Quantum mechanics fundamentally differs from classical physics in its treatment of measurement. In contrast to the deterministic evolution described by the Schrödinger equation, quantum measurement is stochastic and non-unitary, collapsing the wavefunction. This seemingly disruptive process is actually essential for several core protocols in quantum information science, including quantum teleportation, error correction, and measurement-based quantum computation. These protocols leverage quantum measurements and the classical processing of their outcomes to construct specific structures of quantum information within spacetime. Remarkably, such structures can also emerge spontaneously from random sequences of unitary interactions and measurements in so-called 'monitored' circuits. These circuits, comprising both unitary gates and controlled projective measurements, are predicted to exhibit distinct non-equilibrium phases characterized by their entanglement structure: a volume law (extensive entanglement) or an area law (limited entanglement), depending on the measurement rate or strength. Experimental study of these measurement-induced entanglement phenomena has been challenging due to limitations in system size and the stochastic nature of measurement. The detection of these phenomena requires either computationally expensive post-selection of measurement outcomes or advanced data-processing techniques. Furthermore, implementing these models often necessitates mid-circuit measurements, which are typically problematic on superconducting processors due to their relatively long duration compared to coherence times. This research tackles these experimental difficulties to study measurement-induced quantum information phases.

Previous theoretical work predicted distinct non-equilibrium phases in monitored quantum circuits, characterized by volume-law or area-law entanglement scaling, depending on measurement strength. Early experimental studies were limited by system size or utilized efficiently simulatable gates. Recent advances proposed and implemented more scalable approaches using classical simulation and possibly active feedback to analyze entanglement structures. However, the exponential scaling of post-selection in standard approaches limited their application to smaller systems. The use of spacetime duality mappings has also been explored to create more experimentally convenient implementations of monitored circuits, exploiting the interchangeability of space and time in the absence of causality. This allows for analysis of quantum information networks in multiple ways, for example, mapping 1D monitored circuits to 2D shallow unitary circuits with only final measurements.

This research overcomes the challenges of studying measurement-induced entanglement phenomena on noisy quantum processors by employing a space-time duality mapping to avoid mid-circuit measurements. This mapping transforms a 1D monitored circuit into a 2D shallow unitary circuit where measurements are performed only at the end. The effective measurement rate is controlled by adjusting the depth of the 2D circuit and the number of measured qubits. The researchers used random quantum circuits composed of iSWAP-like and random single-qubit rotation unitaries on a grid of qubits. They then post-selected measurement outcomes of a subset of qubits, leaving behind a 1D chain of unmeasured qubits for entanglement entropy measurement. To overcome the scalability issues of post-selection, a novel decoding protocol was developed that correlates quantum readout data and the classical measurement record to construct a hybrid quantum-classical order parameter. This protocol focuses on the entanglement of a single 'probe' qubit, conditioned on measurement outcomes, serving as a proxy for the entanglement phase of the entire system. Classical simulation then eliminates the need for computationally expensive post-selection, making the approach sample-efficient. This decoding protocol computes the Bloch vector of the probe qubit conditional on the measurement record and defines an order parameter (ζ) based on the Bloch vector's direction relative to the Bloch sphere's equator. A proxy entropy (S_proxy) is then calculated from this order parameter. The researchers applied this decoding method to 2D shallow circuits on a 70-qubit superconducting processor, varying the number of qubits (N) and gate density (ρ). Noise mitigation techniques were also employed to estimate behavior in the absence of noise, by defining a normalized order parameter and proxy entropy. The experiment involved running 2,000 random circuit instances with 1,000 shots each, systematically varying system size and gate density to explore the phase transition.

The experiments revealed distinct area-law and volume-law entanglement phases, consistent with theoretical predictions. A phase transition was observed, tunable by adjusting either the circuit depth or gate density. The entanglement entropy scaling was subextensive in the area-law phase and approximately linear in the volume-law phase, characterized by the behavior of the second Renyi mutual information between qubit subsystems. The sensitivity to noise served as a useful order parameter, distinguishing the phases. In the disentangling phase, the probe qubit's entanglement was localized, while in the entangling phase, it was sensitive to noise across the entire system. The decoded order parameter (ζ) and proxy entropy (S_proxy) showed distinct behaviors across the phases, with S_proxy exhibiting a rapid decay in the disentangling phase and a plateau in the entangling phase. The data revealed a crossover between the phases at an intermediate gate density, which shifted to higher densities with increasing system size, demonstrating the instability of the phases to noise in the thermodynamic limit. The measurement-induced teleportation behavior was also observed where quantum information travels beyond the limits imposed by unitary dynamics, indicated by the persistence of entanglement with increasing decoding radius. Analysis of the experimental data (at a gate density of 1 and circuit depth of 5) suggested a limit on the practical system size for this type of experiment, due to the exponential suppression of the decoded signal with increasing numbers of qubits. Extrapolation indicated that qubit arrays larger than approximately 150 qubits would become excessively entangled with the environment for detectable signatures of the ideal entanglement structure, suggesting a limit of roughly 12x12 qubit arrays at current noise levels.

The findings demonstrate the experimental realization and characterization of measurement-induced entanglement phases on a noisy NISQ processor. The use of space-time duality and the novel decoding protocol allowed the observation of these phases on significantly larger systems than previously possible. The observation of distinct phases and the utilization of noise as a diagnostic tool highlight the potential of leveraging hardware limitations for useful insights. The results confirm theoretical predictions and provide a more comprehensive understanding of measurement-induced quantum phases. The identified limitations due to noise in large-scale quantum processors suggest a practical upper bound on the system size achievable with current technology, stressing the necessity for advancements in noise reduction for scaling up these types of experiments.

This research successfully demonstrated the observation of measurement-induced entanglement phases on a 70-qubit superconducting processor. The novel decoding protocol, combined with space-time duality, enabled the study of these phases at a scale previously inaccessible. The findings reveal the crucial role of noise in these systems and highlight the need for further advancements in reducing noise levels to scale these experiments further. Future research directions include exploring more efficient decoders, investigating other types of noise, and testing this approach on other quantum computing platforms.

The study was limited by the noise present in the NISQ processor, resulting in an observed phase transition that was shifted from its theoretical value and an upper bound on the practically achievable system size. The decoding protocol, while significantly improving scalability, is still reliant on classical simulation, which could become computationally challenging for significantly larger circuits. Furthermore, the study focused primarily on a specific type of random circuit; investigating other circuit architectures and measurement patterns could provide further insights.