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6.2 The circuit model of quantum computation
The quantum Turing machine model is not practical when it comes to actual
algorithm construction, and just as in the classical case, it is a theoretical con-
struct far from real computer design. The model is difficult to work with since
thestateofthecomputerisasuperpositionofnotjustthedataonthetape,but
alsothe headpositionandthe internalconfiguration. This leadsto trickyques-
tionsaboutthehaltingofthemachine,asdifferentbranchesofthecomputation
may take different number of steps to complete their respective computations.
[43].
The quantumcircuitmodel is ageneralizationofthe classicalcircuitmodel.
Insteadofbits,qubits aretransmittedinthewires. Theclassicallogicgatesare
replacedbyquantumgatesrepresentedbyunitaryoperators. Inthismodel,the
state of the computer is a superposition of the data only. The actual wiring of
the circuit and the number of gates applied to the data are treated classically.
1Theclassicalcasecould beseenascorresponding toallthecoefficients except onebeing
zero,thenon-zeroonebeingequalto1.
124
The model is formulated in terms of unitary computation matrices, that
given arbitraryn-qubit input vectors,produce the desired n-qubit output vec-
tors. Algorithm construction amounts to composing such matrices out of sim-
pler, primitive matrices acting on just a few qubits at a time. There are two
importantquestions;findingauniversalsetofbuildingblocks,i.e. programming
primitives, and finding methods for efficient algorithm construction.
The circuit model is, however, subject to certain limitations. Its classical
counterpartis the reversible logic circuit described in chapter 2. A logic circuit
computes a fixed function for a given range of input, say n bits. If the output
is required for data beyond this range, the circuit must be extended to m > n
input bits. In principle, an algorithm is needed for this, or put differently, an
algorithm is needed to generate the uniform circuit family C computing
n }∞n=1
{
the required function for arbitrary length input. The task of assembling the
uniform circuit family cannot be performed by another circuit [25]. Therefore,
initself,thecircuitmodelisnotacompletecomputationalmodel,andthesame
is true for quantum circuits.
Therefore,wemustrefinethenotionofalgorithmconstructioninthe circuit
model, to providing a uniform circuit family for the problem at hand.
C
Figure 6.1: A general circuit.
Since each wire carry a two-state quantum bit, an n-qubit circuit C per-
n
forms a unitary operation represented by a 2n 2n unitary matrix U .
Cn
×
If all the input wires are used for data, then a given circuit performs one
and the same algorithm on the data. Thus, the program is hardwired into
the circuit, and in this sense, the circuit is not a general purpose computer.
But nothing prevents us from considering some of the inputs as supplying a
program, or rather an instruction, to be carried out on the rest of the input,
whichis then the data proper.2 Universalityin the contextofthe circuitmodel
will be discussed below.
2Of course, on a certain level of abstraction one need not make any distinction between
dataandprogram.
125
6.2.1 Gates and wires
An abstract quantum circuit is built out of wires and gates.3 The wires carry
the qubits between the gates, from outputs to inputs. The qubit processing
takes place in the gates. There is to be no feedback wires. The number of
output wires and input wires are equal for individual gates as well as for the
complete circuit.
G
n wires n wires
Figure 6.2: A generic quantum gate.
AgenericquantumgateGperformstheunitaryoperation ψ =U ψ .
out G in
| i | i
TheN N unitarymatrixU representingann-qubitgatewithN =2nbelong
Gn
×
to the Lie group U(N).
6.2.2 General notation
A general one-qubit unitary gate is represented by a 2 2 unitary matrix
×
u u
U = 00 01 , (6.11)
(cid:18)u 10 u 11(cid:19)
andlikewise,n-qubitgatesarerepresentedby2n 2nmatriceswherethematrix
×
elements are denoted by u with indices ranging from 0 to 2n 1.4 For fixed
ij
−
index i, u are row vectors, and for fixed index j, u are column vectors. In
ij ij
eitherview,unitarityforthematrixisequivalenttoorthonormalityoftheserow
and column vectors respectively. This is a property of unitary matrices that is
often useful precisely when deciding unitarity.
These matrices are realizations of unitary operators U in some orthonormal
basis, most often in the computational basis. Graphically they are represented