3.3 Multi-qubit systems and tensor products

Multi-qubit systems are the backbone of quantum computing, combining multiple qubits to create complex quantum states. These systems enable powerful algorithms and entanglement, offering exponential computational power compared to classical computers.

Tensor products are crucial for describing multi-qubit states, combining individual qubit spaces into larger state spaces. This mathematical tool allows for compact representation of joint states and entanglement, simplifying the description of quantum systems.

Multi-qubit Systems

Multi-qubit systems in quantum computing

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• Multi-qubit systems consist of multiple qubits, the fundamental building blocks of quantum computers, each capable of being in a superposition of $|0\rangle$ and $|1\rangle$ states (Bloch sphere representation)
• Multi-qubit systems enable the implementation of complex quantum algorithms (Shor's algorithm, Grover's search) and allow for the creation of entangled states, a key resource in quantum computing where the states of qubits are correlated
• The number of states in a multi-qubit system grows exponentially with the number of qubits, providing exponential computational power compared to classical computers (quantum supremacy, quantum advantage)

Tensor products for multi-qubit states

• Tensor products, denoted by the symbol $\otimes$, are mathematical operations used to describe multi-qubit states by combining the state spaces of individual qubits to create a larger state space (Hilbert space)
• Tensor products allow for the representation of joint states of multiple qubits and enable the description of entangled states, which cannot be factored into individual qubit states (Bell states, GHZ states)
• Tensor products provide a compact notation for multi-qubit state vectors, simplifying the mathematical representation of quantum systems (Dirac notation, bra-ket notation)

Constructing and Transforming Multi-qubit States

Construction of multi-qubit state vectors

• The state vector of a multi-qubit system is obtained by taking the tensor product of individual qubit states, e.g., for a two-qubit system with qubits $A$ and $B$, the state vector is given by $|\psi\rangle_{AB} = |\psi\rangle_A \otimes |\psi\rangle_B$
• Computational basis states for multi-qubit systems are obtained by taking tensor products of single-qubit computational basis states ($|0\rangle$ and $|1\rangle$), e.g., for a two-qubit system, the computational basis states are $|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$
• Superposition states in multi-qubit systems are represented as linear combinations of computational basis states, with coefficients indicating the amplitudes of each basis state, e.g., $|\psi\rangle_{AB} = \alpha|00\rangle + \beta|01\rangle + \gamma|10\rangle + \delta|11\rangle$, where $|\alpha|^2 + |\beta|^2 + |\gamma|^2 + |\delta|^2 = 1$ (normalization condition)

Multi-qubit transformations with tensor products

• Multi-qubit transformations are achieved by applying quantum gates, which are unitary operators that act on the state vector, to the multi-qubit state vector (quantum circuit model)
• Tensor product of single-qubit gates involves applying a single-qubit gate to one qubit in a multi-qubit system, e.g., applying the Hadamard gate ($H$) to qubit $A$ in a two-qubit system: $H_A \otimes I_B$, where $I$ is the identity operator
• Controlled gates are multi-qubit gates that apply a single-qubit operation to a target qubit based on the state of a control qubit, e.g., the Controlled-NOT (CNOT) gate applies an $X$ gate (bit flip) to the target qubit if the control qubit is in the $|1\rangle$ state
• Tensor product of multi-qubit gates involves combining multiple multi-qubit gates to perform complex transformations, e.g., applying a CNOT gate followed by a controlled-Z gate (phase flip): $(CNOT \otimes I) \cdot (CZ \otimes I)$