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The decomposition of Λ (U) can be generalized to more than 2 control qubits.
2
In fact, for any 2 2 unitary operator U, the controlled operation Λ (U)
n 1
canbe implemente× d using only Θ(n2)elementary operations. Thereare s− everal
such constructions [54, 39]. The one below is from [54].
Consider the gate construction in figure 6.17, which is an example with
n=5.
c
1
c
2
c
3
=
c
4
c
5
U V V+ V
Figure 6.17: Simulation of a Λ (U) operation.
5
This is obviously a generalization of the decomposition of Λ (U) and the
2
proof is similar with V an operator such that V2 =1.
To estimate the complexity of this decomposition, we need a construction
of the generalizedToffoli gate Λ (X). There are constructions of these gates
n 2
−
141
of order Θ(n) in the number of Toffoli gates and elementary gates used. The
reader is referred to [54] for details, a paper that contains a wealth of gate
constructions.
The overall complexity C(n 1) of Λ (U) can now be estimated. The
n 1
− −
cost of simulating the gates Λ (V) and Λ (V ) is constant, and the complexity
1 1 †
of Λ (X) is Θ(n). Then, the complexity of Λ (V) is C(n 2) by applying
n 2 n 2
− − −
the constructionoffigure 6.17recursively. This yields a recurrenceequationfor
the cost C(n)
C(n 1)=C(n 2)+Θ(n).
− −
Since (n 1)2 (n 2)2 is in Θ(n), it is clear that C(n) is in Θ(n2).7
− − −
6.2.11 Decomposing general two-level unitary operation
on n-qubit states
Let U be a two-level unitary matrix acting on an n qubit state. The two levels
involved can be any two directions in the full space of the state. Suppose the
twodirectionsaregivenbythecomputationalbasisstates s = s s s and
1 2 n
| i | ··· i
t = t t t respectively. These two directions can differ in between 1 and
1 2 n
| i | ··· i
n places. In order to write U in terms of a single qubit operation, swap oper-
ations must be applied to yield two directions that differ in precisely one place
corresponding to one particular qubit. The swap operations can be performed
by generalized Λ (X) operations. An example will clarify the situation.
n 1
−
Consider a 3 qubit computer and a particular two level matrix U
1 0 0 0 0 0 0 0
0 a 0 0 0 0 b 0
 
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
U = .
0 0 0 0 1 0 0 0
 
0 0 0 0 0 1 0 0
 
0 c 0 0 0 0 d 0
 
0 0 0 0 0 0 0 1
 
Denote by u the two-dimensional matrix
a b
U = .
′ (cid:18)c d(cid:19)
The matrix U acts non-triviallyonthe computationalbasisstates 001 and
| i
110 . Using so called Gray coding, the binary number 001 can be transformed
| i
by one-bit flips into 110 through the steps (they are not unique)
001 000 010 110
→ → →
The first two steps can be performed by the circuit
7This solution is a particular solution. A more careful treatment of this equation shows
that the solution tothe homogenous equation, whichingeneral yields exponential behavior,
inthiscasegivesasolutionc1n=c,aconstant.
142
0 0 0
0 0 1
1 0 0
Figure 6.18: Implementation of example Gray code transformation.
The idea is to use generalizedcontrolled NOT gates to transform 001 into