Nana Liu | Shanghai Jiao Tong University (original) (raw)

Papers by Nana Liu

We present a simplified analog quantum simulation protocol for preparing quantum states that embe... more 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 ddimensional 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.

arXiv:2405.00673 , 2024

Matrix geometric means between two positive definite matrices can be defined equivalently from di... more Matrix geometric means between two positive definite matrices can be defined equivalently from distinct perspectives -- as solutions to certain nonlinear systems of equations, as points along geodesics in Riemannian geometry, and as solutions to certain optimisation problems.
This diversity already suggests the potential for varied applications, as well as acting as a bridge between different domains. Here we devise new quantum subroutines to efficiently prepare quantum unitary operators that embed the standard matrix geometric mean and its generalisations called the weighted matrix geometric mean. This enables the construction of solutions to the algebraic Riccati equation, which is an important class of nonlinear systems of equations that appears in machine learning, optimal control, estimation, and filtering.
Using these subroutines,
we present a new class of quantum learning algorithms called quantum geometric mean metric learning. This has applications in efficiently finding the best distance measure and solving classification problems in the weakly supervised limit and for anomaly detection, for both classical and quantum problems. We also show how our method can be generalised to a particular p^th-order system of nonlinear equations.
These quantum subroutines for matrix geometric means are also useful in other areas of quantum information. For example, we show how to use them in the estimation of geometric Renyi relative entropies and the Uhlmann fidelity by means of the Fuchs--Caves observable.
In particular, our quantum algorithms for estimating the Uhlmann and Matsumoto fidelities have optimal dependence on the precision.
Finally, we provide a BQP-complete problem based on matrix geometric means that can be solved by our subroutines, thus characterising their computational capability.

Physical Review A

We study a recently introduced simple method [S. Jin, N. Liu, and Y. Yu, Quantum simulation of pa... more We study a recently introduced simple method [S. Jin, N. Liu, and Y. Yu, Quantum simulation of partial differential equations via Schrödingerisation, arXiv:2212.13969] for solving general linear partial differential equations with quantum simulation. This method converts linear partial differential equations into a Hamiltonian system, using a simple transformation called the warped phase transformation. Here we provide a more-in-depth technical discussion and expand on this approach in a more detailed and pedagogical way. We apply this to examples of partial differential equations, including heat, convection, Fokker-Planck, linear Boltzmann, and Black-Scholes equations. This approach can also be extended to general linear partial differential equations, including the Vlasov-Fokker-Planck equation and the Liouville representation equation for nonlinear ordinary differential equations. Extension to higher-order time derivatives is also possible.

arXiv (Cornell University), Apr 2, 2023

Quantum dynamics, typically expressed in the form of a time-dependent Schrödinger equation with a... more Quantum dynamics, typically expressed in the form of a time-dependent Schrödinger equation with a Hermitian Hamiltonian, is a natural application for quantum computing. However, when simulating quantum dynamics that involves the emission of electrons, it is necessary to use artificial boundary conditions (ABC) to confine the computation within a fixed domain. The introduction of ABCs alters the Hamiltonian structure of the dynamics, and existing quantum algorithms can not be directly applied since the evolution is no longer unitary. The current paper utilizes a recently introduced Schrödingerisation method [JLY22a, JLY22b] that converts non-Hermitian dynamics to a Schrödinger form, for the artificial boundary problems. We implement this method for three types of ABCs, including the complex absorbing potential technique, perfectly matched layer methods, and Dirichlet-to-Neumann approach. We analyze the query complexity of these algorithms, and perform numerical experiments to demonstrate the validity of this approach. This helps to bridge the gap between available quantum algorithms and computational models for quantum dynamics in unbounded domains.

arXiv (Cornell University), Dec 30, 2022

We study a new method-called Schrödingerisation introduced in [25]-for solving general linear par... more We study a new method-called Schrödingerisation introduced in [25]-for solving general linear partial differential equations with quantum simulation. This method converts linear partial differential equations into a 'Schrödingerised' or Hamiltonian system, using a new and simple transformation called the warped phase transformation. Here we provide more in-depth technical discussions and expand on this approach in a more detailed and pedagogical way. We apply this to more examples of partial differential equations, including heat, convection, Fokker-Planck, linear Boltzmann and Black-Scholes equations. This approach can also be extended to Schrödingerise general linear partial differential equations, including the Vlasov-Fokker-Planck equation and the Liouville representation equation for nonlinear ordinary differential equations.

arXiv (Cornell University), Dec 28, 2022

We present a simple new way-called Schrödingerisation-to simulate general linear partial differen... more We present a simple new way-called Schrödingerisation-to simulate general linear partial differential equations via quantum simulation. Using a simple new transform, referred to as the warped phase transformation, any linear partial differential equation can be recast into a system of Schrödinger's equations-in real time-in a straightforward way. This can be seen directly on the level of the dynamical equations without more sophisticated methods. This approach is not only applicable to PDEs for classical problems but also those for quantum problems-like the preparation of quantum ground states, Gibbs states and the simulation of quantum states in random media in the semiclassical limit.

arXiv (Cornell University), Feb 9, 2022

We investigate time complexities of finite difference methods for solving the high-dimensional li... more We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, the high-dimensional linear hyperbolic equation and the multiscale hyperbolic heat system with quantum algorithms (hence referred to as the "quantum difference methods"). For the heat and linear hyperbolic equations we study the impact of explicit and implicit time discretizations on quantum advantages over the classical difference method. For the multiscale problem, we find the time complexity of both the classical treatment and quantum treatment for the explicit scheme scales as O(1/ε), where ε is the scaling parameter, while the scaling for the multiscale Asymptotic-Preserving (AP) schemes does not depend on ε. This indicates that it is still of great importance to develop AP schemes for multiscale problems in quantum computing.

Social Science Research Network, 2023

arXiv (Cornell University), May 4, 2023

This paper explores the feasibility of quantum simulation for partial differential equations (PDE... more This paper explores the feasibility of quantum simulation for partial differential equations (PDEs) with physical boundary or interface conditions. Semi-discretisation of such problems does not necessarily yield Hamiltonian dynamics and even alters the Hamiltonian structure of the dynamics when boundary and interface conditions are included. This seemingly intractable issue can be resolved by using a recently introduced Schrödingerisation method [JLY22a,JLY22b]-it converts any linear PDEs and ODEs with non-Hermitian dynamics to a system of Schrödinger equations, via the so-called warped phase transformation that maps the equation into one higher dimension. We implement this method for several typical problems, including the linear convection equation with inflow boundary conditions and the heat equation with Dirichlet and Neumann boundary conditions. For interface problems we study the (parabolic) Stefan problem, linear convection, and linear Liouville equations with discontinuous and even measure-valued coefficients. We perform numerical experiments to demonstrate the validity of this approach, which helps to bridge the gap between available quantum algorithms and computational models for classical and quantum dynamics with boundary and interface conditions.

arXiv: 2308.00646, 2023

Quantum simulators were originally proposed for simulating one partial differential equation (PDE... more Quantum simulators were originally proposed for simulating one partial differential equation (PDE) in particular-Schrödinger'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 Schrödingerisation, 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.

arXiv (Cornell University), Jul 13, 2017

With the rise of quantum technologies, it is necessary to have practical and preferably nondestru... more With the rise of quantum technologies, it is necessary to have practical and preferably nondestructive methods to measure and read-out from such devices. A current line of research towards this has focussed on the use of ancilla systems which 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. We propose the use of continuous variable qumodes as ancillary probes, and show that the interaction Hamiltonian can be fully characterised and directly sampled from measurements of the qumode alone. We suggest how such probes may also be used to determine thermodynamical properties, including reconstruction of the partition function. We show that the method is robust to realistic experimental imperfections such as finite-sized measurement bins and squeezing, and discuss how such probes are already feasible with current experimental setups.

arXiv (Cornell University), Apr 6, 2023

Quantum simulation is known to be capable of simulating certain dynamical systems in continuous t... more Quantum simulation is known to be capable of simulating certain dynamical systems in continuous time-Schrödinger's equations being the most direct and well-known-more efficiently than classical simulation. Any linear dynamical system can in fact be transformed into a system of Schrödinger's equations via a method called Schrödingerisation [1]. Building on the observation that iterative methods in linear algebra, and more generally discrete linear dynamical systems, can be viewed as discrete time approximations of dynamical systems which evolve continuously in time, we can apply the Schrödingerisation technique. Thus quantum simulation can be directly applied to the continuous-time limits of some of the simplest iterative methods. This applies to general (explicit) iterative schemes or discrete linear dynamical systems. In particular, we introduce the quantum Jacobi and quantum power methods for solving the quantum linear systems of equations and for estimating the maximum eigenvector and eigenvalue of a matrix respectively. The proposed quantum simulation can be performed on either discrete-variable quantum systems or on hybrid continuousvariable and discrete-variable quantum systems. This framework provides an interesting alternative method to solve linear algebra problems using quantum simulation.

Physical Review Research, 2021

ArXiv:2304.02865, 2023

arxiv.:2305.02710, 2023

arXiv (Cornell University), Feb 15, 2022

We construct quantum algorithms to compute physical observables of nonlinear PDEs with M initial ... more We construct quantum algorithms to compute physical observables of nonlinear PDEs with M initial data. Based on an exact mapping between nonlinear and linear PDEs using the level set method, these new quantum algorithms for nonlinear Hamilton-Jacobi and scalar hyperbolic PDEs can be performed with a computational cost that is independent of M , for arbitrary nonlinearity. Depending on the details of the initial data, it can also display up to exponential advantage in both the dimension of the PDE and the error in computing its observables. For general nonlinear PDEs, quantum advantage with respect to M is possible in the large M limit.

Cornell University - arXiv, Sep 22, 2022

Most problems in uncertainty quantification, despite its ubiquitousness in scientific computing, ... more Most problems in uncertainty quantification, despite its ubiquitousness in scientific computing, applied mathematics and data science, remain formidable on a classical computer. For uncertainties that arise in partial differential equations (PDEs), large numbers M >> 1 of samples are required to obtain accurate ensemble averages. This usually involves solving the PDE M times. In addition, to characterise the stochasticity in a PDE, the dimension L of the random input variables is high in most cases, and classical algorithms suffer from curse-of-dimensionality. We propose new quantum algorithms for PDEs with uncertain coefficients that are more efficient in M and L in various important regimes, compared to their classical counterparts. We introduce transformations that transfer the original d-dimensional equation (with uncertain coefficients) into d + L (for dissipative equations) or d + 2L (for wave type equations) dimensional equations (with certain coefficients) in which the uncertainties appear only in the initial data. These transformations also allow one to superimpose the M different initial data, so the computational cost for the quantum algorithm to obtain the ensemble average from M different samples is then independent of M , while also showing potential advantage in d, L and precision ǫ in computing ensemble averaged solutions or physical observables.

SSRN Electronic Journal

We construct quantum algorithms to compute the solution and/or physical observables of nonlinear ... more 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, 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.

Quantum

Solving the time-dependent Schrödinger equation is an important application area for quantum algo... more Solving the time-dependent Schrödinger equation is an important application area for quantum algorithms. We consider Schrödinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter ℏ, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schrödinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of ℏ and the precision ε are obtained. It is found that the number of required qubits, m, scales only logarithmically with respect to ℏ. When the solution has bounded derivatives up to order ℓ, the symmetric Trotting method has gate complexity O((εℏ)−12polylog(ε−32ℓℏ−1−12ℓ)), provided that the diagonal unitary operators in the pseudo-spectral ...

SSRN Electronic Journal

We investigate time complexities of finite difference methods for solving the high-dimensional li... more We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, the high-dimensional linear hyperbolic equation and the multiscale hyperbolic heat system with quantum algorithms (hence referred to as the "quantum difference methods"). Our detailed analyses show that for the heat and linear hyperbolic equations the quantum difference methods provide exponential speedup over the classical difference method with respect to the spatial dimension. For the multiscale problem, the time complexity of both the classical treatment and quantum treatment for the explicit scheme scales as O(1/ε), where ε is the scaling parameter, while the scaling for the Asymptotic-Preserving (AP) schemes does not depend on ε. This indicates that it is still of great importance to develop AP schemes for multiscale problems in quantum computing.

arXiv:2405.00673 , 2024

Physical Review A

arXiv (Cornell University), Apr 2, 2023

arXiv (Cornell University), Dec 30, 2022

arXiv (Cornell University), Dec 28, 2022

arXiv (Cornell University), Feb 9, 2022

Social Science Research Network, 2023

arXiv (Cornell University), May 4, 2023

arXiv: 2308.00646, 2023

arXiv (Cornell University), Jul 13, 2017

arXiv (Cornell University), Apr 6, 2023

Physical Review Research, 2021

ArXiv:2304.02865, 2023

arxiv.:2305.02710, 2023

arXiv (Cornell University), Feb 15, 2022

Cornell University - arXiv, Sep 22, 2022

SSRN Electronic Journal

Quantum

SSRN Electronic Journal

We investigate time complexities of finite difference methods for solving the high-dimensional li... more We investigate time complexities of finite difference methods for solving the high-dimensional linear heat equation, the high-dimensional linear hyperbolic equation and the multiscale hyperbolic heat system with quantum algorithms (hence referred to as the "quantum difference methods"). Our detailed analyses show that for the heat and linear hyperbolic equations the quantum difference methods provide exponential speedup over the classical difference method with respect to the spatial dimension. For the multiscale problem, the time complexity of both the classical treatment and quantum treatment for the explicit scheme scales as O(1/ε), where ε is the scaling parameter, while the scaling for the Asymptotic-Preserving (AP) schemes does not depend on ε. This indicates that it is still of great importance to develop AP schemes for multiscale problems in quantum computing.

This thesis is an exploration of the power of photonic resources, as viewed from several differen... more This thesis is an exploration of the power of photonic resources, as viewed from several different but related perspectives. They range from quantum computation, precision parameter estimation to the thermodynamics of relativistic quantum systems, as applied to cosmology in particular. The use of photonic states allows us to address several important questions about the resources required in quantum mechanical processes.

In chapter 1, we propose a new quantum computational model, called the `power of one qumode', that relies mainly on a single-mode photonic squeezed state. In particular, we show the amount of squeezing can quantitatively relate the resource requirements of factoring to the problem of finding the trace of large unitary matrices, a result with consequences for understanding how powerful quantum computation really is. Furthermore, we can connect squeezing to other known resources like precision, energy, qudit dimensionality and qubit number, which is a useful stepping stone to finding the resources that enable quantum computation.

In chapter 2, we exploit the quantum mechanical properties of photonic states for use in precision parameter estimation of general linear optical processes, which is useful for a diverse number of applications, from characterising an unknown process in a photonic quantum computer to biological imaging. We introduce a formalism that quantifies this improvement in precision. We also provide conditions under which one can easily check for photonic states that are optimal to use in this context, which is a potentially important result for future experimental efforts.

In chapter 3, we explore the connection between two-mode squeezed states, commonly used in quantum optics, and relativistic quantum processes, in particular in cosmology. Using this connection, we apply recently developed tools from the thermodynamics of quantum systems perturbed far from equilibrium to address an old question of entropy production in cosmology from a surprising new angle.