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arXiv:2108.04274 · Physical review. B./Physical review. B

Decodable hybrid dynamics of open quantum systems with Z2 symmetry

Yaodong Li, Matthew P. A. Fisher

Published Dec 4, 2023• 31 citations
arXivHTMLPDF

Abstract

Within a toy model of open quantum systems, this work discusses the relation between measurement-induced phase transitions (MIPTs) of quantum entanglement and quantum error correction (QEC), which also has a phase transition at an ``error threshold''. In particular, the authors show how concepts from QEC can facilitate the observation of MIPTs and, conversely, how insights from MIPTs can inform the design of decoding algorithms.

DOI
10.1103/physrevb.108.214302

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Within a toy model of open quantum systems, this work discusses the relation between measurement-induced phase transitions (MIPTs) of quantum entanglement and quantum error correction (QEC), which also has a phase transition at an ``error threshold''. In particular, the authors show how concepts from QEC can facilitate the observation of MIPTs and, conversely, how insights from MIPTs can inform the design of decoding algorithms.
We analyze the dynamics of entanglement entropy in a generic quantum many-body open system from the perspective of quantum information and error corrections. We introduce a random unitary circuit model with intermittent projective measurements, in which the degree of information scrambling by the unitary and the rate of projective measurements are independently controlled. This model displays two stable phases, characterized by the volume-law and area-law scaling entanglement entropy in steady states. The transition between the two phases is understood from the point of view of quantum error correction: the chaotic unitary evolution protects quantum information from projective measurements that act as errors. A phase transition occurs when the rate of errors exceeds a threshold that depends on the degree of information scrambling. We confirm these results using numerical simulations and obtain the phase diagram of our model. Our work shows that information scrambling plays a crucial role in understanding the dynamics of entanglement in an open quantum system and relates the entanglement phase transition to changes in quantum channel capacity.
Quantum error correction (QEC) is the cornerstone of fault‑tolerant quantum computation, enabling logical qubits to be protected against physical noise by encoding them into highly entangled states. In this paper we propose that the very ability of a QEC code to protect against errors is itself subject to a phase transition. We formulate a theory of the Quantum Error Correction Phase Transition (QECP), in which the system undergoes a sharp change from a regime of successful error suppression to a regime where logical errors proliferate, as a control parameter – such as the physical error rate, the code distance, or the connectivity of the underlying hardware – is varied. Drawing analogies to conventional symmetry‑breaking and topological phase transitions, we construct a statistical mechanical model that captures the competition between error‑correction resources and noise. We analyze the critical behavior, derive scaling relations, and discuss implications for the design of scalable quantum architectures. The existence of a QECP provides a new physical lens through which to view quantum information processing and suggests that fault‑tolerance thresholds may be understood as critical points in an extended parameter space.
This is a conference proceeding in the framework of workshop “OpenQMBP2023” * at Institute Pascal (Orsay, France) and associated with the lecture given by Prof. Ehud Altman. We provide a comprehensive analysis of recent results in the context of measurement-induced phase transitions (MIPT) in quantum systems, with a particular focus on hybrid quantum circuits as a model system in one-dimension. Recent resutls, demonstrate how varying the rate of projective measurements can induce phase transitions, resulting in abrupt changes in the properties of the entanglement. The interplay between unitary evolution and measurement processes can be investigated, through mappings to classical statistical models and the application of replica field theory techniques. Starting from a low-entangled state, there can be three regimes characterized by different dynamics of bipartite entanglement entropies for a portion of the system: high-rate measurements leading to rapid entanglement saturation (area law), low-rate measurements allowing linear entanglement growth (up to volume law), and a critical rate at which entanglement grows logarithmically. Finally, we present results on the non-local effects of local measurements by ex-amining the field theory of critical ground states in Tomonaga-Luttinger liquids.
Monitored many-body quantum systems can exhibit a measurement-induced phase transition (MIPT) between entangling and disentangling dynamical phases. Proposed approaches to study the MIPT experimentally typically rely on a classical decoding process, but the complexity of this decoding generally grows exponentially with the system size unless the dynamics is restricted to a fine-tuned set of unitary operators. In this work we overcome this difficulty in the context of tree-shaped quantum circuits. We construct a hybrid circuit with Haar-random unitary operators, and we show that the MIPT can be detected without postselection using a simple decoding process whose complexity grows linearly with the number of qubits. The tree structure also enables a complete theoretical description of the MIPT and all its critical properties. We experimentally realize the MIPT on a trapped-ion quantum computer and show that the results are precisely described by theory without the need for error mitigation. To overcome the exponential complexity of measurement induced phase transition (MIPT), the authors propose a protocol which can detect MIPT easily on a tree-like quantum circuit. They experimentally realize the MIPT on a trapped-ion quantum computer and show that the results are precisely described by theory without the need for error mitigation.
Quantum error correction (QEC) codes protect quantum information from errors due to decoherence. Many of them also serve as prototypical models for exotic topological quantum matters. Investigating the behavior of the QEC codes under decoherence sheds light on not only the codes' robustness against errors but also new out-of-equilibrium quantum phases driven by decoherence. The phase transitions, including the error threshold, of the decohered QEC codes can be probed by the systems' R\'enyi entropies SR{S}_{R}SR​ with different R\'enyi indices RRR. In this paper, we study the general construction of the statistical models that characterize the R\'enyi entropies of QEC codes decohered by Pauli noise. We show that these statistical models can be organized into a ``tapestry'' woven by rich duality relations among them. For Calderbank-Shor-Steane (CSS) codes with bit-flip and phase-flip errors, we show that each R\'enyi entropy is captured by a pair of dual statistical models with randomness. For $R=2,3,\ensuremath{\infty}$, there are additional dualities that map between the two error types, relating the critical bit-flip and phase-flip error rates of the decoherence-induced phase transitions in the CSS codes. For CSS codes with an ``ememem symmetry'' between the XXX-type and the ZZZ-type stabilizers, the dualities with $R=2,3,\ensuremath{\infty}$ become self-dualities with super-universal self-dual error rates. These self-dualities strongly constrain the phase transitions of the code signaled by ${S}_{R=2,3,\ensuremath{\infty}}$. For general stabilizer codes decohered by generic Pauli noise, we also construct the statistical models that characterize the systems' entropies and obtain general duality relations between Pauli noise with different error rates.
Probing a quantum system disrupts its state in a phenomenon known as backaction. In this work we investigate the conditions for this disruption to result in a phase transition in the information dynamics of a monitored system. We introduce a framework that captures a wide range of experiments encompassing probes comprised of projective measurements and probes which more generally transfer quantum information from the system to a quantum computer. The latter case is relevant to the recently introduced quantum-enhanced experiments in which quantum postprocessing is performed on a quantum computer and which can offer an exponential sampling advantage over projective measurements and classical postprocessing. Our framework explicitly considers the effects of an environment using a model of unitary evolution which couples the system, apparatus, and environment. Information dynamics is investigated using the R\'enyi and von Neumann entropies of the evolving state, and we construct a replica theory for studying these quantities. We identify the possible replica symmetries an experiment can possess and discuss the meaning of their spontaneous symmetry breaking. In particular, we identify a minimum subgroup whose spontaneous symmetry breaking results in an entanglement transition. This symmetry is only possible when the information in the apparatus is as informative about the dynamics of the system as the information transferred to the environment. We call this requirement the information exchange symmetry and quantify it by a relationship between the entropies. We then introduce a generalized notion of spontaneous symmetry breaking such that the entanglement transition can be understood as the spontaneous breaking of the information exchange symmetry and without referring to the replica theory. Information exchange symmetry breaking is then shown to generalize the phenomenology of the measurement-induced phase transition (MIPT). We apply this theory to a quantum-enhanced experiment probing the unitary dynamics of a Haar brickwork quantum circuit, and we determine that the universality class of the critical point is distinct from that for the MIPT. This notion of information exchange symmetry breaking generalizes the MIPT and provides a framework for understanding the dynamics of quantum information in quantum-enhanced experiments.