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it was shown that almost any n-bit gate with n 2 is universal. The question
≥
as to which gates are not universal was clarified in [51], a result which we will
return to below.
Single qubit and CNOT universality
Relying on the fact that the Deutsch gate Λ (U) is universal, and the gate
2
constructions of section6.2.10, where it was shownthat any Λ (U) gate can be
2
decomposedintermsofsinglequbitandCNOTgates,itfollowsthatsinglequbit
and CNOT gates are universal for quantum computation. This is approximate
universality, since the Deutsch gate is approximately universal.
Howeverit is possible to show thatsingle qubit andCNOT gates arein fact
exactly universal. This stronger results follows from the exact decomposition
of a general N-dimensional unitary matrix in terms of 2-dimensional unitaries
acting on two-dimensional subspaces. Relying on the construction in section
6.2.11, this can be simulated with single-qubit gates and CNOT-gates.
6.2.16 Discrete sets of gates
Finally we arrive at a discussion on finite sets of practical one qubit gates to-
gether with CNOT gates. Relying on the previous discussion, it now suffices to
effectively approximate general one-qubit gates with a discrete set of one-qubit
gates.
The standard set of gates consists of the Hadamard gate H, the π/8 gate T
and the phase gate S [53]. The phase gate is not really needed for universality,
but it is needed in order to perform the approximations fault tolerantly, an
important topic that will be briefly mentioned below.
The proof that the gates H, π/8 and T are sufficient to approximate any
single qubit gate, is quite complicated. It will not be repeated here, the reader
is instead referred to [39].
147
Efficiency of the gate constructions
In order to estimate the efficiency of this construction, consider an n-qubit
circuit approximating a U(2n) operation. Suppose the circuit requires f(n)
gates,mostofwhichneednotbeintheuniversalsetofgatessothatmostofthem
must be approximated. Suppose furthermore that the total tolerance to errors
is required to be ǫ. Then each gate must be approximated to within an error
δ =ǫ/f(n) since the errors adds at most linearly. The overall efficiency is thus
determinedbytheefficiencywithwhichasinglequbitgatecanbeapproximated.
Naively,onewouldsuspectthatroughlyΘ(1/δ)gatesfromthediscretesetwould
be needed, resulting in an overall efficiency of Ω(f(n)/δ)=Ω(f(n)/(ǫ/f(n))=
Ω(f(n)2/ǫ).
However, according to the Solovay-Kitaev theorem (which unfortunately is
too complicatedto provehere)a generalsingle qubit gatecanbe approximated
within an error δ using (logc(1/δ)) gates from the discrete set. The constant
O
c is number approximatelyequalto 2. Therefore,the overallefficiency becomes
(f(n)logc(f(n)/δ)),whichisapolylogarithmicincreaseinthenumberofgates
O
as compared to original circuit.
What about the function f(n)? An upper bound for this function can be
calculated based on the gate constructions in sections 6.2.11 and 6.2.14. From
the decomposition of an arbitrary U(2n) matrix in terms of two-level unitaries
(section 6.2.14), we know that it takes 2n(2n 1)/2 gates, i.e. (4n). Then,
− O
accordingtosection6.2.11,eachsuchtwo-levelunitarygate(actingonthestate
space of n qubits) can be implemented using (n2) single qubit and CNOT
O
gates. Therefore, the function f(n) is in (n24n).
O
Putting everything together we see that the overall complexity is
(n24nlogc(n24n/δ)), which is exponential in n. But the exponential behav-
O
ior arises not from the approximation of single qubit gates by discrete single
qubit gates, but from the complexity of breaking down general U(2n) matrices
in terms of two-level matrices. Thus, fast quantum algorithms cannot rely on
naive universality constructions.
6.2.17 General results
The particular universality results reviewed to above are beautifully subsumed
under a general theorem [51]. In order to state the theorem, the notion of an
imprimitive gate must be defined.
A two-qubit gate V is said to be primitive if it maps decomposable states
intodecomposablestates,i.e. if x and y arequbits, thenthere arequbits u
| i | i | i
and v such that V x y = u v . V is imprimitive if it is not primitive.
| i | i| i | i| i
There is a simple condition to determine whether a gate is primitive or
not. Let P be an operator that swaps the two qubits in a product state, i.e.
P x y = y x . Then it can be shownthat V is primitive if and only if V can
| i| i | i| i
be expressed as S T or as (S T)P for some single qubit gates S and T, so
⊗ ⊗
that V acts as V x y =S x T y or as V x y =S y T x .
| i| i | i⊗ | i | i| i | i⊗ | i
148
Theorem
Given a two-qubit gate V, the following conditions are equivalent
the collection of all single qubit gates together with V is approximately
•
universal
the collectionofallsingle qubitgatestogetherwith V is exactlyuniversal
•
V is imprimitive
•
Of course, a discrete set of single qubit gates can never yield exact univer-
sality.
149
Bibliography
[1] J. Gruska, Quantum Computing, McGraw-Hill, London 1999.
[2] A. Steane, Quantum Computing, Rep. Prog. Phys. 61 (1998) 117-173.
[3] A.GalindoandM.A.Mart´in-Delgado,Information andcomputation: Clas-