量程 DynamicsQuantum Dynamics

量程机制很大程度上是对量程动态的调查,旨在了解初始量程状态 $ \ket{\psi (0)} $ 如何随时间而变化(有关 Dirac 表示法的详细信息,请参阅有关量子计算的概念文档)。Quantum mechanics is largely the study of quantum dynamics, which seeks to understand how an initial quantum state $\ket{\psi(0)}$ varies over time (see the conceptual docs on quantum computing for more info on Dirac notation). 具体来说,在此初始条件下,将会发现量程状态、dynamical 系统的发展时间和规范,并会寻找量程状态 $ \ket{\psi (t)} $。Specifically, given this initial condition a quantum state, an evolution time and a specification of the quantum dynamical system, a quantum state $\ket{\psi(t)}$ is sought. 在继续解释量子 dynamics 之前,采取一步操作并考虑传统 dynamics 会很有用,因为它可让你深入了解量程机制与传统动态的不同之处。Before proceeding to explain quantum dynamics, it is useful to take a step back and think about classical dynamics since it provides insights into how quantum mechanics is really not that different from classical dynamics.

在传统 dynamics 中,我们从牛顿的第二个定律来判断,微粒的位置根据 $F (x,t) = ma = m\frac {\ dd ^ 2} {\dd t ^ 2} {x} (t) $,其中 $F (x,t) $ 为强制,$m $ 为大容量,$a $ 为加速。In classical dynamics, we know from Newton's second law of motion that the position of a particle evolves according to $F(x,t)=ma=m\frac{\dd^2}{\dd t^2}{x}(t)$, where $F(x,t)$ is the force,$m$ is the mass and $a$ is the acceleration. 然后,假设初始位置 $x (0) $、演化时间 $t $,以及对该粒子执行操作的力的说明,则我们可以通过对由 $x (t) $ 牛顿的方程式公式给定的差异公式来查找 $x (t) $。Then, given an initial position $x(0)$, evolution time $t$, and description of the forces that act on the particle, we can then find $x(t)$ by solving the differential equation given by Newton's equations for $x(t)$. 以这种方式指定力就是一个难题。Specifying the forces in this way is a bit of a pain. 因此,我们经常会根据系统的潜在能量来表达力,这为我们提供 $-\ partial_x V (x,t) = m \frac{\dd ^ 2} {\dd t ^ 2} {x} (t) $。So we often express the forces in terms of the potential energy of the system, which gives us $-\partial_x V(x,t) = m \frac{\dd^2}{\dd t^2}{x}(t)$. 因此,对于粒子,系统的 dynamics 只能由潜在能源函数、粒子质量和演化时间来指定。Thus, for a particle, the dynamics of the system are specified only by the potential energy function, the particle mass, and the evolution time.

对于超出 $F = ma $ 的传统 dynamics,通常会引入更广泛的语言。A broader language is often introduced for classical dynamics that goes beyond $F=ma$. 一种表述(特别适用于量子机制)是 Hamiltonian 的机制。One formulation, which is particularly useful in quantum mechanics, is Hamiltonian mechanics. 在 Hamiltonian 机制中,系统和(通用化)位置和 momenta 的总能量为描述任意传统对象的运动所需的所有信息。In Hamiltonian mechanics, the total energy of a system and the (generalized) positions and momenta give all the information needed to describe the motion of an arbitrary classical object. 具体而言,让 $f (x,p,t) $ 作为系统的一般化位置 $x $ 和 momenta $p $,并允许 $H (x,p,t) $ 为 Hamiltonian 函数。Specifically, let $f(x,p,t)$ be some function of the generalized positions $x$ and momenta $p$ of a system and let $H(x,p,t)$ be the Hamiltonian function. 例如,如果采用 $f (x,p,t) = x (t) $ and $H (x,p,t) = p ^ 2 (t)/2m-V (x,t) $,则我们将恢复上述 Newtonian dynamics 情况。For example, if we take $f(x,p,t)= x(t)$ and $H(x,p,t)=p^2(t)/2m - V(x,t)$, then we would recover the above case of Newtonian dynamics. 在通用性下,我们将 \begin{align} \frac{d}{dt} f & = \ partial_t f-(\ partial_x H \ partial_p f + \ partial_p H \ partial_x f)\\ & \defeq \ partial_t f + \{f,H\}。In generality, we then have that \begin{align} \frac{d}{dt} f &= \partial_t f- (\partial_x H\partial_p f + \partial_p H\partial_x f)\\ &\defeq \partial_t f + \{f,H\}. 此处的 \end{align} $\{f,H\} $ 称为泊松形括号,由于它在定义 dynamics 中扮演的中心角色,它在经典 dynamics 中显示为普遍。\end{align} Here $\{f,H\}$ is called the Poisson bracket and appears ubiquitously in classical dynamics because of the central role it plays in defining dynamics.

可以使用完全相同的语言描述量程 dynamics。Quantum dynamics can be described using exactly the same language. Hamiltonian 或总能量会完全指定任何闭合的量程系统的动态。The Hamiltonian, or total energy, completely specifies the dynamics of any closed quantum system. 但是,这两个理论之间的差别很大。There are, however, some substantial differences between the two theories. 在传统的机制中 $x $ 和 $p $ 只是数字,而在量程结构中则不是这样。In classical mechanics $x$ and $p$ are just numbers, whereas in quantum mechanics they are not. 的确,它们甚至不是在上下班: $xp \ne px $。Indeed, they do not even commute: $xp \ne px$.

描述这些非上下班途中对象的正确的数学概念是一个运算符,在这种情况下,$x $ 和 $p $ 只能采用一组离散的值,这两个值与矩阵的概念相符。The right mathematical concept to describe these non-commuting objects is an operator, which in cases where $x$ and $p$ can only take a discrete set of values coincides with the concept of a matrix. 因此,为了简单起见,我们假定量程系统是有限的,因此可以使用向量和矩阵进行描述。Thus for simplicity, we will assume that our quantum system is finite so that it can be described using vectors and matrices. 我们进一步要求这些矩阵是 Hermitian 的(也就是说,矩阵的共轭换位与原始矩阵相同)。We further require that these matrices be Hermitian (meaning that the conjugate transpose of the matrix is the same as the original matrix). 这可以确保这些矩阵的本征值是真实值;我们要做的一个条件是,当我们测量的数量类似于无法返回虚数的位置时。This guarantees that the eigenvalues of the matrices are real-valued; a condition which we impose to ensure that when we measure a quantity like position that we don't get back out an imaginary number.

正如物中的位置和势头势头需要用运算符替换的一样,Hamiltonian 函数需要替换为运算符。Just as the analogues of position and momentum in quantum mechanics need to be replaced by operators, the Hamiltonian function needs to be similarly replaced by an operator. 例如,对于可用空间中的粒子,已 $H (x,p) = p ^ 2/2m $,而在量程机制中,Hamiltonian 运算符 $ \hat{H} $ 是 $ \hat{H} = \hat{p} ^ 2/2m $,其中 $ \hat{p} $ 是动力运算符。For example, for a particle in free space we have that $H(x,p) = p^2/2m$ whereas in quantum mechanics the Hamiltonian operator $\hat{H}$ is $\hat{H}= \hat{p}^2/2m$ where $\hat{p}$ is the momentum operator. 从这一角度来看,从经典到量子 dynamics,只涉及到用运算符替换普通 dynamics 中使用的变量。From this perspective, going from classical to quantum dynamics merely involves replacing the variables used in ordinary dynamics with operators. 通过将普通的古典 Hamiltonian 转换为量子语言来构造 Hamiltonian 运算符后,我们可以通过 \begin{表达任意量子机械数量(即,量子机械运算符) $ \hat{f} (t) $ 的动态。align} \frac{d}{dt} \hat{f} = \ partial_t \hat{f} + [\hat{f},\hat{H}],\end{align} 其中 $ [f,H] = fH-Hf $ 称为 commutator。Once we have constructed the Hamiltonian operator by translating the ordinary classical Hamiltonian over to quantum language, we can express the dynamics of an arbitrary quantum mechanical quantity (i.e. quantum mechanical operator) $\hat{f}(t)$ via \begin{align} \frac{d}{dt} \hat{f} = \partial_t \hat{f} + [\hat{f},\hat{H}], \end{align} where $[f,H] = fH -Hf$ is known as the commutator. 此表达式与上面给出的传统表达式的不同之处在于,泊松形括号 $\{f,H\} $ 替换为 $f $ 和 $H $ 之间的 commutator。This expression is exactly like the classical expression given above with the difference that the Poisson bracket $\{f,H\}$ being replaced with the commutator between $f$ and $H$. 这一过程采用传统的 Hamiltonian,并使用它来查找量程 Hamiltonian,在量程术语中称为规范量化。This process of taking a classical Hamiltonian and using it to find a quantum Hamiltonian is known in quantum jargon as canonical quantization.

哪些操作员 $f $ 是最感兴趣的?What operators $f$ are we most interested in? 此问题的答案取决于要解决的问题。The answer to this depends on the problem that we want to solve. 最有用的要查找的数量或许是量程状态运算符,如前面的概念文档中所讨论的那样,可以使用它来提取想要了解 dynamics 的所有内容。Perhaps the most useful quantity to find is the quantum state operator, which as discussed in the earlier conceptual documentation can be used to extract everything that we would like learn about the dynamics. 执行此操作(并将结果简化为一个具有纯状态的情况)后,将找到量程状态的 Schrödinger 公式 "\begin{align} i \ partial_t \ket{\psi (t)} = \hat{H} (t) \ket{\psi (t)}"。After doing this (and simplifying the result to the case where one has a pure state), the Schrödinger equation for the quantum state is found \begin{align} i\partial_t \ket{\psi(t)} = \hat{H}(t) \ket{\psi(t)}. \end{align}\end{align}

尽管此公式可能不如上面给出的直观,但它可能是了解如何模拟量程或传统计算机上的量子 dynamics 的最简单的表达式。This equation, though perhaps less intuitive than that given above, yields perhaps the simplest expression for understanding how to simulate quantum dynamics on either a quantum or classical computer. 这是因为,可以采用以下形式表示差异公式的解决方案(对于 Hamiltonian 为常量的情况,在 $t $) \begin{align} \ket{\psi (t)} = e ^ {-iHt} \ket{\psi (0)} 中。This is because the solution to the differential equation can be expressed in the following form (for the case where the Hamiltonian is constant in $t$) \begin{align} \ket{\psi(t)} = e^{-iHt}\ket{\psi(0)}. \end{align} 此处 $e ^ {-iHt} $ 是单一矩阵。\end{align} Here $e^{-iHt}$ is a unitary matrix. 这意味着,存在可用于执行它的量程线路,因为量程计算机可以非常接近任何单一矩阵。This means that there exists a quantum circuit that can be designed to perform it because quantum computers can closely approximate any unitary matrix. 这是一种查找量程线路来实现量程时间演化运算符的行为,$e ^ {-iHt} $ 就是所谓的量程模拟,或特别 dynamical 的量程模拟。This act of finding a quantum circuit to implement the quantum time evolution operator $e^{-iHt}$ is what is often called quantum simulation, or in particular dynamical quantum simulation.