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We present a mathematical theory of metastable pure states in closed many-body quantum systems with finite-dimensional Hilbert space. Given a Hamiltonian, a pure state is defined to be metastable when all sufficiently local operators either stabilize the state, or raise its average energy. We prove that short-range entangled metastable states are necessarily eigenstates (scars) of a perturbatively close Hamiltonian. Given any metastable eigenstate of a Hamiltonian, in the presence of perturbations, we prove the presence of prethermal behavior: local correlation functions decay at a rate bounded by a time scale nonperturbatively long in the inverse metastability radius, rather than Fermi's Golden Rule. Inspired by this general theory, we prove that the lifetime of the false vacuum in certain $d$-dimensional quantum models grows at least as fast as $\exp(\epsilon^{-d} \log^{-3} \epsilon^{-1} )$, where $\epsilon\rightarrow 0$ is the relative energy density of the false vacuum; up to logarithms, this lower bound matches, for the first time, explicit calculations using quantum field theory. We identify metastable states at finite energy density in the PXP model, along with exponentially many metastable states in "helical" spin chains and the two-dimensional Ising model. Our inherently quantum formalism reveals precise connections between many problems, including prethermalization, robust quantum scars, and quantum nucleation theory, and applies to systems without known semiclassical and/or field theoretic limits., Comment: 26+34 pages; 6+3 figures

We propose a class of metrological resource states whose quantum Fisher information scales optimally in both system size and noise rate. In these states, qubits are partitioned into sensing groups with relatively large correlations within a group but small correlations between groups. The states are obtainable from local Hamiltonian evolution, and we design a metrologically optimal and efficient measurement protocol utilizing time-reversed dynamics and single-qubit local measurements. Using quantum domino dynamics, we also present a protocol free of the time-reversal step that has an estimation error roughly twice the best possible value., Comment: 7 + 17 pages, 3 pages

We provide a polynomial-time classical algorithm for noisy quantum circuits. The algorithm computes the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble (e.g. the computational basis). Our approach is based upon the intuition that noise exponentially damps non-local correlations relative to local correlations. This enables one to classically simulate a noisy quantum circuit by only keeping track of the dynamics of local quantum information. Our algorithm also enables sampling from the output distribution of a circuit in quasi-polynomial time, so long as the distribution anti-concentrates. A number of practical implications are discussed, including a fundamental limit on the efficacy of noise mitigation strategies: any quantum circuit for which error mitigation is efficient must be classically simulable., Comment: 11 pages, 3 figures + 22 page Supplementary Information

The $N$-qubit Greenberger-Horne-Zeilinger (GHZ) state is an important resource for quantum technologies. We consider the task of GHZ encoding using all-to-all interactions, which prepares the GHZ state in a special case, and is furthermore useful for quantum error correction and enhancing the rate of qubit interactions. The naive protocol based on parallelizing CNOT gates takes $\mathrm{O}(1)$-time of Hamiltonian evolution. In this work, we propose a fast protocol that achieves GHZ encoding with high accuracy. The evolution time $\mathrm{O}(\log^2N/N)$ almost saturates the theoretical limit $\Omega(\log N/N)$. Moreover, the final state is close to the ideal encoded one with high fidelity $> 1-10^{-3}$, up to large system sizes $N\lesssim 2000$. The protocol only requires a few stages of time-independent Hamiltonian evolution; the key idea is to use the data qubit as control, and to use fast spin-squeezing dynamics generated by e.g. two-axis-twisting., Comment: 6+3 pages, 3+2 figures. Comments are welcome

We prove the existence of extensive many-body Hamiltonians with few-body interactions and a many-body mobility edge: all eigenstates below a nonzero energy density are localized in an exponentially small fraction of "energetically allowed configurations" within Hilbert space. Our construction is based on quantum perturbations to a classical low-density parity check code. In principle, it is possible to detect this eigenstate localization by measuring few-body correlation functions in efficiently preparable mixed states., Comment: 5+8 pages, 1+0 figures

We bound the time necessary to implement the shift unitary on a one-dimensional ring, both using local Hamiltonians and those with power-law interactions. This time is constrained by the Frobenius light cone; hence we prove that shift unitaries are parametrically harder to implement than preparing long-range Bell pairs for sufficiently small power-law exponent. We note an intriguing similarity between the proof of our results, and the hardness of preparing symmetry-protected topological states with symmetry-preserving Hamiltonians., Comment: 17 pages, 1 figure

Bilevel optimization has gained significant attention in recent years due to its broad applications in machine learning. This paper focuses on bilevel optimization in decentralized networks and proposes a novel single-loop algorithm for solving decentralized bilevel optimization with a strongly convex lower-level problem. Our approach is a fully single-loop method that approximates the hypergradient using only two matrix-vector multiplications per iteration. Importantly, our algorithm does not require any gradient heterogeneity assumption, distinguishing it from existing methods for decentralized bilevel optimization and federated bilevel optimization. Our analysis demonstrates that the proposed algorithm achieves the best-known convergence rate for bilevel optimization algorithms. We also present experimental results on hyperparameter optimization problems using both synthetic and MNIST datasets, which demonstrate the efficiency of our proposed algorithm.

We discuss the generic slowing down of quantum dynamics in low energy density states of spatially local Hamiltonians. Beginning with quantum walks of a single particle, we prove that for certain classes of Hamiltonians (deformations of lattice-regularized $H\propto p^{2k}$), the ``butterfly velocity" of particle motion at low energies has an upper bound that must scale as $E^{(2k-1)/2k}$, as expected from dimensional analysis. We generalize these results to obtain bounds on the typical velocities of particles in many-body systems with repulsive interactions, where for certain families of Hubbard-like models we obtain similar scaling., Comment: 12 pages, 0 figures

Large language models (LLMs) have demonstrated remarkable performance on a variety of natural language tasks based on just a few examples of natural language instructions, reducing the need for extensive feature engineering. However, most powerful LLMs are closed-source or limited in their capability for languages other than English. In this technical report, we present Baichuan 2, a series of large-scale multilingual language models containing 7 billion and 13 billion parameters, trained from scratch, on 2.6 trillion tokens. Baichuan 2 matches or outperforms other open-source models of similar size on public benchmarks like MMLU, CMMLU, GSM8K, and HumanEval. Furthermore, Baichuan 2 excels in vertical domains such as medicine and law. We will release all pre-training model checkpoints to benefit the research community in better understanding the training dynamics of Baichuan 2., Comment: Baichuan 2 technical report. Github: https://github.com/baichuan-inc/Baichuan2

We show that it is possible to perform Heisenberg-limited metrology on GHZ-like states, in the presence of generic spatially local, possibly strong interactions during the measurement process. An explicit protocol, which relies on single-qubit measurements and feedback based on polynomial-time classical computation, achieves the Heisenberg limit. In one dimension, matrix product state methods can be used to perform this classical calculation, while in higher dimensions the cluster expansion underlies the efficient calculations. The latter approach is based on an efficient classical sampling algorithm for short-time quantum dynamics, which may be of independent interest., Comment: 30+7 pages, 3+0 figures. Presented at the 27th Annual Conference on Quantum Information Processing (QIP 2024) as a talk. Accepted for publication in Quantum

The computational complexity of simulating quantum many-body systems generally scales exponentially with the number of particles. This enormous computational cost prohibits first principles simulations of many important problems throughout science, ranging from simulating quantum chemistry to discovering the thermodynamic phase diagram of quantum materials or high-density neutron stars. We present a classical algorithm that samples from a high-temperature quantum Gibbs state in a computational (product state) basis. The runtime grows polynomially with the number of particles, while error vanishes polynomially. This algorithm provides an alternative strategy to existing quantum Monte Carlo methods for overcoming the sign problem. Our result implies that measurement-based quantum computation on a Gibbs state can provide exponential speed up only at sufficiently low temperature, and further constrains what tasks can be exponentially faster on quantum computers., Comment: 4+4 pages; 0+1 figure

We review the mathematical speed limits on quantum information processing in many-body systems. After the proof of the Lieb-Robinson Theorem in 1972, the past two decades have seen substantial developments in its application to other questions, such as the simulatability of quantum systems on classical or quantum computers, the generation of entanglement, and even the properties of ground states of gapped systems. Moreover, Lieb-Robinson bounds have been extended in non-trivial ways, to demonstrate speed limits in systems with power-law interactions or interacting bosons, and even to prove notions of locality that arise in cartoon models for quantum gravity with all-to-all interactions. We overview the progress which has occurred, highlight the most promising results and techniques, and discuss some central outstanding questions which remain open. To help bring newcomers to the field up to speed, we provide self-contained proofs of the field's most essential results., Comment: review article. 93 pages, 10 figures, 1 table. v2: minor changes

Quantum entanglement is a particularly useful characterization of topological orders which lack conventional order parameters. In this work, we study the entanglement in topologically ordered states between two arbitrary spatial regions, using two distinct mixed-state entanglement measures: the so-called "computable cross-norm or realignment" (CCNR) negativity, and the more well-known partial-transpose (PT) negativity. We first generally compute the entanglement measures: We obtain general expressions both in (2+1)D Chern-Simons field theories under certain simplifying conditions, and in the Pauli stabilizer formalism that applies to lattice models in all dimensions. While the field-theoretic results are expected to be topological and universal, the lattice results contain nontopological/nonuniversal terms as well. This raises the important problem of continuum-lattice comparison which is crucial for practical applications. When the two spatial regions and the remaining subsystem do not have triple intersection, we solve the problem by proposing a general strategy for extracting the topological and universal terms in both entanglement measures. Examples in the (2+1)D $Z_2$ toric code model are also presented. In the presence of trisection points, however, our result suggests that the subleading piece in the PT negativity is not topological and depends on the local geometry of the trisections, which is in harmonics with a technical subtlety in the field-theoretic calculation., Comment: Published version, 22 pages, 9 captioned figures

We calculate the amount of entanglement shared by two intervals in the ground state of a (1+1)-dimensional conformal field theory (CFT), quantified by an entanglement measure $\mathcal{E}$ based on the computable cross norm (CCNR) criterion. Unlike negativity or mutual information, we show that $\mathcal{E}$ has a universal expression even for two disjoint intervals, which depends only on the geometry, the central charge c, and the thermal partition function of the CFT. We prove this universal expression in the replica approach, where the Riemann surface for calculating $\mathcal{E}$ at each order n is always a torus topologically. By analytic continuation, result of n=1/2 gives the value of $\mathcal{E}$. Furthermore, the results of other values of n also yield meaningful conclusions: The n=1 result gives a general formula for the two-interval purity, which enables us to calculate the Renyi-2 N-partite information for N<=4 intervals; while the $n=\infty$ result bounds the correlation function of the two intervals. We verify our findings numerically in the spin-1/2 XXZ chain, whose ground state is described by the Luttinger liquid., Comment: 5+2 pages, 3+2 figures; published version

We calculate the quantum Lyapunov exponent $\lambda_L$ and butterfly velocity $v_B$ in the dilute Bose gas at temperature $T$ deep in the Bose-Einstein condensation phase. The generalized Boltzmann equation approach is used for calculating out-of-time ordered correlators, from which $\lambda_L$ and $v_B$ are extracted. At very low temperature where elementary excitations are phonon-like, we find $\lambda_L\propto T^5$ and $v_B\sim c$, the sound velocity. At relatively high temperature, we have $\lambda_L\propto T$ and $v_B\sim c(T/T_*)^{0.23}$. We find $\lambda_L$ is always comparable to the damping rate of a quasiparticle, whose energy depends suitably on $T$. The chaos diffusion constant $D_L=v_B^2/\lambda_L$, on the other hand, differs from the energy diffusion constant $D_E$. We find $D_E\ll D_L$ at very low temperature and $D_E\gg D_L$ otherwise., Comment: 11 pages, 4 figures, published version

We prove that prethermalization is a generic property of gapped local many-body quantum systems, subjected to small perturbations, in any spatial dimension. More precisely, let $H_0$ be a Hamiltonian, spatially local in $d$ spatial dimensions, with a gap $\Delta$ in the many-body spectrum; let $V$ be a spatially local Hamiltonian consisting of a sum of local terms, each of which is bounded by $\epsilon \ll \Delta$. Then, the approximation that quantum dynamics is restricted to the low-energy subspace of $H_0$ is accurate, in the correlation functions of local operators, for stretched exponential time scale $\tau \sim \exp[(\Delta/\epsilon)^a]$ for any $a<1/(2d-1)$. This result does not depend on whether the perturbation closes the gap. It significantly extends previous rigorous results on prethermalization in models where $H_0$ was frustration-free. We infer the robustness of quantum simulation in low-energy subspaces, the existence of athermal ``scarred" correlation functions in gapped systems subject to generic perturbations, the long lifetime of false vacua in symmetry broken systems, and the robustness of quantum information in non-frustration-free gapped phases with topological order., Comment: 5+35 pages; 1+4 figures, 1+0 tables. v2: added some results, improved presentation

The speed of light $c$ sets a strict upper bound on the speed of information transfer in both classical and quantum systems. In nonrelativistic quantum systems, the Lieb-Robinson Theorem imposes an emergent speed limit $v \hspace{-0.2mm} \ll \hspace{-0.2mm} c$, establishing locality under unitary evolution and constraining the time needed to perform useful quantum tasks. We extend the Lieb-Robinson Theorem to quantum dynamics with measurements. In contrast to the expectation that measurements can arbitrarily violate spatial locality, we find at most an $(M \hspace{-0.5mm} +\hspace{-0.5mm} 1)$-fold enhancement to the speed $v$ of quantum information, provided the outcomes of measurements in $M$ local regions are known. This holds even when classical communication is instantaneous, and extends beyond projective measurements to weak measurements and other nonunitary channels. Our bound is asymptotically optimal, and saturated by existing measurement-based protocols. We tightly constrain the resource requirements for quantum computation, error correction, teleportation, and generating entangled resource states (Bell, GHZ, quantum-critical, Dicke, W, and spin-squeezed states) from short-range-entangled initial states. Our results impose limits on the use of measurements and active feedback to speed up quantum information processing, resolve fundamental questions about the nature of measurements in quantum dynamics, and constrain the scalability of a wide range of proposed quantum technologies., Comment: 24 pages main text + 61 pages SM; main text significantly expanded, includes new Lieb-Robinson bounds, additional discussion, more clarity on main results