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arXiv:2309.16262

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

Junpeng Hu, Jin, Shi, Nana Liu, Lei Zhang

Published Sep 28, 2023• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2309.16262

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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)V(t)=\exp(-At)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.
ABSTRACT Nagy's unitary dilation theorem in the 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 Schrödingerization approach and demonstrate its viability as an alternative dilation technique. This approach is applicable to operators in the form of , which arises in wide‐ranging applications, particularly in solving linear ordinary and partial differential equations. Importantly, the Schrödingerization 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 Schrödingerization procedure which makes it suitable for analog quantum computing. Furthermore, by discretizing continuous variables, the Schrödingerization method can also be effectively employed in finite‐dimensional scenarios suitable for qubit‐based quantum computing.
Abstract 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. Designing quantum algorithms for time-dependent Hamiltonian is generally much more complicated than time-independent Hamiltonian problems, due to the challenge in handling the time-ordering; moreover, almost all existing such algorithms are developed for qubit-based quantum devices. Here we propose an alternative formalism based on Sambe-Howland’s clock 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 Schrödingerisation, 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.
Quantum simulation is 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 (Jin et al. 2022. Quantum simulation of partial differential equations via Schrödingerisation. ( https://arxiv.org/abs/2212.13969 ) and Jin et al. 2023. Phys. Rev. A 108 , 032603. ( doi:10.1103/PhysRevA.108.032603 )). We show how Schrödingerisation allows quantum simulation to be directly used for the simulation of continuous-time versions of general (explicit) iterative schemes or discrete linear dynamical systems. In particular, we use this new method to solve linear systems of equations and for estimating the maximum eigenvector and eigenvalue of a matrix, respectively. This method is applicable using either discrete-variable quantum systems or on hybrid continuous-variable and discrete-variable quantum systems. This framework provides an interesting alternative to solve linear algebra problems using quantum simulation.
Abstract 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 (they 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 (CV) 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 ordinary differential equations. 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 CV 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.
Recently, Jin et al. proposed a quantum simulation technique for any linear partial differential equations (PDEs), called Schrödingerisation [1-3]. In this paper, the Schrödingerisation technique for quantum simulation is expanded to any nonlinear PDEs by combining it with the homotopy analysis method (HAM) [4-6]. The HAM can transfer a nonlinear PDE into a series of linear PDEs with guaranteeing convergence of the series. In this way, any nonlinear PDEs can be solved by quantum simulation using a quantum computer. For simplicity, we call the procedure “HAM-Schrödingerisation quantum algorithm”. Quantum computing is a groundbreaking technique. Hopefully, the “HAM-Schrödingerisation quantum algorithm” can open a door to highly efficient simulation of complicated turbulent flows by means of quantum computing in future.
There is a powerful mathematical method for simulating quantum dynamics that goes beyond finite qubit representations called infinite-dimensional Hamiltonian systems. These systems make it possible to describe quantum fields, open quantum systems, and hybrid designs more accurately by simulating unbounded operators, spectral decompositions, and continuous-variable interactions. Quantum algorithms for partial differential equations, many-body physics, and error-tolerant models of infinite Hilbert spaces are all areas where they are useful. Adding Hamiltonians with infinite dimensions to computer systems makes it possible to use both classical approximation and quantum hardware. This method makes it easier to scale up, improves accuracy when dynamics aren’t simple, and sets the stage for more advanced robust quantum computing models in chemistry, information sciences, and physics.