Quantum phenomena often arise from the interaction between a quantum system and its classical environment. Quantum simulators, programmable arrays of qubits, allow for the controlled study of this interaction through measurement operations. Measurements typically reduce entanglement within the simulator, but the exact behavior of entanglement under progressive measurements is largely unknown for many quantum states. This research aims to explore this behavior, specifically focusing on the potential emergence of criticality—a sharp transition between distinct phases of entanglement—during sequential qubit measurements. The Born rule dictates the probabilistic nature of measurement outcomes, a fundamental aspect of quantum mechanics constantly at play in macroscopic systems. Quantum simulators, however, provide unique control over the interaction with the environment, enabling detailed investigation of the impact of progressive measurements on a system's quantum characteristics. Ideal quantum simulators in a uniformly random state exhibit volume-law entanglement, where the entanglement entropy scales linearly with the number of qubits. Intuitively, measuring individual qubits should linearly decrease entanglement. However, deviations from this linearity can reveal critical behavior, indicating phase transitions in entanglement. This research faces experimental challenges: imperfections in quantum processors limit the fidelity of quantum gates and measurements; measuring entanglement entropy is computationally demanding for large systems; and theoretical models often lack accuracy in experimental settings. This study overcomes these challenges by using a specifically designed ensemble of quantum circuits on a quantum simulator to program and experimentally observe an entanglement phase transition, using up to 48 superconducting qubits, to generate and analyze volume-law states.

Previous research has shown evidence of entanglement phase transitions in various settings. Studies have explored transitions in random ensembles of quantum circuits subject to continuous measurement, revealing entanglement phase transitions influenced by the measurement rate. Other work demonstrated transitions in models with topological order. However, detecting and characterizing these transitions experimentally remains challenging due to limitations in current quantum hardware and the computational complexity of measuring entanglement entropy. Existing theoretical models, often based on random circuit ensembles, offer only approximate descriptions in experimentally relevant scenarios. This paper builds upon these prior works by providing a controlled experimental setup with an exact mapping to a known classical model, thus allowing for more precise characterization and observation of the entanglement phase transition.

The researchers employed a quantum simulator consisting of an array of superconducting qubits. To induce an entanglement phase transition, they implemented quantum circuits designed to generate highly entangled states. These circuits are based on previous theoretical work on entanglement phase transitions and solve a system of R linear equations on L Boolean variables using R+L=N qubits. The system is represented by the matrix equation Bx = y mod 2, where B is an R × L Boolean matrix, x is a vector of Boolean variables, and y is a vector of parities. The qubits are organized into two registers: a 'variable' register with L qubits and a 'parity' register with R qubits. The initial state is |ψin⟩ = |+⟩⊗L ⊗ |0⟩⊗R. The output state |ψout⟩ is an entangled superposition of solutions x for each possible y. The entanglement entropy S is quantified using the entanglement entropy, which counts the number of bits of entanglement. The total entanglement between variable and parity qubits is approximately log₂(Ny) = rank(B), where Ny is the number of possible vectors y. The authors' approach hinges on an exact mapping between the entanglement entropy of their prepared quantum states and the ground state entropy of a classical 3-spin model. This mapping is crucial for efficiently measuring entanglement in their experimental setup. By choosing a specific distribution for populating the matrix B (selecting three distinct variables uniformly at random for each equation), they establish an exact correspondence between their quantum states and the ground states of the unfrustrated 3-spin model. This model exhibits a phase transition at a critical measurement ratio αc ≈ 0.918, separating a paramagnetic phase (α < αc) and a spin glass phase (α > αc). In the paramagnetic phase, the rank(BM) = M = Lα, where BM are the rows of B corresponding to the M measured parity qubits, and the entanglement entropy scales linearly with L and α. In the spin glass phase, rank(BM) < M, and entanglement decreases more slowly than linearly with increasing α. Measuring parity qubits collapses the output state. In the paramagnetic phase, each measurement decreases entanglement by one bit, whereas in the spin glass phase, some parity qubits are already determined by previous measurements, and further measurements do not affect entanglement. The entanglement entropy is efficiently measured using the spin glass order parameter q(BM) = <(1/L)∑i=1L(-1)xi>, where the average is over all solutions x for a given BM and yout,M, and xi is the ith variable in x. They derive an identity that connects this order parameter to the entanglement entropy. The order parameter is zero in the paramagnetic phase and jumps to a finite value at αc, indicating the transition to the spin glass phase. Experiments were conducted using the ibm_washington, ibmq_brooklyn, and ibm_hanoi quantum processors with up to 48 qubits. Circuit optimization techniques were employed to minimize the number of gates, reducing errors. Error mitigation techniques focusing on measurement results satisfying BMx = yout were implemented to further improve the accuracy of results. Finite-size scaling analysis was used to determine the critical measurement ratio αc,exp and critical exponent νexp from the experimental data.

The experimental results strongly support the theoretical predictions. The order parameter q, experimentally determined as a function of the measurement ratio α, clearly shows a transition between the paramagnetic and spin glass phases at a critical point αc,exp = 0.95 ± 0.06, which is consistent with the theoretical value of αc ≈ 0.918. The transition becomes sharper with increasing system size, as expected from spin glass physics. Finite-size scaling analysis yields an experimental critical exponent νexp = 2.5 ± 0.5. The observed spin glass order parameter is directly linked to the entanglement entropy, providing a method for efficiently measuring entanglement in the quantum simulator. The study demonstrates that measurements alone can trigger an entanglement phase transition to a spin glass phase, even without imposing specific symmetries on the system. The spin glass states implemented are a subset of stabilizer states, an important class of states for quantum computation. The authors note that similar phenomena may exist in more general classes of states, suggesting the broader relevance of this type of entanglement criticality.

This research demonstrates that partial measurements of quantum states can induce intricate entanglement phenomena, including the emergence of a spin glass phase. The observed vitrification of qubits due to measurement highlights the significant impact of the classical environment on quantum systems. The results have implications for quantum computation, as entanglement is a crucial resource. The precise control over entanglement dynamics demonstrated here provides valuable insights into the behavior of quantum systems under measurement. The exact correspondence between the entanglement entropy and the spin glass order parameter provides an efficient method to measure entanglement, overcoming a significant hurdle in experimental quantum information science. The findings suggest that similar entanglement criticality might be observed in more general monitored quantum systems, warranting further investigation.

This study experimentally demonstrates the emergence of entanglement criticality and spin glass behavior in a quantum simulator through consecutive qubit measurements. This work showcases an effective experimental approach to characterize entanglement using the spin glass order parameter, and the findings extend the understanding of entanglement dynamics in monitored quantum systems. Future research could investigate the generality of these findings in different types of quantum systems and explore the potential implications for fault-tolerant quantum computation.

The experiments were performed on existing noisy intermediate-scale quantum (NISQ) processors. While error mitigation techniques were employed, the presence of noise may still affect the results. The specific choice of the 3-spin model for the mapping might limit the generalizability of the findings to other types of quantum systems. The sample size used for calculating the order parameter was limited to enhance efficiency, possibly influencing the accuracy, particularly for small system sizes. Further experiments with improved hardware and larger sample sizes would strengthen the conclusions.