Qubit

A qubit is not to be confused with a cubit, which is an ancient measure of length.

A quantum bit, or qubit (sometimes qbit) is a unit of quantum information. That information is described by a state vector in a 2-level quantum mechanical system which is formally equivalent to a two-dimensional vector space over the complex numbers.

Benjamin Schumacher discovered a way of interpreting quantum states as information. He came up with a way of compressing the information in a state, and storing the information on a smaller number of states. This is now known as Schumacher compression. Schumacher is also credited with inventing the term qubit.

Bit vs. Qubit

A bit is the base of computer information. Regardless of its physical representation, it is always read as either a 0 or a 1. An analogy to this is a light switch - the down position can represent 0 (normally equated to off) and the up position can represent 1 (normally equated to on).

A qubit has some similarities to a classical bit, but is overall very different. Like a bit, a qubit can have only two possible values - normally a 0 or a 1. The difference is that whereas a bit must be either 0 or 1, a qubit can be 0, 1, or a superposition of both.

Representation

The states a qubit may be measured in are known as basis states (or vectors). As is the tradition with any sort of quantum states, Dirac, or bra-ket notation is used to represent them.

This means that the two computational basis states are conventionally written as $|0\rangle$ and $|1\rangle$ (pronounced: 'ket 0' and 'ket 1').

Qubit States

A pure qubit state is a linear superposition of those two states. This means that the qubit can be represented as a linear combination of $|0\rangle$ and $|1\rangle$:

$|\psi \rangle =\alpha |0\rangle +\beta |1\rangle ,\,$

where α and β are probability amplitudes and can in general be complex.

When we measure this qubit in the standard basis, the probability of outcome $|0\rangle$ is $|\alpha |^{2}$ and the probability that the outcome is $|1\rangle$ is $|\beta |^{2}$. Because the absolute squares of the amplitudes equate to probabilities, it follows that α and β must be constrained by the equation

$|\alpha |^{2}+|\beta |^{2}=1\,$

simply because this ensures you must measure either one state or the other.

The state space of a single qubit register can be represented geometrically by the Bloch sphere. This is a two dimensional space which has an underlying geometry of the surface of a sphere. This essentially means that the single qubit register space has two local degrees of freedom. An n-qubit register space has 2ⁿ⁺¹ − 2 degrees of freedom. This is much larger than 2n, which is what one would expect classically with no entanglement. The reason for this difference is that a qubit can be represented by any point on the surface of the sphere, while a classical bit can only be represented by the very top or very bottom of the sphere.

Measurement

Because of quantum mechanics, any measurement of a quantum system inevitably alters the system. Much like Schrödinger's cat, a qubit can exist in more than one state, but measuring that qubit causes that superposition to collapse into one state or the other, according to the probabilities mentioned above.

Obviously, if measurement of the state collapses it into one of the basis states, it becomes very hard to measure the precise amplitudes α and β, or their corresponding probabilities. If one seeks to find these amplitudes, they may recreate the superposition and make multiple measurements. Other methods of finding the amplitudes without disrupting the superpositioned qubit are being studied, but have proven very difficult to implement.

Entanglement

An important distinguishing feature between a qubit and a classical bit is that multiple qubits can exhibit quantum entanglement. Entanglement is a nonlocal property that allows a set of qubits to express higher correlation than is possible in classical systems. Take, for example, two entangled qubits in the Bell state

$|\Phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|00\rangle +|11\rangle )$

(Note that in this state, there are equal probabilities of measuring $|00\rangle$ and $|11\rangle$). Imagine that these two entangled qubits are separated, with one each given to Alice and Bob. Alice makes a measurement of her qubit, obtaining - with equal probabilities - either $|0\rangle$ or $|1\rangle$. Because of the qubits' entanglement, Bob must now get the exact same measurement as Alice, i.e. if she measured a $|0\rangle$, Bob must measure the same, as $|00\rangle$ is the only state where Alice's qubit is a $|0\rangle$.

Entanglement also allows multiple states (such as are the Bell state mentioned above) to be acted on simultaneously, unlike classical bits that can only have one value at a time. Entanglement is a necessary ingredient of any quantum computation that cannot be done efficiently on a classical computer.

The use of entanglement in quantum computing has been referred to as "quantum parallelism", and offers a possible explanation for the power of quantum computing: because the state of the computer can be in a quantum superposition of many different classical computational paths, these paths can all proceed concurrently.

Quantum Register

A number of entangled qubits taken together is a qubit register. Quantum computers perform calculations by manipulating qubits within a register.

Variations of the Qubit

Similar to the qubit, a qutrit is a unit of quantum information in a 3-level quantum system. This is analagous to the unit of classical information trit. The term "Qudit" is used to denote a unit of quantum information in a d-level quantum system.

External links

• An update on qubits in the Jan 2005 issue of Scientific American
• An update on qubits in the Oct 2005 issue of Scientific American
• The organization cofounded by one of the pioneers in quantum computation, David Deutsch