Quantum register

In quantum computing, a quantum register is a system comprising multiple qubits. It is the quantum analogue of the classical processor register. Quantum computers perform calculations by manipulating qubits within a quantum register.

In quantum computing, a quantum register is a system comprising multiple qubits. It is the quantum analogue of the classical processor register. Quantum computers perform calculations by manipulating qubits within a quantum register. The Hilbert space, \mathcal{H} , in which the data is stored in a quantum register is given by \mathcal{H} = \mathcal{H_{n-1}}\otimes\mathcal{H_{n-2}}\otimes\ldots\otimes\mathcal{H_0} where \otimes is the tensor product. The number of dimensions of the Hilbert spaces depends on what kind of quantum systems the register is composed of. Qubits are 2-dimensional complex spaces ( \mathbb{C}^2 ), while qutrits are 3-dimensional complex spaces ( \mathbb{C}^3 ), etc. For a register composed of N number of d-dimensional (or d-level) quantum systems we have the Hilbert space \mathcal{H}=(\mathbb{C}^d)^{\otimes N} = \underbrace{\mathbb{C}^d \otimes \mathbb{C}^d \otimes \dots \otimes \mathbb{C}^d }_{N\text{ times}} \cong \mathbb{C}^{d^N}. The registers quantum state vector \psi of this d^N -dimensional Hilbert space can in the bra-ket notation be written as a linear combination of some set of orthogonal basis vectors labeled |0\rangle to |d^N-1\rangle, as |\psi\rangle = \sum_{k=0}^{d^N-1} a_k|k\rangle = a_0|0\rangle + a_1|1\rangle + \dots + a_{d^N-1}|d^N-1\rangle. Such linear combinations are in quantum mechanics called superpositions and the values a_k are probability amplitudes. Because of the Born rule and the 2nd axiom of probability theory, \sum_{k=0}^{d^N-1} |a_k|^2 = 1, so the possible state space of the register is the surface of the unit sphere in \mathbb{C}^{d^N}. Examples: The quantum state vector of a 5-qubit register is a unit vector in \mathbb{C}^{2^5}=\mathbb{C}^{32}. A register of four qutrits similarly is a unit vector in \mathbb{C}^{3^4}=\mathbb{C}^{81}. If we consider two pure qubits in superpositions |a_0\rangle=\frac{1}{\sqrt2}(|0\rangle + |1\rangle) and |a_1\rangle=\frac{1}{\sqrt2}(|0\rangle - |1\rangle) , using the quantum register definition |a\rangle=|a_{0}\rangle\otimes|a_{1}\rangle = \frac{1}{2}(|00\rangle - |01\rangle + |10\rangle - |11\rangle) it follows that it is capable of storing all the possible values (by having non-zero probability amplitude for all outcomes) spanned by two qubits simultaneously.

Further reading

Text derived from https://en.wikipedia.org/w/index.php?oldid=1362511748 under CC BY-SA 4.0; modifications have been made.