Superposition and Entanglement of Qubits in Quantum Computers

Difference Between Bits and Qubits

Bits and qubits are both units of information, but they have fundamentally different properties. Bits are the basic unit of information in classical computing, while qubits are the basic unit of information in quantum computing.

Bits can have only two values: 0 or 1. This means that a bit can represent a single binary value, such as "on" or "off", "true" or "false", "high" or "low". Bits are stored in digital circuits and manipulated using logical gates.

Qubits, on the other hand, can have multiple values simultaneously. This is because qubits are subject to the laws of quantum mechanics, which allow them to exist in a superposition of states. A qubit can be in a superposition of 0 and 1, meaning that it can represent both values at the same time. This property of superposition is what gives qubits their power.

Qubits are stored in quantum circuits and manipulated using quantum gates. Quantum gates are designed to exploit the properties of superposition and entanglement to perform complex calculations.

The difference between bits and qubits is one of the key differences between classical computing and quantum computing. Classical computers can only perform calculations on bits, while quantum computers can perform calculations on qubits. This means that quantum computers have the potential to solve problems that are intractable for classical computers.

In quantum computers, information is encoded in qubits. The number of qubits in a quantum computer determines its computing power. A quantum computer with 10 qubits can perform calculations that would be impossible for a classical computer with 1024 bits.

As the number of qubits in a quantum computer increases, its computing power increases exponentially. This means that a quantum computer with 20 qubits would be millions of times faster than a classical computer with 20,000 bits.

The ability of qubits to exist in a superposition of states is what gives quantum computers their potential to solve problems that are intractable for classical computers. For example, quantum computers could be used to factor large numbers, break encryption codes, and design new drugs.

The tensor product of basis vectors for a two-qubit system can be written as:

|00⟩ = |0⟩ ⊗ |0⟩ |01⟩ = |0⟩ ⊗ |1⟩ |10⟩ = |1⟩ ⊗ |0⟩ |11⟩ = |1⟩ ⊗ |1⟩

Where |0⟩ and |1⟩ are the basis vectors for a single qubit. The tensor product of two vectors is the vector that contains all possible combinations of the two vectors. For example, |00⟩ is the vector that contains the two combinations |0⟩ ⊗ |0⟩ and |1⟩ ⊗ |1⟩.

The tensor product of basis vectors is important in quantum computing because it allows us to represent the states of multiple qubits. For example, the state |11⟩ represents the state where both qubits are in the |1⟩ state.

Superposition of Qubits in Quantum Computers

Superposition is a fundamental property of quantum mechanics that allows quantum particles to exist in multiple states at the same time. This is in contrast to classical physics, where particles can only exist in one state at a time. Classical physics is a successful description of the macroscopic world, but it breaks down when applied to the microscopic world. This is because classical physics assumes that objects have well-defined positions and momenta, but this is not the case for microscopic objects. For this reason, classical physics is sometimes called "non-relativistic physics" to distinguish it from relativistic physics and quantum mechanics, which are more accurate descriptions of the microscopic world.

Fig., Representation of Bit and Q-bit

In quantum computing, qubits are the basic unit of information. Qubits can exist in a superposition of both 0 and 1, which means that they can be both 0 and 1 at the same time. This is what allows quantum computers to perform calculations that are impossible for classical computers.

For example, a quantum computer could be used to solve the traveling salesman problem, which is a problem that is considered to be NP-complete. NP-complete problems are a class of computationally difficult problems. They are called NP-complete because they are in the complexity class NP, which stands for "nondeterministic polynomial time". This means that there exists a nondeterministic algorithm that can solve the problem in polynomial time. However, a nondeterministic algorithm is an algorithm that can guess the correct answer to the problem. The problem of travelling salesman is so difficult that it would take a classical computer billions of years to solve, but a quantum computer could solve it in a fraction of a second.

The superposition of qubits is a very fragile state, and it can be easily disturbed by noise. This is one of the challenges that need to be addressed before quantum computers can be widely used. However, as quantum computing technology continues to develop, it is likely that these challenges will be overcome.

The states in a quantum bit, or qubit, are in superposition, which can be expressed mathematically as:

|ψ⟩ = α|0⟩ + β|1⟩

Where:

• |ψ⟩ is the state of the qubit
• α and β are complex numbers that represent the probability of the qubit being in state |0⟩ and state |1⟩, respectively
• |0⟩ and |1⟩ are the basis states of the qubit, which represent the states where the qubit is 0 and 1, respectively

The sum of |α|² and |β|² must always be equal to 1, which represents the fact that the qubit must be in one of the two states |0⟩ or |1⟩.

|α|² + |β|² = 1

In a complex two-dimensional vector space, a qubit can be represented as a unit vector.

This means that a qubit can exist between the |0⟩ and |1⟩ states in a continuum of states, or in other words, it can exist in a superposition of both states at the same time.

When a qubit is measured, the superposition collapses and the qubit is found to be in either state |0⟩ or |1⟩ with a probability given by the absolute value of the square of the coefficients of the vector.

This property of quantum bits, called superposition, allows them to perform calculations that are impossible for classical computers. For example, quantum computers can be used to factor large numbers, break encryption codes, and design new drugs.

Let a qubit is in a state

1/(√3) |0⟩ + √(2/3) |1⟩

This equation represents the state of a qubit that is in a superposition of the |0⟩ and |1⟩ states.

The probability of the qubit being in the |0⟩ state is |1/(√3)|² or 1/3 or 33%, and the probability of the qubit being in the |1⟩ state is |√(2/3)|² or 2/3 or 66%.

The equation also shows that the qubit is a unit vector in a complex two-dimensional vector space. This means that the qubit can exist between the |0⟩ and |1⟩ states in a continuum of states, or in other words, it can exist in a superposition of both states at the same time.

When a qubit is measured, the superposition collapses and the qubit is found to be in either state |0⟩ or |1⟩ with a probability given by the absolute value of the square of the coefficients of the vector. In this case, the qubit has a 33% chance of being measured as |0⟩ and a 66% chance of being measured as |1⟩.

The superposition of qubits is one of the key properties that allows quantum computers to perform calculations that are impossible for classical computers. For example, quantum computers can be used to factor large numbers, break encryption codes, and design new drugs.

Entanglement of Q-bit States

Entanglement is a physical phenomenon in which the states of two or more particles are correlated in such a way that they cannot be described independently of each other, even when the particles are separated by a large distance.

For example, if two electrons are entangled, and one electron is measured to be in a spin-up state, the other electron must be in a spin-down state, regardless of how far apart they are. This is because the two electrons share a single wavefunction, and any change to the state of one electron instantaneously affects the state of the other electron.

The equation for two qubits in an entangled state is:

|ψ⟩ = 1/√2 (|00⟩ + |11⟩)

This equation represents the state of two qubits that are in a superposition of the |00⟩ and |11⟩ states. The |00⟩ state represents the case where both qubits are in the |0⟩ state, and the |11⟩ state represents the case where both qubits are in the |1⟩ state.

The probability of the two qubits being in the |00⟩ state is |1/√2|² = 1/2, and the probability of the two qubits being in the |11⟩ state is also |1/√2|² = 1/2.

When the two qubits are measured, they will collapse into either the |00⟩ state or the |11⟩ state, with equal probability. However, the important thing to note is that the two qubits will always collapse into the same state, regardless of which qubit is measured first.

This property of entanglement is what makes it so powerful for quantum computing. For example, quantum computers can use entanglement to perform calculations that are impossible for classical computers.

Here are some examples of how superposition of qubits can be used in quantum computers:

• Quantum cryptography: Quantum computers can be used to create unbreakable encryption keys. This is because the superposition of qubits makes it impossible to eavesdrop on a quantum communication channel without disturbing the key.
• Quantum simulation: Quantum computers can be used to simulate the behavior of molecules and materials. This could help to accelerate the development of new drugs and materials.
• Quantum machine learning: Quantum computers can be used to train machine learning models that are more powerful than classical machine learning models. This could lead to new breakthroughs in artificial intelligence.

The superposition of qubits is a powerful property of quantum mechanics that has the potential to revolutionize many industries. As quantum computing technology continues to develop, it is likely that we will see even more amazing applications of this technology.