In the previous post, we have sketched the basic ideas behind NMR based quantum computation. In this post, we will discuss single qubits and single qubit operations in more depth. The rotating frame of reference In NMR based quantum computing, quantum gates are realized by applying oscillating magnetic fields to our probe. As an oscillating … Continue reading Single qubit NMR based quantum computation

# Tag: Math

# Quantum error correction: an introduction to toric codes

While playing with the IBM Q experience in some of my recent posts, we have seen that real qubits are subject to geometric restrictions - two-qubit gates cannot involve arbitrary qubits, but only qubits that are in some sense neighbors. This suggests that efficient error correction codes need to tie to the geometry of the … Continue reading Quantum error correction: an introduction to toric codes

# Quantum error correction with stabilizer codes

In our previous discussion of quantum error correction, we have assumed that quantum gates can act on any two physical qubits. In reality, however, this is not true - only nearby qubits and interact, and our error correction needs to take the geometric arrangements of the qubits into account. The link between these geometric constraints … Continue reading Quantum error correction with stabilizer codes

# Into the quantum lab – first steps with IBMs Q experience

Even though physical implementations of quantum computers make considerable progress, it is not likely that you will have one of them under your desk in the next couple of years. Fortunately, some firms like IBM and Rigetti have decided to make some of their quantum devices available only so that you can play with them. … Continue reading Into the quantum lab – first steps with IBMs Q experience

# The EM algorithm and Gaussian mixture models – part II

In this post, I will discuss the general form of the EM algorithm to obtain a maximum likelihood estimator for a model with latent variables. First, let us describe our model. We suppose that we are given some joint distribution of a random variable X (the observed variables) and and random variable Z (the latent … Continue reading The EM algorithm and Gaussian mixture models – part II

# The EM algorithm and Gaussian mixture models – part I

In the last few posts on machine learning, we have looked in detail at restricted Boltzmann machines. RBMs are a prime example for unsupervised learning - they learn a given distribution and are able to extract features from a data set, without the need to label the data upfront. However, there are of course many … Continue reading The EM algorithm and Gaussian mixture models – part I

# Why you need statistics to understand neuronal networks

When I tried to learn about neuronal networks first, I did what probably most of us would do - I started to look for tutorials, blogs etc. on the web and was surprised by the vast amount of resources that I found. Almost every blog or webpage about neuronal networks has a section on training … Continue reading Why you need statistics to understand neuronal networks

# The Metropolis-Hastings algorithm

In this post, we will investigate the Metropolis-Hastings algorithm, which is still one of the most popular algorithms in the field of Markov chain Monte Carlo methods, even though its first appearence (see [1]) happened in 1953, more than 60 years in the past. It does for instance appear on the CiSe top ten list … Continue reading The Metropolis-Hastings algorithm

# Recurrent and ergodic Markov chains

Today, we will look in more detail into convergence of Markov chains - what does it actually mean and how can we tell, given the transition matrix of a Markov chain on a finite state space, whether it actually converges. So suppose that we are given a Markov chain on a finite state space, with … Continue reading Recurrent and ergodic Markov chains

# Finite Markov chains

In this post, we will look in more detail into an important class of Markov chains - Markov chains on finite state spaces. Many of the subtleties that are present when studying Markov chains in general state spaces do not appear in the finite case, while most of the key ideas and features of Markov … Continue reading Finite Markov chains