The discrete Fourier transform can sample an unknown periodic function to find its period. This is called period finding and has a wide range of applications, including factoring of numbers into primes.

 Widely considered one of the most important numerical algorithms ever developed.

Reduces complexity of computing DFT from \(O(n^2)\) to \(O(nlogn)\) where \(n\) is the data size. Uses a divide and conquer strategy.

However, there are cases where FFT is not efficient enough, like in analyzing MRI data (where approximations must be used), or in period finding (like factoring into prime numbers).

Computes DFT of \(2^n\) elements in \(O(n^2)\) time where FFT would take \(O(n2^n)\) time.

This speedup allows for efficient algorithms for factoring, discrete logarithm, and the hidden abelian subgroup problem.

However, in many cases this is not a problem since we are interested in the expected value of an operator over the \(2^n\) Fourier coefficients. This is the case in period finding where we can extract the period \(r\) by applying an operator to the Fourier coefficients.

Electrons have a property called spin, which behaves like angular momentum and induces a magnetic field that can be measured (think right hand rule in a circuit).

An electron can either have spin \(\frac12\) up or down, which is the binary classification we use to encode "0" or "1".

For a qubit in state \(|\psi>\), we represent this superposition mathematically as follows:

$$|\psi>  = a|0> +  b|1>$$

where \(a, b\) are complex numbers called probability amplitudes.

A system of \(n\) qubits is then a superposition of all \(2^n\) possible states (i.e. \(|00...0>, |00...1>, ..., |11...1>\)).

Dirac Notation

\(|\psi>  = a|00> +  b|01> +  c|10> +  d|11>\)

Vector Notation

$$\vec{\psi} = {\begin{pmatrix} a, b,  c,  d \end{pmatrix}}^T$$

Given a state

$$|\psi> = a_0|0> +  a_1|1> +  a_2|2> + ... + a_n|n>$$

the probability \(P(x)\) that we will measure a state \(|x>\) is the complex modulo squared of the associated amplitude: 

$$P(x) = |a_x|^2 = Real(a_x)^2 + Imaginary(a_x)^2$$

Since all probabilities must sum to one, \(\|\vec{\psi}\| = 1\)

Defn: A matrix \(U\) is unitary if \(UU^{\dagger} = I\).

Note: A matrix is equivalently unitary if it preserves the inner product of vectors (\(<\vec{x}, \vec{y}>  =  <U\vec{x}, U\vec{y}>\)).

Hadamard Gate takes basis states to a balanced superposition, and upon a second application, takes them back to their original state.

The phase shift gate leaves \(|0\rangle\) unchanged and takes

\(|1\rangle \rightarrow e^{i \phi}|1\rangle\).

So it does not affect the probability of measuring different states, it only shifts the phase.

Defn: The Fourier transform \(\hat{f}(\omega)\) of a function \(f(t) \) is

$$\hat{f}(\omega) = \int_{-\infin}^{\infin} f(t) e^{-i \omega t} dt$$

where \(\hat{f}(\omega)\) returns the associated complex amplitude with a given frequency \(\omega\).

Defn: The inverse Fourier transform of \(\hat{f}(\omega)\) of is

$$f(t) = \int _{-\infin}^{\infin} \hat{f}(\omega) e^{i \omega t} d \omega$$

When the underlying signal is periodic, we can use the discrete Fourier transform instead of the discrete-time Fourier transform, which cannot be computed efficiently in general.

The output \(y_0, ..., y_N-1\) is the sequence of complex coefficients where \(y_k\) is the contribution from \(e^{i \omega k}\) to the signal.

...sounds like a unitary transformation!

Let \(N = 2^n\) be the number of samples. If we apply the QFT on a register of \(n\) qubits in the state

$$|x>  = x_0|0> + ... + x_{N-1}|N-1>$$ and get the state

$$|y>  = QFT|x>  = y_0|0> + ... + y_{N-1}|N-1>$$

we compute the DFT from \(\vec{x}\) to \(\vec{y}\).

The leading \(\frac{1}{\sqrt{N}}\) is to normalize the output state so that the QFT is unitary.

Defn: The inverse quantum Fourier transform from \(y_0, ..., y_N-1\) to \(x_0, ..., x_N-1\) is:

$$x_j  = \frac{1}{\sqrt{N}} \sum_{k=0}^{N-1} y_k e^{-i \omega jk}$$

First we find

We will use two operations to perform the QFT:

1. Hadamard Gate
2. Controlled Phase Gate

Similarily we will take the QFT of the odd and even parts of the input sequence, and then multiply the odd terms by the phase \(e^{iwj}\).

2. Recombine - Since \(QFT_2 = H\) for a single qubit, after computing the QFT on the \(n-1\) most significant bits of that subproblem, we recombine by applying \(H\) on the least significant bit of that subproblem.

We apply this phase shift on all \(n-1\) most significant qubits.

We define \(\hat{R_k}\) to be the phase shift applied if the term is odd. The shift is applied on the \(k^{th}\) most significant bit of that subproblem:

Many public-key cryptography schemes, including RSA, rely upon the assumption that integer factorization is difficult.

A 2,048 bit RSA key is computationally intractable on a classical computer, but with a large enough quantum computer it could be broken.

The algorithm has two steps:

1. Turn factoring into a period finding problem (done classically)
2. Find the period using the QFT

Runs in \(O(logN)^2(loglogN)(logloglogN))\) for integer of size \(N\).

Shor's algorithm solves the HSP on finite Abelian groups.

It is still an open question if efficient quantum algorithms exist for non-Abelian groups like the symmetric group (graph isomorphism) and dihedral group (shortest vector problem).

Algorithm published in 2009, but first general-purpose implementation of the algorithm appeared in 2018 (Zhao et. al).

Uses quantum phase (eigenvalue) estimation which looks for an eigenvalue given an eigenvector and a unitary operator. Quantum phase estimation relies upon the inverse quantum Fourier transform as a subroutine.

Rebentrost et al. show that HLL can be used to provide an exponential speedup on quantum support vector machines.

In 2018, Zhao et. al developed an algorithm and included an implementation using HLL to perform Bayesian training of deep neural networks with exponential speedup.