Introduction to quantum computing and quantum information ... · Introduction to quantum computing and quantum information science Iustin Ouatu Pembroke College, University of Oxford,

Introduction to quantum computing and quantum information science

Iustin Ouatu

Pembroke College, University of Oxford, Oxford, United Kingdom

I have spent approximately 2 months at the Theoretical Physics Department at the National Insti-

tute for Physics and Nuclear Engineering, located in Bucharest, Romania, working with Dr. Radu

Ionicioiu, Cambridge PhD (1999) in Quantum Gravity. I was introduced in the world of quantum

computing and quantum information by being guided to study from textbooks, scientific articles,

lecture courses. The studied material tends to be vast, ranging from ideas of Theoretical Computer

Science, abstract mathematics (linear algebra, group theory), to Electrical Engineering and Quantum

Mechanics. In the end, I have attended an workshop for Quantum Cryptography and Information

held in Bucharest, given by Caltech Postdoc Alexandru Gheorghiu. In these 2 months I realized I

want to pursue a PhD in the field of Quantum Information and Quantum Computing and I am truly

satisfied that I was able to dig in these fields due to this Rokos STEM Foundation Funded Internship.

I would like to say thank you to all those who facilitated me to be funded by this scheme. Above

all, I would like to say thank you to the donor, Christopher Rokos, MA in Mathematics, Pembroke

College ’89.

I. INTRODUCTION, BACKGROUND AND MOTIVATION

The world is currently witnessing the second quantum revolution. It all goes back to Heisenberg, Dirac, Schr

ö dinger or Einstein, the founding fathers of quantum mechanics, who created the most well-tested scientific

theory of all times. Back then, there was mystery and uncertainty about the correctness of such a revolution-

ary theory. That was the first quantum revolution. Now, researchers possess the ability to use the precise

laws of quantum mechanics to develop quantum technologies. It is all about creatively using superposition,

entanglement, non-locality and randomness, the defining features of quantum mechanics, in order to come up

with innovations which have a direct impact on economy and society as a whole. Quantum technologies will

be integrated along with the existing classical technologies up to the point at which quantum supremacy will

prevail.

Many areas of the modern world will change. The classical idea of computation will be shifted towards

a mix of both classical and quantum computation, a point in time at which quantum computers will coexist

with classical computers. This will help to solve essential problems in nowadays world exponentially faster:

optimization of a delivery chain or code breaking, for example. In medicine, new drugs will be created due

to the superior power of quantum computers over classical ones. There is going to be a revolution in material

science. Users will browse on a super-secure quantum internet and will send and receive messages which

will be encrypted using quantum security keys. The entire area of sensing/imaging will benefit from the recent

developments of quantum technologies: the quantum telescope, the quantum microscope and the quantum radar

will be the state-of-the-art machines used in industry.

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A lot of interest in quantum technologies can be seen across the world. Australia, Canada, China, EU, Japan,

Singapore, US, they all have very strong quantum research programmes and invest huge amounts of money into

them in order to win the so-called ’quantum race for supremacy’. EU has understood that a developed society

needs to invest in quantum technologies in order to be able to compete at the highest possible level, so it has

launched a 1 billion euros quantum research programme: Quantum Flagship. It is also the case for China, who

have achieved a milestone when they launched the first quantum satellite : Micius - this helped China to become

the first nation to quantum mechanically encrypt a video conference, leading the way to a world of quantum

mechanical communication. They currently invest 10 billion dollars in a big Quantum Research Facility. In

industry, many major companies invest money to become the first to have the first working universal quantum

computer: IBM [1], Google [2], Microsoft [3], Intel [4], Toshiba [5]. ESA [6], NASA [7] are also interested in

quantum technologies.

II. QUANTUM MECHANICS FORMALISM

In this section, a short introduction to the formalism of quantum mechanics is given. The theory of quantum

mechanics is based on postulates.

The first postulate describes the arena in which a quantum system lives and how its current state is de-

scribed. Each quantum system has an associated wavefunction |ψ〉, encapsulating all the information one can

extract about the current state of that system. It is a vector living in a Hilbert space H , that is, a complex vector

space equipped with an inner product. All the vectors are normed to one :

∀ |ψ〉 ∈ H ||ψ|| = 〈ψ|ψ〉 = 1 (1)

The second postulate describes the arena in which composite quantum systems live. Every wavefunction

associated with a composite quantum system lives in a Hilbert space of higher dimensionality than its compo-

nents: H12 = H1 ⊗H2 and dimH12 = dimH1 + dimH2.

The third postulate describes the dynamics of a quantum system: what is the state of a quantum system at

a future moment in time? It evolves according to a unitary matrix U:

t0 −→ t1 (2)

|ψinitial〉 −→ |ψfinal〉 = U |ψinitial〉 (3)

such that UU † = U †U = 1

Finally, the fourth postulate presents how a system responds when a measurement is made on it.

A quantum measurement is described by a set of measurement operators Mm which act on the state space

of the system being measured. If |ψ〉 is measured, the probability that result m occurs is given by :

p(m) = 〈ψ|M †mMm |ψ〉 (4)

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The state of the system after the measurement is:

Mm |ψ〉√〈ψ|M †mMm |ψ〉

(5)

The measurement operators in the set Mm satisfy the completeness relation (equivalently, the probabilities sum

up to 1): ∑m

M †mMm = 1 (6)

Measurements split up in general measurements, projective measurements and POVM measure-

ments(positive operator-valued measure).

When it comes to projective measurements, one usually only specifies a complete set of orthogonal projec-

tors Pm satisfying∑

m Pm = 1 and PmPm′ = δmm′Pm. This is sufficient to describe measurements which

are repeatable. This formalism is usually studied in undergraduate level quantum mechanics.

However, when one uses a silvered screen to measure the position of a photon, the photon is absorbed and

it is impossible to repeat the measurement again. Thus, a more general approach is needed and the elegant

formalism of general measurements is used.

POVM measurements can be understood as a special case of the general measurements, providing the easiest

way to study the statistics of the measurement when the state of the system after the experiment is not important

to the physicist.

It is known that non-orthogonal states cannot be distinguished (see [8]). A simple application of POVM

might be the ability to distinguish with certainty between non-orthogonal states some of the time, while the

other part of the time gaining no information at all about the states. To specify a POVM, one has to list a set of

positive valued operators Em such that∑

mEm = 1. The probability to obtain result m is then given by :

p(m) = 〈ψ|Em |ψ〉.

The measurement problem is of fundamental importance in quantum mechanics and it is a the core of

quantum information and quantum computation.

Using these postulates, together with electrical engineering, mathematics and computer science, researchers

have built many applications which are based on the beauty of quantum mechanics.

III. QUANTUM COMPUTING

A. Preliminaries

Whenever one thinks of an algorithm, that sequence of operations can be related to how a Turing Machine

works.

Alan Turing (OBE FRS) was a British Computer Scientist, most well known for formalizing the idea of

an algorithm and computation as a whole using a theoretical model called the Turing Machine. The Turing

Machine consists of four elements: a program, a finite state control, a tape, a read-write tape-head.

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The finite state control is a collection of internal states : q1, q2, q3, ..., qm with m being a fixed constant.

Two important additional states are the halting state: qh and the starting state: qs.

The tape is a vector (1-dimensional object) which extends to infinity, with square-blocks called tape squares.

In each box of the vector, an element from an alphabet (a finite collection of distinct symbols) is written.

The read-write tape-head is a pointer to the current state of the Turing Machine.

The idea is that the execution of the algorithm causes the read-write tape-head to move through the tape,

at the same time changing the current state of the finite state control. The finite state control starts in the state

qs and the tape-head is initially located at the leftmost square of the tape. The program is an ordered list of

program lines and, in broad terms, it causes the tape-head to move. The algorithm terminates when the current

state is the halt-state qh and the output is then the entry at which the tape-head points.

The Turing Machine model of computation completely describes every operation performed on a computer.

Church and Turing independently put forward the so called Church-Turing thesis, an essential statement in

computer science (see [9]):

’The class of functions computable by a Turing machine corresponds exactly to the class of functions which

we would naturally regard as being computable by an algorithm.’

This is an idealized model and while it offers huge insights into theoretical computer science, there is another

way of thinking about computers.

An alternative model which describes the idea of computation is the Circuit model. It is an approach more

feasible for understanding quantum computers as they are physical things and not theoretical, infinite in size,

machines. Using bits, wires and logic gates, every operation on a modern computer can be expressed using a

circuit.

Not all of the well known classical gates can be implemented on a quantum computer. A quantum system

evolves unitarily (see 2), so non-invertible gates are not suitable for quantum computation. Also, the FANOUT

gate violates the no-cloning theorem There are restrictions on the circuits so that one does not get unreasonable

results.

The link between the Turing model of computation and the circuit model of computation is made via circuit

families. A circuit family is uniform if there is an algorithm running on a Turing Machine which generates a

description of that family. In short, it shows an electrical engineer how to connect the wires and logic gates to

perform the algorithm.

It is known and has been shown in computer science that functions computable on uniform circuit families

are those functions computable on Turing Machines.

Quantum computers provide speed-up over classical computers on certain tasks. A method to quantize the

resources used by a computer to perform a task is needed.

Asymptotic notation or essential behaviour of a function shows how that function behaves when relating to

infinity. For example : 10n+ 20ln(n) + constant goes like n an is thus polynomial in n.

Formally, f(n) is O(g(n)) if ∃(n0, c) for which for n > n0 it is the case that f(n) ≤ cg(n).

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Formally, f(n) is Ω(g(n)) if ∃(n0, c) for which for n > n0 it is the case that f(n) ≥ cg(n).

Formally, f(n) is Θ(g(n)) iff f(n) is both O(g(n)) and Ω(g(n)).

For example, the function 2n is Ω(n3) because for sufficiently large n, n3 ≤ 2n.

In short, if a computer uses resources bounded by a polynomial in n then the problem it solves is said to

be easy, tractable, feasible. If a computer uses resources which grow faster than any polynomial in n, then the

problem it tries to solve is said to be hard, intractable, infeasible.

The above notations lead to complexity classes. Two very important complexity classes are P and NP. P is

a subset of NP. The problems in P are solvable in polynomial times.

The problems in NP are hard for a computer. Given an yes instance of the problem, the computer can easily

verify the validity of it. On contrary, the other direction cannot be efficiently computed. An example is prime

factorization of a number. This requires exponential time to be solved on a computer and most cryptographic

protocols are based on the inability of a modern computer to factorize huge (512 bits) numbers. It is easy to

perform multiplication to see if the given decomposition leads to the required number. However, it is hard to

factorize that number on our own. Quantum computers aim at solving problems from NP. It is not known up

until know which type of problems quantum computers will generally solve. It theoretical computer science it

is agreed that the class of problems solvable by a quantum computer in polynomial time is called BQP. Still,

one cannot exactly relate this complexity class to NP problems.

I found it fascinating to study about theoretical computer science and see how much thinking is behind our

modern-day applications.

B. Quantum gates

Quantum gates are the analogous to the classical gates operating on bits. However, they act on qubits,

the fundamental piece of information in quantum information science. A qubit is a vector with norm 1 living

in a two dimensional complex vector space equipped with an inner product, i.e. a Hilbert space. It can be

represented by any two level system (levels are labelled by |0〉 and |1〉). While two complex numbers need

four real parameters to be fully specified, only two real parameters are needed to fully specify a qubit. A

|ψ〉 = α |0〉+ β |1〉 is constrained by 〈ψ|ψ〉 = 1 and the fact that the overall phase does not matter in quantum

mechanics. That is, the measurement statistics an experimentalist obtains are the same irrespective to the

overall phase. This can be understood by the fact that one measures probabilities (squared absolute values)

and |eiφ|2 = 1. Two degrees of freedom are eliminated and a qubit is represented using two real parameters.

A visualization of a qubit which gives valuable insights how it evolves is given by its representation on the

Bloch’s sphere.

A point on the Bloch’s sphere can be parametrized using two angles (real values):

|ψ〉 = cos (θ

2) |0〉+ eiφ sin (

θ

2) |1〉 (7)

6

FIG. 1: Bloch’s sphere. Any point on the sphere can be parametrized using two real values, θ and φ

Any operation on a qubit must preserve the norm and thus one can only rotate the qubit on the sphere. Only

allowed operations are unitary (2x2 unitary matrices in this case).

Some operations are:

X =

0 1

1 0

Y =

0 −i

i 0

Z =

1 0

0 −1

(8)

H =1√2

1 1

1 −1

S =

1 0

0 i

T =

1 0

0 exp(iπ/4)

Pφ =

1 0

0 eiφ

These are the building blocks of a quantum circuit. Some of these gates form universal sets. They are

universal for quantum computation, that is, any circuit can be simulated to sufficient precision using a finite

number of gates from an universal set. As proved in [10], the set H,Pφ, CNOT is universal for quantum

computation (for the action of CNOT gate, refer to III B 1).

It is important to distinguish rotations about the three axes x, y, z:

Rx(θ) = e−iθX/2 =

cosθ

2−i sin

θ

2

−i sinθ

2cos

θ

2

(9)

Ry(θ) = e−iθY/2 =

cosθ

2− sin

θ

2

sinθ

2cos

θ

2

(10)

7

Rz(θ) = e−iθZ/2 =

e−iθ/2 0

0 eiθ/2

(11)

Using these three rotations, then there exists four real numbers α, β, γ, δ which completely describe any

unitary operation U on a single qubit:

U = eiαRz(β)Ry(γ)Rz(δ) (12)

This leads to another form of the decomposition of any unitary operation on a single qubit:

U = eiαAXBXC (13)

with ABC = 1. This will be important when considering controlled operations (III B 1). These matrices act

on one qubit. For example, the X gate is the flip gate:

X

1

0

=

0

1

(14)

and

X

0

1

=

1

0

(15)

There exist higher dimensionality gates as well, acting on multiple qubits.

1. Controlled operations

A controlled operation is a gate for which its action depends on the value of some control bits. Controlled

operations involve two or more qubits. It is important to distinguish between target and control qubits. The

gate acts on the target qubits, conditioned on the values of the control qubits. The most trivial example is the

CNOT gate. It is depicted in figure 2a. It flips the value of the target qubit iff the value of the control qubit is

set to |1〉. An important application involving CNOT gates is the SWAP gate. It swaps the values of the two

qubits it acts on:

A controlled operation U conditioned on n control qubits is written as : Cn(U). It can be shown ([10]) that

for unitary V with V 2 = U then a C2(U) can be decomposed into:

An important case of C2(U) is the Toffoli gate : C2(X). For this, V =(1− i)(I + iX)

2(see 4).

Putting all this together, the Fredkin swaps the two target qubits iff the control qubit is set to |1〉. Its matrix

8

(a) CNOT-gate (b) SWAP-gate.

FIG. 2: Quantum Gates

FIG. 3: C2(U)-gate, with V 2 = U

representation is:

Fredkin =

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 0

0 0 0 0 0 0 0 1

(16)

and can be pictured as :

It is very well understood the fact that complex n qubit controlled gates can be decomposed into simpler,

elementary operations. For example, for ∀ unitary U2×2, a C2(U) can be simulated with only 14 elementary

operations [10].

It is interesting that a discrete set of gates cannot be used to implement an arbitrary unitary operation exactly,

but only to approximate it.

However, as errors add up linearly at most when in a sequence of gates, the Solovay-Kitaev theorem (as

stated in [11]) gives a powerful result:

”Let G be an instruction set for SU(d), and let a desired accuracy ε > 0 be given. There is a constant c > 0

such that for any U ∈ SU(d) there exists a finite sequence S of gates from G of length O(logc(1

ε

)and such

that d(U, S) < ε.”

9

FIG. 4: C2(U)-gate

A direct application of this is that for a circuit of m CNOT and single-qubit unitary operations a set of

O(m logc(m

ε)) is required to approximate it to accuracy ε.

Single-qubit gates and CNOT’s can implement any two-level unitary operation. Two-level unitary gates are

universal for quantum computation. This is because ∀ d × d unitary matrix U can be written as:

U = V1....Vk (17)

with k 6d(d− 1)

2and Vi being two-level unitary matrices

For a complete, formal description of the behaviour of controlled operations, one can consult [10].

Manipulation of quantum gates is easier when working with their block diagonal forms:

CNOT = diag (I,X) =

I 0

O X

(18)

C(Z) = diag (I, Z) (19)

CNOT = (I ⊗H)C(Z) (I ⊗H)

CNOT = diag (H,H)C(Z)diag (H,H)

CNOT = diag (H,H)diag (I, Z)diag (H,H)

CNOT = diag (HIH,HZH)

CNOT = diag (I,X)

Equipped with all this information, the main algorithms of quantum computing can be understood.

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C. Quantum Algorithms

There are three classes of quantum algorithms which provide an advantage over classical computation.

The first class is based on Quantum Fourier Transform Algorithm (will be discussed in III C 1). This

algorithm does not provide speed-up when transforming classical data, but can be creatively used and it is the

building block of algorithms which solve problems untracktable (exponentially hard) on normal computers,

thus providing a massive speed-up: Shor’s Algorithm and Discrete Logarithm Problem are example of such

instances.

The second class is based on Quantum Search Algorithm (will be discussed in III C 2). The main idea is

that it implies an oracle, a somehow mysterious concept of a black-box capable of anything in order to help

solving a particular problem. Researchers have the difficult task to come up with a practical implementation

of this black-box and up until now real applications have been developed (i.e. it is possible to think of such

black-boxes).

The third class is based on Quantum Simulation Algorithms (will not be discussed in this report). Here a

quantum computer simulates a quantum system. A recent breakthrough of this kind is the success of the D-

WAVE probabilistic quantum computer to efficiently simulate how a complex quantum system behaves when

placed in a variable magnetic field, giving rise to interesting phase transitions between paramagnetic, anti-

ferromagnetic and spin-glass phases. (for scientifically presented results, see [12]).

1. Quantum Fourier Transform

It is well known that for an input set x0, x1, ..., xN−1, the classical Fourier transform produces an output set

y0, y1, ..., yN−1, with

yk =1√N

N−1∑j=0

xje2πijkN (20)

Quantum mechanically, an input set of states |0〉 , |1〉 , ..., |N − 1〉 is transformed according to:

|j〉 → 1√N

N−1∑k=0

e2πijkN |k〉 (21)

Alternatively,

N−1∑j=0

xj |j〉 →N−1∑k=0

yk |k〉 (22)

This transformation is unitary : 〈j|U †U |j〉 = 1.

The Quantum Fourier Transform (QFT) for a state |000...0〉 with n qubits starting in state |0〉 is then |+〉⊗n,

where |+〉 =|0〉+ |1〉√

2.

The following notation is defined:

0.jljl+1...jm → jl2

+jl+1

22+jl+2

23+ ...+

jm2m−l+1

(23)

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It can be then shown that the QFT of |j〉 = |j1j2j3...jn〉 is:(|0〉+ e2πi0.jn|1〉√

2

)(|0〉+ e2πi0.jn−1jn|1〉

√2

)...

(|0〉+ e2πi0.j1j2...jn|1〉√

2

)(24)

To implement the QFT in practice, a gate Rk is defined:

Rk =

1 0

0 exp(2πi

2k)

(25)

An efficient algorithm to perform QFT is then:

FIG. 5: QFT

The image shows the need for n + (n − 1) + (n − 2) + ... + 1 =n(n− 1)

2. In addition, one has to add

restoring gates for the qubits, to return them in the original order in which they entered the quantum circuit.

This is done usingn

2SWAP gates, for a total of

3n

2CNOT gates. In total, Θ(n2) gates are used for QFT.

The classical version of Fast Fourier Transform uses Θ(n2n) gates. In conclusion, the QFT is more efficient,

providing polynomial speed!

Apart from this, the QFT is of fundamental importance in different applications which lead to the famous

Shor’s algorithm: phase estimation, order finding (uses phase estimation) and prime factorization (uses

order finding).

These algorithms will not be discussed here, but, at the time of writing, the author is familiar with them and

understands their behaviour.

2. Grover Search

The second class of quantum algorithms which provide speed-up over classical algorithms is the search

algorithms. This offers an alternative approach to searching problem. Supposing a list of N objects indexed

by x ranging from 0 to N − 1, one can define a function f(x) such that f(x) = 0 if x is not a solution to the

problem, while f(x) = 1 if x is a solution to the problem.

The main idea of the search algorithms is the usage of an oracle. This helps identifying the solutions of the

problem. It is a black-box which performs the following unitary operation: |x〉 |q〉 → |x〉 |q ⊕ f(x)〉 To check

if x is a solution, the oracle is applied to the initial state |x〉 |0〉.

12

Sometimes is more convenient to apply the oracle as it follows:

|x〉(|0〉 − |1〉√

2

)→ (−1)f(x) |x〉

(|0〉 − |1〉√

2

)(26)

which can be restated without writing the |−〉 state:

|x〉 → (−1)f(x) |x〉 (27)

The Grover subroutine is summarized below:

1) Start the system as |0〉⊗n

2) Apply Hadamard transform on all qubits : H⊗n

3)

|x〉 → −(−1)δx0 |x〉 (28)

4) Apply Hadamard transform on all qubits: H⊗n

It is schematically depicted in 6.

FIG. 6: Grover subroutine

Steps 2,3,4 have the cumulated effect of:

H⊗n(2 |0〉 〈0| − 1)H⊗n = 2 |ψ〉 〈ψ| − 1 (29)

where

|ψ〉 =1√N

N−1∑x=0

|x〉 (30)

Thus, the Grover subroutine G is:

G = (2 |ψ〉 〈ψ| − 1)O (31)

With this in mind, the search algorithm circuit is as below:

As an addition, the search algorithm can be beautifully understood using a geometric argument.

13

FIG. 7: Search algorithm circuit

Defining two sums, one iterating through the solutions of the problem and the other iterating through the

values which are not solutions of the problem (′

represents the solutions) :

α =1√

N −M

′′∑x

|x〉 (32)

β =1√M

′∑x

|x〉 (33)

the initial state |ψ〉 can be written as:

|ψ〉 =

√N −MN

|α〉+

√M

N|β〉 (34)

Applying the oracle O is equivalent to a reflection about |α〉:

O(a |α〉+ b |β〉) = a |α〉 − b |β〉 (35)

Applying 2 |ψ〉 〈ψ| − 1 represents a rotation about |ψ〉. Two reflections⇐⇒rotation.

The whole idea is that repeated applications of G rotate the initial state closer and closer to |β〉. After k

iterations, the state is:

Gk |ψ〉 = cos2k + 1

2θ |α〉+ sin

2k + 1

2θ |β〉 (36)

where

cosθ

2=

√N −MN

(37)

This is helpful because when, at the end, measuring in the computational basis, one obtains a state which form

|β〉 (i.e. a solution of the problem) with high probability (whp).

To rotate |ψ〉 to withinθ

2≤ π

4of |β〉, one can show that O(

√N

M) Grover iterations are needed. This is a

quadratic speed-up over the classical O(N

M).

The quantum search algorithm is unique and interesting because even without knowing the solution to the

problem, the oracle somehow recognises a solution when it sees one. The tricky part is to come up with practical

implementations of the oracle.

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IV. QUANTUM INFORMATION

A very short introduction to the complex, abstract field of quantum information science is given in this

chapter. The writer understands some of the key concepts in quantum information, how to mathematically

model the environment and to use the rules of quantum mechanics to create some practical applications.

Quantum information science takes into account the effects of noise, that is the interaction of an open system

with the environment. In practice, systems are considered to be open, as no real application is perfectly closed

and non-interacting with basically anything.

A. How to visualize quantum noise

The mathematical formalism which governs the understanding of noise in open quantum systems is called

quantum operations. A quantum operation is a map which takes as input a quantum state ρ and returns another,

transformed quantum state ρ′:

E(ρ) = ρ′ (38)

There are three equivalent ways to understand how to deal with noise appearing in open quantum systems.

The first idea is to visualize a quantum operation as a CP-map: The open quantum system is embodied in a

bath, the so-called environment. A unitary operation U is applied on the whole construction (environment plus

the system as a whole). To get the transformed state of the quantum system, one simply discards the bath by

tracing over the environment.

A figure which show this is: 8 ([13]): Mathematically:

FIG. 8: A CP-map. Picture taken from Master Thesis Presentation: Weak Values in Quantum Measurements

written by Yutaka Shikano, Dept. of Physics, Tokyo Institute of Technology

E(ρ) = trenv[U(ρ⊗ ρenv)U †

](39)

15

Let|ek〉

be a basis for the environment space. It is sufficient to model the environment as starting in a

pure state |0〉 = |e0〉 :

ρenv = |e0〉 〈e0| (40)

This is because when the experimentalist prepares the states, it usually tries to undo all the previous correlations

between the environment and the system. A product state of the bath and the system is created, making the

above assumption adequate. Equation 39 becomes:

E(ρ) =∑k

〈ek|U[ρ⊗ |e0〉 〈e0|

]U † |ek〉

E(ρ) =∑k

EkρE†k

with

Ek = 〈ek|U |e0〉 (41)

Because the quantum operation ε is trace preserving, it follows that:∑k

E†kEk = 1 (42)

When gaining new information about the process that occurs when the map is applied∑k

E†kEk ≤ 1 (43)

With these, the mathematical formalism of quantum operation is complete.

The second approach is the physical interpretation. It offers a different view on quantum operations. It

implies the principle of implicit measurement : the state of the system does not change when a measurement on

the environment is made. Supposing result k occurs on the environment, the state of the open quantum system

is :

ρk ∼ trenv(|ek〉 〈ek|U(ρ⊗ (|e0〉 〈e0|)U † |ek〉 〈ek|

)= 〈ek|U(ρ⊗ |e0〉 〈e0|)U † |ek〉 =

EkρE†k

Normalizing:

ρk =EkρE

†k

Tr(EkρE†k)

(44)

The action of the noise can be understood as taking the initial state of the system ρ to one of the states in the

setEkρE

†k

Tr(EkρE

†k

) with probability p(k) = Tr(EkρE

†k

).

The third approach is by stating physically-motivated axioms which govern the behaviour of quantum

operations:

16

A1: Tr(E(ρ)) represents the probability that the process represented by E occurs. 0 ≤ Tr(E(ρ)) ≤ 1

A2: E is a convex linear map:

∀pi

E(∑

i

piρi)

=∑i

E(ρi)

A3: E is a completely positive map

E(A) > 0 ∀A

(1⊗ E)(A) > 0 ∀A ∈ RQ1

(any A in the product space of any random system R added to Q1)

There is a theorem which links the axioms to the previously encountered operator-sum form of a map:

The map E satisfies A1, A2, A3 iff E(ρ) =∑

k EkρE†k with

∑iE†iEi ≤ 1.

The proof can be found in any textbook of quantum information and will not be discussed here, see [8] or

[14].

All quantum operations modify the topology of Bloch’s sphere. For example, the bit-flip operation is de-

scribed by: nothing happens with probability p, a bit is flipped (X gate is applied) with probability (1− p):

E0 =√p 1 (45)

E1 =√

1− p X (46)

Its action is depicted in figure 9:

FIG. 9: Geometrical modification of the Bloch sphere

Something more complex is the loss of energy from a quantum system. This is called amplitude damping

and represents a very important quantum channel. Supposing an optical mode with either zero or one pho-

ton a |0〉 + b |1〉, this is coupled to the environment by introducing a beamsplitter in the way of the ’flying

photons’. The superposition mode is coupled to another single optical mode by the unitary transformation

B = exp(θ(a†b − ab†), where b, b† are the operators for the second mode. For the theoretical description of

17

the beamsplitter Hamiltonian, see [15]. Supposing the environment starts in |0〉, the result after the passage

through the beamsplitter is:

B |0〉(a |0〉+ b |1〉) = aB |00〉+ bB |01〉 = (47)

a |00〉+ bBa† |00〉 = (48)

a |00〉+ bBa†B†B |00〉 = (49)

a |00〉+ b(a†cosθ + b†sinθ

)|00〉 = (50)

a |00〉+ b(cosθ |01〉+ sinθ |10〉

)(51)

Here, the second index in the kets represents the system. Applying the partial trace operation with respect to

the first index (discarding the environment), the quantum operation of amplitude damping is obtained:

EAD(ρ) = E0ρE†0 + E1ρE

†1 (52)

with

E0 =

1 0

0√

1− γ

(53)

and

E0 =

0√γ

0 0

(54)

and

γ = sin2θ (55)

The action of this channel can be understood as it follows: E0 leaves |0〉 unchanged and reduces (as cos θ ≤ 1)

the amplitude of |1〉. The environment ’perceives’ the system more likely to be in |0〉 than in |1〉. Here no

energy is dissipated to the environment E1 changes |1〉 to |0〉 : in practice, this is represented by losing a

photon from the system to the environment. Here energy is dissipated. It can be generalized to channels which

do not only leave the state |0〉 invariant alone. It is then called "generalized amplitude damping".

B. How close two quantum states are? How accurate is a quantum channel?

Having in mind the quantum information as quantum states being modified by quantum operations, it is

important to understand the performance of such channels. That is, how well the information is preserved

under the action of quantum channels? Starting from concepts of classical information theory, a very short

introduction to how distances are measured in quantum information theory is given below.

There are many ways to quantify how close are two classical probability distributions. Three of them are:

1)The Kolmogorov distance

18

2)The Kantorovich-Rubinstein distance

3)The bounded-Lipschitz distance

A particularly useful measure is the first one, the Kolmogorov distance, named after a Russian mathemati-

cian with hugely important contributions in probability theory . It is defined as:

D(px, qx

)=

1

2

∑x

|px − qx| (56)

It has the following properties:

1)it is symmetric in its arguments

2)it respects the triangle inequality

D(x, z) ≤ D(x, y) +D(y, z) (57)

1),2) combined give the Kolmogorov distance the property of being a metric. Another useful quantization of

how close together are two probability distributions is the Fidelity:

By definition:

Fidelity(px, qx

)=∑x

√pxqx (58)

This has the physical interpretation that it is the inner product between two vectors (px and qx) lying on the unit

sphere. These vectors lie on the unit sphere because their norm is equal to 1 (the sum of probabilities from a

probability distribution has to sum to 1:∑

x (√px)2).

Moving to the quantum realm, the distance between two quantum states is defined as:

D(ρ, σ) =1

2Tr|ρ− σ| (59)

where |A| =√A†A.

For [σ, ρ] = 0, the two quantum states are diagonal in the same basis, that is they can be written as:

ρ =∑i

ri |i〉 〈i|

σ =∑i

si |i〉 〈i|

If this is the case, the quantum distance becomes a classical distance between two eigenvalues of the states:

D(ρ, σ) =1

2Tr|

∑i

(ri − si

)|i〉 〈i|

= D(ri, si)

Using the Bloch’s sphere representation of a quantum state, the distance between the states can be viewed

as:

ρ =1+ ~r · ~σ

2

σ =1+ ~s · ~σ

2

19

D(ρ, σ) =1

2Tr|ρ− σ| = 1

2tr|(~r − ~s) · ~σ| 1

2

As(~r − ~s

)· ~σ has eigenvalues ±|~r − ~s|,

tr|(~r − ~s) · σ| = 2|~r − ~s| (60)

Thus,

D(ρ, σ) =1

2|~r − ~s| (61)

Also, the trace distance is preserved under unitary operations:

D(ρ, σ) = D(UρU †, UσU †) (62)

It can be shown that it is a metric on the space of density operators:

1) D(ρ, σ) = 0 iff ρ = σ,

2) D(·, ·) is symmetric in its arguments and

3) it respects the triangle identity: D(ρ, τ) ≤ D(ρ, σ) +D(σ, τ).

The above can lead to the fact that trace preserving quantum operations are contractive, that is:

D(E(ρ), E(σ)

)≤ D(ρ, σ) (63)

For completeness, one can also list the fact that the trace distance possesses strong convexity and thus is

jointly convex in its inputs.

V. CONCLUSION AND ACKNOWLEDGEMENTS

Throughout this report, I have extensively used ideas from the following books: [8], [14]. As I found out

from more than one researcher in quantum computing/information, these are the two main books one starts

with when studying about these topics. I am glad I have managed to go through many topics presented in these

two books. None of the results cited above are my own and more in depth-references can be found at the end

of every chapter in [8].

I have given a short introduction to the ideas I have played around with during my internship at the Depart-

ment of Theoretical Physics of the National Institute of Physics and Nuclear Engineering (NIPNE), Bucharest,

Romania. As this is intended to be a short report, it is somehow understood that the size of this document has

not to be too big. Thus, I only cited some results, omitting steps and not giving complete proofs. I hope that a

vague idea of my work and how I spent the 2 months can be seen from this report and is clear to the reader.

I have truly enjoyed studying about quantum computing and quantum information. I have seen what it

means to work in a theoretical physics department and I have benefited from insights given by researchers who

published in international journals. This is definitely going to be useful for me in the long run.

I would like to say thank you to all the persons who have facilitated me to obtain this fund. I would like to

express my gratitude to the donor, Christopher Charles Rokos, MA in Mathematics, Pembroke College ’89.

20

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