T2: 2. Quantum Circuits Flashcards
Give the phase shift matrix S
Identity; bottom right i.
Give the phase shift matrix T
Identity; bottom right e^iπ/4
Give the Hadamard gate matrix H
1/sqrt(2) All ones; bottom right -1
What action does Hadamard have on basis states |0⟩ and |1⟩
|0⟩ →|+⟩
|1⟩ →|-⟩
How many unitaries does it take to represent a generic U acting on n qubits.
N=2^n
(1/2)N(N-1)
What kinds of classical circuits have quantum counterparts and why?
Reversible circuits. Since these can be expressed as unitary transformations, which by definition, have an inverse.
Which single-bit classical gates have quantum analogues? Why?
The identity and NOT gate; these are reversible and so we can express them as unitary matrices.
How can we express an arbitrary unitary operator U?
U=e^iα R_n(θ)
Where R_n(θ) is a rotation about the n axis.
What are the relations between S,T, and Z
S^2 = Z and T^2=S
What is special about the CNOT in a quantum circuit?
It creates entanglement.
Define ‘multiply controlled unitaries’
Unitaries which act if all other qubits are 1.
On what kind of subspace does a multiply controlled unitary act.
Two states which differ by a single bit, with all other bits =1.
Define gray code
The process of using NOTs and CNOTs to prepare a state
How does an arbitrary unitary act on the Bloch sphere?
It preserves the length of r, so acts as a rotation.
What is the basis for Z?
Computational basis
