following state vector inH⊗2: Quantum bit:qubit
|φiq:= α|0i+β|1i
Where |α|2+|β|2 = 1,α,β ∈ C. For n qubits, thetensor product 2.1of individual qubits defines the quantum state i. e. :
|Ψi=α1|00 . . . 0i+. . .+αn|11 . . . 1i
Such that∑i|αi|2=1forαi ∈C.
A primitive model of quantum computation isQuantum Circuit Model.
In this model each computational process consists of a number of Quantum Circuits
Quantum Gates, operating on qubits. Quantum gates can have arity
one or more e. g. Pauli operators (see Figure 2.1) have arity one and
controlled CNot, Toffoli have arity two and three respectively. The
Pauli gateXoperates as the quantum not gate.ZandYgates change
the phase of a qubit state. CNot gate consists of a control qubit and
2.3 q ua n t u m c o m p u tat i o n 23 X = 0 1 1 0 ! , Z= 1 0 0 −1 ! , Y= 0 −i i 0 ! , I = 1 0 0 1 !
Figure2.1: Matrix representation of arity-1quantum operations
CNot= 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 , Toffoli= 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0
Figure2.2: Matrix representation of arity-2,3quantum operations gate on the target qubit. Finally, Toffoli gate has two controlled qubits
and depending on both of them applies X gate on the third (target)
qubit. Remarkably,any classical circuit can be replaced by an equiva-
lent circuit with only Toffoli gates.
In the quantum circuit model, measurement is done using theMea-
surement gate. This gate performsgeneral measurementi. e. in the stan-
dard basis. The outcome of measurement gate is a classical piece of
information and permanently changed quantum state. It should be
noted that often we need to apply a special case general measure-
ment orprojective measurement(i. e. in other basis than standard basis
(see [84, p 87]). In quantum circuit model, this kind of measurement
can be achieved using series of unitary gates prior the measurement
2.3 q ua n t u m c o m p u tat i o n 24
|ψi • H •
|0i
|0i H • X Z |ψi
Figure2.3: Teleportation Circuit
Using a discreteset of quantum gates, any quantum circuit can be
approximated to an arbitrary precision. In other words, similar to
the set of classical gates {AND,OR, NOT}, there is a set of quan-
tum gates capable of approximatinguniversalquantum computation.
For example, Solovay-Kitaev theorem [84, p 617] states that a circuit
with marbitrary unitaries, can be approximated for any eusing only O(m logc(m/e))gates from the universal set:
{Hadamard,Phase,CNOT,π/8}
Quantum circuits usually are depicted with wires as for qubits and
boxes for quantum gates (e. g. X ). Double wire appears after
quantum measurement, denoting classical outcome of measurement
gate (depicted by ). Controlled gates are shown with circle
for control qubit and dots for target qubits. For example the circuit
in the Figure 2.3 shows how to perform quantum Teleportation [17],
using Pauli and measurement gates. We shall present Teleportation
protocol in the Chapter6
Main quantum algorithms such as Shor’s algorithms [95] for fac-
toring integer and discrete logarithm are presented using quantum
circuits. From complexity point of view, these algorithms belong to
2.4 q ua n t u m i n f o r m at i o n 25 family of polynomial sized quantum circuits with bounded error prob-
ability. The aforementioned Solovay-Kitaev theorem gives a uniform
construction of quantum circuits. A remarkable result in quantum
complexity theory connects quantum computing with classical com-
puting and states thatBQP⊆PSPACE[19]. More details onquantum
computational complexity can be found in [19].
2.4 q ua n t u m i n f o r m at i o n
In the previous section we have presented closed QIP system. In re-
ality, we need to deal with noises, and in fact this is a challenging
part in the implementation of quantum systems.Quantum Information
Theoryinvestigates thedynamicsofopen quantum systems(i. e. systems
which are affected by noises). The success of quantum information
theory follows from the progresses which have been made during
development ofQuantum Error CorrectionandQuantum Fault Tolerant
computation. In this section, the main results in quantum informa-
tion theory are briefly presented. We start from the model of quan-
tum error corrections, then we look into the quantum fault tolerant
computation. The case studies related to these areas are presented in
6. Finally, we introduce the notion ofoverlapof quantum states, arises
in many applications of quantum information theory. In particular,
by computing fidelitybetween two quantum states, we can measure
2.4 q ua n t u m i n f o r m at i o n 26 tive way of testing equality of quantum states and therefore suggests
another method for implementingequality testin Chapter5. Quantum Error Model
In quantum error correction theory, errors or noises are considered
to be superoperators E, acting on density operators. This model can
describe dynamics of open quantum systems that are weakly or even
strongly coupled with environment, assuming that the effect of er-
rors/noises are quantum operations themselves. A useful feature of
this model is that it describes changes to quantum states, discretely,
making it more convenient for formal analysis.
Let ρ be a density operator corresponding to the states which we
want to protect with error correcting code and letE be as above. Let
C denote a quantum error correcting code i. e. a subspace of a larger
Hilbert state than theρ’s initial Hilbert space (some examples of such
codes are given in Chapter 6). Then an error correction protocol is
successful if there is a superoperator R, such that it can correct E to
retrieveρ:
(R ◦ E)(ρ)∝ρ (2.5)
The Equation 2.4 is in its most general form, however in our case
studies we have performed perfect recovery, that is to say we substi-
tute∝with equality. The following theorem [84, p 436] formalises the
conditions needed for error correction protocols:
Theorem2.3 For a quantum code C, and a error correction protocol P