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as well as a finite number of output bits.
2. The output from C is denoted by C (x) and is defined for all binary
n n
numbers x of at most n bits of length.
3. If m<n and x is at most m bits in length then C (x) =C (x). This is
m n
the consistency requirement.
What prevents the circuit model fromyielding aneffective method for com-
putingeveryfunctionwhatsoeveristhefactthattheredoesnotexistaneffective
method to construct the circuits in the family for every number n.
By a uniform circuit family we mean a consistent circuit family for which
there does exist analgorithm,for example running ona Turing machine, which
computes a description of the circuit for every number n. In this way, the
uniform circuit model is by definition equivalent to the other models of compu-
tation.
Fromthisweseeafundamentaldifferencebetweenthecircuitmodelandthe
Turingmachine model. Oncea Turingmachine is programmed,it willcompute
the values of the function for every input number for which it halts. A given
circuitwithagivennumberofinputbitsonlycomputesthe functionforafinite
range of values. Beyond this range of values, a new circuit (in the family) is
demanded.
Another way of looking at the fact that non-uniform circuits compute all
functionsisthatthenon-uniformcircuitmodelcannotbefinitelydescribed. The
listofcircuitsisinfiniteandwehavenofinitewaytogeneratethelistofcircuits.
It therefore falls outside the characterization of a finitely defined algorithm. It
is clear that circuits can be built to compute any computable function, but in
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this way one gets specialized circuit families for each computational task. In
order to get a universal model of computation, the circuit must be wired to
perform a standard set of instructions on input data, the computational task
itself being supplied as a program. This is the way an ordinary von Neumann
architecture digital computer works in a (fetch instruction, fetch data, execute
instruction, save data) cycle. The circuit must thus be clocked and in this way
computational steps are introduced.
Thecircuitmodelisusefultodescribequantumcomputation,butinitselfit
is rather awkward. What makes an algorithm for an infinite set of instances of
a problem useful is the fact that once the algorithm is known, it permits us to
obtain new knowledge. If we do not know the value of a computable function
foracertainargument,runthe algorithmto findthatvalueout! Usingcircuits,
newmembersofthecircuitfamilymustbecomputedinordertogetnewvalues
of the function.
In a sense, this is a reflection of the fact that the circuits really just furnish
formulas for the function values, not algorithms. If there is an explicit formula
for a function, no algorithm is needed to compute the values.
2.4.2 Reversible gates
The classical logical gates are all irreversible except for the NOT gate. This
meansthatthevaluesoftheinputbitscannotbeinferredfromthevaluesofthe
output bits. Just one example illustrates the point. If an AND gate outputs
0, there is no way to know which of the possible input combinations 00, 01 or
10 resulted in the output.
In [5] and [10] it was shown that an irreversible logical operation has to
dissipate a certainminimum amountof energy. Onthe other hand, a reversible
logicaloperationdoesnothavetodissipateanyenergy. This leadto aninterest
in reversible computations, and this was also one of the initial motivations be-
hindresearchintoquantumcomputation,sincetheevolutionofclosedquantum
systems are inherently reversible.
Thefirstrequirementforagatetobereversibleisthatthenumberofoutput
gates equals the number of input gates. A simple example is the CNOT gate.
It has two input bits and two output bits.
x x'
y y'
Figure 2.5: The CNOT gate.
The function computed by this gate is shown in the ’truth’ table below.
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x y x’ y’
0 0 0 0
0 1 0 1
1 0 1 1
1 1 1 0
The name CNOT stands for controlled-NOT. The first input bit x con-
trols the second input bit in such a way that when x = 0 , the second output
bit y =y, andwhen x=1, then y =NOTy. The firstoutput bit x is always
′ ′ ′
equaltox. Notethatthe gatecanalsoberegardedasageneralizationofXOR
since y =xXORy.
′
The fact that the CNOT-gate is reversible can be seen in two ways. First,
by simply inspecting the truth table, it is clear that knowing x and y , x and
′ ′
y can be uniquely derived. Secondly, if a second CNOT is connected after the
first CNOT, the total effect will be same as just to unit wires, i.e. CNOT is
its own inverse.
The functional relations between inputs and outputs can thus be written
x =x
′ .
(cid:26)y =x y
′
⊕
The CNOT gate can be used to make a copy of an input bit. If y in the
truth table is fixed to 0, both x and y are equal to x.
′ ′
x y x’ y’
0 0 0 0
1 0 1 1
This can also be seen as a FANOUT.
AnotherreversiblegateistheToffoligate. Ithasthreeinputwiresandthree
output wires.
x x
y y