Hilbert Space Delocalization under Random Unitary Circuits.

The unitary dynamics of a quantum system initialized in a selected basis state yield, generically, a state that is a superposition of all the basis states. This process, associated with the quantum information scrambling and intimately tied to the resource theory of coherence, may be viewed as a gradual delocalization of the system's state in the Hilbert space. This work analyzes the Hilbert space delocalization under the dynamics of random quantum circuits, which serve as a minimal model of the chaotic dynamics of quantum many-body systems. We employ analytical methods based on the replica trick and Weingarten calculus to investigate the time evolution of the participation entropies which quantify the Hilbert space delocalization. We demonstrate that the participation entropies approach, up to a fixed accuracy, their long-time saturation value in times that scale logarithmically with the system size. Exact numerical simulations and tensor network techniques corroborate our findings.

Error-Resilience Phase Transitions in Encoding-Decoding Quantum Circuits.

Understanding how errors deteriorate the information encoded in a many-body quantum system is a fundamental problem with practical implications for quantum technologies. Here, we investigate a class of encoding-decoding random circuits subject to local coherent and incoherent errors. We analytically demonstrate the existence of a phase transition from an error-protecting phase to an error-vulnerable phase occurring when the error strength is increased. This transition is accompanied by Rényi entropy transitions and by onset of multifractal features in the system. Our results provide a new perspective on storing and processing quantum information, while the introduced framework enables an analytic understanding of a dynamical critical phenomenon in a many-body system.

Entanglement Growth and Minimal Membranes in (d+1) Random Unitary Circuits.

Understanding the nature of entanglement growth in many-body systems is one of the fundamental questions in quantum physics. Here, we study this problem by characterizing the entanglement fluctuations and distribution of a (d+1)-dimensional qubit lattice evolved under a random unitary circuit. Focusing on Clifford gates, we perform extensive numerical simulations of random circuits in 1≤d≤4 dimensions. Our findings demonstrate that properties of growth of bipartite entanglement entropy are characterized by the roughening exponents of a d-dimensional membrane in a (d+1)-dimensional elastic medium.

Controlling Entanglement at Absorbing State Phase Transitions in Random Circuits.

Many-body unitary dynamics interspersed with repeated measurements display a rich phenomenology hallmarked by measurement-induced phase transitions. Employing feedback-control operations that steer the dynamics toward an absorbing state, we study the entanglement entropy behavior at the absorbing state phase transition. For short-range control operations, we observe a transition between phases with distinct subextensive scalings of entanglement entropy. In contrast, the system undergoes a transition between volume-law and area-law phases for long-range feedback operations. The fluctuations of entanglement entropy and of the order parameter of the absorbing state transition are fully coupled for sufficiently strongly entangling feedback operations. In that case, entanglement entropy inherits the universal dynamics of the absorbing state transition. This is, however, not the case for arbitrary control operations, and the two transitions are generally distinct. We quantitatively support our results by introducing a framework based on stabilizer circuits with classical flag labels. Our results shed new light on the problem of observability of measurement-induced phase transitions.

Universal Behavior beyond Multifractality of Wave Functions at Measurement-Induced Phase Transitions.

We investigate the structure of many-body wave functions of 1D quantum circuits with local measurements employing the participation entropies. The leading term in system size dependence of participation entropy indicates a model-dependent multifractal scaling of the wave functions at any nonzero measurement rate. The subleading term contains universal information about measurement-induced phase transitions and plays the role of an order parameter, being constant nonzero in the error-correcting phase and vanishing in the quantum Zeno phase. We provide robust numerical evidence investigating a variety of quantum many-body systems and provide an analytical interpretation of this behavior expressing the participation entropy in terms of partition functions of classical statistical models in 2D.

Constraint-Induced Delocalization.

We study the impact of quenched disorder on the dynamics of locally constrained quantum spin chains, that describe 1D arrays of Rydberg atoms in both the frozen (Ising-type) and dressed (XY-type) regime. Performing large-scale numerical experiments, we observe no trace of many-body localization even at large disorder. Analyzing the role of quenched disorder terms in constrained systems we show that they act in two, distinct and competing ways: as an on-site disorder term for the basic excitations of the system, and as an interaction between excitations. The two contributions are of the same order, and as they compete (one towards localization, the other against it), one does never enter a truly strong disorder, weak interaction limit, where many-body localization occurs. Such a mechanism is further clarified in the case of XY-type constrained models: there, a term which would represent a bona fide local quenched disorder term acting on the excitations of the clean model must be written as a series of nonlocal terms in the unconstrained variables. Our observations provide a simple picture to interpret the role of quenched disorder that could be immediately extended to other constrained models or quenched gauge theories.

Polynomially Filtered Exact Diagonalization Approach to Many-Body Localization.

Polynomially filtered exact diagonalization method (POLFED) for large sparse matrices is introduced. The algorithm finds an optimal basis of a subspace spanned by eigenvectors with eigenvalues close to a specified energy target by a spectral transformation using a high order polynomial of the matrix. The memory requirements scale better with system size than in the state-of-the-art shift-invert approach. The potential of POLFED is demonstrated examining many-body localization transition in 1D interacting quantum spin-1/2 chains. We investigate the disorder strength and system size scaling of Thouless time. System size dependence of bipartite entanglement entropy and of the gap ratio highlights the importance of finite-size effects. We discuss possible scenarios regarding the many-body localization transition obtaining estimates for the critical disorder strength.

Thouless Time Analysis of Anderson and Many-Body Localization Transitions.

Spectral statistics of disordered systems encode Thouless and Heisenberg timescales, whose ratio determines whether the system is chaotic or localized. We show that the scaling of the Thouless time with the system size and disorder strength is very similar in one-body Anderson models and in disordered quantum many-body systems. We argue that the two parameter scaling breaks down in the vicinity of the transition to the localized phase, signaling a slowing-down of dynamics.

Fidelity susceptibility in Gaussian random ensembles.

The fidelity susceptibility measures the sensitivity of eigenstates to a change of an external parameter. It has been fruitfully used to pin down quantum phase transitions when applied to ground states (with extensions to thermal states). Here, we propose to use the fidelity susceptibility as a useful dimensionless measure for complex quantum systems. We find analytically the fidelity susceptibility distributions for Gaussian orthogonal and unitary universality classes for arbitrary system sizes. The results are verified by a comparison with numerical data.

Tunneling-induced restoration of the degeneracy and the time-reversal symmetry breaking in optical lattices.

We study the ground-state properties of bosons loaded into the p band of a one-dimensional optical lattice. We show that the phase diagram of the system is substantially affected by the anharmonicity of the lattice potential. In particular, for a certain range of tunneling strength, the full many-body ground state of the system becomes degenerate. In this region, an additional symmetry of the system, namely, the parity of the occupation number of the chosen orbital, is spontaneously broken. The state with a nonvanishing staggered angular momentum, which breaks the time-reversal symmetry, becomes the true ground state of the system.