# The Quantum Bit

The best place to start our journey through quantum computing is to recall how classical computing works and try to extend it. Since our final quantum computing model will be a circuit model, we should informally discuss circuits first.

A circuit has three parts: the “inputs,” which are bits (either zero or one); the “gates,” which represent the lowest-level computations we perform on bits; and the “wires,” which connect the outputs of gates to the inputs of other gates. Typically the gates have one or two input bits and one output bit, and they correspond to some logical operation like AND, NOT, or XOR.

A simple example of a circuit. The V’s are “OR” and the Λ’s are “AND.” Image source: Ryan O’Donnell

If we want to come up with a different model of computing, we could start regular circuits and generalize some or all of these pieces. Indeed, in our motivational post we saw a glimpse of a probabilistic model of computation, where instead of the inputs being bits they were probabilities in a probability distribution, and instead of the gates being simple boolean functions they were linear maps that preserved probability distributions (we called such a matrix “stochastic”).

Rather than go through that whole train of thought again let’s just jump into the definitions for the quantum setting. In case you missed last time, our goal is to avoid as much physics as possible and frame everything purely in terms of linear algebra.

## Qubits are Unit Vectors

The generalization of a bit is simple: it’s a unit vector in $\mathbb{C}^2$. That is, our most atomic unit of data is a vector $(a,b)$ with the constraints that $a,b$ are complex numbers and $|a|^2 + |b|^2 = 1$. We call such a vector a qubit.

A qubit can assume “binary” values much like a regular bit, because you could pick two distinguished unit vectors, like $(1,0)$ and $(0,1)$, and call one “zero” and the other “one.” Obviously there are many more possible unit vectors, such as $\frac{1}{\sqrt{2}}(1, 1)$ and $(-i,0)$. But before we go romping about with what qubits can do, we need to understand how we can extract information from a qubit. The definitions we make here will motivate a lot of the rest of what we do, and is in my opinion one of the major hurdles to becoming comfortable with quantum computing.

A bittersweet fact of life is that bits are comforting. They can be zero or one, you can create them and change them and read them whenever you want without an existential crisis. The same is not true of qubits. This is a large part of what makes quantum computing so weird: you can’t just read the information in a qubit! Before we say why, notice that the coefficients in a qubit are complex numbers, so being able to read them exactly would potentially encode an infinite amount of information (in the infinite binary expansion)! Not only would this be an undesirably powerful property of a circuit, but physicists’ experiments tell us it’s not possible either.

So as we’ll see when we get to some algorithms, the main difficulty in getting useful quantum algorithms is not necessarily figuring out how to compute what you want to compute, it’s figuring out how to tease useful information out of the qubits that otherwise directly contain what you want. And the reason it’s so hard is that when you read a qubit, most of the information in the qubit is destroyed. And what you get to see is only a small piece of the information available. Here is the simplest example of that phenomenon, which is called the measurement in the computational basis.

Definition: Let $v = (a,b) \in \mathbb{C}^2$ be a qubit. Call the standard basis vectors $e_0 = (1,0), e_1 = (0,1)$ the computational basis of $\mathbb{C}^2$. The process of measuring $v$ in the computational basis consists of two parts.

1. You observe (get as output) a random choice of $e_0$ or $e_1$. The probability of getting $e_0$ is $|a|^2$, and the probability of getting $e_1$ is $|b|^2$.
2. As a side effect, the qubit $v$ instantaneously becomes whatever state was observed in 1. This is often called a collapse of the waveform by physicists.

There are more sophisticated ways to measure, and more sophisticated ways to express the process of measurement, but we’ll cover those when we need them. For now this is it.

Why is this so painful? Because if you wanted to try to estimate the probabilities $|a|^2$ or $|b|^2$, not only would you get an estimate at best, but you’d have to repeat whatever computation prepared $v$ for measurement over and over again until you get an estimate you’re satisfied with. In fact, we’ll see situations like this, where we actually have a perfect representation of the data we need to solve our problem, but we just can’t get at it because the measurement process destroys it once we measure.

Before we can talk about those algorithms we need to see how we’re allowed to manipulate qubits. As we said before, we use unitary matrices to preserve unit vectors, so let’s recall those and make everything more precise.

## Qubit Mappings are Unitary Matrices

Suppose $v = (a,b) \in \mathbb{C}^2$ is a qubit. If we are to have any mapping between vector spaces, it had better be a linear map, and the linear maps that send unit vectors to unit vectors are called unitary matrices. An equivalent definition that seems a bit stronger is:

Definition: A linear map $\mathbb{C}^2 \to \mathbb{C}^2$ is called unitary if it preserves the inner product on $\mathbb{C}^2$.

Let’s remember the inner product on $\mathbb{C}^n$ is defined by $\left \langle v,w \right \rangle = \sum_{i=1}^n v_i \overline{w_i}$ and has some useful properties.

• The square norm of a vector is $\left \| v \right \|^2 = \left \langle v,v \right \rangle$.
• Swapping the coordinates of the complex inner product conjugates the result: $\left \langle v,w \right \rangle = \overline{\left \langle w,v \right \rangle}$
• The complex inner product is a linear map if you fix the second coordinate, and a conjugate-linear map if you fix the first. That is, $\left \langle au+v, w \right \rangle = a \left \langle u, w \right \rangle + \left \langle v, w \right \rangle$ and $\left \langle u, aw + v \right \rangle = \overline{a} \left \langle u, w \right \rangle + \left \langle u,v \right \rangle$

By the first bullet, it makes sense to require unitary matrices to preserve the inner product instead of just the norm, though the two are equivalent (see the derivation on page 2 of these notes). We can obviously generalize unitary matrices to any complex vector space, and unitary matrices have some nice properties. In particular, if $U$ is a unitary matrix then the important property is that the columns (and rows) of $U$ form an orthonormal basis. As an immediate result, if we take the product $U\overline{U}^\text{T}$, which is just the matrix of all possible inner products of columns of $U$, we get the identity matrix. This means that unitary matrices are invertible and their inverse is $\overline{U}^\text{T}$.

Already we have one interesting philosophical tidbit. Any unitary transformation of a qubit is reversible because all unitary matrices are invertible. Apparently the only non-reversible thing we’ve seen so far is measurement.

Recall that $\overline{U}^\text{T}$ is the conjugate transpose of the matrix, which I’ll often write as $U^*$. Note that there is a way to define $U^*$ without appealing to matrices: it is a notion called the adjoint, which is that linear map $U^*$ such that $\left \langle Uv, w \right \rangle = \left \langle v, U^*w \right \rangle$ for all $v,w$. Also recall that “unitary matrix” for complex vector spaces means precisely the same thing as “orthogonal matrix” does for real numbers. The only difference is the inner product being used (indeed, if the complex matrix happens to have real entries, then orthogonal matrix and unitary matrix mean the same thing).

Definition: single qubit gate is a unitary matrix $\mathbb{C}^2 \to \mathbb{C}^2$.

So enough with the properties and definitions, let’s see some examples. For all of these examples we’ll fix the basis to the computational basis $e_0, e_1$. One very important, but still very simple example of a single qubit gate is the Hadamard gate. This is the unitary map given by the matrix

$\displaystyle \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$

It’s so important because if you apply it to a basis vector, say, $e_0 = (1,0)$, you get a uniform linear combination $\frac{1}{\sqrt{2}}(e_1 + e_2)$. One simple use of this is to allow for unbiased coin flips, and as readers of this blog know unbiased coins can efficiently simulate biased coins. But it has many other uses we’ll touch on as they come.

Just to give another example, the quantum NOT gate, often called a Pauli X gate, is the following matrix

$\displaystyle \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$

It’s called this because, if we consider $e_0$ to be the “zero” bit and $e_1$ to be “one,” then this mapping swaps the two. In general, it takes $(a,b)$ to $(b,a)$.

As the reader can probably imagine by the suggestive comparison with classical operations, quantum circuits can do everything that classical circuits can do. We’ll save the proof for a future post, but if we want to do some kind of “quantum AND” operation, we get an obvious question. How do you perform an operation that involves multiple qubits? The short answer is: you represent a collection of bits by their tensor product, and apply a unitary matrix to that tensor.

We’ll go into more detail on this next time, and in the mean time we suggest checking out this blog’s primer on the tensor product. Until then!

# A Motivation for Quantum Computing

Quantum mechanics is one of the leading scientific theories describing the rules that govern the universe. It’s discovery and formulation was one of the most important revolutions in the history of mankind, contributing in no small part to the invention of the transistor and the laser.

Here at Math ∩ Programming we don’t put too much emphasis on physics or engineering, so it might seem curious to study quantum physics. But as the reader is likely aware, quantum mechanics forms the basis of one of the most interesting models of computing since the Turing machine: the quantum circuit. My goal with this series is to elucidate the algorithmic insights in quantum algorithms, and explain the mathematical formalisms while minimizing the amount of “interpreting” and “debating” and “experimenting” that dominates so much of the discourse by physicists.

Indeed, the more I learn about quantum computing the more it’s become clear that the shroud of mystery surrounding quantum topics has a lot to do with their presentation. The people teaching quantum (writing the textbooks, giving the lectures, writing the Wikipedia pages) are almost all purely physicists, and they almost unanimously follow the same path of teaching it.

Scott Aaronson (one of the few people who explains quantum in a way I understand) describes the situation superbly.

There are two ways to teach quantum mechanics. The first way – which for most physicists today is still the only way – follows the historical order in which the ideas were discovered. So, you start with classical mechanics and electrodynamics, solving lots of grueling differential equations at every step. Then, you learn about the “blackbody paradox” and various strange experimental results, and the great crisis that these things posed for physics. Next, you learn a complicated patchwork of ideas that physicists invented between 1900 and 1926 to try to make the crisis go away. Then, if you’re lucky, after years of study, you finally get around to the central conceptual point: that nature is described not by probabilities (which are always nonnegative), but by numbers called amplitudes that can be positive, negative, or even complex.

The second way to teach quantum mechanics eschews a blow-by-blow account of its discovery, and instead starts directly from the conceptual core – namely, a certain generalization of the laws of probability to allow minus signs (and more generally, complex numbers). Once you understand that core, you can then sprinkle in physics to taste, and calculate the spectrum of whatever atom you want.

Indeed, the sequence of experiments and debate has historical value. But the mathematics needed to have a basic understanding of quantum mechanics is quite simple, and it is often blurred by physicists in favor of discussing interpretations. To start thinking about quantum mechanics you only need to a healthy dose of linear algebra, and most of it we’ve covered in the three linear algebra primers on this blog. More importantly for computing-minded folks, one only needs a basic understanding of quantum mechanics to understand quantum computing.

The position I want to assume on this blog is that we don’t care about whether quantum mechanics is an accurate description of the real world. The real world gave an invaluable inspiration, but at the end of the day the mathematics stands on its own merits. The really interesting question to me is how the quantum computing model compares to classical computing. Most people believe it is strictly stronger in terms of efficiency. And so the murky depths of the quantum swamp must be hiding some fascinating algorithmic ideas. I want to understand those ideas, and explain them up to my own standards of mathematical rigor and lucidity.

So let’s begin this process with a discussion of an experiment that motivates most of the ideas we’ll need for quantum computing. Hopefully this will be the last experiment we discuss.

## Shooting Photons and The Question of Randomness

Does the world around us have inherent randomness in it? This is a deep question open to a lot of philosophical debate, but what evidence do we have that there is randomness?

Here’s the experiment. You set up a contraption that shoots photons in a straight line, aimed at what’s called a “beam splitter.” A beam splitter seems to have the property that when photons are shot at it, they will be either be reflected at a 90 degree angle or stay in a straight line with probability 1/2. Indeed, if you put little photon receptors at the end of each possible route (straight or up, as below) to measure the number of photons that end at each receptor, you’ll find that on average half of the photons went up and half went straight.

The triangle is the photon shooter, and the camera-looking things are receptors.

If you accept that the photon shooter is sufficiently good and the beam splitter is not tricking us somehow, then this is evidence that universe has some inherent randomness in it! Moreover, the probability that a photon goes up or straight seems to be independent of what other photons do, so this is evidence that whatever randomness we’re seeing follows the classical laws of probability. Now let’s augment the experiment as follows. First, put two beam splitters on the corners of a square, and mirrors at the other two corners, as below.

The thicker black lines are mirrors which always reflect the photons.

This is where things get really weird. If you assume that the beam splitter splits photons randomly (as in, according to an independent coin flip), then after the first beam splitter half go up and half go straight, and the same thing would happen after the second beam splitter. So the two receptors should measure half the total number of photons on average.

But that’s not what happens. Rather, all the photons go to the top receptor! Somehow the “probability” that the photon goes left or up in the first beam splitter is connected to the probability that it goes left or up in the second. This seems to be a counterexample to the claim that the universe behaves on the principles of independent probability. Obviously there is some deeper mystery at work.

## Complex Probabilities

One interesting explanation is that the beam splitter modifies something intrinsic to the photon, something that carries with it until the next beam splitter. You can imagine the photon is carrying information as it shambles along, but regardless of the interpretation it can’t follow the laws of classical probability.

The simplest classical probability explanation would go something like this:

There are two states, RIGHT and UP, and we model the state of a photon by a probability distribution $(p, q)$ such that the photon has a probability $p$ of being in state RIGHT a probability $q$ of being in state UP, and like any probability distribution $p + q = 1$. A photon hence starts in state $(1,0)$, and the process of traveling through the beam splitter is the random choice to switch states. This is modeled by multiplication by a particular so-called stochastic matrix (which just means the rows sum to 1)

$\displaystyle A = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix}$

Of course, we chose this matrix because when we apply it to $(1,0)$ and $(0,1)$ we get $(1/2, 1/2)$ for both outcomes. By doing the algebra, applying it twice to $(1,0)$ will give the state $(1/2, 1/2)$, and so the chance of ending up in the top receptor is the same as for the right receptor.

But as we already know this isn’t what happens in real life, so something is amiss. Here is an alternative explanation that gives a nice preview of quantum mechanics.

The idea is that, rather than have the state of the traveling photon be a probability distribution over RIGHT and UP, we have it be a unit vector in a vector space (over $\mathbb{C}$). That is, now RIGHT and UP are the (basis) unit vectors $e_1 = (1,0), e_2 = (0,1)$, respectively, and a state $x$ is a linear combination $c_1 e_1 + c_2 e_2$, where we require $\left \| x \right \|^2 = |c_1|^2 + |c_2|^2 = 1$. And now the “probability” that the photon is in the RIGHT state is the square of the coefficient for that basis vector $p_{\text{right}} = |c_1|^2$. Likewise, the probability of being in the UP state is $p_{\text{up}} = |c_2|^2$.

This might seem like an innocuous modification — even a pointless one! — but changing the sum (or 1-norm) to the Euclidean sum-of-squares (or the 2-norm) is at the heart of why quantum mechanics is so different. Now rather than have stochastic matrices for state transitions, which are defined they way they are because they preserve probability distributions, we use unitary matrices, which are those complex-valued matrices that preserve the 2-norm. In both cases, we want “valid states” to be transformed into “valid states,” but we just change precisely what we mean by a state, and pick the transformations that preserve that.

In fact, as we’ll see later in this series using complex numbers is totally unnecessary. Everything that can be done with complex numbers can be done without them (up to a good enough approximation for computing), but using complex numbers just happens to make things more elegant mathematically. It’s the kind of situation where there are more and better theorems in linear algebra about complex-valued matrices than real valued matrices.

But back to our experiment. Now we can hypothesize that the beam splitter corresponds to the following transformation of states:

$\displaystyle A = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix}$

We’ll talk a lot more about unitary matrices later, so for now the reader can rest assured that this is one. And then how does it transform the initial state $x =(1,0)$?

$\displaystyle y = Ax = \frac{1}{\sqrt{2}}(1, i)$

So at this stage the probability of being in the RIGHT state is $1/2 = (1/\sqrt{2})^2$ and the probability of being in state UP is also $1/2 = |i/\sqrt{2}|^2$. So far it matches the first experiment. Applying $A$ again,

$\displaystyle Ay = A^2x = \frac{1}{2}(0, 2i) = (0, i)$

And the photon is in state UP with probability 1. Stunning. This time Science is impressed by mathematics.

Next time we’ll continue this train of thought by generalizing the situation to the appropriate mathematical setting. Then we’ll dive into the quantum circuit model, and start churning out some algorithms.

Until then!

[Edit: Actually, if you make the model complicated enough, then you can achieve the result using classical probability. The experiment I described above, while it does give evidence that something more complicated is going on, it does not fully rule out classical probability. Mathematically, you can lay out the axioms of quantum mechanics (as we will from the perspective of computing), and mathematically this forces non-classical probability. But to the best of my knowledge there is no experiment or set of experiments that gives decisive proof that all of the axioms are necessary. In my search for such an experiment I asked this question on stackexchange and didn’t understand any of the answers well enough to paraphrase them here. Moreover, if you leave out the axiom that quantum circuit operations are reversible, you can do everything with classical probability. I read this somewhere but now I can’t find the source 😦

One consequence is that I am more firmly entrenched in my view that I only care about quantum mechanics in how it produced quantum computing as a new paradigm in computer science. This paradigm doesn’t need physics at all, and apparently the motivations for the models are still unclear, so we just won’t discuss them any more. Sorry, physics lovers.]