Caterina Mora, Hans Briegel
We define the algorithmic complexity of a quantum state relative to a given precision parameter, and give upper bounds for various examples of states. We also establish a connection between the entanglement of a quantum state and its algorithmic complexity.
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Algorithmic Design of Heralded Linear Optical Circuits for Multipartite Entanglement
该研究将预示光子电路设计转化为图搜索问题，利用线性量子图缩减搜索空间，自动构建并优化电路，为多种多体纠缠资源态提供高效设计方案。
Entanglement accelerates quantum simulation
本研究探讨了量子纠缠与量子模拟的关系，建立了基于纠缠熵的更紧算法误差上界，并开发了结合测量装置的自适应模拟算法，证明纠缠可加速量子算法。
In this paper we give a definition for the Kolmogorov complexity of a pure quantum state. In classical information theory the algorithmic complexity of a string is a measure of the information needed by a universal machine to reproduce the string itself. We define the complexity of a quantum state by means of the classical description complexity of an (abstract) experimental procedure that allows us to prepare the state with a given fidelity. We argue that our definition satisfies the intuitive idea of complexity as a measure of ``how difficult'' it is to prepare a state. We apply this definit…
We present a simple communication complexity problem where three parties benefit from sharing bound entanglement. This demonstrates that entanglement distillability of the shared state is not necessary in order to surpass classical communication complexity.
We extend algorithmic information theory to quantum mechanics, taking a universal semicomputable density matrix (``universal probability'') as a starting point, and define complexity (an operator) as its negative logarithm. A number of properties of Kolmogorov complexity extend naturally to the new domain. Approximately, a quantum state is simple if it is within a small distance from a low-dimensional subspace of low Kolmogorov complexity. The von Neumann entropy of a computable density matrix is within an additive constant from the average complexity. Some of the theory of randomness translat…
Entanglement is the defining characteristic of quantum mechanics. Bipartite entanglement is characterized by the von Neumann entropy. Entanglement is not just described by a number, however; it is also characterized by its level of complexity. The complexity of entanglement is at the root of the onset of quantum chaos, universal distribution of entanglement spectrum statistics, hardness of a disentangling algorithm and of the quantum machine learning of an unknown random circuit, and universal temporal entanglement fluctuations. In this paper, we numerically show how a crossover from a simple …
We consider the problem of remote state preparation recently studied in several papers. We study the communication complexity of this problem, in the presence of entanglement and in the scenario of single use of the channel.
We prove a general lower bound on the bounded-error entanglement-assisted quantum communication complexity of Boolean functions. The bound is based on the concept that any classical or quantum protocol to evaluate a function on distributed inputs can be turned into a quantum communication protocol. As an application of this bound, we give a very simple proof of the statement that almost all Boolean functions on n+n bits have linear communication complexity, even in the presence of unlimited entanglement.
A long sequence of tosses of a classical coin produces an apparently random bit string, but classical randomness is an illusion: the algorithmic information content of a classically-generated bit string lies almost entirely in the description of initial conditions. This letter presents a simple argument that, by contrast, a sequence of bits produced by tossing a quantum coin is, almost certainly, genuinely (algorithmically) random. This result can be interpreted as a strengthening of Bell's no-hidden-variables theorem, and relies on causality and quantum entanglement in a manner similar to Bel…
Entanglement Scaling and Problem Structure in Quantum Approximate and Adiabatic Optimization Algorithms
研究量子近似优化算法（QAOA）在MaxCut问题中的纠缠标度，发现次优参数训练会改变纠缠轮廓，而高性能优化器下纠缠标度与费米子高斯态一致，并与绝热量子计算对比揭示不同标度行为。
We consider a variation of the multi-party communication complexity scenario where the parties are supplied with an extra resource: particles in an entangled quantum state. We show that, although a prior quantum entanglement cannot be used to simulate a communication channel, it can reduce the communication complexity of functions in some cases. Specifically, we show that, for a particular function among three parties (each of which possesses part of the function's input), a prior quantum entanglement enables them to learn the value of the function with only three bits of communication occurri…
An outstanding problem in quantum computing is the calculation of entanglement, for which no closed-form algorithm exists. Here we solve that problem, and demonstrate the utility of a quantum neural computer, by showing, in simulation, that such a device can be trained to calculate the entanglement of an input state, something neither an algorithmic quantum computer nor a classical neural net can do.
We propose a probabilistic two-party communication complexity scenario with a prior nonmaximally entangled state, which results in less communication than that is required with only classical random correlations. A simple all-optical implementation of this protocol is presented and demonstrates our conclusion.
Quantum Entanglement Phase Transitions and Computational Complexity: Insights from Ising Models
本文构建二维二分簇态并测量体量子比特，发现未测量的一维边界态在特定条件下会发生从体积律到面积律的纠缠相变，并揭示了该相变与Ising配分函数计算复杂性之间的关联。
Thermalization of finite complexity and its application to heat bath algorithmic cooling
本文提出基于碰撞模型的有限复杂度热操作，通过迭代协议实现亚浴温冷却，并通过引入量子比特机器提升冷却效率。
Algorithmic Idealism III: "Algorithmic State" Formulation of Quantum Mechanics
本研究提出算法唯心主义框架，将量子力学重新诠释为计算过程，通过智能体-环境反馈循环建模现实，以信息结构解释核心量子现象，提出新公设并解决相关哲学问题，为基础物理学提供新视角。
Sample Complexity for Embedded Multipartite Entanglement Witness via Pauli and Clifford Classical Shadows
本研究通过经典阴影协议探讨了多部纠缠见证的样本复杂度，发现随着见证的全局性增强，从保罗到克利福德的性能转变。
We discuss the Bennett-Brassard 1984 (BB84) quantum key distribution protocol in the light of quantum algorithmic information. While Shannon's information theory needs a probability to define a notion of information, algorithmic information theory does not need it and can assign a notion of information to an individual object. The program length necessary to describe an object, Kolmogorov complexity, plays the most fundamental role in the theory. In the context of algorithmic information theory, we formulate a security criterion for the quantum key distribution by using the quantum Kolmogorov …
Entanglement complexity of the Rokhsar-Kivelson-sign wavefunctions
本文利用保真度、纠缠谱、稳定子 Rényi 熵和解纠缠算法等工具，刻画 Rokhsar–Kivelson 符号波函数在体积律与次体积律相之间的纠缠复杂性转变。
For technological purposes and theoretical curiosity, it is very interesting to have a building block that produces a considerable amount of entanglement between on-demand sites through a simple control of a few sites. Here, we consider permanently-coupled spin networks and study entanglement generation between qubit pairs to find low-complexity structures capable of generating considerable entanglement between various qubit pairs. We find that in axially symmetric networks the generated entanglement between some qubit pairs is rather larger than generic networks. We show that in uniformly-cou…
We extend the notion of Gacs quantum algorithmic entropy, originally formulated for finitely many qubits, to infinite dimensional quantum spin chains and investigate the relation of this extension with two quantum dynamical entropies that have been proposed in recent years. Further, we prove an extension of Brudno's theorem in quantum spin chains with shift dynamics.
In this article we use the idea of algorithmic complexity (AC) to study various cosmological scenarios, and as a means of quantizing the gravitational interaction. We look at 5D and 7D cosmological models where the Universe begins as a higher dimensional Planck size spacetime which fluctuates between Euclidean and Lorentzian signatures. These fluctuations are governed by the AC of the two different signatures. At some point a transition to a 4D Lorentzian signature Universe occurs, with the extra dimensions becoming ``frozen'' or non-dynamical. We also apply the idea of algorithmic complexity …
Quantum entanglement is an indispensable resource for many significant quantum information processing tasks. However, because of the noise in quantum channels, it is difficult to distribute quantum entanglement over a long distance in practice. A solution for this challenge is the quantum repeater which can extend the distance of entanglement distribution. In this scheme, the time consumption of classical communication and local operations takes an important place in perspective of time efficiency. Motivated by this observation, we exploit the basic quantum repeater scheme in perspectives of n…
We prove a trade-off relation between the entanglement cost and classical communication complexity of causal order structure of a protocol in distributed quantum information processing. We consider an implementation of a class of two-qubit unitary gates by local operations and classical communication (LOCC) assisted with shared entanglement, in an information theoretical scenario of asymptotically many input pairs and vanishingly small error. We prove the trade-off relation by showing that (i) one ebit of entanglement per pair is necessary for implementing the unitary by any two-round protocol…
We present an application of a genetic algorithmic computational method to the optimization of the concurrence measure of entanglement for the cases of one dimensional chains, as well as square and triangular lattices in a simple tight-binding approach in which the hopping of electrons is much stronger than the phonon dissipation
In this research we present a quantum circuit for estimating algorithmic complexity using the coding theorem method. This accelerates inferring algorithmic structure in data for discovering causal generative models. The computation model is restricted in time and space resources to make it computable in approximating the target metrics. The quantum circuit design based on our earlier work that allows executing a superposition of automata is presented. As a use-case, an application framework for protein-protein interaction ontology based on algorithmic complexity is proposed. Using small-scale …
We shed new light on entanglement measures in multipartite quantum systems by taking a computational-complexity approach toward quantifying quantum entanglement with two familiar notions--approximability and distinguishability. Built upon the formal treatment of partial separability, we measure the complexity of an entangled quantum state by determining (i) how hard to approximate it from a fixed classical state and (ii) how hard to distinguish it from all partially separable states. We further consider the Kolmogorovian-style descriptive complexity of approximation and distinction of partial …
This paper employs a powerful argument, called an algorithmic argument, to prove lower bounds of the quantum query complexity of a multiple-block ordered search problem in which, given a block number i, we are to find a location of a target keyword in an ordered list of the i-th block. Apart from much studied polynomial and adversary methods for quantum query complexity lower bounds, our argument shows that the multiple-block ordered search needs a large number of nonadaptive oracle queries on a black-box model of quantum computation that is also supplemented with advice. Our argument is also …
Edge of entanglement in non-ergodic states: a complexity parameter formulation
本文用复矩阵多参数 Gaussian 系综刻画二分基下的非遍历纯态，提出复杂度参数并揭示纠缠熵随子系统尺寸的普适标度与临界区域。
We propose a class of generalizations of the geometric entanglement for pure states by exploiting the matrix product state formalism. This generalization is completely divested from the notion of separability and can be freely tuned as a function of the bond dimension to target states which vary in entanglement complexity. We first demonstrate its value in a toy spin-1 model where, unlike the conventional geometric entanglement, it successfully identifies the AKLT ground state. We then investigate the phase diagram of a Haldane chain with uniaxial and rhombic anisotropies, revealing that the g…
Recently the theory of communication developed by Shannon has been extended to the quantum realm by exploiting the rules of quantum theory. This latter stems on complex vector spaces. However complex (as well as real) numbers are just idealizations and they are not available in practice where we can only deal with rational numbers. This fact naturally leads to the question of whether the developed notions of capacities for quantum channels truly catch their ability to transmit information. Here we answer this question for the quantum capacity. To this end we resort to the notion of semi-comput…
Sharp complexity phase transitions generated by entanglement
该研究定量关联了量子系统中的纠缠与模拟计算复杂性，证明了在 $k$--正则图态的单量子比特测量中，随正则性参数变化，纠缠与复杂性在 $k=3$ 和 $k=n-3$ 处发生尖锐相变，并揭示了模拟复杂性的对偶性。
In recent years, the entanglement spectra of quantum states have been identified to be highly valuable for improving our understanding on many problems in quantum physics, such as classification of topological phases, symmetry-breaking phases, and eigenstate thermalization, etc. However, it remains a major challenge to fully characterize the entanglement spectrum of a given quantum state. An outstanding problem is whether the difficulty is intrinsically technical or fundamental? Here using the tools in computational complexity, we perform a rigorous analysis to pin down the counting complexity…
Maximum Separation of Quantum Communication Complexity With and Without Shared Entanglement
研究提出一类输入规模为n的问题，其在有纠缠辅助的量子通信模型中无需通信即可解决，但在无先共享纠缠的双向量子通信模型中需要Ω(n)的量子比特通信。该结果展示了量子通信复杂度在有无共享纠缠时的最大分离，并首次给出了无共享纠缠情况下的下界。
Does gravity constrain computation? We study this question using the AdS/CFT correspondence, where computation in the presence of gravity can be related to non-gravitational physics in the boundary theory. In AdS/CFT, computations which happen locally in the bulk are implemented in a particular non-local form in the boundary, which in general requires distributed entanglement. In more detail, we recall that for a large class of bulk subregions the area of a surface called the ridge is equal to the mutual information available in the boundary to perform the computation non-locally. We then argu…
We study the dynamical generation of entanglement for a two-body interacting system, starting from a separable coherent state. We show analytically that in the quasiclassical regime the entanglement growth rate can be simply computed by means of the underlying classical dynamics. Furthermore, this rate is given by the Kolmogorov-Sinai entropy, which characterizes dynamical complexity of classical motion. Our results, illustrated by numerical simulations on a model of coupled rotators, establish in the quasiclassical regime a link between the generation of entanglement, a purely quantum phenome…
The Cost of Nonlocality: A Dynamical Performance Equation of Energy-Entanglement-Complexity
本研究通过统一量子速度极限与Lieb-Robinson界，建立能量-纠缠性能方程，揭示局域相互作用系统中非局域纠缠生成的物理代价与性能权衡，并提供性能瓶颈诊断工具。
The complexity of entanglement embezzlement
本研究通过通用盗用族研究纠缠盗用的电路复杂性，建立通用下界，证明电路复杂性是完美盗用的物理障碍，并推导出一维临界系统的指数增长下界。
We investigate the evolution of complexity and entanglement following a quench in a one-dimensional topological system, namely the Su-Schrieffer-Heeger model. We demonstrate that complexity can detect quantum phase transitions and shows signatures of revivals; this observation provides a practical advantage in information processing. We also show that the complexity saturates much faster than the entanglement entropy in this system, and we provide a physical argument for this. Finally, we demonstrate that complexity is a less sensitive probe of topological order, compared with measures of enta…
Decoupling Spectral Gaps from Entanglement Complexity in Quantum Annealing
本研究证明量子退火中谱隙标度与纠缠复杂性可解耦，通过反转传统相变中的配对关系，指出纠缠复杂性不足以衡量退火难度，需探索捕捉希尔伯特空间几何的量子资源。
We consider free fermion systems in arbitrary dimensions and represent the occupation pattern of each eigenstate as a classical binary string. We find that the Kolmogorov complexity of the string correctly captures the scaling behavior of its entanglement entropy (EE). In particular, the logarithmically-enhanced area law for EE in the ground state and the volume law for EE in typical highly excited states are reproduced. Since our approach does not require bipartitioning the system, it allows us to distinguish typical and atypical eigenstates directly by their intrinsic complexity. We reveal t…
It is often claimed that one cannot locate a notion of causation in fundamental physical theories. The reason most commonly given is that the dynamics of those theories do not support any distinction between the past and the future, and this vitiates any attempt to locate a notion of causal asymmetry -- and thus of causation -- in fundamental physical theories. I argue that this is incorrect: the ubiquitous generation of entanglement between quantum systems grounds a relevant asymmetry in the dynamical evolution of quantum systems. I show that by exploiting a connection between the amount of e…
We define a new criterion for selecting a specific minimal entanglement purification of given mixed states in generic quantum states using the entanglement of purification. We then propose that its holographic dual is the state living on the boundary of the entanglement wedge in the surface-state correspondence. Finally, we make some remarks about the relationship between this and the complexity equals action proposals.
Interplay of entanglement structures and stabilizer entropy in spin models
本文研究了自旋模型中纠缠结构与非稳定子性之间的关系，通过分析纠缠谱和稳定子熵，揭示了它们作为量子相变标志的作用。
Krylov complexity, mode-resolved complexity and entanglement entropy across phase transitions in the non-Hermitian extended Su-Schrieffer-Heeger model
研究含次近邻跳跃和虚交错化学势的扩展SSH模型相变，用Krylov扩展复杂度和纠缠熵分析两种动力学协议，表明其可分别有效表征相变，还引入动量分辨复杂度揭示转变细节。
Scrambling of Entanglement from Integrability to Chaos: Bootstrapped Time-Integrated Spread Complexity
通过时间积分量建立扩散复杂度与量子动力学保真度的新解析关系，并利用自举Rosenzweig-Porter系综展示积分扩散复杂度在不同遍历区域对纠缠加扰行为的精细分辨能力。
We introduce "binding complexity", a new notion of circuit complexity which quantifies the difficulty of distributing entanglement among multiple parties, each consisting of many local degrees of freedom. We define binding complexity of a given state as the minimal number of quantum gates that must act between parties to prepare it. To illustrate the new notion we compute it in a toy model for a scalar field theory, using certain multiparty entangled states which are analogous to configurations that are known in AdS/CFT to correspond to multiboundary wormholes. Pursuing this analogy, we show t…
We study the universality of scaling of entanglement in Shor's factoring algorithm and in adiabatic quantum algorithms across a quantum phase transition for both the NP-complete Exact Cover problem as well as the Grover's problem. The analytic result for Shor's algorithm shows a linear scaling of the entropy in terms of the number of qubits, therefore difficulting the possibility of an efficient classical simulation protocol. A similar result is obtained numerically for the quantum adiabatic evolution Exact Cover algorithm, which also shows universality of the quantum phase transition the syst…
Random Clifford circuits doped with non Clifford gates exhibit transitions to universal entanglement spectrum statistics[1] and quantum chaotic behavior. In [2] we proved that the injection of $O(n)$ non Clifford gates into a $n$-qubit Clifford circuit drives the transition towards the universal value of the purity fluctuations. In this paper, we show that doping a Clifford circuit with $O(n)$ single qubit non Clifford measurements is both necessary and sufficient to drive the transition to universal fluctuations of the purity.
Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that "hides" initially localized quantum information. In this work, we lay the mathematical foundations of studying randomness complexities beyond scrambling by entanglement properties. We do so by analyzing the generalized (in particular Rényi) entanglement entropies of designs, i.e. ensembles of unitary channels or pure states that mimic the uniformly random distribution (given by the Haar measure) up to certain moments. A main collective conclusion is that the Rényi entang…
In this article, we investigate the quantum circuit complexity and entanglement entropy in the recently studied black hole gas framework using the two-mode squeezed states formalism written in arbitrary dimensional spatially flat cosmological Friedmann-Lema$\hat{i}$tre-Robertson-Walker (FLRW) background space-time. We compute the various complexity measures and study the evolution of these complexities by following two different prescriptions viz. Covariant matrix method and Nielsen's method. Independently, using the two-mode squeezed states formalism we also compute the Rényi and Von-Neumann …
Entanglement Complexity in Many-body Systems from Positivity Scaling Laws
本文基于正性缩放定律提出多体系统纠缠复杂度的计算框架，证明了在特定条件下纠缠复杂度与阶数p呈多项式关系，为验证多体方法的计算效率提供了理论依据。
In this work, we make a connection between two seemingly different problems. The first problem involves characterizing the properties of entanglement in the ground state of gapped local Hamiltonians, which is a central topic in quantum many-body physics. The second problem is on the quantum communication complexity of testing bipartite states with EPR assistance, a well-known question in quantum information theory. We construct a communication protocol for testing (or measuring) the ground state and use its communication complexity to reveal a new structural property for the ground state entan…
Entanglement is usually quantified by von Neumann entropy, but its properties are much more complex than what can be expressed with a single number. We show that the three distinct dynamical phases known as thermalization, Anderson localization, and many-body localization are marked by different patterns of the spectrum of the reduced density matrix for a state evolved after a quantum quench. While the entanglement spectrum displays Poisson statistics for the case of Anderson localization, it displays universal Wigner-Dyson statistics for both the cases of many-body localization and thermaliza…
Motivated by holographic complexity proposals as novel probes of black hole spacetimes, we explore circuit complexity for thermofield double (TFD) states in free scalar quantum field theories using the Nielsen approach. For TFD states at t = 0, we show that the complexity of formation is proportional to the thermodynamic entropy, in qualitative agreement with holographic complexity proposals. For TFD states at t > 0, we demonstrate that the complexity evolves in time and saturates after a time of the order of the inverse temperature. The latter feature, which is in contrast with the results of…
Decoherence, Entanglement Negativity and Circuit Complexity for Open Quantum System
本文研究两类开放量子系统，对比复杂度、线性熵和纠缠负度的饱和时间，发现完全混合态下两类复杂度均饱和且饱和时间短于线性熵，Caldeira Leggett模型中线性熵与纠缠负度饱和时间同量级。
Operator entanglement growth quantifies complexity of cellular automata
本文提出用矩阵乘积算子奇异值谱的熵增长（算子纠缠熵）作为定量度量，对经典确定性元胞自动机的复杂性进行分类和动力学行为刻画。
This study clarifies the proper criteria to assess the modeling capacity of a general tensor model. The work analyze the problem based on the study of tensor ranks, which is not a well-defined quantity for higher order tensors. To process, the author introduces the separability issue to discuss the Cannikin's law of tensor modeling. Interestingly, a connection between entanglement studied in information theory and tensor analysis is established, shedding new light on the theoretical understanding for modeling capacity problems.
Stabilizer Entropy and entanglement complexity in the Sachdev-Ye-Kitaev model
研究Sachdev-Ye-Kitaev模型中退相干熵与纠缠复杂性的相互作用，揭示SYK4的复杂性与SYK2的非普遍性。
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