Quantum Computing Basics

Basic Concepts

What is a Qubit?

Qubits (“cue-bit”) are the quantum analog of bits. While a classical bit can only be in one of two states (0 or 1), a qubit can exist in a superposition of both states simultaneously. They are the basic unit of quantum computing.

A general pure qubit state is expressed as:

$$|\psi ⟩=\alpha |0⟩+\beta |1⟩$$

Where:

• α and β are complex probability amplitudes, $|\alpha {|}^2+|\beta {|}^2=1$
• |0⟩ and |1⟩ are the computational basis states

Key Properties of a Qubit

• Superposition: Unlike classical bits, qubits can exist in superposition. Superposition describes a quantum system that exists in multiple states simultaneously. It is in state $|0⟩$ with amplitude $\alpha$ and state $|1⟩$ with amplitude $\beta$.
• Measurement: When measured, a qubit collapses to either $|0⟩$ or $|1⟩$ with probabilities $|\alpha {|}^2$ and $|\beta {|}^2$ respectively. Thus, $|\alpha {|}^2$ and $|\beta {|}^2=1$. The probability amplitudes determine the probability of measuring the qubit in either state when a measurement is made. The act of measuring fundamentally changes the state. If the outcome of the measurement is 0 then after the measurement the qubit is actually in the $|0⟩$ state and if the outcome is 1 the new state is $|1⟩$.
• Entanglement: Qubits can be entangled with other qubits, creating correlated quantum states.

Example

Consider the qubit in the state $\frac{1}{\sqrt{2}}|0⟩+\frac{1}{\sqrt{2}}|1⟩$.

If we measure, the probability of measuring $|0⟩$ is $|\frac{1}{\sqrt{2}}{|}^2=\frac{1}{2}$, and the probability of measuring $|1⟩$ is also $|\frac{1}{\sqrt{2}}{|}^2=\frac{1}{2}$.

If the outcome happens to be $|0⟩$, our qubit will collapse to the state $|0⟩$. Now, if we measure again, the probability of measuring $|0⟩$ is $|1{|}^2=1$, while the probability of measuring $|1⟩$ is $|0{|}^2=0$. This shows that the act of measuring has fundamentally changed our qubit.

Now remember that the coefficients in the superposition can be complex. For example, we might have a qubit in the state $\frac{1}{3}|0⟩+\frac{2+2i}{3}|1⟩$.

The probability of measuring $|0⟩$ is ${\left|\frac{1}{3}\right|}^2=\frac{1}{9}$, and the probability of measuring $|1⟩$ is ${\left|\frac{2+2i}{3}\right|}^2=\frac{(4+4)}{9}=\frac{8}{9}$.

Quantum Gates

Now that we know about qubits, how do we use them to perform computations? We manipulate qubits by applying operations, such as gates or measurements to them. The gates we use are similar to those found in classical computation such as “AND”, “OR”, and “NOT”. Some of the quantum gates we use are the “NOT” (or “X”) gate, rotations around the x, y and z axes “RX”, “RY”, and “RZ”, and the “Hadamard” gate. Quantum gates are used to entangle qubits, put them into superposition, change the outcome probabilities of measurements, and more. When a measurement happens, any superposition that might have existed is said to collapse into a classical state.

References

Berkeley C191 Course Notes