We show that a bipartite Gaussian quantum system interacting with an external Gaussian environment may possess a unique Gaussian entangled stationary state and that any initial state converges towards this stationary state. We discuss dependence of entanglement on temperature and interaction strength and show that one can find entangled stationary states only for low temperatures and weak interactions.
We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our method employs an iterable cost function incorporating a linear combination of overlap history states. The proposed algorithm is not only efficient in time complexity, but has lower memory costs compared to classical methods. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation cost scales economically with the number of time steps. As a proof of concept, we apply our algorithm to solve a range of fractional partial differential equations commonly encountered in engineering applications, such as the sub-diffusion equation, the non-linear Burgers' equation and a coupled diffusive epidemic model. We assess quantum hardware performance under realistic noise conditions, further validating the practical utility of our algorithm.
In recent years, the neural-network quantum states method has been investigated to study the ground state and the time evolution of many-body quantum systems. Here we expand on the investigation and consider a quantum quench from the paramagnetic to the anti-ferromagnetic phase in the tilted Ising model. We use two types of neural networks, a restricted Boltzmann machine and a feed-forward neural network. We show that for both types of networks, the projected time-dependent variational Monte Carlo (p-tVMC) method performs better than the non-projected approach. We further demonstrate that one can use K-FAC or minSR in conjunction with p-tVMC to reduce the computational complexity of the stochastic reconfiguration approach, thus allowing the use of these techniques for neural networks with more parameters.
The eigenstate thermalization hypothesis (ETH) describes the properties of diagonal and off-diagonal matrix elements of local operators in the eigenenergy basis. In this work, we propose a relation between (i) the singular behaviour of the off-diagonal part of ETH at small energy differences, and (ii) the smooth profile of the diagonal part of ETH as a function of the energy density. We establish this connection from the decay of the autocorrelation functions of local operators, which is constrained by the presence of local conserved quantities whose evolution is described by hydrodynamics. We corroborate our predictions with numerical simulations of two non-integrable spin-1 Ising models, one diffusive and one super-diffusive, which we perform using dynamical quantum typicality up to 18 spins.
Gaussian quantum Markov semigroups are the natural non-commutative extension of classical Ornstein-Uhlenbeck semigroups. They arise in open quantum systems of bosons where canonical non-commuting random variables of positions and momenta come into play. If there exits a faithful invariant density we explicitly compute the optimal exponential convergence rate, namely the spectral gap of the generator, in non-commutative $L^2$ spaces determined by the invariant density showing that the exact value is the lowest eigenvalue of a certain matrix determined by the diffusion and drift matrices. The spectral gap turns out to depend on the non-commutative $L^2$ space considered, whether the one determined by the so-called GNS or KMS multiplication by the square root of the invariant density. In the first case, it is strictly positive if and only if there is the maximum number of linearly independent noises. While, we exhibit explicit examples in which it is strictly positive only with KMS multiplication. We do not assume any symmetry or quantum detailed balance condition with respect to the invariant density.
Algorithms developed to solve many-body quantum problems, like tensor networks, can turn into powerful quantum-inspired tools to tackle problems in the classical domain. In this work, we focus on matrix product operators, a prominent numerical technique to study many-body quantum systems, especially in one dimension. It has been previously shown that such a tool can be used for classification, learning of deterministic sequence-to-sequence processes and of generic quantum processes. We further develop a matrix product operator algorithm to learn probabilistic sequence-to-sequence processes and apply this algorithm to probabilistic cellular automata. This new approach can accurately learn probabilistic cellular automata processes in different conditions, even when the process is a probabilistic mixture of different chaotic rules. In addition, we find that the ability to learn these dynamics is a function of the bit-wise difference between the rules and whether one is much more likely than the other.
Xinfang Zhang, Zhihao Wu, Gregory A. L. White, Zhongcheng Xiang, Shun Hu, Zhihui Peng, Yong Liu, Dongning Zheng, Xiang Fu, Anqi Huang, Dario Poletti, Kavan Modi, Junjie Wu, Mingtang Deng, Chu Guo The development of fault-tolerant quantum processors relies on the ability to control noise. A particularly insidious form of noise is temporally correlated or non-Markovian noise. By combining randomized benchmarking with supervised machine learning algorithms, we develop a method to learn the details of temporally correlated noise. In particular, we can learn the time-independent evolution operator of system plus bath and this leads to (i) the ability to characterize the degree of non-Markovianity of the dynamics and (ii) the ability to predict the dynamics of the system even beyond the times we have used to train our model. We exemplify this by implementing our method on a superconducting quantum processor. Our experimental results show a drastic change between the Markovian and non-Markovian regimes for the learning accuracies.
We consider a class of games between two competing players that take turns acting on the same many-body quantum register. Each player can perform unitary operations on the register, and after each one of them acts on the register the energy is measured. Player A aims to maximize the energy while player B to minimize it. This class of zero-sum games has a clear second mover advantage if both players can entangle the same portion of the register. We show, however, that if the first player can entangle a larger number of qubits than the second player (which we refer to as having quantum advantage), then the second mover advantage can be significantly reduced. We study the game for different types of quantum advantage of player A versus player B and for different sizes of the register, in particular, scenarios in which absolutely maximally entangled states cannot be achieved. In this case, we also study the effectiveness of using random unitaries. Last, we consider mixed initial preparations of the register, in which case the player with a quantum advantage can rely on strategies stemming from the theory of ergotropy of quantum batteries.
Interspersing unitary dynamics with local measurements results in measurement-induced phases and transitions in many-body quantum systems. When the evolution is driven by a local Hamiltonian, two types of transitions have been observed, characterized by an abrupt change in the system size scaling of entanglement entropy. The critical point separates the strongly monitored area-law phase from a volume law or a sub-extensive, typically logarithmic-like one at low measurement rates. Identifying the key ingredients responsible for the entanglement scaling in the weakly monitored phase is the key purpose of this work. For this purpose, we consider prototypical one-dimensional spin chains with local monitoring featuring the presence/absence of U(1) symmetry, integrability, and interactions. Using exact numerical methods, the system sizes studied reveal that the presence of interaction is always correlated to a volume-law weakly monitored phase. In contrast, non-interacting systems present sub-extensive scaling of entanglement. Other characteristics, namely integrability or U(1) symmetry, do not play a role in the character of the entanglement phase.
Digital quantum computers have the potential to study the dynamics of complex quantum systems. Nonequilibrium open quantum systems are, however, less straightforward to be implemented. Here we consider a collisional model representation of the nonequilibrium open dynamics for a boundary-driven XXZ spin chain, with a particular focus on its steady states. More specifically, we study the interplay between the accuracy of the result versus the depth of the circuit by comparing the results generated by the corresponding master equations. We study the simulation of a boundary-driven spin chain in regimes of weak and strong interactions, which would lead in large systems to diffusive and ballistic dynamics, considering also possible errors in the implementation of the protocol. Last, we analyze the effectiveness of digital simulation via the collisional model of current rectification when the XXZ spin chains are subject to non-uniform magnetic fields.
Recently there has been an intense effort to understand measurement induced transitions, but we still lack a good understanding of non-Markovian effects on these phenomena. To that end, we consider two coupled chains of free fermions, one acting as the system of interest, and one as a bath. The bath chain is subject to Markovian measurements, resulting in an effective non-Markovian dissipative dynamics acting on the system chain which is still amenable to numerical studies in terms of quantum trajectories. Within this setting, we study the entanglement within the system chain, and use it to characterize the phase diagram depending on the ladder hopping parameters and on the measurement probability. For the case of pure state evolution, the system is in an area law phase when the internal hopping of the bath chain is small, while a non-area law phase appears when the dynamics of the bath is fast. The non-area law exhibits a logarithmic scaling of the entropy compatible with a conformal phase, but also displays linear corrections for the finite system sizes we can study. For the case of mixed state evolution, we instead observe regions with both area, and non-area scaling of the entanglement negativity. We quantify the non-Markovianity of the system chain dynamics and find that for the regimes of parameters we study, a stronger non-Markovianity is associated to a larger entanglement within the system.
Different neural network architectures can be unsupervisedly or supervisedly trained to represent quantum states. We explore and compare different strategies for the supervised training of feed forward neural network quantum states. We empirically and comparatively evaluate the performance of feed forward neural network quantum states in different phases of matter for variants of the architecture, for different hyper-parameters, and for two different loss functions, to which we refer as \emphmean-squared error and \emphoverlap, respectively. We consider the next-nearest neighbor Ising model for the diversity of its phases and focus on its paramagnetic, ferromagnetic, and pair-antiferromagnetic phases. We observe that the overlap loss function allows better training of the model across all phases, provided a rescaling of the neural network.
We study the Hamiltonian dynamics of a many-body quantum system subjected to periodic projective measurements which leads to probabilistic cellular automata dynamics. Given a sequence of measured values, we characterize their dynamics by performing a principal component analysis. The number of principal components required for an almost complete description of the system, which is a measure of complexity we refer to as PCA complexity, is studied as a function of the Hamiltonian parameters and measurement intervals. We consider different Hamiltonians that describe interacting, non-interacting, integrable, and non-integrable systems, including random local Hamiltonians and translational invariant random local Hamiltonians. In all these scenarios, we find that the PCA complexity grows rapidly in time before approaching a plateau. The dynamics of the PCA complexity can vary quantitatively and qualitatively as a function of the Hamiltonian parameters and measurement protocol. Importantly, the dynamics of PCA complexity present behavior that is considerably less sensitive to the specific system parameters for models which lack simple local dynamics, as is often the case in non-integrable models. In particular, we point out a figure of merit that considers the local dynamics and the measurement direction to predict the sensitivity of the PCA complexity dynamics to the system parameters.
We study the quantum dynamics of a many-body system subject to coherent evolution and coupled to a non-Markovian bath. We propose a technique to unravel the non-Markovian dynamics in terms of quantum jumps, a connection that was so far only understood for single-body systems. We develop a systematic method to calculate the probability of a quantum trajectory, and formulate it in a diagrammatic structure. We find that non-Markovianity renormalizes the probability of realizing a quantum trajectory, and that memory effects can be interpreted as a perturbation on top of the Markovian dynamics. We show that the diagrammatic structure is akin to that of a Dyson equation, and that the probability of the trajectories can be calculated analytically. We then apply our results to study the measurement-induced entanglement transition in random unitary circuits. We find that non-Markovianity does not significantly shift the transition, but stabilizes the volume law phase of the entanglement by shielding it from transient strong dissipation.
Belief propagation is a well-studied algorithm for approximating local marginals of multivariate probability distribution over complex networks, while tensor network states are powerful tools for quantum and classical many-body problems. Building on a recent connection between the belief propagation algorithm and the problem of tensor network contraction, we propose a block belief propagation algorithm for contracting two-dimensional tensor networks and approximating the ground state of $2D$ systems. The advantages of our method are three-fold: 1) the same algorithm works for both finite and infinite systems; 2) it allows natural and efficient parallelization; 3) given its flexibility it would allow to deal with different unit cells. As applications, we use our algorithm to study the $2D$ Heisenberg and transverse Ising models, and show that the accuracy of the method is on par with state-of-the-art results.
The restricted Boltzmann machine (RBM) has been successfully applied to solve the many-electron Schr$\ddot{\text{o}}$dinger equation. In this work we propose a single-layer fully connected neural network adapted from RBM and apply it to study ab initio quantum chemistry problems. Our contribution is two-fold: 1) our neural network only uses real numbers to represent the real electronic wave function, while we obtain comparable precision to RBM for various prototypical molecules; 2) we show that the knowledge of the Hartree-Fock reference state can be used to systematically accelerate the convergence of the variational Monte Carlo algorithm as well as to increase the precision of the final energy.
OTOC has been used to characterize the information scrambling in quantum systems. Recent studies showed that local conserved quantities play a crucial role in governing the relaxation dynamics of OTOC in non-integrable systems. In particular, slow scrambling of OTOC is seen for observables that has an overlap with local conserved quantities. However, an observable may not overlap with the Hamiltonian, but with the Hamiltonian elevated to an exponent larger than one. Here, we show that higher exponents correspond to faster relaxation, although still algebraic, and with exponents that can increase indefinitely. Our analytical results are supported by numerical experiments.
Out-of-time ordered correlators (OTOCs) help characterize the scrambling of quantum information and are usually studied in the context of nonintegrable systems. In this work, we compare the relaxation dynamics of OTOCs in interacting integrable and nonintegrable spin-1/2 XYZ chains in regimes without a classical counterpart. In both kinds of chains, using the presence of symmetries such as $U(1)$ and supersymmetry, we consider regimes in which the OTOC operators overlap or not with the Hamiltonian. We show that the relaxation of the OTOCs is slow (fast) when there is (there is not) an overlap, independently of whether the chain is integrable or nonintegrable. When slow, we show that the OTOC dynamics follows closely that of the two-point correlators. We study the dynamics of OTOCs using numerical calculations, and gain analytical insights from the properties of the diagonal and of the off-diagonal matrix elements of the corresponding local operators in the energy eigenbasis.
Digital quantum computers have the potential to simulate complex quantum systems. The spin-boson model is one of such systems, used in disparate physical domains. Importantly, in a number of setups, the spin-boson model is open, i.e. the system is in contact with an external environment which can, for instance, cause the decay of the spin state. Here we study how to simulate such open quantum dynamics in a digital quantum computer, for which we use one of IBM's hardware. We consider in particular how accurate different implementations of the evolution result as a function of the level of noise in the hardware and of the parameters of the open dynamics. For the regimes studied, we show that the key aspect is to simulate the unitary portion of the dynamics, while the dissipative part can lead to a more noise-resistant simulation. We consider both a single spin coupled to a harmonic oscillator, and also two spins coupled to the oscillator. In the latter case, we show that it is possible to simulate the emergence of correlations between the spins via the oscillator.
Dynamical decoupling (DD) refers to a well-established family of methods for error mitigation, comprising pulse sequences aimed at averaging away slowly evolving noise in quantum systems. Here, we revisit the question of its efficacy in the presence of noisy pulses in scenarios important for quantum devices today: pulses with gate control errors, and the computational setting where DD is used to reduce noise in every computational gate. We focus on the well-known schemes of periodic (or universal) DD, and its extension, concatenated DD, for scaling up its power. The qualitative conclusions from our analysis of these two schemes nevertheless apply to other DD approaches. In the presence of noisy pulses, DD does not always mitigate errors. It does so only when the added noise from the imperfect DD pulses do not outweigh the increased ability in averaging away the original background noise. We present breakeven conditions that delineate when DD is useful, and further find that there is a limit in the performance of concatenated DD, specifically in how far one can concatenate the DD pulse sequences before the added noise no longer offers any further benefit in error mitigation.
Neural network quantum states are a promising tool to analyze complex quantum systems given their representative power. It can however be difficult to optimize efficiently and effectively the parameters of this type of ansatz. Here we propose a local optimization procedure which, when integrated with stochastic reconfiguration, outperforms previously used global optimization approaches. Specifically, we analyze both the ground state energy and the correlations for the non-integrable tilted Ising model with restricted Boltzmann machines. We find that sequential local updates can lead to faster convergence to states which have energy and correlations closer to those of the ground state, depending on the size of the portion of the neural network which is locally updated. To show the generality of the approach we apply it to both 1D and 2D non-integrable spin systems.
The calculation of off-diagonal matrix elements has various applications in fields such as nuclear physics and quantum chemistry. In this paper, we present a noisy intermediate scale quantum algorithm for estimating the diagonal and off-diagonal matrix elements of a generic observable in the energy eigenbasis of a given Hamiltonian. Several numerical simulations indicate that this approach can find many of the matrix elements even when the trial functions are randomly initialized across a wide range of parameter values without, at the same time, the need to prepare the energy eigenstates.
We demonstrate a method for finding the decoherence-subalgebra $\mathcal{N}(\mathcal{T})$ of a Gaussian quantum Markov semigroup on the von Neumann algebra $\mathcal{B}(\Gamma(\mathbb{C}^d))$ of all bounded operator on the Fock space $\Gamma(\mathbb{C}^d)$ on $\mathbb{C}^d$. We show that $\mathcal{N}(\mathcal{T})$ is a type I von Neumann algebra $L^\infty(\mathbb{R}^{d_c};\mathbb{C})\bar{\otimes}\mathcal{B}(\Gamma(\mathbb{C}^{d_f}))$ determined, up to unitary equivalence, by two natural numbers $d_c,d_f\leq d$. This result is illustrated by some applications and examples.
The understanding of the emergence of equilibrium statistical mechanics has progressed significantly thanks to developments from typicality, canonical and dynamical, and from the eigenstate thermalization hypothesis. Here we focus on a nonequilibrium scenario in which two nonintegrable systems prepared in different states are locally and non-extensively coupled to each other. Using both perturbative analysis and numerical exact simulations of up to 28 spin systems, we demonstrate the typical emergence of nonequilibrium (quasi-)steady current for weak coupling between the subsystems. We also identify that these currents originate from a prethermalization mechanism, which is the weak and local breaking of the conservation of the energy for each subsystem.
We study the transport and spectral property of a segmented diode formed by an XX $+$ XXZ spin chain. This system has been shown to become an ideal rectifier for spin current for large enough anisotropy. Here we show numerical evidence that the system in reverse bias has three different transport regimes depending on the value of the anisotropy: ballistic, diffusive and insulating. In forward bias we encounter two regimes, ballistic and diffusive. The system in forward and reverse bias shows significantly different spectral properties, with distribution of rapidities converging towards different functions. In the presence of dephasing the system becomes diffusive, rectification is significantly reduced, the relaxation gap increases and the spectral properties in forward and reverse bias tend to converge. For large dephasing the relaxation gap decreases again as a result of Quantum Zeno physics.
The relaxation of out-of-time-ordered correlators (OTOCs) has been studied as a mean to characterize the scrambling properties of a quantum system. We show that the presence of local conserved quantities typically results in, at the fastest, an algebraic relaxation of the OTOC provided (i) the dynamics is local and (ii) the system follows the eigenstate thermalization hypothesis. Our result relies on the algebraic scaling of the infinite-time value of OTOCs with system size, which is typical in thermalizing systems with local conserved quantities, and on the existence of finite speed of propagation of correlations for finite-range-interaction systems. We show that time-independence of the Hamiltonian is not necessary as the above conditions (i) and (ii) can occur in time-dependent systems, both periodic or aperiodic. We also remark that our result can be extended to systems with power-law interactions.
Recent years have seen tremendous progress in the theoretical understanding of quantum systems driven dissipatively by coupling them to different baths at their edges. This was possible because of the concurrent advances in the models used to represent these systems, the methods employed, and the analysis of the emerging phenomenology. Here we aim to give a comprehensive review of these three integrated research directions. We first provide an overarching view of the models of boundary-driven open quantum systems, both in the weak and strong coupling regimes. This is followed by a review of state-of-the-art analytical and numerical methods, both exact, perturbative and approximate. Finally, we discuss the transport properties of some paradigmatic one-dimensional chains, with an emphasis on disordered and quasiperiodic systems, the emergence of rectification and negative differential conductance, and the role of phase transitions, and we give an outlook on further research options.
The interplay between interaction, disorder, and dissipation has shown a rich phenomenology. Here we investigate a disordered XXZ spin chain in contact with a bath which, alone, would drive the system towards a highly delocalized and coherent Dicke state. We show that there exist regimes for which the natural orbitals of the single-particle density matrix of the steady state are all localized in the presence of strong disorders, either for weak interaction or strong interaction. We show that the averaged steady-state occupation in the eigenbasis of the open system Hamiltonian could follow an exponential decay for intermediate disorder strength in the presence of weak interactions, while it is more evenly spread for strong disorder or for stronger interactions. Last, we show that strong dissipation increases the coherence of the steady states, thus reducing the signatures of localization. We capture such signatures of localization also with a concatenated inverse participation ratio which simultaneously takes into account how localized are the eigenstates of the Hamiltonian, and how close is the steady state to an incoherent mixture of different energy eigenstates.
Understanding the intricate properties of one-dimensional quantum systems coupled to multiple reservoirs poses a challenge to both analytical approaches and simulation techniques. Fortunately, density matrix renormalization group-based tools, which have been widely used in the study of closed systems, have also been recently extended to the treatment of open systems. We present an implementation of such method based on state-of-the-art matrix product state (MPS) and tensor network methods, that produces accurate results for a variety of combinations of parameters. Unlike most approaches, which use the time-evolution to reach the steady-state, we focus on an algorithm that is time-independent and focuses on recasting the problem in exactly the same language as the standard Density Matrix Renormalization Group (DMRG) algorithm, initially put forward by M. C. Bañuls et al. in Phys. Rev. Lett. 114, 220601 (2015). Hence, it can be readily exported to any of the available DMRG platforms. We show that this implementation is suited for studying thermal transport in one-dimensional systems. As a case study, we focus on the XXZ quantum spin chain and benchmark our results by comparing the spin current and magnetization profiles with analytical results. We then explore beyond what can be computed analytically. Our code is freely available on github at https://www.github.com/heitorc7/oDMRG.
We show how to learn structures of generic, non-Markovian, quantum stochastic processes using a tensor network based machine learning algorithm. We do this by representing the process as a matrix product operator (MPO) and train it with a database of local input states at different times and the corresponding time-nonlocal output state. In particular, we analyze a qubit coupled to an environment and predict output state of the system at different time, as well as reconstruct the full system process. We show how the bond dimension of the MPO, a measure of non-Markovianity, depends on the properties of the system, of the environment and of their interaction. Hence, this study opens the way to a possible experimental investigation into the process tensor and its properties.
Finding the precise location of quantum critical points is of particular importance to characterise quantum many-body systems at zero temperature. However, quantum many-body systems are notoriously hard to study because the dimension of their Hilbert space increases exponentially with their size. Recently, machine learning tools known as neural-network quantum states have been shown to effectively and efficiently simulate quantum many-body systems. We present an approach to finding the quantum critical points of the quantum Ising model using neural-network quantum states, analytically constructed innate restricted Boltzmann machines, transfer learning and unsupervised learning. We validate the approach and evaluate its efficiency and effectiveness in comparison with other traditional approaches.
In XXZ chains, spin transport can be significantly suppressed when the interactions in the chain and the bias of the dissipative driving are large enough. This phenomenon of negative differential conductance is caused by the formation of two oppositely polarized ferromagnetic domains at the edges of the chain. Here we show that this many-body effect, combined with a non-uniform magnetic field, can allow a high degree of control of the spin current. In particular, by studying all the possible combinations of a dichotomous local magnetic field, we found that a configuration in which the magnetic field points up for half of the chain and down for the other half, can result in giant spin-current rectification, for example up to $10^8$ for a system with $8$ spins. Our results show clear indications that the rectification can increase with the system size.
We study the interplay between interactions and finite-temperature dephasing baths. We consider a double well with strongly interacting bosons coupled, via the density, to a bosonic bath. Such a system, when the bath has infinite temperature and instantaneous decay of correlations, relaxes with an emerging algebraic behavior with exponent 1/2. Here we show that, because of the finite-temperature baths and of the choice of spectral densities, such an algebraic relaxation may occur for a shorter duration and the characteristic exponent can be lower than 1/2. These results show that the interaction-induced impeding of relaxation is stronger and more complex when the bath has finite temperature and/or nonzero timescale for the decay of correlations.
The high fidelity generation of strongly entangled states of many particles, such as cat states, is a particularly demanding challenge. One approach is to drive the system, within a certain final time, as adiabatically as possible, in order to avoid the generation of unwanted excitations. However, excitations can be generated also by the presence of dissipative effects such as dephasing. Here we compare the effectiveness of Local Adiabatic and the FAst QUasi ADiabatic protocols in achieving a high fidelity for a target superposition state both with and without dephasing. In particular we consider trapped ions set-ups in which each spin interacts with all the others with the uniform coupling strength or with a power-law coupling. In order to mitigate the effects of dephasing, we complement the adiabatic protocols with dynamical decoupling and we test its effectiveness. The protocols we study could be readily implemented with state-of-the-art techniques.
Neural-network quantum states have shown great potential for the study of many-body quantum systems. In statistical machine learning, transfer learning designates protocols reusing features of a machine learning model trained for a problem to solve a possibly related but different problem. We propose to evaluate the potential of transfer learning to improve the scalability of neural-network quantum states. We devise and present physics-inspired transfer learning protocols, reusing the features of neural-network quantum states learned for the computation of the ground state of a small system for systems of larger sizes. We implement different protocols for restricted Boltzmann machines on general-purpose graphics processing units. This implementation alone yields a speedup over existing implementations on multi-core and distributed central processing units in comparable settings. We empirically and comparatively evaluate the efficiency (time) and effectiveness (accuracy) of different transfer learning protocols as we scale the system size in different models and different quantum phases. Namely, we consider both the transverse field Ising and Heisenberg XXZ models in one dimension, and also in two dimensions for the latter, with system sizes up to 128 and 8 x 8 spins. We empirically demonstrate that some of the transfer learning protocols that we have devised can be far more effective and efficient than starting from neural-network quantum states with randomly initialized parameters.
Recent works have shown that generic local Hamiltonians can be efficiently inferred from local measurements performed on their eigenstates or thermal states. Realistic quantum systems are often affected by dissipation and decoherence due to coupling to an external environment. This raises the question whether the steady states of such open quantum systems contain sufficient information allowing for full and efficient reconstruction of the system's dynamics. We find that such a reconstruction is possible for generic local Markovian dynamics. We propose a recovery method that uses only local measurements; for systems with finite-range interactions, the method recovers the Lindbladian acting on each spatial domain using only observables within that domain. We numerically study the accuracy of the reconstruction as a function of the number of measurements, type of open-system dynamics and system size. Interestingly, we show that couplings to external environments can in fact facilitate the reconstruction of Hamiltonians composed of commuting terms.
Chu Guo, Yong Liu, Min Xiong, Shichuan Xue, Xiang Fu, Anqi Huang, Xiaogang Qiang, Ping Xu, Junhua Liu, Shenggen Zheng, He-Liang Huang, Mingtang Deng, Dario Poletti, Wan-Su Bao, Junjie Wu Recent advances on quantum computing hardware have pushed quantum computing to the verge of quantum supremacy. Random quantum circuits are outstanding candidates to demonstrate quantum supremacy, which could be implemented on a quantum device that supports nearest-neighbour gate operations on a two-dimensional configuration. Here we show that using the Projected Entangled-Pair States algorithm, a tool to study two-dimensional strongly interacting many-body quantum systems, we can realize an effective general-purpose simulator of quantum algorithms. This technique allows to quantify precisely the memory usage and the time requirements of random quantum circuits, thus showing the frontier of quantum supremacy. With this approach we can compute the full wave-function of the system, from which single amplitudes can be sampled with unit fidelity. Applying this general quantum circuit simulator we measured amplitudes for a $7\times 7$ lattice of qubits with depth $1+40+1$ and double-precision numbers in 31 minutes using less than $93$ TB memory on the Tianhe-2 supercomputer.
Quantum many body physics simulations with Matrix Product States can often be accelerated if the quantum symmetries present in the system are explicitly taken into account. Conventionally, quantum symmetries have to be determined before hand when constructing the tensors for the Matrix Product States algorithm. In this work, we present a Matrix Product States algorithm with an adaptive $U(1)$ symmetry. This algorithm can take into account of, or benefit from, $U(1)$ or $Z_2$ symmetries when they are present, or analyze the non-symmetric scenario when the symmetries are broken without any external alteration of the code. To give some concrete examples we consider an XYZ model and show the insight that can be gained by (i) searching the ground state and (ii) evolving in time after a symmetry-changing quench. To show the generality of the method, we also consider an interacting bosonic system under the effect of a symmetry-breaking dissipation.
The use of two-site Lindblad dissipator to generate thermal states and study heat transport raised to prominence since [J. Stat. Mech. (2009) P02035] by Prosen and Žnidarič. Here we propose a variant of this method based on detailed balance of internal levels of the two site Hamiltonian and characterize its performance. We study the thermalization profile in the chain, the effective temperatures achieved by different single and two-site observables, and we also investigate the decay of two-time correlations. We find that at a large enough temperature the steady state approaches closely a thermal state, with a relative error below 1% for the inverse temperature estimated from different observables.
We study the heat and spin transport properties in a ring of interacting spins coupled to heat baths at different temperatures. We show that interactions, by inducing avoided crossings, can be a mean to tune both the total heat current flowing between the ring and the baths, and the way it flows through the system. In particular, we recognize three regimes in which the heat current flows clockwise, counter-clockwise, and in parallel. The temperature bias between the baths also induces a spin current within the ring, whose direction and magnitude can be tuned by the interaction. Lastly, we show how the ergotropy of the nonequilibrium steady state can increase significantly near the avoided crossings.
This work reports the functioning of a single atom energy-conversion device, operating either as a quantum engine or a refrigerator, coupled to a quantum load. The "working fluid" is comprised of two optical levels of a single ion, and the load is one vibrational mode of the same ion cooled down to the quantum regime. The energy scales of these two modes differ by 9 orders of magnitude. We realize cyclic energy transfers between the working fluid and the quantum load, either increasing or decreasing the population of the vibrational mode. This is achieved albeit the interaction between the load and the working fluid leads to a significant population redistribution and quantum correlations between them. The performance of the engine cycles as a function of several parameters is examined, and found to be in agreement with theory. We specifically look at the ergotropy of the load, which indicates the amount of energy stored in the load that can be extracted with a unitary process. We show that ergotropy rises with the number of engine cycles despite an increase in the entropy of the load. Our experiment represents the first fully quantum 4-stroke energy-conversion device operating with a generic coupling to a quantum load.
We study the rectification of heat current in an XXZ chain segmented in two parts. We model the effect of the environment with Lindblad heat baths. We show that, in our system, rectification is large for strong interactions in half of the chain and if one bath is at cold enough temperature. For the numerically accessible chain lengths, we observe that the rectification increases with the system size. We gain insight in the rectification mechanism by studying two-time correlations in the steady state. The presence of interactions also induces a strong nonlinear response to the temperature difference, resulting in superlinear and negative differential conductance regimes.
We study non-interacting fermionic systems dissipatively driven at their boundaries, focusing in particular on the case of a non-number-conserving Hamiltonian, which for example describes an $XY$ spin chain. We show that despite the lack of number conservation, it is possible to convert the problem of calculating the normal modes of the master equations and their corresponding rapidities, into diagonalizing simply an $L\times L$ tridiagonal bordered $2-$Toeplitz matrix, where $L$ is the size of the system. Such structure of matrix allows us to further reduce the problem into solving a scalar trigonometric non-linear equation for which we also show, in the case of an Ising chain, exact analytical explicit, and system size independent, solutions.
Many-body quantum systems present a rich phenomenology which can be significantly altered when they are in contact with an environment. In order to study such setups, a number of approximations are usually performed, either concerning the system, the environment, or both. A typical approach for large quantum interacting systems is to use master equations which are local, Markovian, and in Lindblad form. Here, we present an implementation of the Redfield master equation using matrix product states and operators. We show that this allows us to explore parameter regimes of the many-body quantum system and the environment which could not be probed with previous approaches based on local Lindblad master equations. We also show the validity of our results by comparing with the numerical exact thermofield-based chain-mapping approach.
We study the effect of disorder on work exchange associated to quantum Hamiltonian processes by considering an Ising spin chain in which the strength of coupling between spins are randomly drawn from either Normal or Gamma distributions. The chain is subjected to a quench of the external transverse field which induces this exchange of work. In particular, we study the irreversible work incurred by a quench as a function of the initial temperature, field strength and magnitude of the disorder. While presence of weak disorder generally increases the irreversible work generated, disorder of sufficient strength can instead reduce it, giving rise to a disorder induced lubrication effect. This reduction of irreversible work depends on the nature of the distribution considered, and can either arise from acquiring the behavior of an effectively smaller quench for the Normal-distributed spin couplings, or that of effectively single spin dynamics in the case of Gamma distributed couplings.
We present a protocol to selectively decouple, recouple, and engineer effective couplings in mesoscopic dipolar spin networks. In particular, we develop a versatile protocol that relies upon magic angle spinning to perform Hamiltonian engineering. By using global control fields in conjunction with a local actuator, such as a diamond Nitrogen Vacancy center located in the vicinity of a nuclear spin network, both global and local control over the effective couplings can be achieved. We show that the resulting effective Hamiltonian can be well understood within a simple, intuitive geometric picture, and corroborate its validity by performing exact numerical simulations in few-body systems. Applications of our method are in the emerging fields of two-dimensional room temperature quantum simulators in diamond platforms, as well as in dipolar coupled polar molecule networks.
Many types of dissipative processes can be found in nature or be engineered, and their interplay with a system can give rise to interesting phases of matter. Here we study the interplay among interaction, tunneling, and disorder in the steady state of a spin chain coupled to a tailored bath. We consider a dissipation which, in contrast to disorder, tends to generate a homogeneously polarized steady state. We find that the steady state can be highly sensitive even to weak disorder. We also establish that, in the presence of such dissipation, even in the absence of interaction, a finite amount of disorder is needed for localization. Last, we show that for strong disorder the system reveals signatures of localization both in the weakly and strongly interacting regimes.
Nonlinear classical dissipative systems present a rich phenomenology in their "route to chaos", including period-doubling, i.e. the system evolves with a period which is twice that of the driving. However, typically the attractor of a periodically driven quantum open system evolves with a period which exactly matches that of the driving. Here we analyze a manybody open quantum system whose classical correspondent presents period-doubling. We show that by analysing the spectrum of the periodic propagator and by studying the dynamical correlations, it is possible to show the occurrence of period-doubling in the quantum (period-$1$) steady state. We also discuss that such systems are natural candidates for clean Floquet time crystals.
Interactions in quantum systems may induce transitions to exotic correlated phases of matter which can be vulnerable to coupling to an environment. Here, we study the stability of a Bose-Hubbard chain coupled to a bosonic bath at zero and non-zero temperature. We show that only above a critical interaction the chain loses bosons and its properties are significantly affected. The transition is of a different nature than the superfluid-Mott insulator transition and occurs at a different critical interaction. We explain such a stable-unstable transition by the opening of a charge gap. The comparison of accurate matrix product state simulations to approximative approaches that miss this transition reveals its many-body origin.
We study the rectification of spin current in $XXZ$ chains segmented in two parts, each with a different anisotropy parameter. Using exact diagonalization and a matrix product states algorithm we find that a large rectification (of the order of $10^4$) is attainable even using a short chain of $N=8$ spins, when one half of the chain is gapless while the other has large enough anisotropy. We present evidence of diffusive transport when the current is driven in one direction and of a transition to an insulating behavior of the system when driven in the opposite direction, leading to a perfect diode in the thermodynamic limit. The above results are explained in terms of matching of spectrum of magnon excitations between the two halves of the chain.
The interplay between dissipation, interactions and gauge fields opens the possibility to rich emerging physics. Here we focus on a set-up in which the system is coupled at its extremities to two different baths which impose a current. We then study the system's response to a gauge field depending on the filling. We show that while the current induced by the baths has a marked dependence on the magnetic field at low fillings which is significantly reduced close to half-filling. We explain the interplay between interactions, gauge field and dissipation by studying the system's energy spectrum at the different fillings. This interplay also results in the emergence of negative differential conductivity. For this study we have developed a number-conserving treatment which allows a numerical exact treatment of fairly large system sizes, and which can be extended to a large class of systems.
In the context of dissipative systems, we show that for any quantum chaotic attractor a corre- sponding classical chaotic attractor can always be found. We provide with a general way to locate them, rooted in the structure of the parameter space (which is typically bidimensional, accounting for the forcing strength and dissipation parameters). In the cases where an approximate point like quantum distribution is found, it can be associated to exceptionally large regular structures. Moreover, supposedly anomalous quantum chaotic behaviour can be very well reproduced by the classical dynamics plus Gaussian noise of the size of an effective Planck constant $\hbar_{\rm eff}$. We give support to our conjectures by means of two paradigmatic examples of quantum chaos and transport theory. In particular, a dissipative driven system becomes fundamental in order to extend their validity to generic cases.
This is the second part of a work in which we show how to solve a large class of Lindblad master equations for non-interacting particles on $L$ sites. Here we concentrate on fermionic particles. In parallel to part I for bosons, but with important differences, we show how to reduce the problem to diagonalizing an $L \times L$ non-Hermitian matrix which, for boundary dissipative driving of a uniform chain, is a tridiagonal bordered Toeplitz matrix. In this way, both for fermionic and spin systems alike, we can obtain analytical expressions for the normal master modes and their relaxation rates (rapidities) and we show how to construct the non-equilibrium steady state.
This is a work in two parts in which we show how to solve a large class of Lindblad master equations for non-interacting particles on $L$ sites. In part I we concentrate on bosonic particles. We show how to reduce the problem to diagonalizing an $L \times L$ non-Hermitian matrix. In particular, for boundary dissipative driving of a uniform chain, the matrix is a tridiagonal bordered Toeplitz matrix which can be solved analytically for the normal master modes and their relaxation rates (rapidities). In the regimes in which an analytical solution cannot be found, our approach can still provide a speed-up in the numerical evaluation. We use this numerical method to study the relaxation gap at non-equilibrium phase transitions in a boundary driven bosonic ladder with synthetic gauge fields. We conclude by showing how to construct the non-equilibrium steady state.
We introduce a minimalistic quantum motor for coupled energy and particle transport. The system is composed of two spins, each coupled to a different bath and to a particle which can move on a ring consisting of three sites. We show that the energy flowing from the baths to the system can be partially converted to perform work against an external driving, even in the presence of moderate dissipation. We also analytically demonstrate the necessity of coupling between the spins. We suggest an experimental realization of our model using trapped ions or quantum dots.
We present a self-contained engine, which is made of one or more two-level systems, each of which is coupled to a single bath, as well as to a common load composed of a particle on a tilted lattice. We show that the energy and the entropy absorbed by the spins are transferred to the particle thus setting it into upward motion at an average constant speed, even when driven by a single spin connected to a single bath. When considering an ensemble of different spins, the velocity of the particle is larger when the tilt is on resonance with any of the spins' energy splitting. Interestingly, we find regimes where the spins' polarization enters periodic cycles with the oscillation period being determined by the tilt of the lattice.
We present a self-contained operator-based approach to derive the spectrum of trapped ions. This approach provides the complete normal form of the low energy quadratic Hamiltonian in terms of bosonic phonons, as well as an effective free particle degree of freedom for each spontaneously broken spatial symmetry. We demonstrate how this formalism can directly be used to characterize an ion chain both in the linear and the zigzag regimes. In particular we compute, both for the ground state and finite temperature states, spatial correlations, heat capacity and dynamical susceptibility. Last, for the ground state which has quantum correlations, we analyze the amount of energy reduction compared to an uncorrelated state with minimum energy, thus highlighting how the system can lower its energy by correlations.
We study a quantum Otto cycle in which the strokes are performed in finite time. The cycle involves energy measurements at the end of each stroke to allow for the respective determination of work. We then optimize for the work and efficiency of the cycle by varying the time spent in the different strokes and find that the optimal value of the ratio of time spent on each stroke goes through sudden changes as the parameters of this cycle vary continuously. The position of these discontinuities depends on the optimized quantity under consideration such as the net work output or the efficiency.
Quantum systems in contact with an environment display a rich physics emerging from the interplay between dissipative and Hamiltonian terms. Here we focus on the role of the geometry of the coupling between the system and the baths. In the specific we consider a dissipative boundary driven ladder in presence of a gauge field which can be implemented with ion microtraps arrays. We show that, depending on the geometry, the currents imposed by the baths can be strongly affected by the gauge field resulting in non-equilibrium phase transitions. In different phases both the magnitude of the current and its spatial distribution are significantly different. These findings allow for novel strategies to manipulate and control transport properties in quantum systems.
We investigate Landau-Zener processes modeled by a two-level quantum system, with its finite bias energy varied in time and in the presence of a single broadened cavity mode at zero temperature. By applying the hierarchy equation method to the Landau-Zener problem, we computationally study the survival fidelity of adiabatic states without Born, Markov, rotating-wave or other perturbative approximations. With this treatment it also becomes possible to investigate cases with very strong system-bath coupling. Different from a previous study of infinite-time Landau-Zener processes, the fidelity of the time-evolving state as compared with instantaneous adiabatic states shows non-monotonic dependence on the system-bath coupling and on the sweep rate of the bias. We then consider the effect of applying a counter-diabatic driving field, which is found to be useful in improving the fidelity only for sufficiently short Landau-Zener processes. Numerically exact results show that different counter-diabatic driving fields can have much different robustness against environment effects. Lastly, using a case study we discuss the possibility of introducing a dynamical decoupling field in order to eliminate the decoherence effect of the environment and at the same time to retain the positive role of a counter-diabatic field. Our work indicates that finite-time Landau-Zener processes with counter-diabatic driving offer a fruitful test bed to understand controlled adiabatic processes in open systems.
Unitary processes allow for the transfer of work to and from Hamiltonian systems. However, to achieve nonzero power for the practical extraction of work, these processes must be performed within a finite time, which inevitably induces excitations in the system. We show that depending on the time scale of the process and the physical realization of the external driving employed, the use of counterdiabatic quantum driving to extract more work is not always effective. We also show that by virtue of the two-time energy measurement definition of quantum work, the cost of counterdiabatic driving can be significantly reduced by selecting a restricted form of the driving Hamiltonian that depends on the outcome of the first energy measurement. Lastly, we introduce a measure, the exigency, that quantifies the need for an external driving to preserve quantum adiabaticity which does not require knowledge of the explicit form of the counterdiabatic drivings, and can thus always be computed. We apply our analysis to systems ranging from a two-level Landau-Zener problem to many-body problems, namely, the quantum Ising and Lipkin-Meshkov-Glick models.
We study energy transport in a chain of quantum harmonic and anharmonic oscillators where the anharmonicity is induced by interaction between local vibrational states of the chain. Using adiabatic elimination and numerical simulations with matrix product states, we show how strong interactions significantly slow down the relaxation dynamics (with the emergence of a new time scale) and can alter the properties of the steady state. We also show that steady state properties are completely different depending on the order in which the limits of infinite time and infinite interaction are taken.
We study the role of quantum statistics in the performance of Otto cycles. First, we show analytically that the work distributions for bosonic and fermionic working fluids are identical for cycles driven by harmonic trapping potentials. Subsequently, in the case of non-harmonic potentials, we find that the interplay between different energy level spacings and particle statistics strongly affects the performances of the engine cycle. To demonstrate this, we examine three trapping potentials which induce different (single particle) energy level spacings: monotonically decreasing with the level number, monotonically increasing, and the case in which the level spacing does not vary monotonically.
Two-time correlations are a crucial tool to probe the dynamics of many-body systems. We use these correlation functions to study the dynamics of dissipative quantum systems. Extending the adiabatic elimination method, we show that the correlations can display two distinct behaviors, depending on the observable of interest: a fast exponential decay, with a timescale of the order of the dissipative coupling, or a much slower dynamics. We apply this formalism to bosons in a double well subjected to phase noise. While the single-particle correlations decay exponentially, the density-density correlations display slow aging dynamics. We also show that the two-time correlations of dissipatively engineered quantum states can evolve in a drastically different manner compared to their Hamiltonian counterparts.
We study the performance of a quantum Otto cycle driven by trapping potentials of the form $V_t(x) \sim x^{2q}$. This family of potentials possesses a simple scaling property which allows for analytical insights into the efficiency and work output of the cycle. We show that, while both the mean work output and the efficiency of two Otto cycles in different trapping potentials can be made equal, the work probability distribution will still be strongly affected by the difference in structure of the energy levels. Lastly, we perform a comparison of quantum Otto cycles in various physically relevant scenarios and find that in certain instances, the efficiency of the cycle is greater when using potentials with larger values of $q$, while, in other cases, with harmonic traps.
We study the dynamics of a strongly interacting bosonic quantum gas in an optical lattice potential under the effect of a dissipative environment. We show that the interplay between the dissipative process and the Hamiltonian evolution leads to an unconventional dynamical behavior of local number fluctuations. In particular we show, both analytically and numerically, the emergence of an anomalous diffusive evolution in configuration space at short times and, at long times, an unconventional dynamics dominated by rare events. Such rare events, common in disordered and frustrated systems, are due here to strong interactions. This complex two-stage dynamics reveals information on the level structure of the strongly interacting gas.
We study the dynamics of bosonic atoms in a double well potential under the influence of dissipation. The main effect of dissipation is to destroy quantum coherence and to drive the system towards a unique steady state. We study how the atom-atom interaction affects the decoherence process. We use a systematic approach considering different atomic densities. We show that, for two atoms, the interaction already strongly suppresses decoherence: a phenomenon we refer to as "interaction impeded decoherence". For many atoms, thanks to the increased complexity of the system, the nature of the decoherence process is dramatically altered giving rise to an algebraic instead of exponential decay.
We study how the interplay of dissipation and interactions affects the dynamics of a bosonic many-body quantum system. In the presence of both dissipation and strongly repulsive interactions, observables such as the coherence and the compressibility display three dynamical regimes: an initial exponential variation followed by a power-law regime and finally a slow exponential convergence to their asymptotic values corresponding to the infinite temperature state. These very long-time scales arise as dissipation forces the population of states disfavored by interactions. The long-time, strong coupling dynamics are understood by performing a mapping onto a classical diffusion process displaying non-Brownian behavior. While both dissipation and strong interactions tend to suppress coherence when acting separately, we find that strong interaction impedes the decoherence process generated by the dissipation.
We analyze the quench dynamics of a one-dimensional bosonic Mott insulator and focus on the time evolution of density correlations. For these we identify a pronounced propagation front, the velocity of which, once correctly extrapolated at large distances, can serve as a quantitative characteristic of the many-body Hamiltonian. In particular, the velocity allows the weakly interacting regime, which is qualitatively well described by free bosons, to be distinguished from the strongly interacting one, in which pairs of distinct quasiparticles dominate the dynamics. In order to describe the latter case analytically, we introduce a general approximation to solve the Bose-Hubbard Hamiltonian based on the Jordan-Wigner fermionization of auxiliary particles. This approach can also be used to determine the ground-state properties. As a complement to the fermionization approach, we derive explicitly the time-dependent many-body state in the noninteracting limit and compare our results to numerical simulations in the whole range of interactions of the Bose-Hubbard model.
How fast can correlations spread in a quantum many-body system? Based on the seminal work by Lieb and Robinson, it has recently been shown that several interacting many-body systems exhibit an effective light cone that bounds the propagation speed of correlations. The existence of such a "speed of light" has profound implications for condensed matter physics and quantum information, but has never been observed experimentally. Here we report on the time-resolved detection of propagating correlations in an interacting quantum many-body system. By quenching a one-dimensional quantum gas in an optical lattice, we reveal how quasiparticle pairs transport correlations with a finite velocity across the system, resulting in an effective light cone for the quantum dynamics. Our results open important perspectives for understanding relaxation of closed quantum systems far from equilibrium as well as for engineering efficient quantum channels necessary for fast quantum computations.
Braneshop, and
the US National Science Foundation.
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