From Bits to Qubits

In a standard computer, better known as a classical computer, information is stored in bits with values of either 0 or 1. In an electrical circuit (like the CPU of a computer), a bit can be made with either a low or high voltage, and it is how the machine stores instructions and subsequently performs tasks. In a quantum computer, however, information is stored in quantum bits — “qubits” — which can simultaneously be 0 and 1 and anything in between. This state of being is called superposition (a term from quantum mechanics). Some might say it refers to the uncertainty of an object being in several states at once; however, from a quantum mechanics perspective, superposition is NOT the same as uncertainty because it is not connected to the uncertainty principle. Regardless, it allows calculations on many states at the same time, which has numerous applications in the world of computing.

Qubits are essentially probabilities. All quantum states are normalized (the probability is set equal to 1), to make finding the probability easier, and because a probability of 1 implies 100% likelihood. A qubit can be mathematically represented as a linear combination of states like so:

|Ψ⟩ = α|0⟩ + β|1⟩

with α and β being complex coefficients for a qubit’s superposition. These coefficients indicate the relative probability of finding the qubit in one state or the other. I am using Dirac notation, which has kets (the bar and angle bracket) to represent column vectors. This is from quantum mechanics and is the standard nomenclature for both classical and quantum states.

Certain particles like photons or electrons can be used as qubits for experiments involving quantum circuits and such. Along these lines, qubits can be in arbitrary polarization and have a quantum mechanical property called spin, with spin-up equal to|0⟩ and spin-down equal to |1⟩. Spin is… complex, to say the least, and is something I will (attempt to) summarize in a separate article, although I am far from being knowledgeable on the topic (spoiler alert: no one is). For now, know that it is essentially a particle’s intrinsic angular momentum.

Another important detail to note: qubits can be entangled. Entanglement is when two quantum objects, like electrons or quantum circuit gates, have the same state. There is the exchange of quantum information between these two objects at a distance.

For those in favor of analogies, a qubit is like a spinning coin. Normally the coin is either “heads” or “tails” but for a spinning coin, it’s both and also neither at the same time. To make it one or the other, you must stop it spinning, forcing it to land either heads-up or tails-up. This stopping of motion would be considered a measurement, which makes a qubit collapse into one of its basis states. To reiterate, quantum measurements always measure a single state.

Now that an overview has been provided and some new terms defined, let’s delve deeper into the differences between classical and quantum bits.

Classical Bits

As previously stated, a classical bit (also known as a cbit) has 2 states — 0 or 1. These are referred to as the computational basis states, and can be represented with column vectors like so:

|0⟩ = [ 1 ]
      [ 0 ]
|1⟩ = [ 0 ]
      [ 1 ]

Matrices can be used to represent an operation on these states. For example, if

x = [ 0 1 ]
    [ 1 0 ]

then multiplying x and |0⟩ produces |1⟩ like so:

x|0⟩ = [ 0 1 ][ 1 ] = [ 0 ] = |1⟩
       [ 1 0 ][ 0 ]   [ 1 ]

Therefore, the matrix x is a reversible transformation — flipping the bits of the original state, like a NOT gate in an electrical circuit. For a single cbit, there are only two reversible transformations: the Flip matrix shown above, and the Identity matrix, shown below:

I = [ 1 0 ]
    [ 0 1 ]

Quantum Bits

Now consider a quantum system which can have one of two states:

1. Electron (the ground state and first excited state) with spin up or down
2. Photon with left or right polarization

Formally these two states can be labeled as |0⟩ and |1⟩. The state this system occupies at any given time can be given by the following (which should look familiar as it is the same formula I used to represent a qubit’s superposition):

|Ψ⟩ = α|0⟩ + β|1⟩

It could also be represented with spin-up and spin-down:

|Ψ⟩ = α|↑⟩ + β|↓⟩

And the probability is normalized to 1 so that

⟨Ψ|Ψ⟩ = 1

or

|α|² + |β|² = 1

To take a quick detour, degrees of freedom refers to the number of directions in which a particle (or in this case, a qubit) can move freely. It can also be described as the number of independent coordinates required to completely specify the position and orientation of a particle in space. There are two types of degrees of freedom:

1. Degree of freedom of translational motion: movement along the x, y, or z axes, with the maximum number always being 3
2. Degree of freedom of rotational motion: circular movement dependent on the structure of the particle

There can also be the degree of freedom of vibrational motion — movement like that of a longitudinal wave, also dependent on the structure of the particle — but it is technically translational motion.

With that being said, the normalization of the probability kills one degree of freedom. This isn’t too important to know at the moment, it is just something to consider.

Bloch Sphere

The Bloch Sphere (named after Nobel physics laureate Felix Bloch) can parametrize a qubit with a point on a unit sphere (radius equal to 1). It is also known as the Riemann sphere in the world of mathematics.

As you can see, the |0⟩ is represented by the positive z-axis and the |1⟩ is represented by the negative z-axis, with Ψ being the qubit somewhere on the surface of the sphere.

To prevent duplication, one could allow 0 < θ < π, so the qubit on the Bloch sphere would be represented as such:

|Ψ⟩ = cos(θ/2)|0⟩ + sin(θ/2)e^(iφ)|1⟩

with θ being the colatitude (the difference between 90° and the latitude) with respect to the z-axis which determines the probability to measure, and φ being the longitude with respect to the x-axis which describes the relative phase.

The importance of the Bloch sphere is that it allows for a geometric representation of a qubit and its number of possible states and their phase.

Overview

I hope you have found this overview of bits and qubits to be helpful, albeit a little excessive. I attempted to be thorough while also maintaining some level of simplicity to be understandable to any reader interested in the topic. I feel in actuality the only thing I managed to achieve was to annoy any quantum physicists reading this, but such is life.

For those wondering why even have a computer use qubits, the advantage of the quantum computer is that, as mentioned before, we can perform calculations on multiple states at the same time, which leads to exponential speedup for quantum algorithms. This is why there is talk of quantum computing being used to break encryption and “hack the internet”, but that is a topic for another day.

Also, it should be noted that quantum computers are not UNIVERSALLY faster than classical computers and so they cannot wholly replace them; quantum computers are only faster for certain types of calculations where the superposition can be used for parallel computing, thus leading to exponential speedup.

If you feel there was something I could have elaborated upon in more detail or some important concept I missed, feel free to comment below. I am always eager to improve the accuracy and clarity of my writings.

I have provided a list of sources below which I regularly referenced when writing this article and will continue to reference for future posts. I highly recommend them for those who wish to do further reading on the matter.

References

Nielsen, M. A., & Chuang, I. L. (2000). Quantum computation and quantum information. Cambridge: Cambridge University Press.

Townsend, J. S. (1992). A modern approach to quantum mechanics. New York: McGraw-Hill.

The Qiskit Team. (2020). Learn Quantum Computation using Qiskit. Retrieved September 04, 2020, from https://qiskit.org/textbook/preface.html

Veritasium. (2017, June 17). How Does A Quantum Computer Work? [Video file]. Retrieved September 7, 2020, from https://www.youtube.com/watch?v=g_IaVepNDT4