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Pseudoentangled states are defined by their ability to hide their entanglement structure: they are indistinguishable from random states to any observer with polynomial resources, yet can have much less entanglement than random states. Existing constructions of pseudoentanglement based on phase- and/or subset-states are limited in the entanglement structures they can hide: e.g., the states may have low entanglement on a single cut, on all cuts at once, or on local cuts in one dimension. Here we introduce new constructions of pseudoentangled states based on (pseudo)random tensor networks that affords much more flexibility in the achievable entanglement structures. We illustrate our construction with the simplest example of a matrix product state, realizable as a staircase circuit of pseudorandom unitary gates, which exhibits pseudo-area-law scaling of entanglement in one dimension. We then generalize our construction to arbitrary tensor network structures that admit an isometric realization. A notable application of this result is the construction of pseudoentangled `holographic' states whose entanglement entropy obeys a Ryu-Takayanagi `minimum-cut' formula, answering a question posed in [Aaronson et al., arXiv:2211.00747]., Comment: 5+6 pages, 3 figures. v2: fixed typos and minor issues

``Deep thermalization'' describes the emergence of universal wavefunction distributions in quantum many-body dynamics, appearing on a local subsystem upon measurement of its environment. In this work, we study in detail the effect of continuous internal symmetries and associated conservation laws on deep thermalization. Concretely, we consider quantum spin systems with a $U(1)$ symmetry associated with the conservation of magnetization (or `charge'), and analyze how the choice of initial states (specifically, their degree of charge fluctuations) and the choice of measurement basis (specifically, whether or not it can reveal information about the local charge density) determine the ensuing universal wavefunction distributions. We find a rich set of possibilities. First we focus on the case of a random state of well-defined charge subjected to charge-revealing masurements, and rigorously prove that the projected ensemble approaches a direct sum of Haar ensembles in the charge sectors of the subsystem of interest. We then analytically derive the limiting wavefunction distributions for more general initial states and measurement bases, finding results that include the Haar ensemble, the ``Scrooge ensemble'' (a distortion of the Haar ensemble by a density matrix), and the ``generalized Scrooge ensemble'' (a stochastic mixture of multiple Scrooge ensembles). These represent nontrivial higher-moment generalizations of the Gibbs state, and notably can depend on the entire charge distribution of the initial state, not just its average. Our findings demonstrate a rich interplay between symmetries and the information extracted by measurements, which allows deep thermalization to exhibit a range of universal behaviors far beyond regular thermalization., Comment: 15+15 pages, 8+3 figures

Classical shadows are a versatile tool to probe many-body quantum systems, consisting of a combination of randomised measurements and classical post-processing computations. In a recently introduced version of the protocol, the randomization step is performed via unitary circuits of variable depth $t$, defining the so-called shallow shadows. For sufficiently large $t$, this approach allows one to get around the use of non-local unitaries to probe global properties such as the fidelity with respect to a target state or the purity. Still, shallow shadows involve the inversion of a many-body map, the measurement channel, which requires non-trivial computations in the post-processing step, thus limiting its applicability when the number of qubits $N$ is large. In this work, we put forward a simple approximate post-processing scheme where the infinite-depth inverse channel is applied to the finite-depth classical shadows and study its performance for fidelity and purity estimation. The scheme allows for different circuit connectivity, as we illustrate for geometrically local circuits in one and two spatial dimensions and geometrically non-local circuits made of two-qubit gates. For the fidelity, we find that the resulting estimator coincides with a known linear cross-entropy, achieving an arbitrary small approximation error $\delta$ at depth $t=O(\log (N/\delta))$ (independent of the circuit connectivity). For the purity, we show that the estimator becomes accurate at a depth $O(N)$. In addition, at those depths, the variances of both the fidelity and purity estimators display the same scaling with $N$ as in the case of global random unitaries. We establish these bounds by analytic arguments and extensive numerical computations in several cases of interest. Our work extends the applicability of shallow shadows to large system sizes and general circuit connectivity., Comment: 21 pages, 8 figures; v2: minor revision and references added

We introduce "holographic shadows", a new class of randomized measurement schemes for classical shadow tomography that achieves the optimal scaling of sample complexity for learning geometrically local Pauli operators at any length scale, without the need for fine-tuning protocol parameters such as circuit depth or measurement rate. Our approach utilizes hierarchical quantum circuits, such as tree quantum circuits or holographic random tensor networks. Measurements within the holographic bulk correspond to measurements at different scales on the boundary (i.e. the physical system of interests), facilitating efficient quantum state estimation across observable at all scales. Considering the task of estimating string-like Pauli observables supported on contiguous intervals of $k$ sites in a 1D system, our method achieves an optimal sample complexity scaling of $\sim d^k\mathrm{poly}(k)$, with $d$ the local Hilbert space dimension. We present a holographic minimal cut framework to demonstrate the universality of this sample complexity scaling and validate it with numerical simulations, illustrating the efficacy of holographic shadows in enhancing quantum state learning capabilities., Comment: 20 pages, 9 figures

The dynamics of monitored systems can exhibit a measurement-induced phase transition (MIPT) between entangling and disentangling phases, tuned by the measurement rate. When the dynamics obeys a continuous symmetry, the entangling phase further splits into a fuzzy phase and a sharp phase based on the scaling of fluctuations of the symmetry charge. While the sharpening transition for Abelian symmetries is well understood analytically, no such understanding exists for the non- Abelian case. In this work, building on a recent analytical solution of the MIPT on tree-like circuit architectures (where qubits are repatedly added or removed from the system in a recursive pattern), we study entanglement and sharpening transitions in monitored dynamical quantum trees obeying U (1) and SU (2) symmetries. The recursive structure of tree tensor networks enables powerful analytical and numerical methods to determine the phase diagrams in both cases. In the U (1) case, we analytically derive a Fisher-KPP-like differential equation that allows us to locate the critical point and identify its properties. We find that the entanglement/purification and sharpening transitions generically occur at distinct measurement rates. In the SU (2) case, we find that the fuzzy phase is generic, and a sharp phase is possible only in the limit of maximal measurement rate. In this limit, we analytically solve the boundaries separating the fuzzy and sharp phases, and find them to be in agreement with exact numerical simulations.

The dynamics of quantum entanglement plays a central role in explaining the emergence of thermal equilibrium in isolated many-body systems. However, entanglement is notoriously hard to measure: recent works have introduced a notion of pseudoentanglement describing ensembles of many-body states that, while only weakly entangled, cannot be efficiently distinguished from states with much higher entanglement. This prompts the question: how much entanglement is truly necessary to achieve thermal equilibrium in quantum systems? In this work we address this question by introducing random circuit models of quantum dynamics that, at late times, equilibrate to pseudoentangled ensembles -- a phenomenon we name pseudothermalization. These models replicate all the efficiently observable predictions of thermal equilibrium, while generating only a small amount of entanglement, thus deviating from the "maximum-entropy principle" that underpins thermodynamics. We examine (i) how a pseudoentangled ensemble on a small subsystem spreads to the whole system as a function of time, and (ii) how a pseudoentangled ensemble can be generated from an initial product state. We map the above problems onto a family of classical Markov chains on subsets of the computational basis. The mixing times of such Markov chains are related to the time scales at which the states produced from the dynamics become indistinguishable from Haar-random states at the level of each statistical moment. Based on a combination of rigorous bounds and conjectures supported by numerics, we argue that each Markov chain's relaxation time and mixing time have different asymptotic behavior in the limit of large system size. This is a necessary condition for a cutoff phenomenon: an abrupt dynamical transition to equilibrium. We thus conjecture that our random circuits give rise to asymptotically sharp pseudothermalization transitions., Comment: 16 pages main text + 8 pages appendix. 9 figures. v2: thoroughly updated presentation, added subsection VI.B on 'Implications', added note on recent results on spectral gaps of random circuits

We introduce a general approach to realize quantum states with holographic entanglement structure via monitored dynamics. Starting from random unitary circuits in $1+1$ dimensions, we introduce measurements with a spatiotemporally-modulated density. Exploiting the known critical properties of the measurement-induced entanglement transition, this allows us to engineer arbitrary geometries for the bulk space (with a fixed topology). These geometries in turn control the entanglement structure of the boundary (output) state. We demonstrate our approach by giving concrete protocols for two geometries of interest in two dimensions: the hyperbolic half-plane and a spatial section of the BTZ black hole. We numerically verify signatures of the underlying entanglement geometry, including a direct imaging of entanglement wedges by using locally-entangled reference qubits. Our results provide a concrete platform for realizing geometric entanglement structures on near-term quantum simulators., Comment: 5 pages, 3 figures

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 informational power -- 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 shadows 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 learnability and gives this notion a general operational meaning via classical shadows., Comment: 16+4 pages, 3 figures. v2: fixed error in Fig.1 panel labels. v3: published version

We introduce a classical algorithm for sampling the output of shallow, noisy random circuits on two-dimensional qubit arrays. The algorithm builds on the recently-proposed "space-evolving block decimation" (SEBD) and extends it to the case of noisy circuits. SEBD is based on a mapping of 2D unitary circuits to 1D {\it monitored} ones, which feature measurements alongside unitary gates; it exploits the presence of a measurement-induced entanglement phase transition to achieve efficient (approximate) sampling below a finite critical depth $T_c$. Our noisy-SEBD algorithm unravels the action of noise into measurements, further lowering entanglement and enabling efficient classical sampling up to larger circuit depths. We analyze a class of physically-relevant noise models (unital qubit channels) within a two-replica statistical mechanics treatment, finding weak measurements to be the optimal (i.e. most disentangling) unraveling. We then locate the noisy-SEBD complexity transition as a function of circuit depth and noise strength in realistic circuit models. As an illustrative example, we show that circuits on heavy-hexagon qubit arrays with noise rates of $\approx 2\%$ per CNOT, based on IBM Quantum processors, can be efficiently sampled up to a depth of 5 iSWAP (or 10 CNOT) gate layers. Our results help sharpen the requirements for practical hardness of simulation of noisy hardware., Comment: 12+7 pages, 4+7 figures; v2: published version

We study classical shadows protocols based on randomized measurements in $n$-qubit entangled bases, generalizing the random Pauli measurement protocol ($n = 1$). We show that entangled measurements ($n\geq 2$) enable nontrivial and potentially advantageous trade-offs in the sample complexity of learning Pauli expectation values. This is sharply illustrated by shadows based on two-qubit Bell measurements: the scaling of sample complexity with Pauli weight $k$ improves quadratically (from $\sim 3^k$ down to $\sim 3^{k/2}$) for many operators, while others become impossible to learn. Tuning the amount of entanglement in the measurement bases defines a family of protocols that interpolate between Pauli and Bell shadows, retaining some of the benefits of both. For large $n$, we show that randomized measurements in $n$-qubit GHZ bases further improve the best scaling to $\sim (3/2)^k$, albeit on an increasingly restricted set of operators. Despite their simplicity and lower hardware requirements, these protocols can match or outperform recently-introduced "shallow shadows" in some practically-relevant Pauli estimation tasks., Comment: 8 pages, 3 figures. v2: added discussion of operators with non-contiguous support in 1D, reference to Shor-Laflamme distributions, minor improvements. v3: fixed typos, added some references and clarified some derivations. Accepted in Quantum

The possibility to obtain a three-dimensional representation of a single object with sub-$\mu$m resolution is crucial in many fields, from material science to clinical diagnostics. This is typically achieved through tomography, which combines multiple two-dimensional images of the same object captured at different orientations. However, this serial imaging method prevents single-shot acquisition in imaging experiments at free electron lasers. In the present experiment, we report on a new approach to 3D imaging using extreme-ultraviolet radiation. In this method, two EUV pulses hit simultaneously an isolated 3D object from different sides, generating independent coherent diffraction patterns, resulting in two distinct bidimensional views obtained via phase retrieval. These views are then used to obtain a 3D reconstruction using a ray tracing algorithm. This EUV stereoscopic imaging approach, similar to the natural process of binocular vision, provides sub-$\mu$m spatial resolution and single shot capability. Moreover, ultrafast time resolution and spectroscopy can be readily implemented, a further extension to X-ray wavelengths can be envisioned as well., Comment: Authors Danny Fainozzi and Matteo Ippoliti contributed equally

Measurement has a special role in quantum theory: by collapsing the wavefunction it can enable phenomena such as teleportation and thereby alter the "arrow of time" that constrains unitary evolution. When integrated in many-body dynamics, measurements can lead to emergent patterns of quantum information in space-time that go beyond established paradigms for characterizing phases, either in or out of equilibrium. On present-day NISQ processors, the experimental realization of this physics is challenging due to noise, hardware limitations, and the stochastic nature of quantum measurement. Here we address each of these experimental challenges and investigate measurement-induced quantum information phases on up to 70 superconducting qubits. By leveraging the interchangeability of space and time, we use a duality mapping, to avoid mid-circuit measurement and access different manifestations of the underlying phases -- from entanglement scaling to measurement-induced teleportation -- in a unified way. We obtain finite-size signatures of a phase transition with a decoding protocol that correlates the experimental measurement record with classical simulation data. The phases display sharply different sensitivity to noise, which we exploit to turn an inherent hardware limitation into a useful diagnostic. Our work demonstrates an approach to realize measurement-induced physics at scales that are at the limits of current NISQ processors.

Classical shadows are a powerful method for learning many properties of quantum states in a sample-efficient manner, by making use of randomized measurements. Here we study the sample complexity of learning the expectation value of Pauli operators via ``shallow shadows'', a recently-proposed version of classical shadows in which the randomization step is effected by a local unitary circuit of variable depth $t$. We show that the shadow norm (the quantity controlling the sample complexity) is expressed in terms of properties of the Heisenberg time evolution of operators under the randomizing (``twirling'') circuit -- namely the evolution of the weight distribution characterizing the number of sites on which an operator acts nontrivially. For spatially-contiguous Pauli operators of weight $k$, this entails a competition between two processes: operator spreading (whereby the support of an operator grows over time, increasing its weight) and operator relaxation (whereby the bulk of the operator develops an equilibrium density of identity operators, decreasing its weight). From this simple picture we derive (i) an upper bound on the shadow norm which, for depth $t\sim \log(k)$, guarantees an exponential gain in sample complexity over the $t=0$ protocol in any spatial dimension, and (ii) quantitative results in one dimension within a mean-field approximation, including a universal subleading correction to the optimal depth, found to be in excellent agreement with infinite matrix product state numerical simulations. Our work connects fundamental ideas in quantum many-body dynamics to applications in quantum information science, and paves the way to highly-optimized protocols for learning different properties of quantum states., Comment: (4+eps pages, 4 figures) main text + (11 pages, 5 figures) supplementary material. v2: switched from state-averaged shadow norm to regular shadow norm, added references. v3: added supplemental section S6 on non-contiguous operators. Accepted version

We study the emergence over time of a universal, uniform distribution of quantum states supported on a finite subsystem, induced by projectively measuring the rest of the system. Dubbed deep thermalization, this phenomenon represents a form of equilibration in quantum many-body systems stronger than regular thermalization, which only constrains the ensemble-averaged values of observables. While there exist quantum circuit models of dynamics in one dimension where this phenomenon can be shown to arise exactly, these are special in that deep thermalization occurs at precisely the same time as regular thermalization. Here, we present an exactly-solvable model of chaotic dynamics where the two processes can be shown to occur over different time scales. The model is composed of a finite subsystem coupled to an infinite random-matrix bath through a small constriction, and highlights the role of locality and imperfect thermalization in constraining the formation of such universal wavefunction distributions. We test our analytical predictions against exact numerical simulations, finding excellent agreement., Comment: 18+4 pages, 4 figures. v2: minor changes, accepted in Quantum

We consider a paradigmatic solvable model of topological order in two dimensions, Kitaev's honeycomb Hamiltonian, and turn it into a measurement-only dynamics consisting of stochastic measurements of two-qubit bond operators. We find an entanglement phase diagram that resembles that of the Hamiltonian problem in some ways, while being qualitatively different in others. When one type of bond is dominantly measured, we find area-law entangled phases that protect two topological qubits (on a torus) for a time exponential in system size. This generalizes the recently-proposed idea of Floquet codes, where logical qubits are dynamically generated by a time-periodic measurement schedule, to a stochastic setting. When all types of bonds are measured with comparable frequency, we find a critical phase with a logarithmic violation of the area-law, which sharply distinguishes it from its Hamiltonian counterpart. The critical phase has the same set of topological qubits, as diagnosed by the tripartite mutual information, but protects them only for a time polynomial in system size. Furthermore, we observe an unusual behavior for the dynamical purification of mixed states, characterized at late times by the dynamical exponent $z = 1/2$ -- a super-ballistic dynamics made possible by measurements., Comment: 9+4 pages, 7+2 figures. v2: 10+5 pages, 7+4 figures. Added Appendix E, improved figures

Quantum thermalization in a many-body system is defined by the approach of local subsystems towards a universal form, describable as an ensemble of quantum states wherein observables acquire thermal expectation values. Recently, it was demonstrated that the distribution of these quantum states can also exhibit universal statistics, upon associating each state with the outcome of a local projective measurement of the complementary subsystem. Specifically, this collection of pure quantum states -- called the projected ensemble -- can under certain conditions mimic the behavior of a maximally entropic, uniformly random ensemble, i.e., form a {\it quantum state-design}, representing a ``deeper'' form of quantum thermalization. In this work, we investigate the dynamical process underlying this novel emergent universality. Leveraging a space-time duality mapping for one-dimensional quantum circuits, we argue that the physics of dynamical purification, which arises in the context of monitored quantum systems, constrains the the projected ensemble's approach towards the uniform distribution. We prove that absence of dynamical purification in the space-time dual dynamics (a condition realized in dual-unitary quantum circuits with appropriate initial states and final measurement bases) generically yields exact state-designs for all moments $k$ at the same time, extending previous rigorous results [Ho and Choi, Phys. Rev. Lett. {\bf 128}, 060601 (2022)]. Conversely, we show that, departing from these conditions, dynamical purification can lead to a separation of timescales between the formation of a quantum state-design for moment $k=1$ (regular thermalization) and for high moments $k\gg 1$ (deep thermalization). Our results suggest that the projected ensemble can probe nuanced features of quantum dynamics inaccessible to regular thermalization, such as quantum information scrambling., Comment: 14+2 pages, 6 figures. v2: added Appendix A on monotonicity of design times, redefined v_p in base 2, added references. v3: 17+9 pages, 7+4 figures. Thoroughly reorganized presentation for clarity, many new figures, new Appendix B+C, credits to Daniel Mark for updated Appendix A

Quantum many-body systems display rich phase structure in their low-temperature equilibrium states. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC). Concretely, dynamical phases can be defined in periodically driven many-body localized systems via the concept of eigenstate order. In eigenstate-ordered phases, the entire many-body spectrum exhibits quantum correlations and long-range order, with characteristic signatures in late-time dynamics from all initial states. It is, however, challenging to experimentally distinguish such stable phases from transient phenomena, wherein few select states can mask typical behavior. Here we implement a continuous family of tunable CPHASE gates on an array of superconducting qubits to experimentally observe an eigenstate-ordered DTC. We demonstrate the characteristic spatiotemporal response of a DTC for generic initial states. Our work employs a time-reversal protocol that discriminates external decoherence from intrinsic thermalization, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. In addition, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to study non-equilibrium phases of matter on current quantum processors.

We introduce a novel class of quantum circuits that are unitary along three distinct "arrows of time". These dynamics share some of the analytical tractability of "dual-unitary" circuits, while exhibiting distinctive and richer phenomenology. We find that two-point correlations in these dynamics are strictly confined to three directions in $(1+1)$-dimensional spacetime -- the two light cone edges, $\delta x=\pm v\delta t$, and the static worldline $\delta x=0$. Along these directions, correlation functions are obtained exactly in terms of quantum channels built from the individual gates that make up the circuit. We prove that, for a class of initial states, entanglement grows at the maximum allowed speed up to an entropy density of at least one half of the thermal value, at which point it becomes model-dependent. Finally, we extend our circuit construction to $2+1$ dimensions, where two-point correlation functions are confined to the one-dimensional edges of a tetrahedral light cone -- a subdimensional propagation of information reminiscent of "fractonic" physics., Comment: 11 pages main text + 2 page appendix, 15 figures. v2: added references, fixed typos

The extension of many-body quantum dynamics to the non-unitary domain has led to a series of exciting developments, including new out-of-equilibrium entanglement phases and phase transitions. We show how a duality transformation between space and time on one hand, and unitarity and non-unitarity on the other, can be used to realize steady state phases of non-unitary dynamics that exhibit a rich variety of behavior in their entanglement scaling with subsystem size -- from logarithmic to extensive to \emph{fractal}. We show how these outcomes in non-unitary circuits (that are "spacetime-dual" to unitary circuits) relate to the growth of entanglement in time in the corresponding unitary circuits, and how they differ, through an exact mapping to a problem of unitary evolution with boundary decoherence, in which information gets "radiated away" from one edge of the system. In spacetime-duals of chaotic unitary circuits, this mapping allows us to uncover a non-thermal volume-law entangled phase with a logarithmic correction to the entropy distinct from other known examples. Most notably, we also find novel steady state phases with \emph{fractal} entanglement scaling, $S(\ell) \sim \ell^{\alpha}$ with tunable $0 < \alpha < 1$ for subsystems of size $\ell$ in one dimension. These fractally entangled states add a qualitatively new entry to the families of many-body quantum states that have been studied as energy eigenstates or dynamical steady states, whose entropy almost always displays either area-law, volume-law or logarithmic scaling. We also present an experimental protocol for preparing these novel steady states with only a very limited amount of postselection via a type of "teleportation" between spacelike and timelike slices of quantum circuits., Comment: v2: updated interpretation of volume-law phase, added discussion on breaking unitarity. v3: published version

The dynamics of entanglement in `hybrid' non-unitary circuits (for example, involving both unitary gates and quantum measurements) has recently become an object of intense study. A major hurdle toward experimentally realizing this physics is the need to apply \emph{postselection} on random measurement outcomes in order to repeatedly prepare a given output state, resulting in an exponential overhead. We propose a method to sidestep this issue in a wide class of non-unitary circuits by taking advantage of \emph{spacetime duality}. This method maps the purification dynamics of a mixed state under non-unitary evolution onto a particular correlation function in an associated unitary circuit. This translates to an operational protocol which could be straightforwardly implemented on a digital quantum simulator. We discuss the signatures of different entanglement phases, and demonstrate examples via numerical simulations. With minor modifications, the proposed protocol allows measurement of the purity of arbitrary subsystems, which could shed light on the properties of the quantum error correcting code formed by the mixed phase in this class of hybrid dynamics., Comment: 4 pages main text + 9 page supplemental material. v2: added references, expanded supplement, minor modifications to text

It was recently suggested that the topology of magic-angle twisted bilayer graphene's (MATBG) flat bands could provide a novel mechanism for superconductivity distinct from both weakly-coupled BCS theory and the $d$-wave phenomenology of the high-$T_c$ cuprates. In this work, we examine this possibility using a density matrix renormalization group (DMRG) study of a model which captures the essential features of MATBG's symmetry and topology. Using large scale cylinder-DMRG calculations to obtain the ground state and its excitations as a function of the electron doping, we find clear evidence for superconductivity driven by the binding of electrons into charge-$2e$ skyrmions. Remarkably, this binding is observed even in the regime where the unscreened Coulomb repulsion is by-far the largest energy scale, demonstrating the robustness of this topological, all-electronic pairing mechanism., Comment: 7 + 14 pages, 5 + 12 figures: (v2) Added clarifications and discussion of pair wavefunction

Recent progress in the realm of noisy, intermediate scale quantum (NISQ) devices represents an exciting opportunity for many-body physics, by introducing new laboratory platforms with unprecedented control and measurement capabilities. We explore the implications of NISQ platforms for many-body physics in a practical sense: we ask which {\it physical phenomena}, in the domain of quantum statistical mechanics, they may realize more readily than traditional experimental platforms. As a particularly well-suited target, we identify discrete time crystals (DTCs), novel non-equilibrium states of matter that break time translation symmetry. These can only be realized in the intrinsically out-of-equilibrium setting of periodically driven quantum systems stabilized by disorder induced many-body localization. While precursors of the DTC have been observed across a variety of experimental platforms - ranging from trapped ions to nitrogen vacancy centers to NMR crystals - none have \emph{all} the necessary ingredients for realizing a fully-fledged incarnation of this phase, and for detecting its signature long-range \emph{spatiotemporal order}. We show that a new generation of quantum simulators can be programmed to realize the DTC phase and to experimentally detect its dynamical properties, a task requiring extensive capabilities for programmability, initialization and read-out. Specifically, the architecture of Google's Sycamore processor is a remarkably close match for the task at hand. We also discuss the effects of environmental decoherence, and how they can be distinguished from `internal' decoherence coming from closed-system thermalization dynamics. Already with existing technology and noise levels, we find that DTC spatiotemporal order would be observable over hundreds of periods, with parametric improvements to come as the hardware advances., Comment: v2: added appendices B and C, added Fig.1, expanded discussion. v3: added Fig. 8

Unitary circuits subject to repeated projective measurements can undergo an entanglement phase transition (EPT) as a function of the measurement rate. This transition is generally understood in terms of a competition between the scrambling effects of unitary dynamics and the disentangling effects of measurements. We find that, surprisingly, EPTs are possible even in the absence of scrambling unitary dynamics, where they are best understood as arising from measurements alone. This motivates us to introduce \emph{measurement-only models}, in which the "scrambling" and "un-scrambling" effects driving the EPT are fundamentally intertwined and cannot be attributed to physically distinct processes. This represents a novel form of an EPT, conceptually distinct from that in hybrid unitary-projective circuits. We explore the entanglement phase diagrams, critical points, and quantum code properties of some of these measurement-only models. We find that the principle driving the EPTs in these models is \emph{frustration}, or mutual incompatibility, of the measurements. Suprisingly, an entangling (volume-law) phase is the generic outcome when measuring sufficiently long but still local ($\gtrsim 3$-body) operators. We identify a class of exceptions to this behavior ("bipartite ensembles") which cannot sustain an entangling phase, but display dual area-law phases, possibly with different kinds of quantum order, separated by self-dual critical points. Finally, we introduce a measure of information spreading in dynamics with measurements and use it to demonstrate the emergence of a statistical light-cone, despite the non-locality inherent to quantum measurements., Comment: 14 pages + bibliography and appendices, 10 figures. v2: added section on quantum code properties, added references. v3: substantial changes to the structure of the paper, improved clarity

Periodic driving has emerged as a powerful tool in the quest to engineer new and exotic quantum phases. While driven many-body systems are generically expected to absorb energy indefinitely and reach an infinite-temperature state, the rate of heating can be exponentially suppressed when the drive frequency is large compared to the local energy scales of the system -- leading to long-lived 'prethermal' regimes. In this work, we experimentally study a bosonic cloud of ultracold atoms in a driven optical lattice and identify such a prethermal regime in the Bose-Hubbard model. By measuring the energy absorption of the cloud as the driving frequency is increased, we observe an exponential-in-frequency reduction of the heating rate persisting over more than 2 orders of magnitude. The tunability of the lattice potentials allows us to explore one- and two-dimensional systems in a range of different interacting regimes. Alongside the exponential decrease, the dependence of the heating rate on the frequency displays features characteristic of the phase diagram of the Bose-Hubbard model, whose understanding is additionally supported by numerical simulations in one dimension. Our results show experimental evidence of the phenomenon of Floquet prethermalization, and provide insight into the characterization of heating for driven bosonic systems.

The fractional quantum Hall (FQH) effect was discovered in two-dimensional electron systems subject to a large perpendicular magnetic field nearly four decades ago. It helped launch the field of topological phases, and in addition, because of the quenching of the kinetic energy, gave new meaning to the phrase "correlated matter". Most FQH phases are gapped like insulators and superconductors; however, a small subset with even denominator fractional fillings nu of the Landau level, typified by nu = 1/2, are found to be gapless, with a Fermi surface akin to metals. We discuss our results, obtained numerically using the infinite Density Matrix Renormalization Group (iDMRG) scheme, on the effect of non-isotropic distortions with discrete N-fold rotational symmetry of the Fermi surface at zero magnetic field on the Fermi surface of the correlated nu = 1/2 state. We find that while the response for N = 2 (elliptical) distortions is significant (and in agreement with experimental observations with no adjustable parameters), it decreases very rapidly as N is increased. Other anomalies, like resilience to breaking the Fermi surface into disjoint pieces, are also found. This highlights the difference between Fermi surfaces formed from the kinetic energy, and those formed of purely potential energy terms in the Hamiltonian., Comment: Review of arXiv:1701.07832, arXiv:1704.06265 and arXiv:1706.09470 with some new results. Conference proceedings for "Quantum Fluids and Solids 2019", Edmonton, AB, August 2019

In several recent works it has been proposed that, due to disorder, the experimentally observed nu=5/2 quantum Hall state could be microscopically composed of domains of Pfaffian order along with domains of AntiPfaffian order. We numerically examine the energetics required for forming such domains and conclude that for the parameters appropriate for recent experiments, such domains would not occur., Comment: 5 pages

A fractional quantum Hall (FQH) system with broken rotational symmetry exploits its geometric degree of freedom to minimize its ground state energy. The mass anisotropy of bare particles interacting isotropically is partially inherited by the many-body FQH state, and the extent to which it does so depends on the type of interaction, filling fraction and ground state phase. Using numerical infinite density matrix renormalization group simulations, we investigate the transference of elliptical ($C_2$-symmetric) anisotropy from the band mass of the bare particles to the FQH states, for various power law interactions. We map out the response of FQH states to small anisotropy as a function of power law exponent, filling, and statistics (bosonic or fermionic) of the constituents. Interestingly, we find a non-analyticity in the linear response of the FQH state at a special filling-dependent value of the power law exponent, above which the interaction effectively becomes zero-range (point-like). We also investigate the the effect of $C_4$-symmetric band distortions, where we observe a strikingly different dependence on filling.

We study the quantum Hall plateau transition on rectangular tori. As the aspect ratio of the torus is increased, the two-dimensional critical behavior, characterized by a subthermodynamic number of topological states in a vanishing energy window around a critical energy, changes drastically. In the thin-torus limit, the entire spectrum is Anderson-localized; however, an extensive number of states retain a Chern number $C\neq 0$. We resolve this apparent paradox by mapping the thin-torus quantum Hall system onto a disordered Thouless pump, where the Chern number corresponds to the winding number of an electron's path in real space during a pump cycle. We then characterize quantitatively the crossover between the one- and two-dimensional regimes for large but finite aspect ratio, where the average Thouless conductance also shows anomalous scaling., Comment: 5 pages, 3 figures main text + 5 pages, 5 figures supplement. v2: substantially expanded supplemental material, updated figure 3, corrected minor errors. v3: added references

A two-dimensional electron gas in a high magnetic field displays macroscopically degenerate Landau levels, which can be split into Hofstadter subbands by means of a weak periodic potential. By carefully engineering such a potential, one can precisely tune the number, bandwidths, bandgaps and Chern character of these subbands. This allows a detailed study of the interplay of disorder, interaction and topology in two dimensional systems. We first explore the physics of disorder and single-particle localization in subbands derived from the lowest Landau level, that nevertheless may have a topological nature different from that of the entire lowest Landau level. By projecting the Hamiltonian onto subbands of interest, we systematically explore the localization properties of single-particle eigenstates in the presence of quenched disorder. We then introduce electron-electron interactions and investigate the fate of many-body localization in subbands of varying topological character.

We explore the problem of localization in topological and non-topological nearly-flat subbands derived from the lowest Landau level, in the presence of quenched disorder and short-range interactions. We consider two models: a suitably engineered periodic potential, and randomly distributed point-like impurities. We perform numerical exact diagonalization on a torus geometry and use the mean level spacing ratio $\langle r \rangle$ as a diagnostic of ergodicity. For topological subbands, we find there is no ergodicity breaking in both the one and two dimensional thermodynamic limits. For non-topological subbands, in constrast, we find evidence of an ergodicity breaking transition at finite disorder strength in the one-dimensional thermodynamic limit. Intriguingly, indications of similar behavior in the two-dimensional thermodynamic limit are found, as well. This constitutes a novel, $\textit{continuum}$ setting for the study of the many-body localization transition in one and two dimensions.

We explore a method for regulating 2+1D quantum critical points in which the ultra-violet cutoff is provided by the finite density of states of particles in a magnetic field, rather than by a lattice. Such Landau level quantization allows for numerical computations on arbitrary manifolds, like spheres, without introducing lattice defects. In particular, when half-filling a Landau level with $N=4$ electron flavors, with appropriate interaction anisotropies in flavor space, we obtain a fully continuum regularization of the O(5) non-linear sigma-model with a topological term, which has been conjectured to flow to a deconfined quantum critical point. We demonstrate that this model can be solved by both infinite density matrix renormalization group calculations and sign-free determinantal quantum Monte Carlo. DMRG calculations estimate the scaling dimension of the O(5) vector operator to be in the range $\Delta_V \sim 0.55 - 0.7$ depending on the stiffness of the non-linear sigma model. Future Monte Carlo simulations will be required to determine whether this dependence is a finite-size effect or further evidence for a weakly first-order transition., Comment: 10 pages, 5 figures

We describe a quench protocol that allows the rapid preparation of ground states of arbitrary interacting conformal field theories in $1+1$ dimensions. We start from the ground state of a related gapped relativistic quantum field theory and consider sudden quenches along the space-like trajectories $t^2 - x^2 = T^2_0$ (parameterized by $T_0$) to a conformal field theory. Using only arguments of symmetry and conformal invariance, we show that the post-quench stress-energy tensor of the conformal field theory is uniquely constrained up to an overall scaling factor. Crucially, the $\textit{geometry}$ of the quench necessitates that the system approach the vacuum energy density over all space except the singular lines $x = \pm t$. The above arguments are verified using an exact treatment of the quench for the Gaussian scalar field theory (equivalently, the Luttinger liquid), and numerically for the quantum $O(N)$ model in the large-$N$ limit. Additionally, for the Gaussian theory, we find in fact that even when starting from certain excited states, the quench conserves entropy, and is thus also suitable for rapidly preparing excited states. Our methods serve as a fast, alternative route to reservoir-based cooling to prepare quantum states of interest., Comment: 13 pages, 6 figures

Fractional quantum Hall (FQH) states are known to possess an internal metric degree of freedom that allows them to minimize their energy when contrasting geometries are present in the problem (e.g., electron band mass and dielectric tensor). We investigate the internal metric of several incompressible FQH states by probing its response to band mass anisotropy using infinite DMRG simulations on a cylinder geometry. We test and apply a method to extract the internal metric of a FQH state from its guiding center structure factor. We find that the response to band mass anisotropy is approximately the same for states in the same Jain sequence, but changes substantially between different sequences. We provide a theoretical explanation of the observed behavior of primary states at filling $\nu = 1/m$ in terms of a minimal microscopic model of flux attachment., Comment: 12 pages including references, 14 figures

We investigate the quantum Hall problem in the lowest Landau level in two dimensions, in the presence of an arbitrary number of $\delta$-function potentials arranged in different geometric configurations. When the number of delta functions $N_\delta$ is smaller than the number of flux quanta through the system ($N_\phi$), there is a manifold of $(N_\phi-N_\delta)$ degenerate states at the original Landau level energy. We prove that the total Chern number of this set of states is +1 regardless of the number or position of the $\delta$ functions. Furthermore, we find numerically that, upon the addition of disorder, this subspace includes a quantum Hall transition which is (in a well-defined sense) $\textit{quantitatively}$ the same as that for the lowest Landau level without $\delta$-function impurities, but with a reduced number $N_\phi' \equiv N_\phi-N_\delta$ of magnetic flux quanta. We discuss the implications of these results for studies of the integer plateau transitions, as well as for the many-body problem in the presence of electron-electron interactions., Comment: 11 pages (including appendix and references), 7 figures

We study the effect of band anisotropy with discrete rotational symmetry $C_N$ (where $N\ge 2$) in the quantum Hall regime of two-dimensional electron systems. We focus on the composite Fermi liquid (CFL) at half filling of the lowest Landau level. We find that the magnitude of anisotropy transferred to the composite fermions decreases very rapidly with $N$. We demonstrate this by performing density matrix normalization group calculations on the CFL, and comparing the anisotropy of the composite fermion Fermi contour with that of the (non-interacting) electron Fermi contour at zero magnetic field. We also show that the effective interaction between the electrons after projecting into a single Landau level is much less anisotropic than the band, a fact which does not depend on filling and thus has implications for other quantum Hall states as well. Our results confirm experimental observations on anisotropic bands with warped Fermi contours, where the only detectable effect on the composite Fermi contour is an elliptical distortion ($N = 2$)., Comment: 6 pages + bibliography, 5 figures

We investigate the relation between the Fermi sea (FS) of zero-field carriers in two-dimensional systems and the FS of the corresponding composite fermions which emerge in a high magnetic field at filling $\nu = \frac{1}{2}$, as the kinetic energy dispersion is varied. We study cases both with and without rotational symmetry, and find that there is generally no straightforward relation between the geometric shapes and topologies of the two FSs. In particular, we show analytically that the composite Fermi liquid (CFL) is completely insensitive to a wide range of changes to the zero-field dispersion which preserve rotational symmetry, including ones that break the zero-field FS into multiple disconnected pieces. In the absence of rotational symmetry, we show that the notion of `valley pseudospin' in many-valley systems is generically not transferred to the CFL, in agreement with experimental observations. We also discuss how a rotationally symmetric band structure can induce a reordering of the Landau levels, opening interesting possibilities of observing higher-Landau-level physics in the high-field regime., Comment: 7 pages + references, 7 figures. Added many-body DMRG calculation

We perform density-matrix renormalization group studies of a two-dimensional electron gas in a high magnetic field and with an anisotropic band mass. At half-filling in the lowest Landau level, such a system is a Fermi liquid of composite fermions. By measuring the Fermi surface of these composite fermions, we determine a relationship between the anisotropy of composite fermion dispersion, $\alpha_{CF}$, and the original anisotropy $\alpha_F$ of the fermion dispersion at zero magnetic field. For systems where the electrons interact via a Coulomb interaction, we find $\alpha_{CF}=\sqrt{\alpha_F}$ within our numerical accuracy. The same result has been found concurrently in recent experiments. We also show results with other forms of the electron-electron interaction; this allows us (a) to benchmark our procedure against known exact results and (b) to show that the relationship between the anisotropies is dependent on the form of the interaction., Comment: 4 pages + references, 4 figures

In this work we address the problem of realizing a reliable quantum memory based on zero-energy Majorana modes in the presence of experimental constraints on the operations aimed at recovering the information. In particular, we characterize the best recovery operation acting only on the zero-energy Majorana modes and the memory fidelity that can be therewith achieved. In order to understand the effect of such restriction, we discuss two examples of noise models acting on the topological system and compare the amount of information that can be recovered by accessing either the whole system, or the zero-modes only, with particular attention to the scaling with the size of the system and the energy gap. We explicitly discuss the case of a thermal bosonic environment inducing a parity-preserving Markovian dynamics in which the introduced memory fidelity decays exponentially in time, independent from system size, thus showing the impossibility to retrieve the information by acting on the zero-modes only. We argue, however, that even in the presence of experimental limitations, the Hamiltonian gap is still beneficial to the storage of information., Comment: 18 pages, 7 figures. Updated to published version

We present a novel discussion of the continuous-time quantum error correction introduced by Paz and Zurek in 1998 [Paz and Zurek, Proc. R. Soc. A 454, 355 (1998)]. We study the general Lindbladian which describes the effects of both noise and error correction in the weak-noise (or strong-correction) regime through a perturbative expansion. We use this tool to derive quantitative aspects of the continuous-time dynamics both in general and through two illustrative examples: the 3-qubit and the 5-qubit stabilizer codes, which can be independently solved by analytical and numerical methods and then used as benchmarks for the perturbative approach. The perturbatively accessible time frame features a short initial transient in which error correction is ineffective, followed by a slow decay of the information content consistent with the known facts about discrete-time error correction in the limit of fast operations. This behavior is explained in the two case studies through a geometric description of the continuous transformation of the state space induced by the combined action of noise and error correction., Comment: 14 pages, 10 figures