Quantum computers are predicted to outperform classical ones for solving partial differential equations, perhaps exponentially. Here we consider a prototypical PDE - the heat equation in a rectangular region - and compare in detail the complexities of ten classical and quantum algorithms for solving it, in the sense of approximately computing the amount of heat in a given region. We find that, for spatial dimension $d \ge 2$, there is an at most quadratic quantum speedup using an approach based on applying amplitude estimation to an accelerated classical random walk. However, an alternative approach based on a quantum algorithm for linear equations is never faster than the best classical algorithms.
At the heart of quantum thermodynamics lies a fundamental question about what is genuine "quantum" in quantum heat engines and how to seek this quantumness, so that thermodynamical tasks could be performed more efficiently compared with classical protocols. Here, using the concept of $P$-representability, we define a function called classicality, which quantifies the degree of non-classicality of bosonic modes. This function allows us to explore the role of non-classicality in quantum heat engines and design optimal protocols for work extraction. For two specific cycles, a quantum Otto and a g…
In this paper we elaborate a hybrid classical-quantum framework which allows one to model and solve heat and mass transfer problems occurring in electric contacts. We utilize special functions and Harrow-Hassidim-Lloyd (HHL) quantum algorithm for finding temperature and arc flux functions exactly and approximately for the Stefan type problems. The Stefan type problems we are considering are based on the Generalized Heat Equation with free boundaries. As examples we consider exact and approximate solutions of inverse one-phase and two-phase Stefan problems. An Inverse Generalized One-Phase Stef…
Despite the increasing interest, the research field which studies the concepts of work and heat at quantum level has suffered from two main drawbacks: first, the difficulty to properly define and measure the work, heat and internal energy variation in a quantum system and, second, the lack of experiments. Here, we report a full characterization of the dissipated heat, work and internal energy variation in a two-level quantum system interacting with an engineered environment. We use the IBMQ quantum computer to implement the driven system's dynamics in a dissipative environment. The experimenta…
We propose a random circuit model to analyze the impact of noise on the performance of variational quantum circuits for classical optimization problems. Our model accounts for the propagation of arbitrary single qubit errors through the circuit. We find that even with a small noise rate, the quality of the obtained classical optima is low on average and a single-qubit error rate of $1 / nD$, where $n$ is the number of qubits and $D$ is the circuit depth, is needed for the possibility of a quantum advantage. We estimate that this translates to an error rate lower than $10^{-6}$ using QAOA for c…
Distributed quantum-classical hybrid algorithm for solving K-SAT problem
School of Computer Science and Engineering, Sun Yat-sen University（中山大学计算机学院）；The Guangdong Key Laboratory of Information Security Technology, Sun Yat-sen University（中山大学广东省信息安全技术重点实验室）；Shenzhen Research Institute of Sun Yat-Sen University（中山大学深圳研究院）；School of Physics and Astronomy, Sun Yat-sen University（中山大学物理与天文学院）；Departamento de Matemática, Instituto Superior Técnico, Instituto de Telecomunicações（里斯本高等理工学院数学系，电信研究所）
Driving Quantum Heat Engines Beyond Classical Limits through Multilevel Coherence
Institute for Quantum Science and Engineering, Texas A&M University（德克萨斯农工大学量子科学与工程研究所）；Okinawa Institute of Science and Technology Graduate University（冲绳科学技术大学院大学）；Baylor University（贝勒大学）；Princeton University（普林斯顿大学）
Entanglement sudden death in spatially separated two-mode Gaussian states coupled to local thermal and squeezed thermal baths is studied by mapping the problem to that of the quantum-to-classical transition. Using Simon's criterion concerning the characterisation of classicality in Gaussian states, the time to ESD is calculated by analysing the covariance matrices of the system. The results for the two-mode system at T=0 and T>0 for the two types of bath states are generalised to $n$-modes, and are shown to be similar in nature to the results for the general discrete $n$-qubit system.
A general quantum algorithm for solving a problem is discussed. The number of steps required to solve a problem using this method is independent of the number of cases that has to be considered classically. Hence, it is more efficient than existing classical algorithms or quantum algorithm, which requires O(sqrt(N)) steps.
Variational quantum eigensolver~(VQE) typically optimizes variational parameters in a quantum circuit to prepare eigenstates for a quantum system. Its applications to many problems may involve a group of Hamiltonians, e.g., Hamiltonian of a molecule is a function of nuclear configurations. In this paper, we incorporate derivatives of Hamiltonian into VQE and develop some hybrid quantum-classical algorithms, which explores both Hamiltonian and wavefunction spaces for optimization. Aiming for solving quantum chemistry problems more efficiently, we first propose mutual gradient descent algorithm …
Considering a kicked rotor coupled to a model heat bath both the classical and quantum entropy productions are calculated exactly. Starting with an initial wave packet, the von Neuman entropy as a function of time is determined from the reduced density matrix while the Liouville evolution of the corresponding Husimi distribution provides us with the classical entropy. It is found that both these entropies agree reasonably satisfying the same asymptotic growth law and more importantly both are proportional to the classical Liapounov exponent.
Quantum Brownian motion, described by the Caldeira-Leggett model, brings insights to understand phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classic…
The prosperous development of both hardware and algorithms for quantum computing (QC) potentially prompts a paradigm shift in scientific computing in various fields. As an increasingly active topic in QC, the variational quantum algorithm (VQA) leads a promising direction for solving partial differential equations on Noisy Intermediate Scale Quantum (NISQ) devices. Although a clear perspective on the advantages of QC over classical computing techniques for specific mathematical and physical problems exists, applications of QC in computational fluid dynamics to solve practical flow problems, th…
Identifying computational tasks suitable for (future) quantum computers is an active field of research. Here we explore utilizing quantum computers for the purpose of solving differential equations. We consider two approaches: (i) basis encoding and fixed-point arithmetic on a digital quantum computer, and (ii) representing and solving high-order Runge-Kutta methods as optimization problems on quantum annealers. As realizations applied to two-dimensional linear ordinary differential equations, we devise and simulate corresponding digital quantum circuits, and implement and run a 6$^{\mathrm{th…
Starting from a simple mapping of a generator of local stochastic dynamics to a quantum Hamiltonian, we derive a condition, which allows us to use the quasi-adiabatic evolution and so relate gapped quantum phases with non-equilibrium's. This leads us to a study of invertible matrix product operators. Finally, we present an ansatz for constructing local stochastic dynamics for which the Perron-Frobenius vector has a Matrix Product Representation. Additionally, we get for free that the dynamics satisfy a generalized form of detailed balance.
Under a general framework, shortcuts to adiabatic processes are shown to be possible in classical systems. We then study the distribution function of the work done on a small system initially prepared at thermal equilibrium. It is found that the work fluctuations can be significantly reduced via shortcuts to adiabatic processes. For example, in the classical case probabilities of having very large or almost zero work values are suppressed. In the quantum case negative work may be totally removed from the otherwise non-positive-definite work values. We also apply our findings to a micro Otto-cy…
The modelling of quantum heat transfer processes at the nanoscale is crucial for the development of energy harvesting and molecular electronics devices. Herein, we adopt a mixed quantum-classical description of a device, in which the open subsystem of interest is treated quantum mechanically and the surrounding heat baths are treated in a classical-like fashion. By introducing such a mixed quantum-classical description of the composite system, one is able to study the heat transfer between the subsystem and bath from a closed system point of view, thereby avoiding simplifying assumptions relat…
We numerically demonstrate heat rectification for linear chains of ions in trap lattices with graded trapping frequencies, in contact with thermal baths implemented by optical molasses. To calculate the local temperatures and heat currents we find the stationary state by solving a system of algebraic equations. This approach is much faster than the usual method that integrates the dynamical equations of the system and averages over noise realizations.
The understanding of the underlying dynamical mechanisms which determine the macroscopic laws of heat conduction is a long standing task of non-equilibrium statistical mechanics. A better understanding of the mechanism of heat conduction may lead to potentially interesting applications based on the possibility to control the heat flow. Indeed, different models of thermal rectifiers has been recently proposed in which heat can flow preferentially in one direction. Although these models are far away from a prototype realization, the underlying mechanisms are of very general nature and, as such, …
Quantum measurement is ultimately a physical process, resulting from an interaction between the measured system and a measuring apparatus. Considering the physical process of measurement within a thermodynamic context naturally raises the following question: How can the work and heat be interpreted? In the present paper we model the measurement process for an arbitrary discrete observable as a measurement scheme. Here the system to be measured is first unitarily coupled with an apparatus and subsequently the compound system is objectified with respect to a pointer observable, thus producing de…
When incomplete information about the control parameters is quantified as a prior distribution, a subtle connection emerges between quantum heat engines and their classical analogs. We study the quantum model where the uncertain parameters are the intrinsic energy scales and compare with the classical models where the intermediate temperature is the uncertain parameter. The prior distribution quantifying the incomplete information has the form $π(x)\propto 1/x$ in both the quantum and the classical models. The expected efficiency calculated in near-equilibrium limit approaches the value of one…
For a free particle, the coupling to its environment can be the relevant mechanism to induce quantum behavior as the temperature is lowered. We study general linear environments with a spectral density proportional to ω^s at low frequencies and consider in particular the specific heat of the free damped particle. For super-Ohmic baths with s>=2, a reentrant classical behavior is found. As the temperature is lowered, the specific heat decreases from the classical value of k_B/2, thereby indicating the appearence of quantum effects. However, the classical value of the specific heat is restored a…
We study the heat statistics of a quantum Brownian motion described by the Caldeira-Leggett model. By using the path integral approach, we introduce a novel concept of the quantum heat functional along every pair of Feynman paths. This approach has an advantage of improving our understanding about heat in quantum systems. First, we demonstrate the microscopic reversibility of the system by connecting the heat functional to the forward and its time-reversed probabilities. Second, we analytically prove the quantum-classical correspondence of the heat functional and their statistics, which allows…
We provide an explicit method for solving general markovian master equations for quadratic bosonic Hamiltonians with linear bath operators. As an example we consider a one-dimensional quantum harmonic oscillator chain coupled to thermal reservoirs at both ends of the chain. We derive an analytic solution of the Redfield master equation for homogeneous harmonic chain and recover classical results, namely, vanishing temperature gradient and constant heat current in the thermodynamic limit. In the case of the disordered gapped chains we observe universal heat current scaling independent of the ba…
A ballistic atom pump is a system containing two reservoirs of neutral atoms or molecules and a junction connecting them containing a localized time-varying potential. Atoms move through the pump as independent particles. Under certain conditions, these pumps can create net transport of atoms from one reservoir to the other. While such systems are sometimes called "quantum pumps," they are also models of classical chaotic transport, and their quantum behavior cannot be understood without study of the corresponding classical behavior. Here we examine classically such a pump's effect on energy a…
We study the applicability of quantum algorithms in computational game theory and generalize some results related to Subtraction games, which are sometimes referred to as one-heap Nim games. In quantum game theory, a subset of Subtraction games became the first explicitly defined class of zero-sum combinatorial games with provable separation between quantum and classical complexity of solving them. For a narrower subset of Subtraction games, an exact quantum sublinear algorithm is known that surpasses all deterministic algorithms for finding solutions with probability $1$. Typically, both Nim …
Classical combinations of quantum states for solving banded circulant linear systems
Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Computer Science, National University of Singapore（新加坡国立大学计算机科学系）
The heat engine, a machine that extracts useful work from thermal sources, is one of the basic theoretical constructs and fundamental applications of classical thermodynamics. The classical description of a heat engine does not include coherence in its microscopic degrees of freedom. By contrast, a quantum heat engine might possess coherence between its internal states. Although the Carnot efficiency cannot be surpassed, and coherence can be performance degrading in certain conditions, it was recently predicted that even when using only thermal resources, internal coherence can enable a quantu…
Interesting effects arise in cyclic machines where both heat and ergotropy transfer take place between the energising bath and the system (the working fluid). Such effects correspond to unconventional decompositions of energy exchange between the bath and the system into heat and work, respectively, resulting in efficiency bounds that may surpass the Carnot efficiency. However, these effects are not directly linked with quantumness, but rather with heat and ergotropy, the likes of which can be realised without resorting to quantum mechanics.
This paper is about quantum heat defined as the change in energy of a bath during a process. The presentation takes into account recent developments in classical strong-coupling thermodynamics, and addresses a version of quantum heat which satisfies quantum-classical correspondence. The characteristic function and the full counting statistics of quantum heat are shown to be formally similar. The paper further shows that the method can be extended to more than one bath, e.g. two baths at different temperatures, which opens up the prospect of studying correlations and heat flow. The paper extend…
Quantum-Classical Hybrid Algorithm for Solving the Learning-With-Errors Problem on NISQ Devices
Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）；State Key Laboratory of Low-Dimensional Quantum Physics and Department of Physics, Tsinghua University（清华大学物理系低维量子物理国家重点实验室）；State Key Laboratory of Mathematical Engineering and Advanced Computing（数学工程与先进计算国家重点实验室）；Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数学科学学院物理与应用物理系）
By using ideas and strong results borrowed from the classical moment problem, we show how -under very general conditions- a discrete number of values of the spectral zeta function (associated generically with a non-decreasing sequence of numbers, and not necessarily with an operator) yields all the moments corresponding to the density of states, as well as those of the partition function of the sequence (the two basic quantities that are always considered in a quantum mechanical context). This goes beyond the well known expression of the small-t asymptotic expansion of the heat kernel of an op…
It is shown how information contained in the pairwise correlations (in general, partial) between atoms of a gas can be used to completely convert heat taken from a thermostat into mechanical work in a process of relaxation of the system to its thermal equilibrium state. Both classical correlations and quantum correlations (entanglement) are considered. The amount of heat converted into work is proportional to the entropy defect of the initial state of the system. For fully correlated particles, in the case of entanglement the amount of work obtained per particle is twice as large as in the cas…
In this work, we obtain the numerical temperature field to a thermally developing fluid flow inside parallel plates problem with a quantum computing method. The physical problem deals with the heat transfer of a steady state, hydrodinamically developed and thermally developing fluid flow inside two parallel plates channel subjected to a prescribed constant heat flux. Its solution is formulated numerically with Finite Differences method, where a sequence of linear systems must be solved in order to determine the complete temperature field. Such linear systems are written as discrete unconstrain…
