Guide to Quantum Computing and Schrödingerization

Introduction: Quantum Computing - A New Computing Paradigm Beyond Binary Logic

What is Quantum Computing?

Quantum computing is a novel computational paradigm based on the principles of quantum mechanics. It utilizes non-classical properties such as superposition and entanglement to perform information processing in fundamentally new ways. Traditional computers are like digital switches—each bit can only be either 0 or 1 at any given moment. In contrast, quantum computers are more like dimmer switches, where each quantum bit (qubit) can exist in a superposition of both 0 and 1 simultaneously. This property allows quantum systems to explore a computational state space that is exponentially larger than that of classical systems, thereby offering the potential for exponential parallelism in certain computational tasks.

• Core Differences:

  • Traditional Computer: Based on binary bits (0/1), processes information sequentially or with limited parallelism.

  • Quantum Computer: Based on quantum bits (qubits), processes information using superposition and interference mechanisms.

Evolution of Quantum Computing: From “a Few Qubits” to the Era of 100-Qubit Volume

The development of quantum computing can be seen as a gradual transition from prototype validation to engineered systems:

• Early Stage (1990s - 2000s):f

  • Qubit Count: In the initial phase, only a very limited number of qubits could be controlled. For example, in 1998, IBM demonstrated a quantum computing prototype with just 2 qubits.

  • Technical Challenges: This stage faced significant physical-level challenges. Qubits were extremely sensitive to environmental noise, leading to rapid decoherence and difficulty in maintaining quantum coherence. Additionally, quantum gate operations lacked sufficient precision, making it impossible to run meaningful quantum algorithms.

  • Stage Characteristics: The primary goal was to demonstrate basic controllability and verify that the principles of quantum mechanics could be applied to information processing. This stage marked the transition from theoretical models to physical experimental systems.

• Mid-term Breakthrough (2010s):

  • Qubit Count: In 2016, IBM launched a 5-qubit programmable quantum computer, while Google achieved high-fidelity control of 9 qubits.

  • Technological Progress: Physical platforms such as superconducting qubits and ion traps gradually matured. Improvements in system stability, qubit connectivity, and control precision enabled more practical experimentation. During this period, the concept of Quantum Volume was proposed as a comprehensive metric for evaluating a quantum processor’s effective computational capacity.

    • What is Quantum Volume? Quantum Volume considers not only the number of qubits, but also gate fidelity, qubit connectivity, and circuit depth. It is similar to evaluating a car—not just by engine power, but by its overall performance, including handling and fuel efficiency.

  • Stage Characteristics: Programmable validation, where processors could run small-scale algorithms and serve as testbeds for algorithm design and prototype research.

• Recent Leap (2020s):

  • Qubit Count: In 2023, IBM introduced the 133-qubit “Heron” processor, and Origin Quantum released the 72-qubit “Wukong” chip. In late 2024, China launched the Zuchongzhi-3, a 105-qubit superconducting processor integrating 182 coupled qubits. The system successfully executed an 83-qubit, 32-layer random circuit sampling task. According to estimations, the fastest classical supercomputer would take over 6.4 billion years to simulate the same task—indicating a quantum speedup of approximately 10¹⁵ times for this specific benchmark.

  • Technological Progress: Alongside increasing qubit counts, the overall fidelity, control parallelism, and circuit depth capabilities have steadily improved. Leading platforms can now support the execution of deeper and more complex quantum circuits.

  • Stage Characteristics: The field is entering a phase of system-level optimization and early-stage practical exploration. The research focus is shifting from isolated physical control to system architecture design, quantum workload scheduling, and noise-aware compilation techniques. Quantum computers are gradually evolving into engineered platforms with potential for solving specialized real-world problems.

Why is Quantum Computing important?

Quantum Computing can solve tasks that traditional computers cannot accomplish, such as:

• Breaking Complex Passwords: RSA encryption algorithm may be easily broken by quantum algorithms (such as Shor’s algorithm).

• Accelerate drug development: Simulate molecular structures and quickly filter effective drugs (such as anti-cancer drugs).

• Optimize Transportation and Logistics: Find the optimal solution among millions of routes, saving time and energy.

• Accelerating Scientific Computing and Engineering Applications: Achieving exponential acceleration in linear algebra and partial differential equation solving problems.

Foundations of Quantum Computing

Qubit1

• Popular Explanation: Traditional bits are like the “heads” or “tails” of a coin, being only 0 or 1; qubits are like a spinning coin, existing in a superposition of both heads and tails before it stops.

• Physical Implementation:

  • Superconducting qubits (e.g., IBM, Google):

    • Principle: Utilize the current loop of superconducting materials at extremely low temperatures (close to -273°C) as qubits.

    • Advantages: Easy to integrate, suitable for large-scale expansion; Disadvantages: Requires complex refrigeration equipment.

  • Ion Trap Qubit(e.g., Honeywell):

    • Principle: Use lasers to trap charged atoms (ions), and encode quantum information through the vibrational states of the ions.

    • Advantages: High stability; Disadvantages: Slow operating speed.

  • Photonic qubits (e.g., China’s “Jiuzhang”):

    • Principle: Represent qubits using the polarization or path state of photons.

    • Advantages: Strong anti-interference, suitable for quantum communication; Disadvantages: Difficult to store information.

Quantum state

• Popular explanation: Quantum state is the “state description” of a quantum system, just like the “probability cloud” in weather forecasting, which tells us the possible states and positions of particles. For example, the electron in a hydrogen atom does not have a fixed orbit, and its quantum state describes the probability distribution of the electron’s appearance around the nucleus.

• Mathematical Representation:

  • $|0⟩$and$|1⟩$: The ground states of qubits, representing classical 0 and 1 respectively.

  • $|+⟩$and$|-⟩$: Superposition states of qubits.

  $|+⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩),|-⟩=\frac{1}{\sqrt{2}}(|0⟩-|1⟩)$

  • The ground state of multiple qubits can be represented by tensor products, for example $|\psi ⟩=|0⟩\otimes |1⟩=|01⟩$.

• Superposition State:

  • The quantum state of a single qubit can be expressed as $|\psi ⟩=\alpha |0⟩+\beta |1⟩$, where $|0⟩$ and $|1⟩$ are the ground states, and α and β are complex numbers. $|\alpha {|}^2$ and $|\beta {|}^2$ represent the probability of the state being $|0⟩$ and $|1⟩$ repectively.

  • The superposition state of multiple qubits is a linear combination of the ground states

  $|\psi ⟩={\sum }_{i=0}^{2^n-1}{\alpha }_i|i⟩={\sum }_{i_0,\cdots ,i_{n-1}=0,1}{\alpha }_{i_0\cdots i_{n-1}}|i_0⟩\cdots |i_{n-1}⟩$

Quantum Logic Gates

In Quantum Computing, qubits are the basic carriers of information, and quantum gates are like magical “switches” that manipulate these qubits , whose mathematical representation is a unitary operator . Through the operations of different quantum gates, we can change the state of qubits and achieve various complex computational tasks . Any quantum gate acting on $n$ qubits can be decomposed into a combination of some basic quantum gates . The following details several common quantum gates for you

Single-qubit gate

Two-qubit quantum gate

Three-Qubit Gate

Quantum Measurement and Density Matrix

Quantum Measurement: From Superposition State to Definite State

• Collapse phenomenon: Measurement forces the quantum state to “collapse” from a superposition state to a definite state ($|0⟩$ or $|1⟩$). Just as a spinning coin is stopped by a pat, the superposition state of a qubit will “choose” a definite outcome when measured. The probability of a measurement outcome is determined by the square of the amplitude of the quantum state. If the quantum state is: $|\psi ⟩=\alpha |0⟩+\beta |1⟩$, the probability of measuring 0 is $|\alpha {|}^2$, and the probability of measuring 1 is $|\beta {|}^2$.

• Observed Quantity:

  Quantum measurement corresponds to making observations in physical experiments and is usually represented by an observable quantity. Usually, projective measurements are considered, and the observable $M$ is expressed as:

  $M=\sum \ast i{\lambda }_i{P}_i,P_i^2=P_i,P_i{P}_j=\delta \ast ij.$

  where $P_i$ is the projection operator, which represents projecting the quantum state onto the $i$-th basis state, and at this time, the measurement result ${\lambda }_i$ will be obtained.

Density Matrix

Pure State and Mixed State

• Pure State: Can be described by a single quantum state vector $|\psi ⟩$.

• Mixed State: The system is in a statistical mixture of multiple pure states and cannot be described by a single state vector.

Density Matrix

The density matrix is a tool for describing the state of a quantum system, applicable to both pure and mixed states. For a pure state $|\psi ⟩$, its density matrix is:

$\rho =|\psi ⟩⟨\psi |$

For a mixed state, the density matrix is:

$\rho ={\sum }_i{p}_i|{\psi }_i⟩⟨{\psi }_i|$

where $p_i$ is the probability of the system being in state $|{\psi }_i⟩$.

Example: Derivation of the Density Matrix of a Pure State

• For pure states $|\psi ⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$, then:

$\rho =|\psi ⟩⟨\psi |=\left(\frac{1}{\sqrt{2}}(|0⟩+|1⟩)\right)\left(\frac{1}{\sqrt{2}}(⟨0|+⟨1|)\right)$

After expansion, we obtain:

$\rho =\frac{1}{2}(|0⟩⟨0|+|0⟩⟨1|+|1⟩⟨0|+|1⟩⟨1|)$

Represented by a matrix as:

$\rho =\left[\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ \frac{1}{2}& \frac{1}{2}\end{array}\right]$

Geometric Significance of the Bloch Sphere

• North Pole ($|0⟩$): Indicates that the quantum state is completely in $|0⟩$.

• South Pole ($|1⟩$): Indicates that the quantum state is completely in $|1⟩$.

• Spherical Point: Represents the quantum state in a superposition state of $|0⟩$ and $|1⟩$, for example $|+⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$.

• Points inside the sphere: represent mixed states, where the quantum state is not a pure state but a statistical mixture of multiple pure states.

Parameterization of the Bloch Sphere

The pure state of a single qubit can be expressed as:

$|\psi ⟩=\cos \left(\frac{\theta }{2}\right)|0⟩+e^{i\varphi }\sin \left(\frac{\theta }{2}\right)|1⟩$

Where:

• $\theta$ is the angle between the quantum state vector and the z-axis, called the polar angle, with a range of $[0,\pi ]$, representing the latitude of the quantum state on the Bloch sphere.

• $\varphi$ is the angle between the projection of the quantum state vector on the x-y plane and the x-axis, called the azimuthal angle, with a range of $[0,2\pi )$, representing the longitude of the quantum state on the Bloch sphere.

Example

• Pure state: $|\psi ⟩=\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$ corresponds to the point on thexaxis direction of the Bloch sphere, $\theta =\frac{\pi }{2}$, $\varphi =0$.

• Mixed state: $\rho =\left[\begin{array}{cc}0.6& 0.2\\ 0.2& 0.4\end{array}\right]$ corresponds to a point inside the Bloch sphere.

Quantum State Evolution and Decoherence

Quantum state evolution

Quantum gate operations change the quantum state, thereby altering its probability distribution and density matrix. For example, the Hadamard gate transforms $|0⟩$ into $\frac{1}{\sqrt{2}}(|0⟩+|1⟩)$, with its probability distribution changing from P(0)=1, P(1)=0 to P(0)=0.5, P(1)=0.5.

decoherence process

Decoherence refers to the loss of coherence of a quantum system after interacting with its environment, where the off-diagonal elements of the density matrix gradually decay. Decoherence causes the quantum system to change from a pure state to a mixed state.

Mathematical Description of Decoherence: Suppose the initial state is a pure state $|\psi ⟩=\alpha |0⟩+\beta |1⟩$, and its density matrix is:

$\rho =\left[\begin{array}{cc}|\alpha {|}^2& \alpha {\beta }^{\ast }\\ {\alpha }^{\ast }\beta & |\beta {|}^2\end{array}\right]$

During the decoherence process, the off - diagonal elements $\alpha {\beta }^{\ast }$ and ${\alpha }^{\ast }\beta$ gradually decay, and finally the density matrix becomes:

$\rho =\left[\begin{array}{cc}|\alpha {|}^2& 0\\ 0& |\beta {|}^2\end{array}\right]$

This indicates that the system degenerates from a pure state to a classical mixed state.

Schrödingerization: The “Universal Key” of Quantum Simulation

Introduction: Schrödingerisation is a method that uses quantum simulation to solve linear partial differential equations (PDEs). This method transforms PDEs into Hamiltonian systems, enabling them to be efficiently solved through Quantum Computing. We first introduce the basic concepts of quantum mechanics, especially the physical meaning of the Schrödinger equation.

Schrödinger Equation

Wave Function and Quantum State

• Popular Explanation: The wave function is the “probability cloud” of a quantum system, describing the possible positions and states of particles. For example the electron in a hydrogen atom does not have a fixed orbit, and its wave function shows the probability of the electron appearing around the nucleus.

• Mathematical Form:

  • Time-dependent Schrödinger equation:

  $i\hslash \frac{\partial }{\partial t}|\psi ⟩=\hat{H}|\psi ⟩$

  • $\hat{H}$ is the Hamiltonian, which determines how the system evolves (similar to how Newton’s laws govern the motion of objects).

  • Stationary Schrödinger Equation:

  $\hat{H}|\psi ⟩=E|\psi ⟩$

  • Used to calculate the stable energy of quantum systems (such as atomic energy levels).

Quantum Measurement and Uncertainty

• Measurement Collapse: Measurement destroys the quantum superposition state. For example, measuring a qubit in the state $|+⟩$ results in a random outcome of 0 or 1.

• Heisenberg Uncertainty Principle:

  • It is impossible to simultaneously know precisely the position and velocity of a particle.

  • Mathematical Expression: $\Delta x\Delta p\ge \frac{\hslash }{2}$

Basic Introduction to Schrödingerization Method

Authors: Shi Jin, Nana Liu, Yue Yu

Introduction: Transformation from the Heat Equation to the Schrödinger Equation

• The evolution operator of the Schrödinger equation $U$ is (assuming $\hslash =1$)

$|\psi ⟩=U|{\psi }_0⟩=e^{-i\hat{H}t}|{\psi }_0⟩,U^{†}U=I.$

Since the Hamiltonian$\hat{H}$is a Hermitian operator (${\hat{H}}^{†}=\hat{H}$),$U=e^{-i\hat{H}t}$is a unitary operator.

• The heat equation describes the diffusion behavior of the temperature field, and its mathematical expression is as follows:

${\partial }_{t}u-\Delta u=0,u(0,x)=u_0(x)$

The original equation cannot be directly solved on a Quantum Computer because its time evolution operator is not a unitary matrix, To make it applicable to Quantum Computing, we introduce a new variable $p$, and define a new function:

$v(t,x,p)=e^{-p}u(t,x),p>0$

After the transformation, the equation becomes:

${\partial }_{t}v+{\partial }_p{\Delta }_{x}v=0.$

Perform a Fourier transform on the spatial variables, yielding:

${\partial }_t\hat{v}-|\xi {|}^2{\partial }_p\hat{v}=0,$

where $\xi$ denotes the Fourier dual variable corresponding to $x$. This is a convection equation that propagates to the left in the $p$direction. In the right half $p>0$, the solution of the equation is only related to the initial values in the right half. Therefore, we can arbitrarily set $p<0$ the initial values introduced in the $v(0,x,p)$ part, for example $v(0,x,p)=e^{-|p|}u(0,x),p<0.$

By applying the Fourier transform with respect to the variable $p$, we obtain:

${\partial }_t\hat{v}-i\eta {\Delta }_x\hat{v}=0,$

where $\eta$ denotes the Fourier dual variable corresponding to $p$. So far, we have obtained an equation in Schrödinger form, whose evolution operator is $U=e^{i\hat{H}t},\hat{H}=\eta {\Delta }_x.$

Schrödingerization Method for Linear Differential Equations (qubits)

• In linear ordinary differential equation systems , consider the common form:

$\frac{du}{dt}=Au(t),$

where$A$ is a matrix. To adapt to Quantum Computing, we split $A$ into Hermitian and anti- Hermitian parts:

$A=H_1+i{H}_2,H_1=\frac{A+A^{†}}{2},H_2=\frac{A-A^{†}}{2i}.$

Then through warped phase transformation

$v(t,x,p)=e^{-p}u(t,x),p>0,$

${\partial }_{t}v=-H_1{\partial }_{p}v+i{H}_{2}v,$

and Fourier Transform,

$\hat{v}(t,x,\eta )={\mathcal{F}}_p\{v(t,x,p)\}={\int }_{-\infty }^{\infty }v(t,x,p)e^{i\eta p}dp,$

${\partial }_t\hat{v}=i\eta H_1\hat{v}+i{H}_2\hat{v}\triangleq i\hat{H}\hat{v},$

Convert this ODE into a Hamiltonian system, with the evolution operator being $U=e^{i\hat{H}t},\hat{H}=\eta H_1+H_2.$ enabling direct simulation via a Quantum Computer.

• Discretize the variable $p$ into a grid, and the computational domain $\Omega$ and grid size $\Delta p$ are defined as follows

$\Omega =[-\pi R,\pi R],\Delta p=\frac{2\pi R}{N_p},N_p=2^{n_p}.$

The initial value after introducing variable $p$ is defined as

$v(0,x,p)=g(p)u(0,x),g(p)=\{\begin{array}{rl}& f(p),p\le 0,\\ & e^{-p},p>0,\end{array}$

where $f(0)=1,f(p)\in L^2((-\infty ,0))$. $g(p)$ can be chosen under two conditions: it will rapidly decay as $p$ tends to $-\infty$and has better smoothness. For example, one may choose

$g(p)=\{\begin{array}{rl}& e^p,p\in (-\infty ,-1),\\ & {\mathcal{P}}_{2k+1},p\in [-1,0],\\ & e^{-p},p\in (0,+\infty ),\end{array}$

where ${\mathcal{P}}_{2k+1}$ is a Hermite interpolation and satisfies

Here $0\le k^{\prime }\le k$ is an integer. The initial value after discretization is

$\mathbf{v}(0)=\mathbf{g}\otimes \mathbf{u}(0):=[g(p_0),\cdots ,g(p_{M-1}){]}^T\otimes \mathbf{u}(0).$

The discretized equation is transformed into

$\frac{d}{dt}{\hat{\mathbf{v}}}_{DV}=i(D\otimes H_1+I_{N_p}\otimes H_2){\hat{\mathbf{v}}}_{DV}=i{H}_{DV}{\hat{\mathbf{v}}}_{DV}.$

The original equation is transformed into a Hamiltonian simulation problem, with the evolution operator being $U_{DV}=e^{i{H}_{DV}t}$.

• For linear partial differential equations, we can first discretize them in space to obtain ordinary differential equations. When the equations have different boundary conditions or source terms, we obtain the following non-homogeneous ordinary differential equations

$\frac{du}{dt}=Au(t)+b.$

Introduce a new variable

$w\equiv 1,\tilde{u}=[u;w],\tilde{A}=[A,B],B=\text{diag}(b),$

The original equation is transformed into

$\frac{d\tilde{u}}{dt}=\tilde{A}\tilde{u}(t).$

Then, use the above-mentioned Schrödingerization method to perform quantum simulation.

Complexity Discussion

• Advantages of Quantum Computing in PDE Solving

  • Exponential Acceleration: Compared to classical computing, the computational complexity of quantum algorithms can be reduced tologarithmic level.

  • Hamiltonian Simulation Method: Utilize Trotter-Suzuki Splitting Method or Quantum Singular Value Transformation (QSVT) and Quantum Fourier Transform for time evolution calculations.

• Complexity Estimation: Assuming the goal is to compute the time evolution of the PDE, the complexity estimation is as follows:

  $\mathcal{O}\left(\frac{\Vert \mathbf{u}(0)\Vert }{\Vert \mathbf{u}(t)\Vert }s{\Vert A\Vert }_{\max }|t|{\delta }^{-\frac{3}{2(k+1)}}\right)$

Schrödingerization Method for Continuous Variables (qumodes)

The Schrödingerization method naturally applies to the case of continuous variables. Qumodes correspond to qubits, where the solution $u(t,x)$ is encoded as a continuous-variable quantum state

$|u(t)⟩\propto \int u(t,x)|x⟩,$

where$|x⟩,x\in {\mathbb{R}}^d$is an orthonormal basis in an infinite-dimensional Hilbert Space. When$x$represents the position of the wave function$u(x,t)$, the corresponding operator is$\hat{x}$, and the conjugate operator is the momentum operator$\hat{p}$, satisfying

$\hat{x}|x⟩=x|x⟩,\hat{p}=-i\partial /\partial x,[\hat{x},\hat{p}]=iI$

where$I$ is the identity operator. At this time, for example, the evolution operator of the heat equation$\exp (-At)$ can be encoded as the following unitary operator:

$U_{CV}(t)=\exp (-i{\hat{p}}^2\otimes \hat{\eta }t),$

At this point $\hat{\eta }$ can be chosen as any single-mode operator, for example $\hat{\eta }=\hat{x}$. In this case, there is no need to discretize any variables in the system. This method directly deals with the continuous nature of partial differential equations and does not rely on the details of discretization schemes, thus enabling more accurate simulation of partial differential equations.

When solving partial differential equations previously, after discretizing the differential operator $A$, it can be represented by a finite-dimensional matrix. In the absence of a discretization scheme, the continuous nature of the differential operator can be retained, and at this time $A=H_1+i{H}_2$ acts on qumodes rather than qubits, where $H_1,H_2$ is a Hermitian operator that also acts on qumodes. At this point, $\exp (-At)$ can be embedded into the following unitary operator:

$U_{CV}(t)=\exp (-i(H_1\otimes \hat{\eta }+H_2\otimes I)t).$

In the case of continuous variables, there is no direct analog of sparse access or block access methods to handle infinite-dimensional operators $A$ (whose norms are infinite), so digital quantum simulation algorithms cannot be directly applied. In this case, it is more appropriate to consider analog quantum simulation, i.e., constructing quantum gates through operators that naturally implement appropriate unitary operations, rather than using primitive gates. For example, if one can access $U_1(t)=\exp (-i{H}_1\otimes \hat{\eta }t)$ and $U_2(t)=\exp (-i{H}_2\otimes It)$, and $[H_1,H_2]=0$, then only $U_1(t)U_2(t)$ needs to be implemented. This situation applies when $H_1$ and $H_2$ both act only on $\hat{x}$ (e.g., ODE problems) or only on $\hat{p}$ (e.g., homogeneous partial differential equation problems with only constant coefficients). More complex cases can be analyzed on a case-by-case basis according to specific problems.

Quantum Algorithm for the Semiclassical Schrödinger Equation

Authors: Shi Jin, Xiantao Li, Nana Liu

In the semi-classical regime, due to $\hslash$ being small, there are significant multi-scale phenomena in the system, and traditional numerical methods require extremely fine grids to capture the characteristics of rapid oscillations. The following details how to use quantum algorithms to solve the Schrödinger equation in this scenario.

The semi-classical form of the Schrödinger equation

• In the semi-classical region, the Schrödinger equation is written as:

$i\frac{\partial }{\partial t}|\psi (t)⟩=H|\psi (t)⟩,H=-\frac{{\hslash }^2}{2}\Delta +{\hslash }^{-1}V(x)$

Spatial Discretization and Pseudo-Spectral Method

• Spatial Discretization:

  Let the spatial domain be $[0,L]$, and the uniform grid be:

  $\Delta x=\frac{L}{M},x_j=j\Delta x,j=0,1,\dots ,M-1$

• Pseudo-spectral method:

  Solving spatial derivatives using Fourier series expansion, the Laplace operator can be efficiently computed through Fast Fourier Transform (FFT).

Time Evolution: Strang Splitting Method

• Strang splitting step

  To solve the time evolution problem, the splitting method is used to decompose the overall evolution operator into kinetic and potential energy parts. Its approximate formula is:

  $|{\psi }_{n+1}⟩\approx U_K\left(\frac{\Delta t}{2}\right)U_V(\Delta t)U_K\left(\frac{\Delta t}{2}\right)|{\psi }_n⟩$

• Potential Energy Operator:

$U_V(\Delta t)=\exp \left(-i\frac{\Delta t}{\hslash }V(x)\right)$

• Kinetic energy operator:

$U_K(\Delta t)=\exp \left(i\frac{\Delta t\hslash \Delta }{2}\right)$

Complexity Estimation and Resource Analysis

When the solution has $\ell$ -order continuous derivatives, the quantum gate complexity is estimated as:

$O((\epsilon \hslash {)}^{-1/2}\mathrm{polylog}((\epsilon \hslash {)}^{-3/(2\ell )}{\hslash }^{-1-1/\ell }\left)\right)$

Furthermore, if only physical observables are considered, the time step can be chosen independently of $\hslash$, further reducing the overall complexity.

Quantum Algorithms for Nonlinear Partial Differential Equations

Authors: Shi Jin, Nana Liu

For nonlinear partial differential equations, traditional methods often require approximate linearization, which may introduce significant errors. This scheme uses the level set method to accurately map the nonlinear PDE to a linear PDE, and then combines the Schrödingerization method to implement quantum simulation, thereby preserving the nonlinear characteristics of the original problem.

Level Set Method Mapping

• Mapping Principle: Using the level set method, the solution of the nonlinear PDE is represented as an implicit surface, thereby transforming the nonlinear problem into a linear one without the need for low-order approximation.

• Hamilton-Jacobi equation:

${\partial }_{t}u+H(\nabla u,x)=0,$

Define the level set function $\varphi (t,x,p)$ as follows :

$\varphi (t,x,p=u(t,x))=0,$

$\varphi$ satisfies the following linear Liouville equation:

${\partial }_t\varphi +{\nabla }_{p}H\cdot {\nabla }_x\varphi -{\nabla }_{x}H\cdot {\nabla }_p\varphi =0.$

• Nonlinear scalar first-order hyperbolic equation:

${\partial }_{t}u+F(u)\cdot {\nabla }_{x}u+Q(x,u)=0$

Define the level set function $\varphi (t,x,p)$ as follows:

$\varphi (t,x,p=u(t,x))=0,$

satisfies $\varphi$ the following equation:

${\partial }_t\varphi +F(p)\cdot {\nabla }_x\varphi -Q(x,p){\partial }_p\varphi =0.$

• Schrödingerization: Introduce the mapped linear PDE using auxiliary variables and transform it into the Schrödinger equation .

Quantum Algorithm Construction and Complexity Analysis

• Quantum Algorithm: Construct quantum algorithms on the transformed Schrödinger system to achieve efficient computation of physical observables.

• Complexity Advantage: It is proven that when the initial data volume $M$is large, the consumption of quantum resources (number of gates, qubits) is independent of $M$, and even exponential acceleration can be achieved.

Quantum dynamics simulation with artificial boundary conditions

Authors: Shi Jin, Nana Liu, Xiantao Li, Yue Yu

When simulating quantum dynamics problems in practice, to prevent system information leakage, it is necessary to introduce artificial boundary conditions (ABC) within a finite computational domain. However, ABC will disrupt the unitarity of the system, making it difficult to directly apply traditional quantum simulation methods. This scheme converts non-unitary evolution into the form of the Schrödinger equation through the Schrödingerization method, thereby restoring unitarity.

Mathematical Modeling

Suppose the quantum wave function $\psi (x,t)$ satisfies the following Schrödinger equation:

$i\frac{\partial \psi }{\partial t}=H\psi$

where the Hamiltonian operator$H$ includes kinetic and potential energy terms:

$H=-\frac{1}{2}\Delta +V(x)$

If the computational domain is $[0,L]$, at the boundary $x=L$, set artificial boundary conditions:

$\psi (L,t)=\alpha \psi (L-\Delta x,t)+\beta \frac{\partial \psi }{\partial x}{|}_{x=L}$

Where:

• $\alpha$ and $\beta$ are adjustment parameters used to simulate the boundary conditions of an open system.

• Schrödingerization: Introduce the mapped linear PDE using auxiliary variables and transform it into the Schrödinger equation.

Numerical Verification and Complexity Analysis

• Complexity Analysis:

The complexity of Hamiltonian simulation is

$\mathcal{O}\left(s{\Vert H\Vert }_{\max ,1}\frac{\log (s{\Vert H\Vert }_{\max ,1}/\epsilon )}{\log \log (s{\Vert H\Vert }_{\max ,1}/\epsilon )}\right),{\Vert H\Vert }_{\max ,1}={\int }_0^T{\Vert H\Vert }_{\max }(t)dt.$

• Numerical Verification: Through simulation experiments, it is verified that this method can maintain system stability and achieve high solution accuracy when dealing with artificial boundary conditions.

Quantum Simulation of Discrete Linear Dynamical Systems and Iterative Methods

Authors: Shi Jin, Nana Liu

Discrete Linear Dynamical Systems and Iterative Methods are widely used in numerical computation and engineering applications. Classical iterative methods usually require multiple matrix operations, resulting in high computational costs, while Quantum Computing, through Schrödingerisation technology, can effectively simulate the continuous limit of discrete systems and be applied to solving linear equations and eigenvalue estimation problems.

Continuation of Iterative Methods

• Continuous ODE representation: For example,the iterative method for general linear problems can be expressed as

$y_{k+1}=G{y}_k+g,y_{k+1}-y_k=(G-I)y_k+g$

can be written in continuous-time form:

$\frac{dy(t)}{dt}=(G-I)y+g$

• Schrödingerization: Introduce auxiliary variables into the transformed ordinary differential equation and further transform it into the Schrödinger equation.

Quantum Simulation and Complexity Analysis

• Quantum Algorithm Implementation: Utilize continuous-time quantum simulation technology to replace the matrix multiplication process in classical iteration, achieving more efficient iterative solutions.

• Complexity Discussion: Analyzed the advantages of the algorithm in terms of iterative convergence and quantum resources, and demonstrated that it can achieve lower complexity than classical methods in certain problems.

Quantum Algorithm for Multiscale Partial Differential Equations

Authors: Junpeng Hu, Shi Jin, Lei Zhang

Multiscale Partial Differential Equations (PDEs) are widely used in physics, engineering, and other scientific fields. These equations pose significant challenges for numerical solutions due to their inclusion of multiple spatiotemporal scales. Classical methods require the use of extremely fine grids and time steps to accommodate the multiscale nature, but this leads to high computational costs, especially in high-dimensional problems. Quantum Computing provides a new solution for multiscale PDEs, which can significantly reduce computational complexity in certain cases.

Multiscale Elliptic Equation

Elliptic multi-scale PDEs mainly appear in fields such as conduction problems and seepage problems in composite materials. We consider the following second-order elliptic equation:

$-\nabla \cdot (A(x)\nabla u(x))=f(x),x\in D,u(x)=0,x\in \partial D.$

where the coefficient matrix $A(x)$has high oscillation and may have multiple scales.

• Two-scale homogenization model

For the two-scale case, assume that the coefficient matrix has the following structure:

$A(x):=A(x,x/\epsilon ),$

where $\epsilon$ is the small-scale parameter. Using the homogenization method, we can construct the homogenized equation, whose solution $u_0(x)$ approximates the solution of the original equation:

• Multiscale Homogenization Model

• Schrödingerization: First approximate the linear algebraic system using ordinary differential equations, then introduce auxiliary variables, and convert it into a Schrödinger equation.

Multiscale Parabolic Equation and Wave Equation

Consider the following multi-scale heat conduction equation:

$\frac{\partial u}{\partial t}-\nabla \cdot (A(x/\epsilon )\nabla u)=f(x,t),$

where $A(x/\epsilon )$has the characteristic of rapid oscillation.

• Two-scale homogenization model

• Schrödingerization: First, perform time discretization on the equations of the finite element method, then convert them into ordinary differential equations and apply the Schrödingerization method

• Wave Equation: First, transform it into a first-order equation, and then handle it in the same way as the parabolic equation

Computational Complexity Analysis

• The computational complexity of classical numerical methods is usually $O(N^d)$, where $N$ is the number of grid points, $d$ is the dimension.

• The computational complexity of the Quantum Computing method can be reduced to: $O(\text{polylog}(N))$.

Quantum Simulation of Maxwell’s Equations

Authors: Shi Jin, Nana Liu, Chuwen Ma

Maxwell’s equations describe the spatio-temporal evolution of electromagnetic fields. Traditional numerical methods (such as the Yee algorithm) can well preserve physical properties during the discretization process, but in Quantum Computing, they need to be converted into the form of the Schrödinger equation to adapt to the quantum simulation framework.

Basic Forms of Maxwell’s Equations

• Formula:

Handling boundary conditions and interface problems

• Introduction to Boundary Conditions: Physical boundary conditions (such as perfect conductors and impedance boundaries) require that the electric and magnetic fields satisfy specific constraints. In traditional numerical methods, these conditions are maintained through specific discretization schemes, but they need to be appropriately transformed in quantum simulations.

• Unitary Transformation and Auxiliary Variables: By introducing auxiliary variables and unitary transformation, non-homogeneous boundary conditions are transformed into homogeneous conditions.

Schrödingerization Transformation and Numerical Implementation

• Conversion Steps: Use the warped phase transformation to convert the Maxwell equations into a high-dimensional Schrödinger system, and then combine the Trotter splitting method and Fourier transform to achieve time evolution.

• Numerical Verification: The effectiveness of this method in maintaining magnetic field divergence-free and energy conservation was verified through numerical experiments, while the requirements for quantum gate complexity and qubit resources were also discussed.

More Application Cases

This method is not only applicable to the heat equation and the convection equation, but can also be applied to more complex mathematical models, such as:

• Black-Scholes Equation (Financial Marekt Modeling)

• Fokker-Planck Equation (Stochastic Process)

• Linear Boltzmann Equation (Particle Transport)

• Vlasov-Fokker-Planck Equation (Plasma Dynamics)

• Hamilton-Jacobi Equation (Nonlinear Dynamical System)

Quantum Computing Glossary

References

• Shi Jin, Nana Liu, Yue Yu.(2022).Quantum simulation of partial differential equations via Schrodingerisation: technical details 

• IBM Quantum. (2023). What is quantum computing?

• Shi Jin, Nana Liu, Chuwen Ma.(2023).Quantum simulation of Maxwell’s equations via Schrödingersation 

• Junpeng Hu, Shi Jin, Lei Zhang.(2023).Quantum Algorithms for Multiscale Partial Differential Equations 

• Shi Jin, Nana Liu.(2023).Quantum simulation of discrete linear dynamical systems and simple iterative methods in linear algebra via Schrodingerisation 

• Shi Jin, Nana Liu, Xiantao Li, Yue Yu.(2023).Quantum Simulation for Quantum Dynamics with Artificial Boundary Conditions 

• Shi Jin, Nana Liu.(2022).Quantum algorithms for computing observables of nonlinear partial di erential equations 

• Shi Jin, Xiantao Li, Nana Liu, Yue Yu.(2023).Quantum Simulation for Partial Differential Equations with Physical Boundary or Interface Conditions 

• Shi Jin, Nana Liu, Yue Yu.(2023).Quantum simulation of partial differential equations: Applications and detailed analysis 

• Shi Jin, Xiantao Li, Nana Liu.(2022).On quantum algorithms for the Schrödinger equation in the semi-classical regime