arXiv subjects

Nana Liu

Publications and source records attributed to Nana Liu.

At least 55 records · Page 3Linked to original sources

arXiv2025

Quantum Circuits for the heat equation with physical boundary conditions via Schrodingerisation

This paper explores the explicit design of quantum circuits for quantum simulation of partial differential equations (PDEs) with physical boundary conditions. These equations and/or their discretized forms usually do not evolve via unitary dynamics, thus are not suitable for quantum simulation. Boundary conditions (either time-dependent or independent) make the problem more difficult. To tackle this challenge, the Schrodingerisation method can be employed, which converts linear partial and ordinary differential equations with non-unitary dynamics into systems of Schrodinger-type equations, via the so-called warped phase transformation that maps the equation into one higher dimension. Despite advancements in Schrodingerisation techniques, the explicit implementation of quantum circuits for solving general PDEs, especially with physical boundary conditions, remains underdeveloped. We present two methods for handling the inhomogeneous terms arising from time-dependent physical boundary conditions. One approach utilizes Duhamel's principle to express the solution in integral form and employs linear combination of unitaries (LCU) for coherent state preparation. Another method applies an augmentation to transform the inhomogeneous problem into a homogeneous one. We then apply the quantum simulation technique from [CJL23] to transform the resulting non-autonomous system to an autonomous system in one higher dimension. We provide detailed implementations of these two methods and conduct a comprehensive complexity analysis in terms of queries to the time evolution input oracle.

quant-ph↗

arXiv2024

Quantum Circuits for partial differential equations via Schrödingerisation

Quantum computing has emerged as a promising avenue for achieving significant speedup, particularly in large-scale PDE simulations, compared to classical computing. One of the main quantum approaches involves utilizing Hamiltonian simulation, which is directly applicable only to Schrödinger-type equations. To address this limitation, Schrödingerisation techniques have been developed, employing the warped transformation to convert general linear PDEs into Schrödinger-type equations. However, despite the development of Schrödingerisation techniques, the explicit implementation of the corresponding quantum circuit for solving general PDEs remains to be designed. In this paper, we present detailed implementation of a quantum algorithm for general PDEs using Schrödingerisation techniques. We provide examples of the heat equation, and the advection equation approximated by the upwind scheme, to demonstrate the effectiveness of our approach. Complexity analysis is also carried out to demonstrate the quantum advantages of these algorithms in high dimensions over their classical counterparts.

quant-ph↗

arXiv2024

Schrödingerization based Quantum Circuits for Maxwell's Equation with time-dependent source terms

The Schrödingerisation method combined with the autonomozation technique in \cite{cjL23} converts general non-autonomous linear differential equations with non-unitary dynamics into systems of autonomous Schrödinger-type equations, via the so-called warped phase transformation that maps the equation into two higher dimension. Despite the success of Schrödingerisation techniques, they typically require the black box of the sparse Hamiltonian simulation, suitable for continuous-variable based analog quantum simulation. For qubit-based general quantum computing one needs to design the quantum circuits for practical implementation. This paper explicitly constructs a quantum circuit for Maxwell's equations with perfect electric conductor (PEC) boundary conditions and time-dependent source terms, based on Schrödingerization and autonomozation, with corresponding computational complexity analysis. Through initial value smoothing and high-order approximation to the delta function, the increase in qubits from the extra dimensions only requires minor rise in computational complexity, almost $\log\log {1/\varepsilon}$ where $\varepsilon$ is the desired precision. Our analysis demonstrates that quantum algorithms constructed using Schrödingerisation exhibit polynomial acceleration in computational complexity compared to the classical Finite Difference Time Domain (FDTD) format.

quant-ph↗

arXiv2024

Schrödingerisation based computationally stable algorithms for ill-posed problems in partial differential equations

We introduce a simple and stable computational method for ill-posed partial differential equation (PDE) problems. The method is based on Schrödingerization, introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603], which maps all linear PDEs into Schrödinger-type equations in one higher dimension, for quantum simulations of these PDEs. Although the original problem is ill-posed, the Schrödingerized equations are Hamiltonian systems and time-reversible, allowing stable computation both forward and backward in time. The original variable can be recovered by data from suitably chosen domain in the extended dimension. We will use the backward heat equation and the linear convection equation with imaginary wave speed as examples. Error analysis of these algorithms are conducted and verified numerically. The methods are applicable to both classical and quantum computers, and we also lay out quantum algorithms for these methods. Moreover, we introduce a smooth initialization for the Schrödingerized equation which will lead to essentially spectral accuracy for the approximation in the extended space, if a spectral method is used. Consequently, the extra qubits needed due to the extra dimension, if a qubit based quantum algorithm is used, for both well-posed and ill-posed problems, becomes almost $\log\log {1/\varepsilon}$ where $\varepsilon$ is the desired precision. This optimizes the complexity of the Schrödingerization based quantum algorithms for any non-unitary dynamical system introduced in [S. Jin, N. Liu and Y. Yu, arXiv:2212.13969][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603].

math.NA↗

arXiv2024

Efficient Learning of Continuous-Variable Quantum States

The characterization of continuous-variable quantum states is crucial for applications in quantum communication, sensing, simulation and computing. However, a full characterization of multimode quantum states requires a number of experiments that grows exponentially with the number of modes. Here we propose an alternative approach where the goal is not to reconstruct the full quantum state, but rather to estimate its characteristic function at a given set of points. For multimode states with reflection symmetry, we show that the characteristic function at M points can be estimated using only O(log M ) copies of the state, independently of the number of modes. When the characteristic function is known to be positive, as in the case of squeezed vacuum states, the estimation is achieved by an experimentally friendly setup using only beamsplitters and homodyne measurements.

quant-ph↗

arXiv2024

ClimSim-Online: A Large Multi-scale Dataset and Framework for Hybrid ML-physics Climate Emulation

Modern climate projections lack adequate spatial and temporal resolution due to computational constraints, leading to inaccuracies in representing critical processes like thunderstorms that occur on the sub-resolution scale. Hybrid methods combining physics with machine learning (ML) offer faster, higher fidelity climate simulations by outsourcing compute-hungry, high-resolution simulations to ML emulators. However, these hybrid ML-physics simulations require domain-specific data and workflows that have been inaccessible to many ML experts. As an extension of the ClimSim dataset (Yu et al., 2024), we present ClimSim-Online, which also includes an end-to-end workflow for developing hybrid ML-physics simulators. The ClimSim dataset includes 5.7 billion pairs of multivariate input/output vectors, capturing the influence of high-resolution, high-fidelity physics on a host climate simulator's macro-scale state. The dataset is global and spans ten years at a high sampling frequency. We provide a cross-platform, containerized pipeline to integrate ML models into operational climate simulators for hybrid testing. We also implement various ML baselines, alongside a hybrid baseline simulator, to highlight the ML challenges of building stable, skillful emulators. The data (https://huggingface.co/datasets/LEAP/ClimSim_high-res) and code (https://leap-stc.github.io/ClimSim and https://github.com/leap-stc/climsim-online) are publicly released to support the development of hybrid ML-physics and high-fidelity climate simulations.

cs.LG↗

arXiv2024

Analog quantum simulation of parabolic partial differential equations using Jaynes-Cummings-like models

We present a simplified analog quantum simulation protocol for preparing quantum states that embed solutions of parabolic partial differential equations, including the heat, Black-Scholes and Fokker-Planck equations. The key idea is to approximate the heat equations by a system of hyperbolic heat equations that involve only first-order differential operators. This scheme requires relatively simple interaction terms in the Hamiltonian, which are the electric and magnetic dipole moment-like interaction terms that would be present in a Jaynes-Cummings-like model. For a d-dimensional problem, we show that it is much more appropriate to use a single d-level quantum system - a qudit - instead of its qubit counterpart, and d+1 qumodes. The total resource cost is efficient in d and precision error, and has potential for realisability for instance in cavity and circuit QED systems.

quant-ph↗

arXiv2024

Quantum simulation of the Fokker-Planck equation via Schrodingerization

This paper studies a quantum simulation technique for solving the Fokker-Planck equation. Traditional semi-discretization methods often fail to preserve the underlying Hamiltonian dynamics and may even modify the Hamiltonian structure, particularly when incorporating boundary conditions. We address this challenge by employing the Schrodingerization method-it converts any linear partial and ordinary differential equation with non-Hermitian dynamics into systems of Schrodinger-type equations. We explore the application in two distinct forms of the Fokker-Planck equation. For the conservation form, we show that the semi-discretization-based Schrodingerization is preferable, especially when dealing with non-periodic boundary conditions. Additionally, we analyze the Schrodingerization approach for unstable systems that possess positive eigenvalues in the real part of the coefficient matrix or differential operator. Our analysis reveals that the direct use of Schrodingerization has the same effect as a stabilization procedure. For the heat equation form, we propose a quantum simulation procedure based on the time-splitting technique. We discuss the relationship between operator splitting in the Schrodingerization method and its application directly to the original problem, illustrating how the Schrodingerization method accurately reproduces the time-splitting solutions at each step. Furthermore, we explore finite difference discretizations of the heat equation form using shift operators. Utilizing Fourier bases, we diagonalize the shift operators, enabling efficient simulation in the frequency space. Providing additional guidance on implementing the diagonal unitary operators, we conduct a comparative analysis between diagonalizations in the Bell and the Fourier bases, and show that the former generally exhibits greater efficiency than the latter.

quant-ph↗

arXiv2024

Intrinsic Orbital Angular Momentum Originated from Optical Catastrophe Superposition

Conventionally, intrinsic orbital angular momentum (OAM) is associated with phase vortices. However, our investigation into the propagation dynamics of 2D superimposed catastrophe beams, termed cyclone catastrophe beams (CCBs), reveals that these beams inherently exhibit rotation and possess OAM, distinct from the typical connection to phase vortices. Our observations clearly show these beams rotating during autofocusing propagation and particle manipulation, confirming the presence of OAM. Theoretical calculations affirm that the OAM of these beams is intrinsic and can be adjusted by varying the number of superimposed beams. Furthermore, our interference and phase studies indicate that, although CCBs exhibit phase vortices, they do not rotate around the singularities of phase vortices and their total topological charges are zero. This implies that the manifestation of OAM within CCBs does not rely on nonzero topological charge of the presented phase vortices within CCBs. Especially, eigenstates decomposition analysis illustrates that CCBs can be decomposed as a composite of Laguerre-Gaussian (LG) modes with uneven fidelity, where the topological charges of LG modes align with multiples of the superimposed catastrophe beams but do not equal to the value of the OAM per photon within CCBs, emphasizing the intrinsic OAM within CCBs and the absence of a connection to phase vortices. Our findings not only advance the understanding of the relationship between OAM and phase vortices but also pave the way for different applications of OAM waves, catalyzing their development in optics and other domains.

physics.optics↗

arXiv2023

Analog quantum simulation of partial differential equations

Quantum simulators were originally proposed for simulating one partial differential equation (PDE) in particular - Schrodinger's equation. Can quantum simulators also efficiently simulate other PDEs? While most computational methods for PDEs - both classical and quantum - are digital (PDEs must be discretised first), PDEs have continuous degrees of freedom. This suggests that an analog representation can be more natural. While digital quantum degrees of freedom are usually described by qubits, the analog or continuous quantum degrees of freedom can be captured by qumodes. Based on a method called Schrodingerisation, we show how to directly map D-dimensional linear PDEs onto a (D+1)-qumode quantum system where analog or continuous-variable Hamiltonian simulation on D+1 qumodes can be used. This very simple methodology does not require one to discretise PDEs first, and it is not only applicable to linear PDEs but also to some nonlinear PDEs and systems of nonlinear ODEs. We show some examples using this method, including the Liouville equation, heat equation, Fokker-Planck equation, Black-Scholes equations, wave equation and Maxwell's equations. We also devise new protocols for linear PDEs with random coefficients, important in uncertainty quantification, where it is clear how the analog or continuous-variable framework is most natural. This also raises the possibility that some PDEs may be simulated directly on analog quantum systems by using Hamiltonians natural for those quantum systems.

quant-ph↗

arXiv2023

Quantum simulation for time-dependent Hamiltonians -- with applications to non-autonomous ordinary and partial differential equations

Non-autonomous dynamical systems appear in a very wide range of interesting applications, both in classical and quantum dynamics, where in the latter case it corresponds to having a time-dependent Hamiltonian. However, the quantum simulation of these systems often needs to appeal to rather complicated procedures involving the Dyson series, considerations of time-ordering, requirement of time steps to be discrete and/or requiring multiple measurements and postselection. These procedures are generally much more complicated than the quantum simulation of time-independent Hamiltonians. Here we propose an alternative formalism that turns any non-autonomous unitary dynamical system into an autonomous unitary system, i.e., quantum system with a time-independent Hamiltonian, in one higher dimension, while keeping time continuous. This makes the simulation with time-dependent Hamiltonians not much more difficult than that of time-independent Hamiltonians, and can also be framed in terms of an analogue quantum system evolving continuously in time. We show how our new quantum protocol for time-dependent Hamiltonians can be performed in a resource-efficient way and without measurements, and can be made possible on either continuous-variable, qubit or hybrid systems. Combined with a technique called Schrodingerisation, this dilation technique can be applied to the quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain nonlinear PDEs, with time-dependent coefficients.

quant-ph↗

arXiv2023

Investigation on trapping capability of circular swallowtail beams

Circular swallowtail beams (CSBs) with their remarkable autofocusing capability have garnered significant interests due to their potential applications in optical trapping. This study delves into a comprehensive investigation of the trapping force properties of CSBs. Through a combination of experimental observations and theoretical analysis, we systematically explore the quantitative manipulation of trapping forces by adjusting specific parameters. This detailed investigation provides insights into the trapping force performance and stability of CSBs. Furthermore, the experimental validation of particle trapping using CSBs underscores their effectiveness, emphasizing their significant potential for optical manipulation and trapping applications.

physics.optics↗

arXiv2023

Dilation theorem via Schrödingerisation, with applications to the quantum simulation of differential equations

Nagy's unitary dilation theorem in operator theory asserts the possibility of dilating a contraction into a unitary operator. When used in quantum computing, its practical implementation primarily relies on block-encoding techniques, based on finite-dimensional scenarios. In this study, we delve into the recently devised Schrödingerisation approach and demonstrate its viability as an alternative dilation technique. This approach is applicable to operators in the form of $V(t)=\exp(-At)$, which arises in wide-ranging applications, particularly in solving linear ordinary and partial differential equations. Importantly, the Schrödingerisation approach is adaptable to both finite and infinite-dimensional cases, in both countable and uncountable domains. For quantum systems lying in infinite dimensional Hilbert space, the dilation involves adding a single infinite dimensional mode, and this is the continuous-variable version of the Schrödingerisation procedure which makes it suitable for analog quantum computing. Furthermore, by discretising continuous variables, the Schrödingerisation method can also be effectively employed in finite-dimensional scenarios suitable for qubit-based quantum computing.

quant-ph↗

arXiv2023

Non-destructively probing the thermodynamics of quantum systems with qumodes

Quantum systems are by their very nature fragile. The fundamental backaction on a state due to quantum measurement notwithstanding, there is also in practice often a destruction of the system itself due to the means of measurement. This becomes acutely problematic when we wish to make measurements of the same system at multiple times, or generate a large quantity of measurement statistics. One approach to circumventing this is the use of ancillary probes that couple to the system under investigation, and through their interaction, enable properties of the primary system to be imprinted onto and inferred from the ancillae. Here we highlight means by which continuous variable quantum modes (qumodes) can be employed to probe the thermodynamics of quantum systems in and out of equilibrium, including thermometry, reconstruction of the partition function, and reversible and irreversible work. We illustrate application of our results with the example of a spin-1/2 system in a transverse field.

quant-ph↗

arXiv2023

Robust autofocusing propagation in turbulence

We conducted a comprehensive study on the robust propagation of the same spot-size Gaussian beam (SSGB), same envelope Gaussian beam (SEGB), Circular Airy beam (CAB) and Circular Pearcey beam (CPB) in complex environments. Our findings clearly demonstrate that autofocusing beams exhibit higher stability in propagation compared with the Gaussian beams. To validate our results, we statistically analyze the intensity fluctuation of autofocusing beams and Gaussian beams. The analysis reveals that the intensity fluctuations of autofocusing beams are significantly smaller than that of Gaussian beams. Furthermore, we study the variation of the coherence factor and find that the coherence of autofocusing beams is better than that of Gaussian beams under turbulence. Additionally, we observe the change of the scintillation index (SI) with the propagation distance z, and our results show that autofocusing beams exhibit less oscillation than Gaussian beams, indicating that autofocusing beams propagate in complex environments with less distortion and fewer intensity fluctuation. Overall, our results suggest that autofocusing beams are promising for applications such as optical trapping and manipulation in complex environments. Our study provides valuable insights into the selection of stable beams that exhibit high intensity and high field gradient at the focal position in complex environments.

physics.optics↗

arXiv2023

Quantum simulation of Maxwell's equations via Schrödingersation

We present quantum algorithms for electromagnetic fields governed by Maxwell's equations. The algorithms are based on the Schrödingersation approach, which transforms any linear PDEs and ODEs with non-unitary dynamics into a system evolving under unitary dynamics, via a warped phase transformation that maps the equation into one higher dimension. In this paper, our quantum algorithms are based on either a direct approximation of Maxwell's equations combined with Yee's algorithm, or a matrix representation in terms of Riemann-Silberstein vectors combined with a spectral approach and an upwind scheme. We implement these algorithms with physical boundary conditions, including perfect conductor and impedance boundaries. We also solve Maxwell's equations for a linear inhomogeneous medium, specifically the interface problem. Several numerical experiments are performed to demonstrate the validity of this approach. In addition, instead of qubits, the quantum algorithms can also be formulated in the continuous variable quantum framework, which allows the quantum simulation of Maxwell's equations in analog quantum simulation.

quant-ph↗

arXiv2023

Controllable nonlinear propagation of partially incoherent Airy beams

The self-accelerating beams such as the Airy beam show great potentials in many applications including optical manipulation, imaging and communication. However, their superior features during linear propagation could be easily corrupted by optical nonlinearity or spatial incoherence individually. Here we investigate how the interaction of spatial incoherence and nonlinear propagation affect the beam quality of Airy beam, and find that the two destroying factors can in fact balance each other. Our results show that the influence of coherence and nonlinearity on the propagation of partially incoherent Airy beams (PIABs) can be formulated as two exponential functions that have factors of opposite signs. With appropriate spatial coherence length, the PIABs not only resist the corruption of beam profile caused by self-focusing nonlinearity, but also exhibits less anomalous diffraction caused by the self-defocusing nonlinearity. Our work provides deep insight into how to maintain the beam quality of self-accelerating Airy beams by exploiting the interaction between partially incoherence and optical nonlinearity. Our results may bring about new possibilities for optimizing partially incoherent structured field and developing related applications such as optical communication, incoherent imaging and optical manipulations.

physics.optics↗

arXiv2023

Time complexity analysis of quantum algorithms via linear representations for nonlinear ordinary and partial differential equations

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ordinary differential equations (ODEs) and nonlinear Hamilton-Jacobi equations (HJE) via linear representations or exact mappings between nonlinear ODEs/HJE and linear partial differential equations (the Liouville equation and the Koopman-von Neumann equation). The connection between the linear representations and the original nonlinear system is established through the Dirac delta function or the level set mechanism. We compare the quantum linear systems algorithms based methods and the quantum simulation methods arising from different numerical approximations, including the finite difference discretisations and the Fourier spectral discretisations for the two different linear representations, with the result showing that the quantum simulation methods usually give the best performance in time complexity. We also propose the Schrödinger framework to solve the Liouville equation for the HJE with the Hamiltonian formulation of classical mechanics, since it can be recast as the semiclassical limit of the Wigner transform of the Schrödinger equation. Comparsion between the Schrödinger and the Liouville framework will also be made.

quant-ph↗