Qubit vitrification and entanglement criticality on a quantum simulator

The Born rule dictates that measurement outcomes on quantum states are random with probabilities set by quantum theory; in everyday macroscopic settings, vast numbers of measurements obscure individual quantum effects. In contrast, quantum simulators allow controlled coupling to a classical environment via programmed measurements, enabling detailed study of how quantum characteristics, especially entanglement, evolve under progressive measurement. For random volume-law states, measuring one qubit typically removes about one bit of entanglement, suggesting a linear decay to a classical state after measuring all qubits. However, entanglement can change abruptly with measurements, exhibiting criticality that separates distinct entanglement phases. Prior studies identified measurement-induced entanglement transitions in random circuit ensembles and in models with topological order. Experimentally detecting such criticality is challenging due to hardware errors, the difficulty of measuring entanglement entropy without full tomography, and theoretical models that are approximate in realistic limits. Here, the authors program two entanglement phases and the critical point between them on a superconducting-qubit quantum simulator with up to 48 qubits, implementing an ensemble of circuits that generate volume-law states whose entanglement can be systematically reduced by qubit measurements. They introduce a physical theory mapping entanglement dynamics to spin vitrification (transition to a spin-glass phase), enabling experimental detection of an entanglement vitrification point consistent with spin-glass theory, and showing that measurements alone can trigger entanglement criticality.

The work builds on studies of measurement-driven entanglement transitions in hybrid and measurement-only random quantum circuits, and on monitored systems exhibiting topological order and criticality. Prior approaches often rely on specific symmetries to generate a spin-glass phase and typically characterize steady-state properties. Moreover, most systems lack an efficient, exact link between a spin-glass order parameter and entanglement entropy, complicating experiments due to the need for tomography. The present study differs by not imposing symmetries, establishing an exact relation between a spin-glass order parameter and entanglement entropy, and observing spin-glass order immediately after circuit execution and measurement on existing quantum hardware. It also leverages known results on diluted p-spin models/XORSAT that exhibit a phase transition at a known critical measurement ratio, providing a concrete theoretical benchmark.

Theory and model: The authors encode a system of R linear equations on L Boolean variables using a Boolean matrix B (size R×L) via Bx = y (mod 2), implemented on a quantum circuit with two registers: L variable qubits initialized in |+> and R parity qubits initialized in |0>. The output state is an equal superposition of all solutions x for each parity vector y that yields solutions, with the number of y given by 2^{rank(B)}. The entanglement entropy between variable and parity registers is S ≈ rank(B). Upon measuring M parity qubits (computational basis), with measurement outcomes fixing the corresponding entries of y, the post-measurement entanglement entropy becomes S ≈ L − rank(B_M), where B_M consists of the measured rows; define the measurement ratio α = M/L. Model choice and mapping: Each equation (row) includes exactly three randomly chosen distinct variables, with no repeated equations, producing an exact correspondence to the unfrustrated 3-spin (XORSAT) model. This classical model has a phase transition at α_c ≈ 0.918. For α < α_c (paramagnetic phase), B_M typically has full rank so rank(B_M)=M=Lα and S ≈ L(1−α) (volume law). For α > α_c (spin-glass phase), rank(B_M) < M, so S still scales with L but decreases sublinearly with α. A mapping to a classical spin Hamiltonian identifies the quantum state as a superposition of ground states, enabling predictions and measurements through spin-glass physics. Entanglement measurement via order parameter: Define the spin-glass order parameter q(B_M) = ⟨(1/L) ∑_{i=1}^L (−1)^{x_i}⟩, averaging over solutions x of Bx = y with measured y_out,M. An exact relation (derived in Methods) links q to the entanglement entropy, providing efficient access to entanglement via q. Hardware: Experiments used IBM Quantum superconducting processors (ibm_washington, ibmq_brooklyn, ibm_hanoi), selecting linear qubit chains with low readout and CNOT error; repetition delay 0.00025 s. Circuit optimization: Circuits are built from B_M (first M rows), which requires fewer gates. SWAP gates are decomposed into three CNOTs; gate patterns are optimized so matrix ones require two CNOTs, zeros three. Row operations transform B_M to a more CNOT-efficient form B_α (row echelon and additions to increase ones), preserving solution sets. Parity qubits are measured as soon as they have interacted with all needed variable qubits, reducing unnecessary swaps. Hadamards on variables are delayed until first participation to reduce decoherence. An upper bound for CNOT count N_CNOT(B_α) is provided; largest experiments (L=24, α≥1) used ~600 CNOTs vs ≥1600 without optimization. Error mitigation and sampling: Post-select only measurement shots (x, y_out) satisfying B_α x = y_out, which strongly mitigates errors. For L={8,16,24}, the number of shots per sample was {10,000; 25,000; 750,000}. Data collection: For each random B (with L columns and L α_max rows), for each α in (0, α_max): construct B_M and optimized B_α, build and execute the circuit repeatedly, retain at least 18 valid (x, y_out) pairs passing the constraint test. Order parameter computation: For each B and α, find a reference solution z to map y_out to 0, transform collected solutions x to x' = x + z with B_α x' = 0, remove duplicates to form set X, and compute q(B) by uniformly sampling up to 24 solutions from X. Average q(B) over matrices to obtain q(α). Classical simulations (for comparison) sample up to min(24, N_GS) solutions from the null space of B_M to estimate q. Finite-size scaling: Apply scaling q(α)=f((α−α_c,exp)L^{1/ν_exp}) and minimize a cost function comparing data to local linear interpolants over a grid α_c,exp∈[0.85,1.10] (step 0.001), ν_exp∈[1.5,4.0] (step 0.01), using data within α_c ± 0.5. Uncertainties are extracted from contours at (1+r)C_min with r=0.25, yielding α_c,exp and ν_exp with error bars.

- The experiment programs and observes two entanglement phases (paramagnetic and spin-glass) and a measurement-induced transition between them on up to 48 superconducting qubits (L up to 24 variables, with parity qubits up to match M) using IBM Quantum processors. - Mapping to the unfrustrated 3-spin model predicts a critical measurement ratio α_c ≈ 0.918. Experimentally extracted critical parameters from finite-size scaling are α_c,exp = 0.95 ± 0.06 and ν_exp = 2.5 ± 0.5, in agreement with theory. - In the paramagnetic phase (α < α_c), measurements reduce entanglement by approximately one bit per measured parity qubit: S ≈ L(1−α) (volume law). In the spin-glass phase (α > α_c), rank(B_M) < M, so additional measurements often do not reduce the number of superposed configurations; entanglement decreases more slowly with α, signaling vitrification. - The spin-glass order parameter q(α), which is exactly related to entanglement entropy in this setup, exhibits a sharp onset at α_c and increases toward 1 for larger α; the transition sharpens with increasing L, consistent with critical behavior. Experimental q(α) matches classical simulations (10,000 matrices per α) and shows expected finite-size trends. - The study demonstrates that partial measurements alone can trigger entanglement criticality and realize a spin-glass phase of entanglement inside a quantum processor, with efficient, order-parameter-based entanglement quantification.

The work addresses whether and how measurements can induce critical changes in entanglement structure. By encoding linear constraints that map exactly to a classical 3-spin model, the authors create quantum states whose entanglement dynamics under partial measurement can be predicted and efficiently measured. The observed transition from a paramagnetic entanglement phase (linear entanglement loss per measurement) to a vitrified spin-glass entanglement phase (sublinear entanglement reduction) confirms that measurements alone can drive criticality in many-qubit systems. The ability to access entanglement via a spin-glass order parameter provides a practical route around the intractability of direct entanglement entropy measurements. Compared to prior work, no symmetry constraints are required, there is an exact relation between the order parameter and entanglement entropy, and the spin-glass order appears immediately after circuit execution and partial measurements. The results, obtained on current noisy hardware, suggest that coupling to a classical environment—via measurements—can generically induce critical entanglement phenomena in broader classes of quantum states and monitored systems.

The study demonstrates the experimental realization of measurement-induced entanglement criticality on a quantum simulator. By preparing entangled superpositions corresponding to ground states of an unfrustrated 3-spin model and progressively measuring parity qubits, the system passes through a vitrification point into a spin-glass entanglement phase. An exact correspondence between a spin-glass order parameter and entanglement entropy enables efficient detection of the transition, with measured critical parameters α_c,exp = 0.95 ± 0.06 and ν_exp = 2.5 ± 0.5, consistent with theory. The approach shows that measurements alone can induce nontrivial entanglement phases on existing hardware. Future research could investigate whether similar mechanisms produce different types of nonanalytic entanglement behavior in more general monitored quantum systems and state classes, explore universality classes and robustness beyond the 3-spin/XORSAT mapping, and scale to larger system sizes with improved hardware and error mitigation.

- Hardware noise and gate/readout errors limit circuit depth; experiments rely on significant circuit optimization and post-selection (keeping only shots satisfying B_α x = y_out) for error mitigation. - Finite-size effects are present due to limited system sizes (L up to 24) and lead to artifacts such as deviations of q from zero at small α and dips for small L. - The computation of q uses a finite sample (up to 24) of solutions per instance, introducing sampling noise and biases that contribute to observed artefacts. - The study focuses on a specific ensemble (unfrustrated 3-spin/XORSAT with 3 variables per equation) and measurements in the computational basis; generalization to other models and measurement schemes remains to be tested. - Critical parameters are extracted via finite-size scaling with uncertainties; precise determination may require larger systems and reduced noise.