Introduction to Quantum Physics and Information Processing

Introduction to Quantum Physics and Information Processing

(a) Experimental setup: 1 slit closed

(b) Classically expected result (c) Actual result: interference

Introduction 9

1.2 Properties of Qubits

Quantum systems have certain properties that are counter-intuitive and

completely outside our range of experience in the classical world. These

“weird” properties are best understood as inevitable consequences of axioms

on which quantum laws are based. These axioms have been arrived at af-

ter considerable effort and study of experimental phenomena, and are now

accepted among physicists as the complete theory which describes the real

world at the fundamental level.

The basic mathematical properties of a qubit can be analyzed and studied

independent of the physical system that realizes it. By treating the qubit as

an abstract mathematical entity, we can develop a general theory of quan-

tum information processing. Some of the strange new properties that become

relevant are now discussed.

Superposition and quantum parallelism: The main implication of

states like that of Equation 1.1 is that a single state contains the potential

for the system to be in either basis state. In some sense the system, say an

electron characterized by its spin value, simultaneously exists in both states

until measured. Physically this does not seem to make sense to our classical

minds unless we say that the electron has not decided which of the two possible

states it should be in, until forced into one of them by a measurement.

This feature is exploited in quantum computation to implement what is

called quantum parallelism: an operation that acts on a bit can now simul-

taneously act on both possible values of the bit if the input is a qubit in a

quantum superposition.

Size of computational space: If we want to do an n-bit computation,

Classically the “space” available for computation is of size n. In terms of a

quantum system of n qubits, the number of possible basis states is 2

n

, and

this is the size of the space available for computation. The size of the space of

states available for computation grows exponentially with the number of bits

(Figure 1.5). This is the power we wish to exploit in quantum computation.

Entanglement and quantum correlations: Multiple qubit systems can

exist in superposition states that are known as entangled states. These states

possess intrinsic correlations between the component systems that are dif-

ferent from classical correlations. These correlations can survive even if the

component systems are taken physically far apart from each other. For exam-

ple, 2-qubit states are in general linear superpositions of |00i, |01i, |10i, and

|11i. Look at the state

1

2

(|00i + |11i). In such a state, the first and second

systems are correlated quantum mechanically: the value of the second qubit

is always equal to that of the first qubit, irrespective of what measurement

we make on which bit and when. Such a state is called “entangled” because

of this correlation.

Quantum correlations can be exploited to generate new methods of pro-

10 Introduction to Quantum Physics and Information Processing

FIGURE 1.5: Computational power: quantum vs classical.

cessing, increasing the efficiency by allowing controlled operations to be per-

formed. These correlations are an invaluable resource in quantum information

theory and we will see their basic applications in quantum state teleportation

and secure information transfer over a distance.

Measurement and state collapse: Though a qubit could exist in a su-

perposition of basis states, a measurement of the qubit would give one of the

two basis states alone. Measurement of a quantum system causes it to collapse

into one of the basis states, which destroys the superposition, including any

information that may be encoded in the probability amplitudes. Some au-

thors express this property as a qubit existing in a superposition not having

a definite state. Measurement results can be predicted with 100% certainty in

“definite” states, and the system exists in a basis state. When a system is not

in a definite state, measurement disturbs the system and one can never know

the original state exactly. It is a quantitative and in-depth study of quantum

measurements that has uncovered new laws of quantum information.

Unitary evolution and reversibility: Quantum dynamical laws gov-

erning the evolution of an isolated quantum system are what are known as

unitary evolutions. Thus the functioning of a quantum computer is necessarily

via unitary transformations of the initial quantum state. Unitary operations

are fully reversible and, from a large body of study on the energetics of com-

putation, are said to lead to greater energy efficiency.

No cloning: This is another peculiar property of generic quantum states:

quantum states that are not basis states cannot be perfectly cloned or copied.

The fact that classical states can be copied and kept aside for further process-

ing is often taken for granted. When implementing a function in a classical

circuit, we often send copies of a particular input to different parts of the

circuit. Such an operation is no longer possible in quantum computing. This

changes the way we look at a quantum computation. And on the upside,

this also makes it possible to exchange information in a secure manner since

tapping a quantum line disturbs the system irrevocably!

Introduction 11

These properties lead us naturally to a model of computation often called

the “circuit model,” based on classical logic-gate circuits, of quantum compu-

tation, which is what we will primarily study in this book. However, several

other models of quantum information processing have also evolved, such as

measurement-based computation, continuous-variable computation and adia-

batic evolution. The interested reader may refer to the literature for these.

1.3 Practical Considerations

Theoretically, the examination of the paradigm of quantum computation

has been very promising and exciting. However these considerations need to be

grounded in reality. Pure quantum systems are found at the microscopic scale

and are difficult to access except by special technological means. To initialize

any information process, we must have the means to assign any desired state

to the qubit. Manipulation of the states of an individual qubit requires a high

level of technological ingenuity. We need not just one qubit but large qubit

registers. These may be built out of a collection of non-interacting qubits but

whether such a register can be built for the system at hand brings in questions

of scalability.

In implementing a quantum gate, we would be required to assemble some

means to applying forces on the system in a precise and accurate manner.

These operations would have to be impervious to error. The major prob-

lem in practice with quantum superposition states is that they are extremely

fragile. The slightest interactions would cause a disturbance by which the co-

herence is lost and the prepared system ends up in one of the basis states!

This phenomenon, known in literature as decoherence, is also crucial in un-

derstanding how the classical world emerges from the quantum substrate.

However, the discovery of quantum error correction and the subsequent con-

struction of fault-tolerant computing has infused confidence in the success of

the paradigm despite this issue.

The final big challenge is in interpreting the results of a measurement on

the system. The whole computational process must be set up such that the

end result is one of the basis states so that measurements give definite and

not probabilistic outcomes.

It may indeed be justifiable to ask if quantum computation is just in theory,

a matter of fanciful speculation, or possible in concrete implementation. While

there are technical challenges in the building of a feasible quantum computer,

the actual implementation is not only possible but also a reality. Various

ingenious techniques in quantum physics have been implemented, and newer

ones are being rapidly developed.

In developing a viable physical implementation, a bunch of criteria, first

to be underlined by DiVincenzo [30], are to be satisfied:

12 Introduction to Quantum Physics and Information Processing

1. A robust, error-tolerant system for qubits

2. A method of initializing (preparing initial states)

3. Scalability: quantum systems that must be replicated to larger numbers

to make bigger registers

4. Ability to manipulate individual quantum states: this is the most chal-

lenging engineering task that is required to make the computer work

5. Readout of output: the end result of the computation must be readable,

that is, measurement with unambiguous results.

Several systems have been analyzed with these criteria in mind. In a given

system too, there could be different possible realizations of a qubit. In Ta-

ble 1.1, we list a few such systems to give you an idea of the variety in the

physics that is involved.

TABLE 1.1: Summary of common physical implementations of quantum com-

puting systems.

System Information carrier Method of control

Quantum Optics photon polarization polarizers, half wave

plates, quarter wave

plates

presence of a single photon

in one of two modes

beamsplitters, mirrors,

and non-linear optical

media

Cavity QED two-level atom interacting

with a single photon

phase-shifters, beam split-

ters, and other linear opti-

cal elements

Trapped Ions hyperfine energy levels

and the vibrational modes

of the atom

pulsed laser light to ma-

nipulate the atomic state

Nuclear Magnetic

Resonance (NMR)

nuclear spin states pulsed RF fields in the

presence of a strong exter-

nal magnetic field

Superconducting

Circuits

Cooper-pair box electrostatic gates and

Josephson junctions

flux-coupled SQUID magnetic fields, spin inter-

actions

current-biased junction pulsed microwave fields

Quantum Dots electron spin magnetic fields and volt-

age pulses

charge state electrostatic gates and