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101 which differ in the first qubit, and then apply U to the first qubit con-
′
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ditioned on the second and third qubit, i.e. a generalized Λ (U ) . Finally, the
2 ′
swapoperationsareappliedinreverse. Thefullcircuitthusbecomesaspictured
in figure 6.19.
U'
Figure6.19: Exampleofdecompositionoftwo-levelunitaryoperationon3-qubit
state.
The general case is a straightforwardgeneralization of this procedure.
The number of gates needed to achieve this decomposition can now be cal-
culated. First,atmost2(n 1)generalizedCNOToperationsΛ (X)toswap
n 1
− −
the input state as described above, and then back again. Each such swap op-
eration can be decomposed into (n) single qubit and CNOT gates according
O
to section ??. The same holds for the Λ (U) operation, yielding an overall
n 1
(n2) complexity for this gate constructio− n.
O
These gate constructions will be used in the following sections discussing
universality for the quantum circuit model.
6.2.12 Universal sets of quantum gates
In any computational model,8 there is a choice as to what constitutes the basic
programming primitives. In the non-reversible classical circuit model, NAND-
8ExceptintheGandymachinemodel[52]
143
gatestogether with FANOUT, is a universalsetof gates. In the reversibleclas-
sical circuit model, the Toffoli gate is universal. Another choice is the Fredkin
gate. Using either ofthese gates,any otherlogic operationas well as FANOUT
can be performed.
In the quantum circuit model, the situation is more complicated. The
unitary quantum gates form a continuum and an N N unitary matrix has
×
N(N 1)/2 complex parameters u . The possible sets of universal operations
ij
−
are therefore much richer than in the classical case where the set of gates is
discrete and finite. However, it can be shown that an arbitrary N N unitary
×
matrixU,actingonN-dimensionalvectorspace,canbeexpressedasaproduct
lower-dimensionalunitary matrices acting on subspaces of the vector space. In
fact, U can be decomposed entirely in terms of matrices acting non-trivially
on just two-dimensional subspaces. For a proof see [44] and [39]. Such a con-
struction is however in general not efficient in terms of the number of required
two-dimensionalmatrices,as the number of matrices needed is (N2), or more
O
precisely 2n 1(2n 1). Remember that in this context the dimension of the
−
−
vectorspaces areN =2n.9 Furthermore,the matrices appearingin such a con-
struction might not be possible to realize in a real physical system. What we
need is a discrete set of standard gates out of which any unitary operation can
be composed. Ingeneralwe cannotexpect such a constructionto be exact,but
rather obtained to within a certain approximation.
6.2.13 Exact and approximate universality
The universality results reported in the literature are somewhat bewildering.
In order to navigate among them, two distinctions should be kept in mind.
First, the distinction between exact universality and approximate universality.
Secondly, the distinction between a finite set of standard gates and an infinite
setof gates (parameterizedby some variables). Note at once that a finite set of
discretestandardgatescanonlybeuniversalintheapproximatesense,sincethe
unitary matrices U(N) is a continuous set, while circuits built using a discrete
set of gates can only generate a countable set of matrices.
Thus exact universality requires circuits with an infinite number of discrete
gates, or circuits using a finite number of gates continously parameterized by a
set of variables.
For practical purposes, approximate universality is the important concept,
sinceanyrealquantumcomputermustpresumablebebuiltfromasetofdiscrete
standard gates.
The two universality concepts, exact universality and approximate univer-
sality, are defined below.
9To avoid confusion, keep in mind that the number of wires n is equal to the number of
qubits. Buteachqubitspansa2-dimensionalcomplexvectorspace,thusthelineardimension
of the vectors and matrices is really N =2n and N N respectively. This is explicit when
×
usingthecomputational basisstates.
144
Definition
Let G be a set of gates with r elements,
G= G , ,G , (6.47)
1,n1 r,nr}
{ ···
whereG isagatethatactsonn qubits. Thesetis(approximately)universal
j,nj j
if any unitary operator U acting on n input qubits can be decomposed into a
finite product U of successive actions of gates G so that the error E(U,U)
j,nj
e e
E(U,U) ǫ
≤
where the error is defined as e
max
E(U,V)= (U,V)ψ .
ψ || | i||
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The maximum is taken over the set of all normalized states in the state space