Multi-Qubit Systems

The true power of quantum computing only becomes evident as you increase the number of qubits. Single qubits possess some counter-intuitive features, such as the ability to be in more than one state at a given time. However, if all you had in a quantum computer were single-qubit gates, then a calculator and certainly a classical supercomputer would dwarf its computational power.

Quantum computing power arises, in part, because the dimension of the vector space of quantum state vectors grows exponentially with the number of qubits. This means that while a single qubit can be trivially modeled, simulating a fifty-qubit quantum computation would arguably push the limits of existing supercomputers. Increasing the size of the computation by only one extra qubit?doubles?the memory required to store the state and roughly?doubles?the computational time. This rapid doubling of computational power is why a quantum computer with a relatively small number of qubits can far surpass the most powerful supercomputers of today, tomorrow, and beyond for some computational tasks.

Mathematical representation involves tensor products, state vectors, and composite systems.

Now......

How do we combine multiple qubits?

The first thing you need to know about to do this would be, the?tensor product.

Tensor product

The tensor product is nothing but a means of combining two matrices of arbitrary sizes into a single block matrix. A block matrix is just another larger matrix that combines the two matrices, and this will be indicative of its two components. The block matrix being indicative of the two-component matrices.

The tensor product combines individual qubit states to form a joint state for the composite system.

That having been said, we can use this mathematical operation to combine two or more qubits and the result we get is also a qubit state! Let’s dive deeper into some of the finer details about the tensor product.

Example: If qubit A is in state |0? and qubit B is in state |1?, their joint state is given by |0? ? |1? = |01?.

The tensor product is represented by the symbol ? and it operates on a given vector like so –

If,

In short, the tensor product extends to any number of qubits in the system.

State Vectors

• The joint state of multiple qubits is represented using a state vector.
• For a system with n qubits, the state vector has 2^n complex amplitudes.
• Example: A two-qubit system with qubit A in state |0? and qubit B in state |1? has the state vector |01? = [0, 0, 1, 0].

Composite Systems:

• When qubits are entangled, the state of the composite system cannot be expressed as a simple tensor product.
• Entanglement allows for correlations between qubits that cannot be explained by individual states.
• Example: The Bell state |Φ+? = (|00? + |11?) / √2, where measuring one qubit instantly determines the state of the other.

Quantum Gates and Operations

• Quantum gates operate on multiple qubits to manipulate their joint state.
• Gate operations, such as the Controlled NOT (CNOT) gate, enable entangling and disentangling qubits.
• Example: Applying a CNOT gate to a two-qubit system flips the target qubit's state if the control qubit is |1?.

Quantum Algorithms

• Multiple qubit systems play a crucial role in quantum algorithms like Shor's algorithm and Grover's search algorithm.
• These algorithms harness the power of entanglement and superposition to solve complex problems efficiently.

Measurement and Observables

• Measuring a multiple-qubit system collapses it into one of the possible measurement outcomes.
• The probabilities of measurement outcomes are determined by the state amplitudes and their coefficients.
• Observable quantities, such as expectation values, can be calculated using the state vector and corresponding operators.