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• Codes, i.e. interesting subspaces of H ⊗n . To make quantum computing fault tolerant,
these subspaces should have special properties. In particular, in a physical realization,
the dynamics of the system should suppress errors.
• Unitary operators acting on H ⊗n that preserve these subspaces.
5.5.4 Quantum gates
First let us discuss quantum gates: for quantum computation, we need unitary operators H ⊗n →
H ⊗n to be realized by some time evolution. Unitarity implies reversibility.
Definition 5.5.9
1. A quantum gate on H ⊗n is a unitary map H ⊗n → H ⊗n that acts as the identity on at
least n − 2 tensorands.
2. Consider a fixed finite set {U i } i∈I of quantum gates, i.e. U i ∈ U(H) or U i ∈ U(H ⊗ H),
called a library of quantum gates. Denote by U αβ
i the gate U i acting on the α and β
tensorand resp. Ui
α acting on the α tensorand of H n . A quantum circuit based on this
library is a finite product of Ui
α and U αβ
i . It is a unitary endomorphism of H ⊗n .
3. A library of quantum gates is called universal, if for any n, the subgroup of U(H ⊗n )
generated by all circuits is dense.
Examples 5.5.10.
1. An important examples of a gate is the CNOT gate (controlled not gate) which acts on
two qubits: H ⊗2 → H ⊗2 . The CNOT gate flips the second qubit (the target qubit) if and
only if the first qubit (the control qubit) is 1. Here we write 1 = | ↑〉 and 0 = | ↓〉.
Before
After
Control Target Control Target
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
The resulting value of the second qubit corresponds to the result of a classical XOR gate
while the control qubit is unchanged.
An experimental realization of the CNOT gate was afforded by a single Beryllium ion
in a trap in 1995 with a reliability of 90%. The two qubits were encoded into an optical
state and into the vibrational state of the ion.
2. The relative phase gate H → H acting on one qubit, a popular choice of which is in the
selected basis | ↑〉, | ↓〉: ( )
1 0
0 exp(2πi/5)
Similarly, the three Pauli matrices give rise to so-called Pauli gates acting on a single
qubit.
3. The library consisting of the CNOT gate and the relative phase gate can be shown to be
universal.
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