Dynamics, criticality, and universality
in random quantum circuits

A quantum system, subjected to continued local measurements throughout its bulk, can undergo a dynamical transition between an entangling and a disentangling phase. The universal properties of this measurement-induced transition, and related entanglement transitions, remain a challenge. Some of the complications of low-dimensional systems are removed in the case of all-to-all-coupled quantum circuits. I will first describe a simplified “classical” limit of this problem which is analytically and numerically tractable. This solution also suggests analytical and numerical approaches away from this limit. In particular I will describe a relationship with a tractable entanglement transition in tree tensor networks. Finally I will discuss the field theory approach to measurement and entanglement transitions.

I will consider the non-equilibrium dynamics of a recently introduced class of "statistically solvable” many-body quantum systems: the dual-unitary circuits. These systems furnish minimal solvable models of chaotic many-body quantum systems. I will first discuss the dynamics of initial-state dependent quantities, such as the entanglement entropies and local correlators, showing that it is possible to find (and classify) a family of initial states in MPS form that allow for an exact solution of the dynamics. Then I will discuss the dynamics of an initial-state independent quantity, the so called “local operator entanglement”, that measures the growth of entanglement that the Heisenberg evolution induces in operator space. I will show that one can identify different subclasses of dual-unitary circuits. In particular I will describe a “maximally chaotic” subclass, where the entanglement of local operators grows linearly, and a “dynamically constrained” one, where the entanglement of local operators is bounded.

The inexorable growth of non-local quantum entanglement is the key feature that distinguishes quantum from classical systems. Monitoring an open system (by making projective measurements) can compete against entanglement growth, leading to a many-body quantum Zeno effect. A “hybrid” quantum circuit model consisting of both unitary gates and projective measurements exhibits a quantum dynamical phase transition between a “weak measurement phase” and a “quantum Zeno phase”. Detailed properties of the weak measurement phase - including relations to quantum error correcting codes - and of the critical properties of this novel quantum entanglement transition will be described.

I will introduce the entanglement feature formalism to describe the structure and dynamics of bipartite quantum many-body entanglement. The approach applies to a class of quantum dynamics, called locally scrambled quantum dynamics, where each step of the evolution is followed by random local basis transformations. In this class, the corresponding entanglement dynamics become Markovian and can be systematically described by the entanglement feature approach. I will apply the approach to analyze the self-organized error-correcting volume-law state that emerges in the random unitary circuit with measurements.

Circuits formed from random unitaries and projective measurements have blossomed into a subfield of quantum information (QI). Whence came these circuits? This talk concerns one of the motivations: According to conventional wisdom, quantum phenomena cannot affection cognition much. Fisher postulated a mechanism by which entanglement might enhance coordinated neural firing: Qubits would manifest as the nuclear spins of phosphorus atoms. These nuclei might resist decoherence for days in Posner molecules. Experimental tests of this proposal have begun. If the proposal holds water, how adroitly could Posner systems process QI, and what QI theory can nature motivate? This work established a framework for answering, translating Fisher’s biochemistry into a quantum-computational model and circuit diagrams. These circuits, motivated by biochemistry, are amongst the earliest formed from random unitaries and projective measurements. References 1) Yunger Halpern and Crosson, Ann. Phys. 407, 92-147 (2019). 
https://www.sciencedirect.com/science/article/pii/S0003491618303014 2) Bene Watts, Yunger Halpern, and Harrow, arXiv:1911.09122 (2019). 
https://arxiv.org/abs/1911.09122

In a quantum error correcting code, entanglement survives in subsystems of the code despite the fact that the system is monitored via stabilizer measurements. In these monitored systems, a phase transition exists between the survival of long-range entanglement in the system, and unrecoverable entanglement with the environment due to a high rate of measurement. We experimentally study both error correction and the dynamics of measurement induced critically using a trapped-ion universal quantum processor. Our approach takes advantage of individual optical addressing to achieve operations on a long chain of 171Yb+ ions, resulting in one of the largest academic general-purpose quantum computers. In this talk, we present recent results of implementing a fault-tolerant error correcting code, the Bacon-Shor [[9,1,3]] code. We also describe the protocol to use this computer to realize a purification phase transition in a monitored non-equilibrium many-body quantum system. We present preliminary results showing excellent agreement between simulations and experiments on small system sizes. * This work is supported by the ARO with funding from the IARPA LogiQ program, the NSF STAQ program, the DOE BES and HEP programs, the AFOSR MURI on Quantum Measurement and Verification, and the AFOSR MURI on Interactive Quantum Computation and Communication Protocols. Authors: Crystal Noel, Pradeep Niroula, Laird Egan, Daiwei Zhu, Debopriyo Biswas, Andrew Risinger, Michael Gullans, David Huse, Marko Cetina, and Chris Monroe

Understanding the computational power of noisy intermediate-scale quantum devices is of both fundamental and practical importance to quantum information science. Here, we address the question of whether error-uncorrected noisy quantum computers can provide computational advantage over classical computers. We simulate the real-time dynamics of 1D noisy random quantum circuits via matrix product operators (MPOs) and characterize the computational power of the 1D noisy quantum system by using a metric we call MPO entanglement entropy. We numerically demonstrate that for the two-qubit gate error rates we considered, there exists a characteristic system size above which adding more qubits does not bring about an exponential growth of the cost of classical MPO simulation of 1D noisy systems. The maximum achievable MPO entanglement entropy is bounded by a constant that depends only on the gate error rate, not on the system size.

We identify a phase transition in non-local but few-body, monitored quantum dynamics, in which the separability of the steady-state changes as the rate of local projective measurements is tuned. In one phase, a fraction of the system belongs to a fully-entangled state, one for which no subsystem is in a pure state, while in the second phase, the steady-state resembles a product state over extensively many subsystems. The two phases are sharply distinguished by the extent to which local measurements alone can increase the mutual information of disjoint subsystems. We access this “separability” phase transition in a family of solvable quantum circuit dynamics, from which we find a relation to a mean-field percolation transition, and characterize the differences between these entangled steady-states and a random Page state. We argue that this transition coincides with a change in the computational hardness of classically determining the output probability distribution for the steady-state in a certain basis of product states.

I will discuss a class of hybrid quantum circuits, with random unitaries and projective measurements, which host long-range order in the area law entanglement phase. Our primary example is circuits with unitaries respecting a global Ising symmetry and two competing types of measurements. The phase diagram has an area law phase with spin glass order, which undergoes a direct transition to a paramagnetic phase with volume law entanglement, as well as a critical regime. Using mutual information diagnostics, we find that such entanglement transitions preserving a global symmetry are in new universality classes. We analyze generalizations of such hybrid circuits to higher dimensions, which allow for coexistence of order and volume law entanglement, as well as topological order without any symmetry restrictions.

Recent years have seen a great deal of effort to understand quantum thermalization: the question of whether closed quantum systems, evolving under unitary dynamics, reach a state of thermal equilibrium. In my talk, I will discuss how the presence of conservation laws affects the dynamics of thermalization. First, we investigate the dynamics of quantum entanglement in systems with conservation laws and uncover a qualitative difference between the behavior of the von Neumann entropy and higher Renyi entropies. We argue that the latter generically grow sub-ballistically in systems with diffusive transport. We provide strong evidence for this in both a U(1) symmetric random circuit model and in a paradigmatic non-integrable spin chain, where energy is the sole conserved quantity. Second, we introduce the dissipation-assisted operator evolution (DAOE) method for calculating transport properties of strongly interacting lattice systems in the high temperature regime. DAOE is based on evolving observables in the Heisenberg picture, and applying an artificial dissipation that reduces the weight on non-local operators.

Random quantum circuits are commonly viewed as hard to simulate classically. In some regimes this has been formally conjectured, and there had been no evidence against the more general possibility that for circuits with uniformly random gates, approximate simulation of typical instances is almost as hard as exact simulation. We prove that this is not the case by exhibiting a shallow circuit family with uniformly random gates that cannot be efficiently classically simulated near-exactly under standard hardness assumptions, but can be simulated approximately for all but a superpolynomially small fraction of circuit instances in time linear in the number of qubits and gates. We furthermore conjecture that sufficiently shallow random circuits are efficiently simulable more generally. To this end, we propose and analyze two simulation algorithms. Implementing one of our algorithms numerically, we give strong evidence that it is efficient both asymptotically and, in some cases, in practice. To argue analytically for efficiency, we reduce the simulation of 2D shallow random circuits to the simulation of a form of 1D dynamics consisting of alternating rounds of random local unitaries and weak measurements -- a type of process that has generally been observed to undergo a phase transition from an efficient-to-simulate regime to an inefficient-to-simulate regime as measurement strength is varied. Using a mapping from quantum circuits to statistical mechanical models, we give evidence that a similar computational phase transition occurs for our algorithms as parameters of the circuit architecture like the local Hilbert space dimension and circuit depth are varied. Based on http://arxiv.org/abs/2001.00021 which is joint work with John Napp, Rolando La Placa, Alexander Dalzell, and Fernando Brandao.

A critical goal for the field of quantum computing is "quantum supremacy" -- a demonstration of a quantum computation that is prohibitively hard for classical computers. Besides dispelling any skepticism about the experimental viability of quantum computers, quantum supremacy also provides a test of quantum theory in the realm of high complexity. A leading near-term candidate, put forth and recently implemented experimentally by the Google/UCSB team is sampling from the probability distributions of randomly chosen quantum circuits, called Random Circuit Sampling. In this talk we'll discuss the status of proving the classical hardness of the Random Circuit Sampling proposal. In particular, we’ll discuss hardness results for computing the output probabilities of random quantum circuits (joint work with Adam Bouland, Chinmay Nirkhe and Umesh Vazirani, https://arxiv.org/abs/1803.04402), and very recent extensions to understanding the robustness of these results in the presence of noise.