Nana Liu

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.

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.

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.

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.

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.

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.

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 ...

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. </p

Although quantum computers are capable of solving problems like factoring exponentially faster than the best-known classical algorithms, determining the resources responsible for their computational power remains unclear. An important class of problems where quantum computers possess an advantage is phase estimation, which includes applications like factoring. We introduce a new computational model based on a single squeezed state resource that can perform phase estimation, which we call the power of one qumode. This model is inspired by an interesting computational model known as deterministic quantum computing with one quantum bit (DQC1). Using the power of one qumode, we identify that the amount of squeezing is sufficient to quantify the resource requirements of different computational problems based on phase estimation. In particular, it establishes a quantitative relationship between the resources required for factoring and DQC1. For example, we find the squeezing required to f...

Shi Jin 2, 3 and Nana Liu 3, 4, ∗ Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China Ministry of Education, Key Laboratory in Scientific and Engineering Computing, Shanghai Jiao Tong University, Shanghai 200240, China University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai 200240, China (Dated: February 17, 2022)

Quantum algorithms have the potential to be very powerful. However, to exploit quantum parallelism, some quantum algorithms require an embedding of large classical data into quantum states. This embedding can cost a lot of resources, for instance by implementing quantum random-access memory (QRAM). An important instance of this is in quantum-enhanced machine learning algorithms. We propose a new way of circumventing this requirement by using a classical-quantum hybrid architecture where the input data can remain classical, which differs from other hybrid models. We apply this to a fundamental computational problem called Boolean oracle identification, which offers a useful primitive for quantum machine learning algorithms. Its aim is to identify an unknown oracle amongst a list of candidates while minimising the number of queries to the oracle. In our scheme, we replace the classical oracle with our hybrid oracle. We demonstrate both theoretically and numerically that the success ra...

With the rise of quantum technologies, it is necessary to have practical and preferably non-destructive 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.

Quantum machine learning models have the potential to offer speedups and better predictive accuracy compared to their classical counterparts. However, these quantum algorithms, like their classical counterparts, have been shown to also be vulnerable to input perturbations, in particular for classification problems. These can arise either from noisy implementations or, as a worst-case type of noise, adversarial attacks. In order to develop defense mechanisms and to better understand the reliability of these algorithms, it is crucial to understand their robustness properties in the presence of natural noise sources or adversarial manipulation. From the observation that measurements involved in quantum classification algorithms are naturally probabilistic, we uncover and formalize a fundamental link between binary quantum hypothesis testing and provably robust quantum classification. This link leads to a tight robustness condition that puts constraints on the amount of noise a classifi...

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.

High-dimensional quantum systems are vital for quantum technologies and are essential in demonstrating practical quantum advantage in quantum computing, simulation and sensing. Since dimen-sionality grows exponentially with the number of qubits, the potential power of noisy intermediate-scale quantum (NISQ) devices over classical resources also stems from entangled states in high dimensions. An important family of quantum protocols that can take advantage of high-dimensional Hilbert space are classification tasks. These include quantum machine learning algorithms, witnesses in quantum information processing and certain decision problems. However, due to counter-intuitive geometrical properties emergent in high dimensions, classification problems are vulnerable to adver-sarial attacks. We demonstrate that the amount of perturbation needed for an adversary to induce a misclassification scales inversely with dimensionality. This is shown to be a fundamental feature independent of the details of the classification protocol. Furthermore, this leads to a trade-off between the security of the classification algorithm against adversarial attacks and quantum advantages we expect for high-dimensional problems. In fact, protection against these adversarial attacks require extra resources that scale at least polynomially with the Hilbert space dimension of the system, which can erase any significant quantum advantage that we might expect from a quantum protocol. This has wide-ranging implications in the use of both near-term and future quantum technologies for classification. Quantum technologies promise exciting advantages in quantum computation [1], simulation [2], metrology [3] and cryptography [4]. Even while large-scale and fault tolerant quantum technologies currently remain out of reach, noisy intermediate-scale quantum (NISQ) devices [5] hope to deliver advantages over classical systems in the near-term. Most of these protocols exploit not only unique quantum characteristics like entanglement and superposition, but also high-dimensional Hilbert spaces. Without the latter, no sizeable quantum advantages in either computation, simulation or sensing are expected. One important class of tasks where high-dimensional Hilbert spaces may be advantageous are classification problems. To correctly categorise an object belongs to one of the most common and basic questions asked in science. In the quantum setting, classification problems can appear mainly in one of two ways. Firstly, a quantum protocol can be used for classification problems with classical data in order to gain a quantum advantage in speed or precision. Many quantum-enhanced machine learning algorithms are of this type [6]. Alternatively, one can classify quantum states or processes themselves, in terms of entanglement [7], phases [8] or other many-body behaviour. Quantum learning protocols also belong to the latter category [9]. From these examples, it therefore appears that advantages for classification tasks will become more pronounced as Hilbert-space dimension grows, at least in the absence of noise. However, as we will see, this is * Electronic address: nana.liu@quantumlah.org no longer true when the classification protocols are subject to adversarial perturbations. These are small, often hard-to-detect perturbations of the object to be classified which give rise to deliberate misclassification. This is highly relevant for many classification problems. In the machine learning context in particular, security breaches in the algorithm are not only desirable for adversarial parties, but also made possible as data used for classification are often shared amongst multiple, possibly un-trusted parties [10]. Recent findings in machine learning suggests that even highly successful classification algorithms can be very vulnerable to adversarial perturbations if the dimension of the data is high enough, such as high-resolution image data [11]. Some quantum machine learning algorithms that resist certain kinds of ad-versarial attacks have also been recently developed [12]. However, it is yet unknown what the fundamental limits to adversarial robustness are for quantum classification problems in general. In this paper, we demonstrate that a perturbation by an amount scaling inversely with the dimension of the quantum system to be classified is sufficient to induce a misclassification. Amazingly, this is a fundamental feature of quantum classification originating from a purely geometrical property of high-dimensional spaces, known as the concentration of measure phenomenon [13]. It is independent of the specifics of any particular classification protocol. Furthermore, detection of these small perturbations by existing efficient certification protocols for quantum systems cannot be efficient. For classification problems, otherwise efficient certification protocols must now require resources scaling polynomially with di-mensionality. This is an exponential resource cost in

We present a verifiable and blind protocol for assisted universal quantum computing on continuous-variable (CV) platforms. This protocol is highly experimentally-friendly to the client, as it only requires Gaussian-operation capabilities from the latter. Moreover, the server is not required universal quantum-computational power either, its only function being to supply the client with copies of a single-mode non-Gaussian state. Uni-versality is attained based on state-injection of the server's non-Gaussian supplies. The protocol is automatically blind because the non-Gaussian resource requested to the server is always the same, regardless of the specific computation. Verification, in turn, is possible thanks to an efficient non-Gaussian state fidelity test where we assume identical state preparation by the server. It is based on Gaussian measurements by the client on the injected states, which is potentially interesting on its own. The division of quantum hardware between client and server assumed here is in agreement with the experimental constraints expected in realistic schemes for CV cloud quantum computing.

Anomaly detection is used for identifying data that deviate from 'normal' data patterns. Its usage on classical data finds diverse applications in many important areas like fraud detection, medical diagnoses, data cleaning and surveillance. With the advent of quantum technologies, anomaly detection of quantum data, in the form of quantum states, may become an important component of quantum applications. Machine learning algorithms are playing pivotal roles in anomaly detection using classical data. Two widely-used algorithms are kernel principal component analysis and one-class support vector machine. We find corresponding quantum algorithms to detect anomalies in quantum states. We show that these two quantum algorithms can be performed using resources logarithmic in the dimensionality of quantum states. For pure quantum states, these resources can also be logarithmic in the number of quantum states used for training the machine learning algorithm. This makes these algorithms potentially applicable to big quantum data applications.

With the rise of quantum technologies, it is necessary to have practical and preferably non-destructive 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.

Although quantum computers are capable of solving problems like factoring exponentially faster than the best-known classical algorithms, determining the resources responsible for their computational power remains unclear. An important class of problems where quantum computers possess an advantage is phase estimation, which includes applications like factoring. We introduce a computational model based on a single squeezed state resource that can perform phase estimation, which we call the power of one qumode. This model is inspired by an interesting computational model known as deterministic quantum computing with one quantum bit (DQC1). Using the power of one qumode, we identify that the amount of squeezing is sufficient to quantify the resource requirements of different computational problems based on phase estimation. In particular, we can use the amount of squeezing to quantitatively relate the resource requirements of DQC1 and factoring. Furthermore, we can connect the squeezing to other known resources like precision, energy, qudit dimensionality, and qubit number. We show the circumstances under which they can likewise be considered good resources.

Although quantum computers are capable of solving problems like factoring exponentially faster than the best-known classical algorithms, determining the resources responsible for their computational power remains unclear. An important class of problems where quantum computers possess an advantage is phase estimation, which includes applications like factoring. We introduce a new computational model based on a single squeezed state resource that can perform phase estimation, which we call the power of one qumode. This model is inspired by an interesting computational model known as deterministic quantum computing with one quantum bit (DQC1). Using the power of one qumode, we identify that the amount of squeezing is sufficient to quantify the resource requirements of different computational problems based on phase estimation. In particular, it establishes a quantitative relationship between the resources required for factoring and DQC1. For example, we find the squeezing required to factor has an exponential scaling whereas no squeezing (i.e., a coherent state) is already sufficient to solve the hardest problem in DQC1.

We investigate the thermodynamical properties of quantum fields in curved spacetime. Our approach is to consider quantum fields in curved spacetime as a quantum system undergoing an out-of-equilibrium transformation. The non-equilibrium features are studied by using a formalism which has been developed to derive fluctuation relations and emergent irreversible features beyond the linear response regime. We apply these ideas to an expanding universe scenario, therefore avoiding assumptions on the relation between entropy and quantum matter. We provide a fluctuation theorem which allows us to understand particle production due to the expansion of the universe as an entropic increase. Our results pave the way towards a different understanding of the thermodynamics of relativistic and quantum systems in our universe.

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.