Tag: Quantum discrete integral transform

Quantum Discrete Integral Transform

The integral transform is a powerful mathematical tool that maps a function from its original domain into a new function space through an integration operation (or summation in the discrete case). Typical examples includs the Fourier transform, Laplace transform, Mellin transform, and many others, are indispensable across mathematics, physics, engineering, signal processing, quantum mechanics, and beyond. These transformation often reveals properties of the original function—such as frequency content, growth behavior, or analytic structure—that are more easily analyzed or manipulated in the transformed space. In most cases, the original function can be recovered exactly via an inverse transform.

Mathematically, an integral transform \mathcal{T} is an operator which maps a function {f} into another function \mathcal{T}[f], and two functions do not necessarily have the same domain and range. Usually an integral transform {T} can be written via a corresponding integral kernel {K(p,x)} as

\displaystyle \mathcal{T}[f](p)=\int K(p,x)f(x) dx

Among the vast family of integral transforms, the Fourier transform stands out for its elegance and ubiquity. In the natural units where \hbar = 1, physicists typically define the position-space wave function \psi(x) from its momentum-space counterpart \psi(p) by the (inverse) Fourier transform (chosen to avoid an overall minus sign in the exponent, which is a common convention in quantum Fourier transform):

\mathcal{F}[\psi](x)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{ipx} \psi(p) dp

have the integral kernel {e^{2\pi i xp}/\sqrt{2\pi}}. This single transformation lies at the very heart of quantum mechanics. It converts differential equations into algebraic ones, turns convolutions into products, and recasts purely local position-space information as global momentum content. Momentum-space methods often render seemingly intractable operator equations manifestly solvable.

The discussion that follows extends these classical ideas into the realm of quantum theory through the notion of quantum integral transforms. Since we are concerned with a discrete qubit system, our focus will be on discrete integral transforms. Note that a function {f(x)} can be regarded as a sequence of numbers index by {x} in a continuous index set. The integral transform \mathcal{T}[f] thus looks like a matrix transform with continuous indices. In the discrete context, it turns out that an integral transform becomes a matrix transform, which is us discrete integral transform or simply discrete transform.

Definition 1 (Discrete integral transform) Let {K_{ij}=:K(i,j)} be a matrix with indices {i,j=0,\cdots,N-1} and {\vec{x}=(x_0,\cdots,x_{N-1})^T=:(x(0),\cdots,x(N-1))^{T}} be a vector (discrete function). Then the discrete integral transform is defined as {y(i)=(Kx)(i)=\sum_{j}K(i,j)x(j)} which is in fact a matrix transform

\displaystyle \vec{y}=K\vec{x}. \ \ \ \ \ (1)

Note that we always assume vectors are column vectors. And for the convenience of the generalization we will also assume that {K} is unitary, for which case the invertible discrete integral transform kernel is given by the Hermitian conjugate {K^{\dagger}} of {K}.

For instance, the discrete Fourier transform determined by the kernel matrix {K_{jk}=e^{2\pi i jk/N}/\sqrt{N}} is of the form

\displaystyle y_k=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}e^{2\pi i jk/N}x_j. \ \ \ \ \ (2)

Here we want to mention that in the large {N} limit, the discrete Fourier transform kernel will become the usual Fourier integral kernel and sum is replaced with the integral. This is the reason why we choose {K_{jk}=e^{2\pi i jk/N}/\sqrt{N}} as discrete Fourier transform kernel.

The quantum discrete integral transform is the quantum analogue of discrete integral transform, actually they are exactly the same transform as we will see. Suppose that we have a Hilbert space {\mathcal{H}} with basis states {|0\rangle,\cdots,|N-1\rangle}, for a given discrete integral transform kernel {K_{ij}}, the corresponding quantum discrete integral transform on the basis states is defined as

\displaystyle |j\rangle\overset{QDIT}{\rightarrow}U_K|j\rangle=\sum_{k=0}^{N-1}K_{kj}|k\rangle. \ \ \ \ \ (3)

Then for a state {|\psi\rangle=\sum_{j=0}^{N-1}x_j|j\rangle/\|\mathbf{x}\|},

\displaystyle U_K|\psi\rangle=\frac{1}{\|\mathbf{x}\|}\sum_{j=0}x_jU_K|j\rangle=\frac{1}{\|\mathbf{x}\|}\sum_{k=0}^{N-1}(\sum_{j=0}K_{kj}x_j)|k\rangle=\frac{1}{\|\mathbf{y}\|}\sum_{k=0}^{N-1}y_k|k\rangle \ \ \ \ \ (4)

calculate the discrete integral transform {y_k=\sum_{j=0}^{N-1}K_{kj}x_j}. Notice that {U_K} is unitary matrix, thus {\|\mathbf{x}\|=\|\mathbf{y}\|}. In summary, we have

Definition 2 (Quantum discrete integral transform) Quantum discrete integral transform corresponding to a unitary kernel {K_{kj}} is a unitary transformation {U_K=(K_{kj})}

\displaystyle |j\rangle\overset{QDIT}{\rightarrow}U_K|j\rangle=\sum_{k=0}^{N-1}K_{kj}|k\rangle. \ \ \ \ \ (5)

To carry out the discrete integral transform {y_k=\sum_{j=0}^{N-1}K_{jk}x_j}, we first prepare the state {|\psi\rangle=\sum_{j=0}^{N-1}x_j|j\rangle/\|\mathbf{x}\|} and then apply {U_K} to get {U_K|\psi\rangle}, finally we read out the value {y_k} by measuring in the basis state {|k\rangle}.

To implement a quantum discrete integral transform algorithm, one must design a quantum circuit that realizes the unitary operator $U_K$. This step is generally highly nontrivial and requires careful thought and insightful design. Moreover, the structure of the quantum circuit depends sensitively on the choice of kernel K; different kernels typically lead to very different circuit constructions.