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Mean field type games (MFTGs) describe Nash equilibria between large coalitions: each coalition consists of a continuum of cooperative agents who maximize the average reward of their coalition while interacting non-cooperatively with a finite number of other coalitions. Although the theory has been extensively developed, we are still lacking efficient and scalable computational methods. Here, we develop reinforcement learning methods for such games in a finite space setting with general dynamics and reward functions. We start by proving that MFTG solution yields approximate Nash equilibria in finite-size coalition games. We then propose two algorithms. The first is based on quantization of the mean-field spaces and Nash Q-learning. We provide convergence and stability analysis. We then propose an deep reinforcement learning algorithm, which can scale to larger spaces. Numerical examples on 5 environments show the scalability and the efficiency of the proposed method.

We study a center-of-mass-conserving Brownian complex Sachdev-Ye-Kitaev model with long-range (power-law) interactions characterized by $1/r^\eta$. The kinetic constraint and long-range interactions conspire to yield rich hydrodynamics associated with the conserved charge, which we reveal by computing the Schwinger-Keldysh effective action. Our result shows that charge transport in this system can be subdiffusive, diffusive, or superdiffusive, with the dynamical exponent controlled by $\eta$. We further employ a doubled Hilbert space methodology to derive an effective action for the out-of-time-order correlator (OTOC), from which we obtain the phase diagram delineating regimes where the lightcone is linear or logarithmic. Our results provide a concrete example of a quantum many-body system with kinetic constraint and long-range interactions in which the emergent hydrodynamic modes and OTOC can be computed analytically., Comment: 4.5 + 10 pages, 4 figures

We study defect CFT within the framework of holographic duality, emphasizing the impact of corner contributions. We model distinct conformal defects using interface branes that differ in tensions and are connected by a corner. Employing the relationship between CFT scaling dimensions and Euclidean gravity actions, we outline a general procedure for calculating the anomalous dimensions of defect changing operators at nontrivial cusps. Several analytical results are obtained, including the cusp anomalous dimensions at big and small angles. While $1/\phi$ universal divergence appears for small cusp angles due to the fusion of two defects, more interestingly, we uncover a bubble phase rendered by a near zero angle cusp, in which the divergence is absent., Comment: 4.2 pages + supplemental material, 3 figures

We study the boundary criticality enriched by boundary fermions, which ubiquitously emerge in topological phases of matter, with a focus on topological insulators and topological superconductors. By employing dimensional regularization and bosonization techniques, we uncover several unprecedented boundary universality classes. These include the boundary Gross-Neveu-Yukawa critical point and the special Berezinskii-Kosterlitz-Thouless (BKT) transition, both resulting from the interplay between edge modes and bulk bosons. We present a comprehensive sketch of the phase diagram that accommodates these boundary criticalities and delineate their critical exponents. Additionally, we explore a 1+1D conformal defect decorated with fermions, where a defect BKT transition is highlighted. We conclude with a discussion on potential experimental realizations of these phenomena., Comment: 4.4 pages + supplemental material, 2 figures, 1 table

Depression recognition based on physiological signals such as functional near-infrared spectroscopy (fNIRS) and electroencephalogram (EEG) has made considerable progress. However, most existing studies ignore the complementarity and semantic consistency of multimodal physiological signals under the same stimulation task in complex spatio-temporal patterns. In this paper, we introduce a multimodal physiological signals representation learning framework using Siamese architecture via multiscale contrasting for depression recognition (MRLMC). First, fNIRS and EEG are transformed into different but correlated data based on a time-domain data augmentation strategy. Then, we design a spatio-temporal contrasting module to learn the representation of fNIRS and EEG through weight-sharing multiscale spatio-temporal convolution. Furthermore, to enhance the learning of semantic representation associated with stimulation tasks, a semantic consistency contrast module is proposed, aiming to maximize the semantic similarity of fNIRS and EEG. Extensive experiments on publicly available and self-collected multimodal physiological signals datasets indicate that MRLMC outperforms the state-of-the-art models. Moreover, our proposed framework is capable of transferring to multimodal time series downstream tasks.

The coupling between defects and extended critical degrees of freedom gives rise to the intriguing theory known as defect conformal field theory (CFT). In this work, we introduce a novel family of boundary and interface CFTs by coupling $N$ Majorana chains with SYK$_q$ interactions at the defect. Our analysis reveals that the interaction with $q=2$ constitutes a new marginal defect. Employing a versatile saddle point method, we compute unique entanglement characterizations, including the $g$-function and effective central charge, of the defect CFT. Furthermore, we analytically evaluate the transmission coefficient using CFT techniques. Surprisingly, the transmission coefficient deviates from the universal relation with the effective central charge across the defect at the large $N$ limit, suggesting that our defect CFT extends beyond all known examples of Gaussian defect CFT., Comment: 15 pages, 6 figures

The widely-used Kohn-Sham implementation of density functional theory (DFT) maps a system of interacting electrons onto an auxiliary non-interacting one and is presumably inaccurate for strongly correlated materials. We present a concrete benchmark for DFT by examining the electronic ground state phase diagram of a strongly interacting chain with uneven, non-integer nuclear charges. The interplay between charge imbalance and Coulomb repulsion yields two competing phases, a band insulator and a Mott insulator. By including infinitesimal lattice distortions, DFT stabilizes the intermediate spontaneously dimerized insulator phase that results from this competition. We assess the phase diagram by mapping the bond length and nuclear charge ratio of the chain to the ionic Hubbard model and performing highly accurate density matrix renormalization group calculations. Our comparative study provides foundational insight into the utility of symmetry-broken DFT as a predictive tool for elucidating phase diagrams with competing orders.

Quantum entanglement marks a definitive feature of topological states. However, the entanglement spectrum remains insufficiently explored for topological states without a bulk energy gap. Using a combination of field theory and numerical techniques, we accurately calculate and analyze the entanglement spectrum of gapless symmetry protected topological states in one dimension. We highlight that the universal entanglement spectrum not only encodes the nontrivial edge degeneracy, generalizing the Li-Haldane conjecture to gapless topological states, but also contains the operator content of the underlying boundary conformal field theory. This implies that the bulk wave function can act as a fingerprint of both quantum criticality and topology in gapless symmetry protected topological states. We also identify a symmetry enriched conformal boundary condition that goes beyond the conventional conformal boundary condition., Comment: 4.5 pages with supplemental materials, 10 figures

The presence of quantum noises inherent to real physical systems can strongly impact the physics in hybrid quantum circuits with local random unitaries and mid-circuit measurements. The quantum noises with a size-independent occurring probability can lead to the disappearance of a measurement-induced entanglement phase transition and the emergence of a single area-law phase. In this work, we investigate the effects of quantum noises with size-dependent probabilities $q=p/L^{\alpha}$ where $\alpha$ represents the scaling exponent. We have identified a noise-induced entanglement phase transition from a volume law to a power (area) law in the presence (absence) of measurements as $p$ increases when $\alpha=1$. With the help of an effective statistical model, we reveal that the phase transition is of first-order arising from the competition between two types of spin configurations and shares the same analytical understanding as the noise-induced coding transition. This unified picture further deepens the understanding of the connection between entanglement behavior and the capacity of information protection. When $\alpha \neq 1$, one spin configuration always dominates regardless of $p$ and thus the phase transition disappears. Moreover, we highlight the difference between the effects of size-dependent bulk noise and boundary noises. We validate our analytical predictions with extensive numerical results from stabilizer circuit simulations., Comment: 11 pages, 18 figures

We investigate the evolution of entanglement within an open, strongly coupled system interacting with a heat bath as its environment, in the frameworks of both the doubly holographic model and the Sachdev-Ye-Kitaev (SYK) model. Generally, the entanglement within the system initially increases due to internal interactions; however, it eventually dissipates into the environment. In the doubly holographic setup, we consider an end-of-the-world brane in the bulk to represent an eternal black hole coupled with its radiation and the evolution of the global thermofield double (TFD) state. For small black holes, the reflected entropy between the bipartition exhibits a ramp-plateau-slump behavior, where the plateau arises due to the phase transition of the entanglement wedge cross-section before the Page time. Similarly, the mutual information between the bipartition displays a ramp-slop-stabilizing behavior. In quantum mechanics, we consider a double copy of the SYK-plus-bath system in a global TFD state, resembling an eternal black hole interacting with an environment. The R\'enyi mutual information within the double-copied SYK clusters exhibits a ramp-plateau-slope-stabilizing behavior. The dynamic behaviors of the entanglement quantities observed in these two models are attributable to the competition between the internal interaction of the system and the external interaction with the baths. Our study provides a fine-grained picture of the entanglement dynamics inside black holes before their Page time., Comment: 31 pages, 10 figures, major revision, published version

In the context of measurement-induced entanglement phase transitions, the influence of quantum noises, which are inherent in real physical systems, is of great importance and experimental relevance. In this Letter, we present a comprehensive theoretical analysis of the effects of both temporally uncorrelated and correlated quantum noises on entanglement generation and information protection. This investigation reveals that entanglement within the system follows $q^{-1/3}$ scaling for both types of quantum noises, where $q$ represents the noise probability. The scaling arises from the Kardar-Parisi-Zhang fluctuation with effective length scale $L_{\text{eff}} \sim q^{-1}$. More importantly, the information protection timescales of the steady states are explored and shown to follow $q^{-1/2}$ and $q^{-2/3}$ scaling for temporally uncorrelated and correlated noises, respectively. The former scaling can be interpreted as a Hayden-Preskill protocol, while the latter is a direct consequence of Kardar-Parisi-Zhang fluctuations. We conduct extensive numerical simulations using stabilizer formalism to support the theoretical understanding. This Letter not only contributes to a deeper understanding of the interplay between quantum noises and measurement-induced phase transition but also provides a new perspective to understand the effects of Markovian and non-Markovian noises on quantum computation., Comment: 4.5 pages, 3 figures, and Supplemental Materials

Quantum electrodynamics (QED) is a cornerstone of particle physics and also finds diverse applications in condensed matter systems. Despite its significance, the dynamics of quantum electrodynamics under a quantum quench remains inadequately explored. In this paper, we investigate the nonequilibrium regime of quantum electrodynamics following a global quantum quench. Specifically, a massive Dirac fermion is quenched to a gapless state with an interaction with gauge bosons. In stark contrast to equilibrium (3+1)-dimensional QED with gapless Dirac fermions, where the coupling is marginally irrelevant, we identify a nonequilibrium fixed point characterized by nonFermi liquid behavior. Notably, the anomalous dimension at this fixed point varies with the initial quench parameter, suggesting an interesting quantum memory effect in a strongly interacting system. Additionally, we propose distinctive experimental signatures for nonequilibrium quantum electrodynamics., Comment: 15 pages, 7 figures

We investigate the impact of measuring one subsystem on the holographic complexity of another. While a naive expectation might suggest a reduction in complexity due to the collapse of the state to a trivial product state during quantum measurements, our findings reveal a counterintuitive result: in numerous scenarios, measurements on one subsystem can amplify the complexity of another. We first present a counting argument elucidating this complexity transition in random states. Then, employing the subregion "complexity=volume" (CV) proposal, we identify a complexity phase transition induced by projection measurements in various holographic CFT setups, including CFT vacuum states, thermofield double states, and the joint system of a black hole coupled to a bath. According to the AdS/BCFT correspondence, the post-measurement dual geometry involves an end-of-the-world brane created by the projection measurement. The complexity phase transition corresponds to the transition of the entanglement wedge to the one connected to the brane. In the context of the thermofield double setup, complete projection on one side can transform the other side into a boundary state black hole with higher complexity or a pure AdS with lower complexity. In the joint system of a black hole coupled to a nongraviting bath, where (a part of) the radiation is measured, the BCFT features two boundaries: one for the black hole and the other for the measurement. We construct the bulk dual involving intersecting or non-intersecting branes, and investigate the complexity transition induced by the projection measurement. Notably, for a subsystem that contains the black hole brane, its RT surface may undergo a transition, giving rise to a complexity jump., Comment: 44 pages, 32 figures

A valley filter capable of generating a valley-polarized current is a crucial element in valleytronics, yet its implementation remains challenging. Here, we propose a valley filter made of a graphene bilayer which exhibits a 1D moir\'{e} pattern in the overlapping region of the two layers controlled by heterostrain. In the presence of a lattice modulation between layers, electrons propagating in one layer can have valley-dependent dissipation due to valley asymmetric interlayer coupling, thus giving rise to a valley-polarized current. Such a process can be described by an effective non-Hermitian theory, in which the valley filter is driven by a valley-resolved non-Hermitian skin effect. Nearly 100\% valley-polarization can be achieved within a wide parameter range and the functionality of the valley filter is electrically tunable. The non-Hermitian topological scenario of the valley filter ensures high tolerance against imperfections such as disorder and edge defects. Our work opens a new route for efficient and robust valley filters while significantly relaxing the stringent implementation requirements., Comment: 6 pages, 3 figures

In this paper, we study a holographic description of weak measurements in conformal field theories (CFTs). Weak measurements can be viewed as a soft projection that interpolates between an identity operator and a projection operator, and can induce an effective central charge distinct from the unmeasured CFT. We model the weak measurement by an interface brane, separating different geometries dual to the post-measurement state and the unmeasured CFT, respectively. In an infinite system, the weak measurement is related to ICFT via a spacetime rotation. We find that the holographic entanglement entropy with twist operators located on the defect is consistent in both calculations for ICFT and weak measurements. We additionally calculate the boundary entropy via holographic entanglement as well as partition function. In a finite system, the weak measurement can lead to a rich phase diagram: for marginal measurements the emergent brane separates two AdS geometries, while for irrelevant measurements the post-measurement geometry features an AdS spacetime and a black hole spacetime that are separated by the brane. Although the measurement is irrelevant in the later phase, the post-measurement geometry can realize a Python's lunch., Comment: 81 pages, 36 figures, added a discussion on the thick brane description, updated references

We study the emergence of anticoncentration and approximate unitary design behavior in local Brownian circuits. The dynamics of circuit averaged moments of the probability distribution and entropies of the output state can be represented as imaginary time evolution with an effective local Hamiltonian in the replica space. This facilitates large scale numerical simulation of the dynamics in $1+1d$ of such circuit-averaged quantities using tensor network tools, as well as identifying the various regimes of the Brownian circuit as distinct thermodynamic phases. In particular, we identify the emergence of anticoncentration as a sharp transition in the collision probability at $\log N$ timescale, where $N$ is the number of qubits. We also show that a specific classical approximation algorithm has a computational hardness transition at the same timescale. In the presence of noise, we show there is a noise-induced first order phase transition in the linear cross entropy benchmark when the noise rate is scaled down as $1/N$. At longer times, the Brownian circuits approximate a unitary 2-design in $O(N)$ time. We directly probe the feasibility of quantum error correction by such circuits, and identify a first order transition at $O(N)$ timescales. The scaling behaviors for all these phase transitions are obtained from the large scale numerics, and corroborated by analyzing the spectrum of the effective replica Hamiltonian., Comment: 17+3 pages, 10 figures

Deconfined quantum criticality (DQC) arises from fractionalization of quasi-particles and leads to fascinating behaviors beyond the Landau-Ginzburg-Wilson (LGW) description of phase transitions. The unusual aspects of DQC in equilibrium also suggest exotic phenomena out of equilibrium, which are still largely unexplored. Here we study manifestations of DQC when driving (quantum annealing) a two-dimensional quantum magnet through a critical point separating antiferromagnetic and spontaneously dimerized ground states. Numerical simulations show that the celebrated Kibble-Zurek scaling (KZS) mechanism is inadequate for describing the annealing process. To explain our results, we introduce the concept of dual asymmetric KZS, where a deconfinement time enters in addition to the conventional relaxation time and the scaling also depends on the direction in which the system is driven through the critical point according to a duality principle connecting the topological defects in the two phases. These defects -- spinons in the dimerized phase and space-time hedgehogs in the antiferromagnetic phase -- require a much longer time scale for equilibration than the amplitude of the order parameter. Beyond the new insights into DQC, our scaling approach provides a new window into out-of-equilibrium criticality with multiple length and time scales., Comment: 15 pages, 9 figures

We extend the results of arXiv:2209.12903 by studying local projective measurements performed on subregions of two copies of a CFT${}_2$ in the thermofield double state and investigating their consequences on the bulk double-sided black hole holographic dual. We focus on CFTs defined on an infinite line and consider measurements of both finite and semi-infinite subregions. In the former case, the connectivity of the bulk spacetime is preserved after the measurement. In the latter case, the measurement of two semi-infinite intervals in one CFT or of one semi-infinite interval in each CFT can destroy the Einstein-Rosen bridge and disconnect the bulk dual spacetime. In particular, we find that a transition between a connected and disconnected phase occurs depending on the relative size of the measured and unmeasured subregions and on the specific Cardy state the measured subregions are projected on. We identify this phase transition as an entangled/disentangled phase transition of the dual CFT system by computing the post-measurement holographic entanglement entropy between the two CFTs. We also find that bulk information encoded in one CFT in the absence of measurement can sometimes be reconstructed from the other CFT when a measurement is performed, or can be erased by the measurement. Finally, we show that a purely CFT calculation of the Renyi entropy using the replica trick yields results compatible with those obtained in our bulk analysis., Comment: 55 pages, 22 figures

Krylov complexity measures the spread of the wavefunction in the Krylov basis, which is constructed using the Hamiltonian and an initial state. We investigate the evolution of the maximally entangled state in the Krylov basis for both chaotic and non-chaotic systems. For this purpose, we derive an Ehrenfest theorem for the Krylov complexity, which reveals its close relation to the spectrum. Our findings suggest that neither the linear growth nor the saturation of Krylov complexity is necessarily associated with chaos. However, for chaotic systems, we observe a universal rise-slope-ramp-plateau behavior in the transition probability from the initial state to one of the Krylov basis states. Moreover, a long ramp in the transition probability is a signal for spectral rigidity, characterizing quantum chaos. Also, this ramp is directly responsible for the late-time peak of Krylov complexity observed in the literature. On the other hand, for non-chaotic systems, this long ramp is absent. Therefore, our results help to clarify which features of the wave function time evolution in Krylov space characterize chaos. We exemplify this by considering the Sachdev-Ye-Kitaev model with two-body or four-body interactions., Comment: 40+26 pages, 19 figures, major revision, published version

Variational quantum algorithms (VQAs) hold great potentials for near-term applications and are promising to achieve quantum advantage on practical tasks. However, VQAs suffer from severe barren plateau problem as well as have a large probability of being trapped in local minima. In this Letter, we propose a novel training algorithm with random quantum gate activation for VQAs to efficiently address these two issues. This new algorithm processes effectively much fewer training parameters than the conventional plain optimization strategy, which efficiently mitigates barren plateaus with the same expressive capability. Additionally, by randomly adding two-qubit gates to the circuit ansatz, the optimization trajectories can escape from local minima and reach the global minimum more frequently due to more sources of randomness. In real quantum experiments, the new training algorithm can also reduce the quantum computational resources required and be more quantum noise resilient. We apply our training algorithm to solve variational quantum simulation problems for ground states and present convincing results that showcase the advantages of our novel strategy where better performance is achieved by the combination of mitigating barren plateaus, escaping from local minima, and reducing the effect of quantum noises. We further propose that the entanglement phase transition could be one underlying reason why our RA training is so effective., Comment: 4.5 pages + references + supplemental, 4 figures

Finding new critical states of matter is an important subject in modern many-body physics. Here we study the effect of measurement and postselection on the critical ground state of a Luttinger liquid theory and show that it can lead to qualitatively new critical states. Depending on the Luttinger parameter $K$, the effect of measurement is irrelevant (relevant) at $K>1$ ($K<1$). We reveal that this causes an entanglement transition between two phases, one with logarithmic entanglement entropy for a subregion ($K>1$), and the other with algebraic entanglement entropy ($K<1$). At the critical point $K=1$, the measurement is marginal, and we find new critical states whose entanglement entropy exhibits a logarithmic behavior with a continuous effective central charge as a function of measurement strength. We also performed numerical density matrix renormalization group and fermionic Gaussian state simulations to support our results. We further discuss promising and feasible routes to experimentally realize new critical states in our work., Comment: 4.7 pages + supplemental material, 4 figures; updated variational quantum algorithms

Measurement-induced phase transitions (MIPT) have attracted increasing attention due to the rich phenomenology of entanglement structures and their relation with quantum information processing. Since physical systems are unavoidably coupled to environment, quantum noise needs be considered in analyzing a system with MIPT, which may qualitatively modify or even destroy certain entanglement structure of the system. In this Letter, we investigate the effect of quantum noise modeled by reset quantum channel acting on each site with probability $q$ on MIPT. Based on the numerical results from the Clifford circuits, we show that the quantum noise can qualitatively change the entanglement properties - the entanglement obeys ``area law'' instead of ``volume law'' with projective measurement rate $p

According to holography, entanglement is the building block of spacetime; therefore, drastic changes of entanglement will lead to interesting transitions in the dual spacetime. In this paper, we study the effect of projective measurements on the Sachdev-Ye-Kitaev (SYK) model's thermofield double state, dual to an eternal black hole in Jackiw-Teitelboim (JT) gravity. We calculate the (Renyi-2) mutual information between the two copies of the SYK model upon projective measurement of a subset of fermions in one copy. We propose a dual JT gravity model that can account for the change of entanglement due to measurement, and observe an entanglement wedge phase transition in the von Neumann entropy. The entanglement wedge for the unmeasured side changes from the region outside the horizon to include the entire time reversal invariant slice of the two-sided geometry as the number of measured Majorana fermions increases. Therefore, after the transition, the bulk information stored in the measured subsystem is not entirely lost upon projection in one copy of the SYK model, but rather teleported to the other copy. We further propose a decoding protocol to elucidate the teleportation interpretation, and connect our analysis to the physics of traversable wormholes., Comment: 35 pages + appendices, 21 figures

In this study, we explore the representation mapping from the domain of visual arts to the domain of music, with which we can use visual arts as an effective handle to control music generation. Unlike most studies in multimodal representation learning that are purely data-driven, we adopt an analysis-by-synthesis approach that combines deep music representation learning with user studies. Such an approach enables us to discover \textit{interpretable} representation mapping without a huge amount of paired data. In particular, we discover that visual-to-music mapping has a nice property similar to equivariant. In other words, we can use various image transformations, say, changing brightness, changing contrast, style transfer, to control the corresponding transformations in the music domain. In addition, we released the Vis2Mus system as a controllable interface for symbolic music generation., Comment: Submitted to ICASSP 2023. GitHub repo: https://github.com/ldzhangyx/vis2mus

Holography has taught us that spacetime is emergent and its properties depend on the entanglement structure of the dual theory. In this paper, we describe how changes in the entanglement due to a local projective measurement (LPM) on a subregion $A$ of the boundary theory modify the bulk dual spacetime. We find that LPMs destroy portions of the bulk geometry, yielding post-measurement bulk spacetimes dual to the complementary unmeasured region $A^c$ that are cut off by end-of-the-world branes. Using a bulk calculation in $AdS_3$ and tensor network models of holography, we show that the portions of the bulk geometry that are preserved after the measurement depend on the size of $A$ and the state we project onto. The post-measurement bulk dual to $A^c$ includes regions that were originally part of the entanglement wedge of $A$ prior to measurement. This suggests that LPMs performed on a boundary subregion $A$ teleport part of the bulk information originally encoded in $A$ into the complementary region $A^c$. In semiclassical holography an arbitrary amount of bulk information can be teleported in this way, while in tensor network models the teleported information is upper-bounded by the amount of entanglement shared between $A$ and $A^c$ due to finite-$N$ effects. When $A$ is the union of two disjoint subregions, the measurement triggers an entangled/disentangled phase transition between the remaining two unmeasured subregions, corresponding to a connected/disconnected phase transition in the bulk description. Our results shed new light on the effects of measurement on the entanglement structure of holographic theories and give insight on how bulk information can be manipulated from the boundary theory. They could also be extended to more general quantum systems and tested experimentally, and represent a first step towards a holographic description of measurement-induced phase transitions., Comment: 60 pages + appendices, 27 figures

In this paper, we consider two types of topological solitons in a generalized Born-Infeld-Higgs model. We explore the self-dual structure of the model and prove the existence of planar vortex solutions. Furthermore, we couple the system with the Einstein equations and study the cosmic strings problem over $\mathbb R^{1,1}\times S$, where $S$ is a Riemann surface. We prove the existence of cosmic string solutions when $S$ is noncompact. We also discuss the decay estimates for vortices and cosmic strings at infinity and show that the minimal energy is quantized and depends on the number of vortices and strings, respectively., Comment: 22 pages

We calculate the frame potential for Brownian clusters of $N$ spins or fermions with time-dependent all-to-all interactions. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using a path integral approach. We argue that the $k$th frame potential comes within $\epsilon$ of the Haar value after a time of order $t \sim k N + k \log k + \log \epsilon^{-1}$. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary $k$-design after a time of order $t \sim k N$. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a $k$-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are also analytically tractable., Comment: 20.5 pages, 5 figures

TensorCircuit is an open source quantum circuit simulator based on tensor network contraction, designed for speed, flexibility and code efficiency. Written purely in Python, and built on top of industry-standard machine learning frameworks, TensorCircuit supports automatic differentiation, just-in-time compilation, vectorized parallelism and hardware acceleration. These features allow TensorCircuit to simulate larger and more complex quantum circuits than existing simulators, and are especially suited to variational algorithms based on parameterized quantum circuits. TensorCircuit enables orders of magnitude speedup for various quantum simulation tasks compared to other common quantum software, and can simulate up to 600 qubits with moderate circuit depth and low-dimensional connectivity. With its time and space efficiency, flexible and extensible architecture and compact, user-friendly API, TensorCircuit has been built to facilitate the design, simulation and analysis of quantum algorithms in the Noisy Intermediate-Scale Quantum (NISQ) era., Comment: Whitepaper for TensorCircuit, 43 pages, 11 figures, 9 tables

We study the effect of disorder on the spacetime supersymmetry that is proposed to emerge at the quantum critical point of pair density wave transition in (2+1)D Dirac semimetals and (3+1)D Weyl semimetals. In the (2+1)D Dirac semimetal, we consider three types of disorder, including random scalar potential, random vector potential and random mass potential, while the random mass disorder is absent in the (3+1)D Weyl semimetal. Via a systematic renormalization group analysis, we find that any type of weak random disorder is irrelevant due to the couplings between the disorder potential and the Yukawa vertex. The emergent supersymmetry is thus stable for weak random potentials. Our work will pave the way for exploration supersymmetry in realistic condensed matter systems.