The Qubit

18 Nov 2013

Like most programmers of classical computers, understanding the basics of quantum information and the underlying mechanics can be a bit of a leap. Classical computers are straight forward; they are a known quantity, they are easily measured and observed, and as a by-product: easily programmed.

The Bit

The fundamental operating unit of the classical computer is the bit. This unit provides a way for a classical computer to represent information using only two states: 0 or 1. By combining these two states into larger values, a classical computer is able to describe, store, and reproduce all sorts of information. Classical computers achieve this using binary, or base-two. This means that all information is represented in its binary format. Using the base-two system, one can easily determine exactly how many states can be represented using the following formula:

States = 2^Bits

The number of states is equal to 2 raised to the number of bits.

The Qubit

The quantum bit (qubit - pronounced cue-bit) is the fundamental operating unit of the quantum computer. This unit performs a similar function to its classical counterpart, however it's when we look under the hood that the similarities end. Whilst a classical bit may be in the states 0 or 1, a qubit is actually comprised of the 0 and 1 basis states at the same time. This means that a qubit is actually in two basis states simultaneously, whereas a classical bit may only be in one of two states. This can be a little difficult to swallow for most classical software engineers, especially when we rewrite the above formula to apply to qubits:

States = 2^Qubits

The number of states is equal to also 2 raised to the number of qubits.

So far, so good - both the classical and the quantum systems seem the same.

Superposition

Quantum systems allow for superposition, which is the production of computional basis states from the underlying qubits. This feature is one of quantum computing's immense powers, as it radically reduces the amount of qubits required to represent state information. This feature means that in order to represent n states, in a classical system, you will need n bits, however in a quantum system, you will ultimately need far, far less qubits.

So why is this important?

Well, you certainly get a lot more bang for your buck in terms of how many states can be represented (and therefore processed) simultaneously. Looking at some figures to compare these two systems:

1 bit - 2¹ = 2 states
- Classical Bits required: 1
- Quantum Bits required: 1
2 bits - 2² = 4 states
- Classical Bits required: 2
- Quantum Bits required: 1
4 bits - 2⁴ = 16 states
- Classical Bits required: 4
- Quantum Bits required: 2
8 bits - 2⁸ = 64 states
- Classical Bits required: 8
- Quantum Bits required: 3

There's not much difference at first, in fact, there's really no gain to be made with half a byte, but as the quantum formula is exponential, the more you add, the faster it begins to grow:

64 bits - 2⁶⁴ = 18446744073709551616 states
- Classical Bits required: 64
- Quantum Bits required: 6

A 64 bit classical processor is capable of shifting 64 bits per cycle, however to move the same amount of information per cycle will only require 6 qubits.

The general formula for representing a qubit's state is as follows:

ψ = α|0› + β|1›

Psi equals alpha ket 0 plus beta ket 1.

Note that the pipe/right chevron is referred to as a 'ket', this is referred to as 'ket notation'. The alpha and beta coefficients are important here, as they determine the probability of measuring |0› and |1›.

The qubit is governed by another rule, which is called 'Normalisation'. This rule states that the absolute values of the coefficients, when squared, must always sum to one:

|α|² + |β|² = 1

The absolute value of alpha square plus the absolute value of beta squared equals 1.

This second rule becomes interesting when trying to picture what qubit state information could look like in time.

Let α = 0.8 and let β = 0.6
|α|² + |β|² = 1
=> |0.8|² + |0.6|²
=> 0.64 + 0.36
=> 1

We could read the actual numbers of the coefficient and think of them as percentages. We could then assign those percentages to the qubit basis states. With this in mind, our qubit now looks like this:

ψ = |0› @ 64% + |1› @ 36%

Psi = ket 0 at 64% plus ket 1 at 36%.

Visualising these results as a pie chart can assist with understanding these results. One could think of a qubit as being a distribution of |0› and |1›.

A single qubit

Finally, a single qubit has two basis states that can be superposed in a continuous and therefore infinite number of ways. Herein lies the power of quantum computing.