Random-matrix models of monitored quantum circuits

Abstract

We study the competition between Haar-random unitary dynamics and measurements for unstructured systems of qubits. For projective measurements, we derive various properties of the statistical ensemble of Kraus operators analytically, including the purification time and the distribution of Born probabilities. The latter generalizes the Porter-Thomas distribution for random unitary circuits to the monitored setting and is log-normal at long times. We also consider weak measurements that interpolate between identity quantum channels and projective measurements. In this setting, we derive an exactly solvable Fokker-Planck equation for the joint distribution of singular values of Kraus operators, analogous to the Dorokhov-Mello-Pereyra-Kumar (DMPK) equation modelling disordered quantum wires. We expect that the statistical properties of Kraus operators we have established for these simple systems will serve as a model for the entangling phase of monitored quantum systems more generally.

v2: minor revisions, references added, 35+3 pages, 3 figures

References in corpus (42)

• From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
• Average Entropy of a Subsystem
• Characterizing Quantum Supremacy in Near-Term Devices
• Measurement-Induced Phase Transitions in the Dynamics of Entanglement
• Quantum Zeno Effect and the Many-body Entanglement Transition
• Measurement-driven entanglement transition in hybrid quantum circuits
• Unitary-projective entanglement dynamics
• Quantum Error Correction in Scrambling Dynamics and Measurement-Induced Phase Transition
• Dynamical purification phase transitions induced by quantum measurements
• Entanglement transition in a monitored free fermion chain -- from extended criticality to area law
• Entanglement in a fermion chain under continuous monitoring
• Quantum Supremacy and the Complexity of Random Circuit Sampling
• Entanglement transition from variable-strength weak measurements
• Observation of measurement-induced quantum phases in a trapped-ion quantum computer
• Statistical mechanics of quantum error correcting codes
• Measurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory
• Self-Organized Error Correction in Random Unitary Circuits with Measurement
• Measurement-induced entanglement and teleportation on a noisy quantum processor
• Truncations of random unitary matrices
• Operator scaling dimensions and multifractality at measurement-induced transitions
• Entanglement and dynamics of diffusion-annihilation processes with Majorana defects
• Emergent conformal symmetry in non-unitary random dynamics of free fermions
• Exact Solution for the Distribution of Transmission Eigenvalues in a Disordered Wire and Comparison with Random-Matrix Theory
• A Quantum to Classical Phase Transition in Noisy Quantum Computers
• Recent exact and asymptotic results for products of independent random matrices
• How Dynamical Quantum Memories Forget
• Singular value correlation functions for products of Wishart random matrices
• Cross Entropy Benchmark for Measurement-Induced Phase Transitions
• Dynamical purification and the emergence of quantum state designs from the projected ensemble
• Probing post-measurement entanglement without post-selection
• Universal microscopic correlation functions for products of truncated unitary matrices
• Triviality of quantum trajectories close to a directed percolation transition
• Universal distribution of Lyapunov exponents for products of Ginibre matrices
• From Integrable to Chaotic Systems: Universal Local Statistics of Lyapunov exponents
• Asymptotics of finite system Lyapunov exponents for some random matrix ensembles
• Isotropic Brownian motions over complex fields as a solvable model for May-Wigner stability analysis
• Stability of large complex systems with heterogeneous relaxation dynamics
• Emergence of Calogero family of models in external potentials: Duality, Solitons and Hydrodynamics
• A Dyson Brownian motion model for weak measurements in chaotic quantum systems
• Noisy monitored quantum dynamics of ergodic multi-qubit systems
• An exact formula for the variance of linear statistics in the one-dimensional jellium mode
• Elementary Proof for Asymptotics of Large Haar-Distributed Unitary Matrices

Cited by in corpus (13)

• Hilbert space delocalization under random unitary circuits
• Zero-temperature entanglement membranes in quantum circuits
• A Dyson Brownian motion model for weak measurements in chaotic quantum systems
• Universal Stochastic Equations of Monitored Quantum Dynamics
• Measurement-Induced Spectral Transition
• Measurement-induced entanglement and complexity in random constant-depth 2D quantum circuits
• Topology and Spectrum in Measurement-Induced Phase Transitions
• Deviations from the Porter-Thomas Distribution due to Nonstatistical $\gamma$ Decay below the ${}^{150}$Nd Neutron Separation Threshold
• Efficient circular Dyson Brownian motion algorithm
• Observable Measurement-Induced Transitions
• Symmetry and Topology of Monitored Quantum Dynamics
• Monitored quantum transport: full counting statistics of a quantum Hall interferometer
• Entropy and singular-value moments of products of truncated random unitary matrices