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3.3.3 Single Qubit Operations in a Coupled System
To be able to simulate multi-qubit systems, the vector description A(t) needs
to be expanded to include all possible qubit states. For n qubits this vector has
2 n entries to accommodate all binary combinations of the possible measurement
outcomes. For two qubits, for example, it would take the form:
A(t) = ( a | 00 〉 (t), a | 01 〉 (t), a | 10 〉 (t), a | 11 〉 (t) ) (3.51)
Here, a | xy 〉 is the complex amplitude of state | xy 〉. Again, the overall phase of
the state is arbitrary, allowing us to choose a | 00 〉 to be real. The normalization
requirement now applies to the entire state in the form:
∣ a| 00 〉
∣ ∣
2
+
∣ ∣a| 01 〉
∣ ∣
2
+
∣ ∣a| 10 〉
∣ ∣
2
+
∣ ∣a| 11 〉
∣ ∣
2
= 1 (3.52)
This gives the qubit state six degrees of freedom. In general, an n-qubit state
has 2 n+1 − 2 degrees of freedom. Since each degree of freedom can be used as
a register in a calculation, this leads to an exponential increase in the power of
a quantum computer with its number of bits. Unfortunately, this also makes it
exponentially harder to simulate. To simulate single qubit operations in an n-
qubit system, it is necessary to expand the Pauli matrices σ x , σ y , and σ z to apply
to only one qubit in the set. This is done by forming the Kronecker product of
these matrices with the identity. For example, an X-rotation on the second of four
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