Multiple Qubits

MULTIPLE QUBITS

Concept

Multiple qubits are termed as the combination of many qubits and their quantum state, which in turn will lead to many complex amplitudes.

Explanation

Single qubits are interesting, but individually they offer no computational advantage. So, let us now look into how we can represent multiple qubits, and how these qubits can interact with each other.

Quantum information processing systems in general use multiple qubits. The state of such a system can be described using a new vector space that takes into account the interaction among the qubits.

This vector space is generated by using a special operation known as tensor product. The tensor product is denoted by the symbol . The tensor product combines the smaller vector spaces of the individual qubits and forms a bigger space.

We know, a single bit has two possible states and a qubit state has two complex amplitudes viz., 1 and 0. Similarly two bits will have four possible states i.e.,

Qubits: 00 01 10 11

States: (1) (2) (3) (4)

To describe the state of two qubits, we require four complex amplitudes, i.e, A two‒qubit system has four basis states. These basis states are constructed from the single‒qubit basis |0>, |1> using the following rule

│u>│v>=│u>|v>=│uv>                  ………….(1)

Where u and v are tensor product of two vectors, given by (u,v) ε (0,1)

Here ε meaning is "IT TENDS TO:

For instance, if the states of the two qubits are given by

| Ψ₁ > = α₀ |0 > + α₁ | 1>         …………..(2)

and | Ψ₂ > = β₀ |0 > + β₁ | 1>         …………..(2)

Then the state of composite system is

| Ψ> = | Ψ₁>| Ψ₂>          ………(4)

Substituting equation (2) and (3) in equation (4), we get

| Ψ > = ( α₀ |0 > + α₁ | 1>)  ( β₀ |0 > + β₁ | 1>)            ………..(5)

Using the rule prescribed in equation (1), we can write equation (5) as

 | Ψ > = α₀β₀ |0 > |0 > + α₀β₁|0>|1> + α₁β₀ |1> |0 > + α₁β₁|1>|1>          ………..(5)

States:      (1)                  (2)             (3)                 (4)

From equation (6), we can see that a two‒qubit system has four basis states i.e.,

    …….. (7)

1st State:

The first state in the two - qubit system, i.e., |0>  |0> indicates that the first qubit is in state |0> and the second qubit is also in state |0>.

2nd state:

For the second state, in the two‒qubit system, i.e., |0>  |1>, we can see that the first qubit is in state |0> and the second qubit is in state |1> etc.,

Similarly we can describe the 3rd and 4th states.

Now by, using the rule as in equation (1), we can write equation (6) as

| Ψ > = α₀β₀ |00 >  + α₀β₁|01> +  α₁β₀ |10> + α₁β₁|11>          ………..(8)

It should be noted, that every two‒qubit state cannot be seperated into two single‒qubit states. So, we can take α₀β₀ as C₀; α₀β₁ as C₁; α₁β₀ as C₂; and α₁β₁ as C₃ etc.,

In general, the state of a two‒qubit system i.e., equation (8) has the form

| Ψ > = C₀|00 >  + C₁|01> + C₂|10> + C₃|11>          ………..(9)

where

|C₀|² + |C₁|²  + |C₂|²  + |C₃|²  = 1

Multiple qubits

Thus, a collection of n qubits, which is referred to as a quantum register of size 'n' is called multiple qubits. An n‒qubit register has 2ⁿ basis states, each of the form

| C₀  | C₁  ……….| C_n-1, with C_i ε [0,1].

A basis state can be represented by a number 0 to 2^n‒1.

Example

For example, the binary state 1001 (which is equal to the decimal number 9) in a quantum register of size 4 is denoted by |9>₄.

A quantum register of 'n' qubits can be in any superposition of 2ⁿ states.

i.e., C₀|0> + C₁|1> + C₂|2> + …….. + C_n^n-1|2^n-1>

If we have n qubits, we need to keep track of 2ⁿ complex amplitudes. Thus, these vectors grow exponentially with the number of qubits. This is the reason quantum computers with large numbers of qubits are so difficult to simulate.

Note: A moderm laptop can easily simulate a general quantum state of around 20 qubits, but simultating 100 qubits is too difficult for the largest super computers.