Recently, there is interest in coincident “sharpening” and “learnability” transitions in monitored quantum systems. In the latter, an outside observer's ability to infer properties of a quantum system from measurements undergoes a phase transition. Such transitions appear to be related to the decodability transition in quantum error correction, but the precise connection is not clear. Here, we study these problems under one framework we call the general quantum inference problem. In cases as above where the system has a Markov structure, we say that the inference is on a quantum hidden Markov model. We show a formal connection to classical hidden Markov models and that they coincide for certain setups. For example, we prove this for those involving Haar-random unitaries and measurements. We introduce the notion of Bayes nonoptimality, where parameters used for inference differs from true ones. This allows us to expand the phase diagrams of above models. At Bayes optimality, we obtain an explicit relation between “sharpening” and “learnability” order parameters, explicitly showing that the two transitions coincide. Next, we study concrete examples. We review quantum error correction on the toric and repetition code and their mapping to two-dimensional (2D) random-bond Ising model (RBIM) through our framework. We study the Haar-random U ( 1 ) -symmetric monitored quantum circuit and tree, mapping each to inference models that we call the planted simple symmetric exclusion process and planted , respectively, and expanding the phase diagram to Bayes nonoptimality. For the circuit, we deduce the phase boundary numerically and analytically argue that it is of a single universality class. For the tree, we present an exact solution of the entire phase boundary, which displays re-entrance as does the 2D RBIM. We discuss these phase diagrams, with their interpretations for quantum inference problems and rigorous arguments on their shapes.
Recently, there is interest in coincident “sharpening” and “learnability” transitions in monitored quantum systems. In the latter, an outside observer's ability to infer properties of a quantum system from measurements undergoes a phase transition. Such transitions appear to be related to the decodability transition in quantum error correction, but the precise connection is not clear. Here, we study these problems under one framework we call the general quantum inference problem. In cases as above where the system has a Markov structure, we say that the inference is on a quantum hidden Markov model. We show a formal connection to classical hidden Markov models and that they coincide for certain setups. For example, we prove this for those involving Haar-random unitaries and measurements. We introduce the notion of Bayes nonoptimality, where parameters used for inference differs from true ones. This allows us to expand the phase diagrams of above models. At Bayes optimality, we obtain an explicit relation between “sharpening” and “learnability” order parameters, explicitly showing that the two transitions coincide. Next, we study concrete examples. We review quantum error correction on the toric and repetition code and their mapping to two-dimensional (2D) random-bond Ising model (RBIM) through our framework. We study the Haar-random U ( 1 ) -symmetric monitored quantum circuit and tree, mapping each to inference models that we call the planted simple symmetric exclusion process and planted , respectively, and expanding the phase diagram to Bayes nonoptimality. For the circuit, we deduce the phase boundary numerically and analytically argue that it is of a single universality class. For the tree, we present an exact solution of the entire phase boundary, which displays re-entrance as does the 2D RBIM. We discuss these phase diagrams, with their interpretations for quantum inference problems and rigorous arguments on their shapes.
The Boltzmann distribution encodes our subjective knowledge of the configuration in a classical lattice model, given only its Hamiltonian. If we acquire further information about the configuration from measurement, our knowledge is updated according to Bayes' theorem. We examine the resulting “conditioned ensembles,” finding that they show many new phase transitions and new renormalization-group fixed points. (Similar conditioned ensembles also describe “partial quenches” in which some of the system's degrees of freedom are instantaneously frozen, while the others continue to evolve.) After describing general features of the replica field theories for these problems, we analyze the effect of measurement on illustrative critical systems, including: critical Ising and Potts models, which show surprisingly rich phase diagrams, with RG fixed points at weak, intermediate, and infinite measurement strength; various models involving free fields, XY spins, or flux lines in 2D or 3D; and geometrical models such as polymers or clusters. We also give a formalism for measurement of classical stochastic processes. We use this to make connections with quantum dynamics, in particular with “charge sharpening” in 1D, for which we give a purely hydrodynamic derivation of the known effective field theory. We discuss qualitative differences between RG flows for the above measured systems, described by N → 1 replica limits, and those for disordered systems, described by N → 0 limits. In addition to discussing measurement of critical states, we give a unifying treatment of a family of inference problems for noncritical states. These are related to the Nishimori line in the phase diagram of the random-bond Ising model, and are relevant to various quantum error correction problems. We describe distinct physical interpretations of conditioned ensembles and note interesting open questions.
Monitored quantum dynamics—unitary evolution interspersed with measurements—has recently emerged as a rich domain for phase structure in quantum many-body systems away from equilibrium. Here we study monitored dynamics from the point of view of an eavesdropper who has access to the classical measurement outcomes, but not to the quantum many-body system. We show that a measure of information flow from the quantum system to the classical measurement record—the —undergoes a phase transition in correspondence with the measurement-induced phase transition (MIPT). This transition determines the eavesdropper’s (in)ability to learn properties of an unknown initial quantum state of the system, given a complete classical description of the monitored dynamics and arbitrary classical computational resources. We make this learnability transition concrete by defining classical shadow protocols that the eavesdropper may apply to this problem, and show that the MIPT manifests as a transition in the sample complexity of various shadow-estimation tasks, which become harder in the low-measurement phase. We focus on three applications of interest: Pauli expectation values (where we find the MIPT appears as a point of optimal learnability for typical Pauli operators), many-body fidelity, and global charge in U ( 1 ) -symmetric dynamics. Our work unifies different manifestations of the MIPT under the umbrella of and gives this notion a general operational meaning via classical shadows. Published by the American Physical Society 2024
During a quantum measurement, superpositions of states with different observable properties probabilistically collapse into one with a sharp value of the measured observable. In macroscopic quantum systems, this collapse arises via a continuous measurement-induced phase transition (MIPT) at a critical value of the strength of interaction with the measurement apparatus. MIPTs lie outside established paradigms for equilibrium or nonequilibrium critical phenomena and delineate distinct, stable dynamical and computational phases of matter. Quantum computers enable programmable simulation of the interaction of a measurement apparatus with a dynamical quantum system, to explore MIPT phenomena over a range of system sizes while retaining quantum coherence. Yet, existing experimental protocols rely on fundamentally nonscalable postselection techniques or direct classical simulation of quantum circuits. Here, we report the scalable observation of finite-size scaling evidence for an observable-sharpening MIPT in monitored quantum circuits in a chain of Yb + 171 ions in Quantinuum’s H1-1 trapped-ion quantum processor. By leveraging an equivalent description as a statistical physics problem, we implement scalable classical algorithms to infer the value of the measured observable from a single experimental shot. This technique enables a truly scalable protocol to observe observable-sharpening MIPTs in generic classes of circuits that cannot be directly classically simulated and also provides enhanced means to detect and suppress errors in the quantum simulation. Published by the American Physical Society 2024
Decoherence is ubiquitous, and poses a significant impediment to the observation of quantum phenomena, such as the measurement-induced entanglement phase transition (MIPT). In this work, we study entanglement transitions in quantum circuits on trees, subject to both noise and measurements. We uncover a rich phase diagram that describes the ability of a tree quantum circuit to retain quantum or classical information in the presence of decoherence. By developing a mapping between the dynamics of information on the tree to a classical Markov process -- also defined on the tree -- we obtain exact solutions to the entanglement transitions displayed by the circuit under various noise and measurement strengths. Moreover, we find a host of phenomena, including the MIPT, which are \textit{robust} to decoherence. The analytical tractability facilitated by the method developed in this paper showcases the first example of an exactly solvable noise-robust MIPT, and holds promise for studies on broader, tree-like circuits.
Random projective measurements are an effective tool to model the interlinked phenomenology of monitoring, inferring, and decoding highly entangled states such as quantum error-correcting codes. At the same time, the study of such monitored dynamics has opened a new arena for critical phenomena, renewing longstanding questions in disordered statistical mechanics, including percolation, random-bond models, and Nishimori physics. Here we investigate the robustness of stabilizer quantum error-correcting codes under a single round of independently sampled single-qubit Pauli measurements and demonstrate that their threshold behavior falls into two broad categories - geometric and homological percolation transitions, with the toric code and color code being the principal examples for each class. We establish the distinct character of these two threshold theories by analyzing the logical-support criteria of these codes and by providing numerical estimates of their respective critical exponents obtained from optimized stabilizer simulations of large-scale systems with over one million qubits. Our framework extends to non-CSS, subsystem, higher-dimensional, and Floquet stabilizer codes and allows us to determine their respective threshold phase diagrams. We exemplify how the latter provide quantitative and qualitative guidance for the computationally more demanding problems of tunable learning and maximum-likelihood decoding. As a practical application of our framework, we discuss the robustness of the gross code.
Monitored many-body systems fall broadly into two dynamical phases, "entangling" or "disentangling," separated by a transition as a function of the rate at which measurements are made on the system.Producing an analytical theory of this measurement-induced transition is an outstanding challenge.Recent work made progress in the context of tree tensor networks, which can be related to all-to-all quantum circuit dynamics with forced (postselected) measurement outcomes.So far, however, there are no exact solutions for dynamics of spin-1/2 degrees of freedom (qubits) with "real" measurements, whose outcome probabilities are sampled according to the Born rule.Here we define dynamical processes for qubits, with real measurements, that have a treelike space-time interaction graph, either collapsing or expanding the system as a function of time.The former case yields an exactly solvable measurement transition.We explore these processes analytically and numerically, exploiting the recursive structure of the tree.We compare the case of "real" measurements with the case of "forced" measurements.Both cases show a transition at a nontrivial value of the measurement strength, with the real measurement case exhibiting a smaller entangling phase.Both exhibit exponential scaling of the entanglement near the transition, but they differ in the value of a critical exponent.An intriguing difference between the two cases is that the real measurement case lies at the boundary between two distinct types of critical scaling.On the basis of our results, we propose a protocol for realizing a measurement phase transition experimentally via an expansion process.