, the transition from volume to area law of entanglement occurs with the same exponent that characterizes the phase transition.