Introduction to Quantum Computing and Quantum Information Theory Dan C. Marinescu and Gabriela M....

Introduction to Introduction to Quantum Computing andQuantum Computing and

Quantum Information TheoryQuantum Information Theory

Dan C. Marinescu and Gabriela M. Dan C. Marinescu and Gabriela M. MarinescuMarinescu

Computer Science Department Computer Science Department

University of Central FloridaUniversity of Central Florida

Orlando, Florida 32816Orlando, Florida 32816, , USAUSA

ISCIS Antalya November 5, 2003ISCIS Antalya November 5, 2003 22

AcknowledgmentsAcknowledgments

The material presented is from the bookThe material presented is from the book

Lectures on Quantum ComputingLectures on Quantum Computing by Dan C. Marinescu and Gabriela M. Marinescuby Dan C. Marinescu and Gabriela M. Marinescu

Prentice Hall, 2004Prentice Hall, 2004

Work supported by National Science Foundation Work supported by National Science Foundation grants MCB9527131, DBI0296107,ACI0296035, and grants MCB9527131, DBI0296107,ACI0296035, and EIA0296179.EIA0296179.


ContentsContents I. I. Computing and the Laws of PhysicsComputing and the Laws of Physics II. A Happy Marriage; Quantum Mechanics &II. A Happy Marriage; Quantum Mechanics & ComputersComputers III. Qubits and Quantum GatesIII. Qubits and Quantum Gates IV. Quantum ParallelismIV. Quantum Parallelism V. Deutsch’s AlgorithmV. Deutsch’s Algorithm VI. Bell States, Teleportation, and Dense VI. Bell States, Teleportation, and Dense

CodingCoding VII. SummaryVII. Summary


Technological limitsTechnological limits

For the past two decades we have enjoyed For the past two decades we have enjoyed Gordon Moore’s lawGordon Moore’s law. But all good things may . But all good things may come to an end…come to an end…

We are limited in our ability to increase We are limited in our ability to increase the density and the density and the speed of a computing engine.the speed of a computing engine.

Reliability will also be affectedReliability will also be affected to increase the speed we need increasingly smaller to increase the speed we need increasingly smaller

circuitscircuits (light needs 1 ns to travel 30 cm in vacuum) (light needs 1 ns to travel 30 cm in vacuum) smaller circuits smaller circuits systems consisting only of a few systems consisting only of a few

particles subject to Heissenberg uncertainty particles subject to Heissenberg uncertainty


Energy/operationEnergy/operation

If there is a minimum amount of energy dissipated If there is a minimum amount of energy dissipated to perform an elementary operation, then to perform an elementary operation, then to to increase the speed, thus the number of operations increase the speed, thus the number of operations performed each second, we require a liner increase performed each second, we require a liner increase of the amount of energy dissipated by the deviceof the amount of energy dissipated by the device. .

The computer technology vintage year 2000 The computer technology vintage year 2000 requires some requires some 3 x 103 x 10-18-18 Joules per elementary Joules per elementary operationoperation. .

Even if this limit is reduced say 100-fold we shall Even if this limit is reduced say 100-fold we shall see a 10 (ten) times increase in the amount of see a 10 (ten) times increase in the amount of power needed by devices operating at a speed 10power needed by devices operating at a speed 1033 times larger than the sped of today's devices.times larger than the sped of today's devices.


Power dissipation, circuit density, and Power dissipation, circuit density, and speedspeed

In 1992 Ralph Merkle from Xerox PARC In 1992 Ralph Merkle from Xerox PARC calculated that a calculated that a 1 GHz computer operating at 1 GHz computer operating at room temperature, with 10room temperature, with 101818 gates packed in a gates packed in a volume of about 1 cmvolume of about 1 cm3 3 would dissipate 3 MW of would dissipate 3 MW of powerpower. . A small city with 1,000 homes each using 3 KW would A small city with 1,000 homes each using 3 KW would

require the same amount of power; require the same amount of power; A 500 MW nuclear reactor could only power some 166 A 500 MW nuclear reactor could only power some 166

such circuits.such circuits.


Talking about the heat…Talking about the heat…

The heat produced by a super dense computing engine is The heat produced by a super dense computing engine is proportional with the number of elementary computing proportional with the number of elementary computing circuits, thus, with the volume of the engine. circuits, thus, with the volume of the engine. The heat The heat dissipated grows as the cube of the radius of the device.dissipated grows as the cube of the radius of the device.

To prevent the destruction of the engine we have to To prevent the destruction of the engine we have to remove the heat through a surface surrounding the remove the heat through a surface surrounding the device. Henceforth, device. Henceforth, our ability to remove heat increases our ability to remove heat increases as the square of the radiusas the square of the radius while the amount of heat while the amount of heat increases with the cube of the size of the computing increases with the cube of the size of the computing engine.engine.


Energy consumption of a logic circuit

Speed of individual logic gates

S

E

(a) (b)

Heat removal for a circuit with densely packedlogic gates poses tremendous challenges.


ContentsContents I. Computing and the Laws of PhysicsI. Computing and the Laws of Physics II. II. A Happy Marriage; Quantum Mechanics &A Happy Marriage; Quantum Mechanics & ComputersComputers III. Qubits and Quantum GatesIII. Qubits and Quantum Gates IV. Quantum ParallelismIV. Quantum Parallelism V. Deutsch’s AlgorithmV. Deutsch’s Algorithm VI. Bell States, Teleportation, and Dense VI. Bell States, Teleportation, and Dense



A happy marriage…A happy marriage…

The two greatest discoveries of the 20-th The two greatest discoveries of the 20-th centurycentury quantum mechanicsquantum mechanics stored program computersstored program computers

produced produced quantum computing and quantum quantum computing and quantum information theoryinformation theory


Quantum; Quantum mechanicsQuantum; Quantum mechanics

QuantumQuantum is a Latin word meaning some quantity. In is a Latin word meaning some quantity. In physics it is used with the same meaning as the word physics it is used with the same meaning as the word discretediscrete in mathematics, i.e., some quantity or in mathematics, i.e., some quantity or variable that can take only sharply defined values as variable that can take only sharply defined values as opposed to a continuously varying quantity. The opposed to a continuously varying quantity. The concepts concepts continuumcontinuum and and continuouscontinuous are known from are known from geometry and calculus. For example, on a segment geometry and calculus. For example, on a segment of a line there are infinitely many points, the of a line there are infinitely many points, the segment consists of a continuum of points. This segment consists of a continuum of points. This means that we can cut the segment in half, and then means that we can cut the segment in half, and then cut each half in half, and continue the process cut each half in half, and continue the process indefinitely.indefinitely.

Quantum mechanics is a mathematical model of the Quantum mechanics is a mathematical model of the physical worldphysical world


Heissenberg uncertainty principleHeissenberg uncertainty principle

Heisenberg uncertainty principle says we cannot Heisenberg uncertainty principle says we cannot determine both the position and the momentum of a determine both the position and the momentum of a quantum particle with arbitrary precision. quantum particle with arbitrary precision.

In his Nobel prize lecture on December 11, 1954 Max In his Nobel prize lecture on December 11, 1954 Max Born says about this fundamental principle of Born says about this fundamental principle of Quantum Mechanics : ``... It shows that not only the Quantum Mechanics : ``... It shows that not only the determinism of classical physics must be abandoned, determinism of classical physics must be abandoned, but also the naive concept of reality which looked but also the naive concept of reality which looked upon atomic particles as if they were very small upon atomic particles as if they were very small grains of sand. At every instant a grain of sand has a grains of sand. At every instant a grain of sand has a definite position and velocity. This is not the case definite position and velocity. This is not the case with an electron. If the position is determined with with an electron. If the position is determined with increasing accuracy, the possibility of ascertaining its increasing accuracy, the possibility of ascertaining its velocity becomes less and vice versa.''velocity becomes less and vice versa.''


A revolutionary approach to computing and A revolutionary approach to computing and communicationcommunication

We need to consider a revolutionary rather than We need to consider a revolutionary rather than an evolutionary approach to computing. an evolutionary approach to computing.

Quantum theory does not play only a supporting Quantum theory does not play only a supporting role by prescribing the limitations of physical role by prescribing the limitations of physical systems used for computing and systems used for computing and communication. communication.

Quantum properties such as Quantum properties such as uncertainty, uncertainty, interference, and interference, and entanglemententanglement

form the foundation of a new brand of theory, form the foundation of a new brand of theory, the quantum information theory where the quantum information theory where computational and communication processes computational and communication processes rest upon fundamental physics. rest upon fundamental physics.


Milestones in quantum physicsMilestones in quantum physics 1900 - Max Plank presents the black body radiation theory; 1900 - Max Plank presents the black body radiation theory;

the quantum theory is born. the quantum theory is born. 1905 - Albert Einstein develops the theory of the 1905 - Albert Einstein develops the theory of the

photoelectric effect.photoelectric effect. 1911 - Ernest Rutherford develops the planetary model of 1911 - Ernest Rutherford develops the planetary model of

the atom.the atom. 1913 - Niels Bohr develops the quantum model of the 1913 - Niels Bohr develops the quantum model of the

hydrogen atom.hydrogen atom. 1923 - Louis de Broglie relates the momentum of a particle 1923 - Louis de Broglie relates the momentum of a particle

with the wavelengthwith the wavelength 1925 - Werner Heisenberg formulates the matrix quantum 1925 - Werner Heisenberg formulates the matrix quantum

mechanics.mechanics. 1926 - Erwin Schrodinger proposes the equation for the 1926 - Erwin Schrodinger proposes the equation for the

dynamics of the wave function.dynamics of the wave function.


Milestones in quantum physics (cont’d)Milestones in quantum physics (cont’d)

1926 - Erwin Schrodinger and Paul Dirac show the 1926 - Erwin Schrodinger and Paul Dirac show the equivalence of Heisenberg's matrix formulation and Dirac's equivalence of Heisenberg's matrix formulation and Dirac's algebraic one with Schrodinger's wave function.algebraic one with Schrodinger's wave function.

1926 - Paul Dirac and, independently, Max Born, Werner 1926 - Paul Dirac and, independently, Max Born, Werner Heisenberg, and Pasqual Jordan obtain a complete Heisenberg, and Pasqual Jordan obtain a complete formulation of quantum dynamics.formulation of quantum dynamics.

1926 - John von Newmann introduces Hilbert spaces to 1926 - John von Newmann introduces Hilbert spaces to quantum mechanics.quantum mechanics.

1927 - Werner Heisenberg formulates the uncertainty 1927 - Werner Heisenberg formulates the uncertainty principle.principle.


Milestones in computing and information Milestones in computing and information theorytheory

1936 - Alan Turing dreams up the Universal Turing Machine, 1936 - Alan Turing dreams up the Universal Turing Machine, UTM. UTM.

1936 - Alonzo Church publishes a paper asserting that 1936 - Alonzo Church publishes a paper asserting that `èvery function which can be regarded as computable can `èvery function which can be regarded as computable can be computed by an universal computing machine''.be computed by an universal computing machine''.

1945 - ENIAC, the world's first general purpose computer, the 1945 - ENIAC, the world's first general purpose computer, the brainchild of J. Presper Eckert and John Macauly becomes brainchild of J. Presper Eckert and John Macauly becomes operational.operational.

1946 - A report co-authored by John von Neumann outlines 1946 - A report co-authored by John von Neumann outlines the von Neumann architecture.the von Neumann architecture.

1948 - Claude Shannon publishes `À Mathematical Theory of 1948 - Claude Shannon publishes `À Mathematical Theory of Communication’’.Communication’’.

1953 - The first commercial computer, UNIVAC I.1953 - The first commercial computer, UNIVAC I.


Milestones in quantum computingMilestones in quantum computing 1961 - Rolf Landauer decrees that computation is physical 1961 - Rolf Landauer decrees that computation is physical

and studies heat generation.and studies heat generation. 1973 - Charles Bennet studies the logical reversibility of 1973 - Charles Bennet studies the logical reversibility of

computations. computations. 1981 - Richard Feynman suggests that physical systems 1981 - Richard Feynman suggests that physical systems

including quantum systems can be simulated exactly with including quantum systems can be simulated exactly with quantum computers. quantum computers.

1982 - Peter Beniof develops quantum mechanical models 1982 - Peter Beniof develops quantum mechanical models of Turing machines.of Turing machines.

1984 - Charles Bennet and Gilles Brassard introduce 1984 - Charles Bennet and Gilles Brassard introduce quantum cryptography.quantum cryptography.

1985 - David Deutsch reinterprets the Church-Turing 1985 - David Deutsch reinterprets the Church-Turing conjecture.conjecture.

1993 - Bennet, Brassard, Crepeau, Josza, Peres, Wooters 1993 - Bennet, Brassard, Crepeau, Josza, Peres, Wooters discover quantum teleportation.discover quantum teleportation.

1994 - Peter Shor develops a clever algorithm for factoring 1994 - Peter Shor develops a clever algorithm for factoring large numbers. large numbers.


Deterministic versus probabilistic photon Deterministic versus probabilistic photon behaviorbehavior

(b)(a)

D1

D2

D3

D5

D7

Detector D1

Detector D2

Beam splitter

Incident beam of light

Reflected beam

Transmitted beam


The puzzling nature of lightThe puzzling nature of light

If we start decreasing the intensity of the incident If we start decreasing the intensity of the incident light we observe the granular nature of light. light we observe the granular nature of light. Imagine that we send a single photon. Then either Imagine that we send a single photon. Then either detector D1 or detector D2 will record the arrival of detector D1 or detector D2 will record the arrival of a photon. a photon.

If we repeat the experiment involving a single If we repeat the experiment involving a single photon over and over again we observe that each photon over and over again we observe that each one of the two detectors records a number of events. one of the two detectors records a number of events.

Could there be hidden information, which controls Could there be hidden information, which controls the behavior of a photon? Does a photon carry a the behavior of a photon? Does a photon carry a gene and one with a ``transmit'' gene continues and gene and one with a ``transmit'' gene continues and reaches detector D2 and another with a ``reflect'' reaches detector D2 and another with a ``reflect'' gene ends up at D1gene ends up at D1??


The puzzling nature of light (cont’d)The puzzling nature of light (cont’d)

Consider now a cascade of beam splitters. As Consider now a cascade of beam splitters. As before, we send a single photon and repeat the before, we send a single photon and repeat the experiment many times and count the number experiment many times and count the number of events registered by each detector. of events registered by each detector.

According to our theory we expect the first beam According to our theory we expect the first beam splitter to decide the fate of an incoming photon; splitter to decide the fate of an incoming photon; the photon is either reflected by the first beam the photon is either reflected by the first beam splitter or transmitted by all of them. Thus, only splitter or transmitted by all of them. Thus, only the first and last detectors in the chain are the first and last detectors in the chain are expected to register an equal number of events. expected to register an equal number of events.

Amazingly enough, the experiment shows that all Amazingly enough, the experiment shows that all the detectors have a chance to register an event.the detectors have a chance to register an event.


State descriptionState description

O

1V

(a)

= q

O

1V

(b)

0 0

1 1q1

q0

q1 = q

q045o

30o


A mathematical model to describe the A mathematical model to describe the state of a quantum systemstate of a quantum system

10 10

1|||| 21

20

are complex numbersare complex numbers|,| 10


Superposition and uncertaintySuperposition and uncertainty

In this model a stateIn this model a state

is a superposition of two basis states, “0” and “1”is a superposition of two basis states, “0” and “1” This state is This state is unknown before we make a measurementunknown before we make a measurement.. After we perform a measurement the system is no longer After we perform a measurement the system is no longer

in an uncertain state but it is in one of the two basis states. in an uncertain state but it is in one of the two basis states. is is the probability of observing the outcome “1”the probability of observing the outcome “1” is is the probability of observing the outcome “0”the probability of observing the outcome “0”

10 10

1|||| 21

20

20 ||2

1 ||


Multiple measurementsMultiple measurements

black

white

hard

soft

(a) (b)

black

white

black

white

hard

soft

(c)


Measurements in multiple basesMeasurements in multiple bases

| >

| >

| >

| >

|

0

1

10

0

1


Measurements of superposition statesMeasurements of superposition states The polarization of a photon is described by a unit vector The polarization of a photon is described by a unit vector

on a two-dimensional space with basis | 0 > and | 1>.on a two-dimensional space with basis | 0 > and | 1>. Measuring the polarization is equivalent to projecting the Measuring the polarization is equivalent to projecting the

random vector onto one of the two basis vectors. random vector onto one of the two basis vectors. Source S sends randomly polarized light to the screen; Source S sends randomly polarized light to the screen;

the measured intensity is I. the measured intensity is I. The filter A with vertical polarization is inserted between The filter A with vertical polarization is inserted between

the source and the screen an the intensity of the light the source and the screen an the intensity of the light measured at E is about I/2. measured at E is about I/2.

Filter B with horizontal polarization is inserted between A Filter B with horizontal polarization is inserted between A and E. The intensity of the light measured at E is now 0. and E. The intensity of the light measured at E is now 0.

Filter C with a 45 deg. polarization is inserted between A Filter C with a 45 deg. polarization is inserted between A and B. The intensity of the light measured at E is about and B. The intensity of the light measured at E is about 1 / 8. 1 / 8.


(a)

(b)

S

E

(c)

E

S

A

intensity = I intensity = I/2

(d)

E

S

A B

intensity = 0

(e)

E

S

A B

intensity = I/8

C

0|

1

|


The superposition probability ruleThe superposition probability rule

If an event may occur in two or more If an event may occur in two or more indistinguishable ways then the probability indistinguishable ways then the probability amplitude of the event is the sum of the amplitude of the event is the sum of the probability amplitudes of each case probability amplitudes of each case considered separately (sometimes known considered separately (sometimes known as Feynmann rule).as Feynmann rule).


(a)

O

1V

| 0 >

| 1 >

(b)

V

| 0 >

| 1 >

(c)

1+q

+q

+q

-q

(| t >) (| t >)

(| r >) (| r >)

direction2direction1

Reflecting mirror U

Reflecting mirror L

Detector D1

Detector D2

Source S1

Beam splitter BS2Beam splitter BS1

Source S2

direction1

direction2


The experiment illustrating the The experiment illustrating the superposition probability rulesuperposition probability rule

In certain conditions, we observe experimentally In certain conditions, we observe experimentally that a photon emitted by S1 is always detected that a photon emitted by S1 is always detected by D1 and never by D2 and one emitted by S2 by D1 and never by D2 and one emitted by S2 is always detected by D2 and never by D1.is always detected by D2 and never by D1.

A photon emitted by one of the sources S1 or S2 A photon emitted by one of the sources S1 or S2 may take one of four different paths shown on may take one of four different paths shown on the next slide, depending whether it is the next slide, depending whether it is transmitted, or reflected by each of the two transmitted, or reflected by each of the two beam splitters.beam splitters.


D1S1

(a) - The TT case: the probabilityamplitude is (+q)(+q).

T +q T +q

(b) The RR case: the probabilityamplitude is (+q)(+q).

D2

S1

(c) - The TR case: theprobability amplitude is (+q)(-q).

T +q

R -q

(d) The RT case: the probabilityamplitude is (+q)(+q).

D2

S1

T +q

R +q

D1S1

R +q R +q

direction1 direction2

BS1 BS2

BS1 BS1

BS1

BS2

BS2

BS2

L

L

U

U


A photon coincidence experimentA photon coincidence experiment

Reflecting mirror U

Reflecting mirror L

Detector D1

Detector D2

Source

Beam splitter


A glimpse into the world of quantum A glimpse into the world of quantum computing and quantum information computing and quantum information

theorytheory

Quantum key distributionQuantum key distribution Exact simulation of systems with a very Exact simulation of systems with a very

large state spacelarge state space Quantum parallelismQuantum parallelism


Quantum key distributionQuantum key distribution

To ensure confidentiality, data is often encrypted. To ensure confidentiality, data is often encrypted. The most reliable encryption techniques are based The most reliable encryption techniques are based upon one time pads whereby the encryption key is upon one time pads whereby the encryption key is used for one session only and then discarded. used for one session only and then discarded.

Thus, there exists the need for reliable and effective Thus, there exists the need for reliable and effective methods for the distribution of the encryption keys. methods for the distribution of the encryption keys. The problem rests on the physical difficulty to The problem rests on the physical difficulty to detect the presence of an intruder when detect the presence of an intruder when communicating through a classical communication communicating through a classical communication channel. channel.

To date, secure and reliable methods for To date, secure and reliable methods for cryptographic key distribution have largely eluded cryptographic key distribution have largely eluded the cryptographic community in spite of the cryptographic community in spite of considerable research effort and ingeniousness.considerable research effort and ingeniousness.


Quantum key distribution setup Quantum key distribution setup Alice and Bob are connected via two communication channels, Alice and Bob are connected via two communication channels,

a quantum and a classical one. Eve eavesdrops on both. a quantum and a classical one. Eve eavesdrops on both. The photons prepared by Alice may have vertical/horizontal The photons prepared by Alice may have vertical/horizontal

(VH) or diagonal polarization (DG). (VH) or diagonal polarization (DG). The photons with vertical/horizontal (VH) polarization may be The photons with vertical/horizontal (VH) polarization may be

used to transmit binary information as follows: a photon with used to transmit binary information as follows: a photon with vertical polarization may transmit a 1 while one with a vertical polarization may transmit a 1 while one with a horizontal polarization may transmit a 0. horizontal polarization may transmit a 0.

Similarly, those with diagonal (DG) polarization may transmit Similarly, those with diagonal (DG) polarization may transmit binary information, 1 encoded as a photon with 45 deg. binary information, 1 encoded as a photon with 45 deg. polarization, and 0 encoded as a photon with a 135 deg. polarization, and 0 encoded as a photon with a 135 deg. polarization. polarization.

Bob uses a calcite crystal to separate photons with different Bob uses a calcite crystal to separate photons with different polarization. Shown is the case when the crystal is set up to polarization. Shown is the case when the crystal is set up to separate vertically polarized photons from the horizontally separate vertically polarized photons from the horizontally polarized ones. To perform a measurement in the DG basis the polarized ones. To perform a measurement in the DG basis the crystal is oriented accordingly.crystal is oriented accordingly.


(a)

Vertical Horizontal 45 deg

Vertical/Horizontal (VH) Diagonal (DG)

(b)

(c)Classical communication channel

Quantum communication channel

Classical wiretap

Quantum wiretapPhoton

separationsystem

Source ofpolarizedphotons

Alice Bob

Eve

135 deg


The quantum key distribution The quantum key distribution algorithm of Bennett and Brassard algorithm of Bennett and Brassard

(BB84)(BB84) Alice selects n, the approximate length of the Alice selects n, the approximate length of the

encryption key. Alice generates two random strings a encryption key. Alice generates two random strings a and b, each of length (4+ )n. By choosing and b, each of length (4+ )n. By choosing sufficiently large Alice and Bob can ensure that the sufficiently large Alice and Bob can ensure that the number of bits kept is close to 2n with a very high number of bits kept is close to 2n with a very high probability.probability.

A subset of length n of the bits in string a will be A subset of length n of the bits in string a will be used as the encryption key and the bits in string $b$ used as the encryption key and the bits in string $b$ will be used by Alice to select the basis (VH) or (DG) will be used by Alice to select the basis (VH) or (DG) for each photon sent to Bob.for each photon sent to Bob.


BB84 (cont’d)BB84 (cont’d) Alice encodes the binary information in string Alice encodes the binary information in string aa

based upon the corresponding values of the bits in based upon the corresponding values of the bits in string string bb. .

For example, if the i-th bit of string For example, if the i-th bit of string bb is is 1 then Alice selects Vertical-Horizontal (VH) polarization. 1 then Alice selects Vertical-Horizontal (VH) polarization.

If VH is selected, then If VH is selected, then a 1 in the i-th position of string a 1 in the i-th position of string aa is sent as a photon with vertical is sent as a photon with vertical

polarization (V), and polarization (V), and a $0$ as a photon with horizontal (H) polarization; a $0$ as a photon with horizontal (H) polarization;

0 then Alice selects Diagonal (DG) polarization. If DG is 0 then Alice selects Diagonal (DG) polarization. If DG is selected, then selected, then a 1 in the i-th position of string a 1 in the i-th position of string aa is sent as a photon with a 45 is sent as a photon with a 45

deg. polarization, and deg. polarization, and a $0$ as a photon with 135 deg. polarization. a $0$ as a photon with 135 deg. polarization.

Both Alice and Bob use the same encoding Both Alice and Bob use the same encoding convention for each of the bases.convention for each of the bases.


BB84 (cont’d)BB84 (cont’d)

In turn, Bob picks up randomly (4+ )n bits to form In turn, Bob picks up randomly (4+ )n bits to form a string a string b’.b’. He uses one of the two basis for the He uses one of the two basis for the measurement of each incoming photon in string measurement of each incoming photon in string aa based upon the corresponding value of the bit in based upon the corresponding value of the bit in string string b’.b’.

For example, a 1 in the i-th position of For example, a 1 in the i-th position of b’b’ implies implies that thethat the i i-th photon is measured in the DG basis, -th photon is measured in the DG basis, while a 0 requires that the photon is measured in while a 0 requires that the photon is measured in the VH basis. the VH basis.

As a result of this measurement Bob constructs the As a result of this measurement Bob constructs the string string a’.a’.



Bob uses the classical communication Bob uses the classical communication channel to request the string channel to request the string bb and Alice and Alice responds on the same channel with responds on the same channel with bb. Then . Then Bob sends Alice string Bob sends Alice string b’b’ on the classical on the classical channel.channel.

Alice and Bob keep only those bits in the set Alice and Bob keep only those bits in the set {{a, aa, a’} for which the corresponding bits in ’} for which the corresponding bits in the set {the set {b, bb, b’} are equal. Let us assume that ’} are equal. Let us assume that Alice and Bob keep only 2n bits.Alice and Bob keep only 2n bits.



Alice and Bob perform several tests to determine the Alice and Bob perform several tests to determine the level of noise and eavesdropping on the channel. The level of noise and eavesdropping on the channel. The set of 2n bits is split into two sub-sets of n bits each. set of 2n bits is split into two sub-sets of n bits each. One sub-set will be the One sub-set will be the check bitscheck bits used to estimate the used to estimate the

level of noise and eavesdropping, and level of noise and eavesdropping, and The other consists of the {\it data} bits used for the The other consists of the {\it data} bits used for the

quantum key. quantum key. Alice selects n check bits at random and sends the Alice selects n check bits at random and sends the

positions and values of the selected bits over the positions and values of the selected bits over the classical channel to Bob. Then Alice and Bob compare classical channel to Bob. Then Alice and Bob compare the values of the check bits. If more than say t bits the values of the check bits. If more than say t bits disagree then they abort and re-try the protocol.disagree then they abort and re-try the protocol.


State space dimension of classical and State space dimension of classical and quantum systemsquantum systems

Individual state spaces of n particles combine Individual state spaces of n particles combine quantum mechanically through the tensor product. If quantum mechanically through the tensor product. If X and Y are vectors, then X and Y are vectors, then their tensor product X Y is also a vector, but its their tensor product X Y is also a vector, but its

dimension is:dimension is: dim(X) x dim(Y) dim(X) x dim(Y) while the vector product X x Y has dimension while the vector product X x Y has dimension dim(X)+dim(Y). dim(X)+dim(Y).

For example, if dim(X)= dim(Y)=10, then the tensor For example, if dim(X)= dim(Y)=10, then the tensor product of the two vectors has dimension 100 while product of the two vectors has dimension 100 while the vector product has dimension 20.the vector product has dimension 20.


Quantum computersQuantum computers

In quantum systems the amount of parallelism In quantum systems the amount of parallelism increases exponentially with the size of the increases exponentially with the size of the system, thus with the number of qubits. This system, thus with the number of qubits. This means that the price to pay for an exponential means that the price to pay for an exponential increase in the power of a quantum computer is a increase in the power of a quantum computer is a linear increase in the amount of matter and space linear increase in the amount of matter and space needed to build the larger quantum computing needed to build the larger quantum computing engine.engine.

A quantum computer will enable us to solve A quantum computer will enable us to solve problems with a very large state space.problems with a very large state space.


ContentsContents I. Computing and the Laws of PhysicsI. Computing and the Laws of Physics II. A Happy Marriage; Quantum Mechanics &II. A Happy Marriage; Quantum Mechanics & ComputersComputers III. III. Qubits and Quantum GatesQubits and Quantum Gates IV. Quantum ParallelismIV. Quantum Parallelism V. Deutsch’s AlgorithmV. Deutsch’s Algorithm VI. Bell States, Teleportation, and Dense VI. Bell States, Teleportation, and Dense



One qubitOne qubit Mathematical abstractionMathematical abstraction Vector in a two dimensional complex vector space Vector in a two dimensional complex vector space

(Hilbert space)(Hilbert space) Dirac’s notationDirac’s notation

ket ket column vector column vector bra bra row vector row vector

bra bra dual vector (transpose and complex conjugate) dual vector (transpose and complex conjugate)

|


Ortonormal basisOrtonormal basis

0| 1|

2

10

2

10


One qubitOne qubit

10 10

1|||| 21

20

are complex numbersare complex numbers|,| 10


A bit versus a qubitA bit versus a qubit

A bitA bit Can be in two distinct states, 0 and 1Can be in two distinct states, 0 and 1 A measurement does not affect the stateA measurement does not affect the state

A qubit A qubit can be in state or in state or in any other can be in state or in state or in any other

state that is a linear combination of the basis statestate that is a linear combination of the basis state When we measure the qubit we find itWhen we measure the qubit we find it

in state with probability in state with probability in state with probability in state with probability

0| 1|10 10

0|1|

20 ||2

1 ||


0

1

0

1

(a) One bit (b) One qubit

Superposition states

Basis (logical) state 1

Basis (logical) state 0


Other states of a qubitOther states of a qubit

12

10

2

1

12

30

2

1


A different basis for one qubitA different basis for one qubit

2

10

2

10

2210

22

1010


Qubit measurementQubit measurement0

1

Possible states of one qubit beforethe measurement

The state of the qubit afterthe measurement

p1

p0


The Boch sphere representation of one The Boch sphere representation of one qubitqubit

A qubit in a superposition state is A qubit in a superposition state is represented as a vector connecting the represented as a vector connecting the center of the Bloch sphere with a point on center of the Bloch sphere with a point on its periphery.its periphery.

The two probability amplitudes can be The two probability amplitudes can be expressed using Euler angles. expressed using Euler angles.


b

| 0 >

| 1 >

z

x

y

ψ|

r


b

z

xy


Two qubitsTwo qubits

Represented as vectors in a 2-dimensional Represented as vectors in a 2-dimensional Hilbert space with four basis vectorsHilbert space with four basis vectors

When we measure a pair of qubits we decide When we measure a pair of qubits we decide that the system it is in one of four statesthat the system it is in one of four states

with probabilities with probabilities

11,10,01,00

11,10,01,00

211

210

201

200 ||,||,||,||


Two qubitsTwo qubits

11100100 11100100

1|||||||| 211

210

201

200


Measuring two qubitsMeasuring two qubits

Before a measurement the state of the system Before a measurement the state of the system consisting of two qubits is uncertain (it is given consisting of two qubits is uncertain (it is given by the previous equation and the corresponding by the previous equation and the corresponding probabilities). probabilities).

After the measurement the state is certain, it is After the measurement the state is certain, it is

00, 01, 10, or 11 like in the case of a classical 00, 01, 10, or 11 like in the case of a classical two bit system.two bit system.


Measuring two qubits (cont’d)Measuring two qubits (cont’d)

What if we observe only the first qubit, what What if we observe only the first qubit, what conclusions can we draw?conclusions can we draw?

We expect that the system to be left in an We expect that the system to be left in an uncertain sate, because we did not measure the uncertain sate, because we did not measure the second qubit that can still be in a continuum of second qubit that can still be in a continuum of states. The first qubit can be states. The first qubit can be 0 with probability 0 with probability 1 with probability1 with probability

201

200 ||||

211

210 ||||



Call the post-measurement state when we Call the post-measurement state when we measure the first qubit and find it to be 0.measure the first qubit and find it to be 0.

Call the post-measurement state when we Call the post-measurement state when we measure the first qubit and find it to be 1.measure the first qubit and find it to be 1.

201

200

01000

||||

0100

I

211

210

11011

||||

1101

I

I0

I1



Call the post-measurement state when we Call the post-measurement state when we measure the second qubit and find it to be 0.measure the second qubit and find it to be 0.

Call the post-measurement state when we Call the post-measurement state when we measure the second qubit and find it to be 1.measure the second qubit and find it to be 1.

210

200

10000

||||

1000

II

II0

II1

211

201

11011

||||

1110

II


Bell states - a special state of a pair of Bell states - a special state of a pair of qubitsqubits

If andIf and

When we measure the first qubit we get the post When we measure the first qubit we get the post measurement statemeasurement state

When we measure the second qubit we get the When we measure the second qubit we get the post mesutrement statepost mesutrement state

2

11100 01001

11|1I 00|0

I

00|0II 11|1

II


This is an amazing result!This is an amazing result!

The two measurements are correlated, once we The two measurements are correlated, once we measure the first qubit we get exactly the same measure the first qubit we get exactly the same result as when we measure the second one. result as when we measure the second one.

The two qubits need not be physically The two qubits need not be physically constrained to be at the same location and yet, constrained to be at the same location and yet, because of the strong coupling between them, because of the strong coupling between them, measurements performed on the second one measurements performed on the second one allow us to determine the state of the first.allow us to determine the state of the first.


EntanglementEntanglement

Entanglement is an elegant, almost exact translation Entanglement is an elegant, almost exact translation of the German term Verschrankung used by of the German term Verschrankung used by Schrodinger who was the first to recognize this Schrodinger who was the first to recognize this quantum effect.quantum effect.

An entangled pair is a single quantum system in a An entangled pair is a single quantum system in a superposition of equally possible states. The superposition of equally possible states. The entangled state contains no information about the entangled state contains no information about the individual particles, only that they are in opposite individual particles, only that they are in opposite states. states.

The important property of an entangled pair is that The important property of an entangled pair is that the measurement of one particle influences the state the measurement of one particle influences the state of the other particle. Einstein called that “Spookyof the other particle. Einstein called that “Spooky

action at a distance".action at a distance".


The spinThe spin

In quantum mechanics the intrinsic angular In quantum mechanics the intrinsic angular moment, the moment, the spinspin, is quantized and the values it , is quantized and the values it may take are multiples of the rationalized may take are multiples of the rationalized Planck constant.Planck constant.

The spin of an atom or of a particle is The spin of an atom or of a particle is characterized by the characterized by the spin quantum number sspin quantum number s , , which may assume integer and half-integer which may assume integer and half-integer values. For a given value of s the projection of values. For a given value of s the projection of the spin on any axis may assume 2s + 1 values the spin on any axis may assume 2s + 1 values ranging from - s to $ + s by unit steps, in other ranging from - s to $ + s by unit steps, in other words words the spin is quantizedthe spin is quantized..


More about the spinMore about the spin

There are two classes of quantum particlesThere are two classes of quantum particles fermions - fermions - spin one-halfspin one-half particles such as the particles such as the

electrons. The spin quantum numbers of fermions can electrons. The spin quantum numbers of fermions can be be s=+1/2 and s=+1/2 and s=-1/2s=-1/2

bosons - bosons - spin onespin one particles. The spin quantum particles. The spin quantum numbers of bosons can be numbers of bosons can be s=+1, s=+1, s=0, and s=0, and s=-1s=-1


The Stern-Gerlach experiment with The Stern-Gerlach experiment with hydrogen atomshydrogen atoms

N

SSource ofhydrogen

atomsMagnet

Screen


Physical embodiment of a qubitPhysical embodiment of a qubit

The electron with tow independent spin The electron with tow independent spin values, +1/2 and -1/2values, +1/2 and -1/2

The photon, with tow independent The photon, with tow independent polarizations, horizonatla and verticalpolarizations, horizonatla and vertical


The spin of the electronThe spin of the electron

The electron has spin s = 1 /2 and the The electron has spin s = 1 /2 and the spin projection can assume the values $ + spin projection can assume the values $ + ½ referred to as ½ referred to as spin upspin up, and -1/2 , and -1/2 referred to as referred to as spin downspin down..


Sz

12

h

12

h-

Rn(0)

Rn(180)

(a) (b)


Light and photonsLight and photons

Light is a form of electromagnetic Light is a form of electromagnetic radiation; the wavelength of the radiation radiation; the wavelength of the radiation in the visible spectrum varies from red to in the visible spectrum varies from red to violet.violet.

Light can be filtered by selectively Light can be filtered by selectively absorbing some color ranges and passing absorbing some color ranges and passing through others.through others.

A polarization filter is a partially A polarization filter is a partially transparent material that transmits light transparent material that transmits light of a particular polarization. of a particular polarization.


PhotonsPhotons Photons differ from the spin 1/2 electrons in two Photons differ from the spin 1/2 electrons in two

ways: ways: (1) they are massless and (1) they are massless and (2) have spin $1$.(2) have spin $1$.

A photon is characterized by its A photon is characterized by its vector momentum (the vector momentum determines vector momentum (the vector momentum determines

the frequency) and the frequency) and polarization. polarization.

In the classical theory light is described as having In the classical theory light is described as having an electric field which oscillates either an electric field which oscillates either verticallyvertically, , the light is x-polarized, or the light is x-polarized, or horizontallyhorizontally, the light is , the light is y-polarized in a plane perpendicular to the y-polarized in a plane perpendicular to the direction of propagation, the z-axis.direction of propagation, the z-axis.

The two basis vectors are |h> and |v>The two basis vectors are |h> and |v>


x

z

y

z

y

x

h

v

(a)

(b)

Vertically and horizontally polarized Vertically and horizontally polarized photonsphotons


Polarization filtersPolarization filters

A polarization filter is a partially transparent material A polarization filter is a partially transparent material that transmits light of a particular polarization.that transmits light of a particular polarization.

If we set the axis of a polarization filter to let pass y-If we set the axis of a polarization filter to let pass y-polarized light, then all photons in the state |v> will polarized light, then all photons in the state |v> will be absorbed in the filter and only the photons in state be absorbed in the filter and only the photons in state |h> will pass through. If the axis of the polarization |h> will pass through. If the axis of the polarization filter is set to let pass x-polarized light, then all filter is set to let pass x-polarized light, then all photons in state |h> will be absorbed and only photons in state |h> will be absorbed and only photons in state | v > will pass through.photons in state | v > will pass through.

When the polarization filter is set at angle with When the polarization filter is set at angle with respect to the coordinate system of the incoming respect to the coordinate system of the incoming beam of light the emerging photons are in a beam of light the emerging photons are in a superposition statesuperposition state


x

y| h >

(a)

(b)

x'

y'| h' >

y

x

x

y

z

| h >

| h' >

| h' >

Incidentbeam of light

PF1

PF2

The effect of a polarization filterThe effect of a polarization filter


Communication with entangles Communication with entangles particlesparticles

Even when separated two entangled Even when separated two entangled particles continue to interact with one particles continue to interact with one another.another.

Particle 2 and particle 3 in an anti-Particle 2 and particle 3 in an anti-correlated state (spin up and spin down). correlated state (spin up and spin down).

Then if we measure particle 1 and particle Then if we measure particle 1 and particle 2 and set them in an anti-correlated state, 2 and set them in an anti-correlated state, then particle 1 ends up in the same state then particle 1 ends up in the same state particle 3 was initially. particle 3 was initially.


New York

New York

London

London

particle1

particle1

particle1

particle2 particle3

particle2

particle2

particle3

particle3

particle3particle2particle1

Initial state

Entangle particle2 and particle3

Particle2 and particle3 in an anti-symmetric entangled state

Separate the entangled particles

Entangle particle1 and particle2Measure the entangled system

(particle1, particle2)


Classical gatesClassical gates

Implement Boolean functions.Implement Boolean functions. Are not reversible (invertible). We cannot Are not reversible (invertible). We cannot

recover the input knowing the output. recover the input knowing the output. This means that there is an iretriviable loss of This means that there is an iretriviable loss of

informationinformation


NOT gate

AND gate

x y = NOT(x)

x y z0

111 1 1

0 00 0

0 0

x y z0

111 1

10

0

00

NAND gate

x

yz = (x) AND (y)

x

yz = (x) NAND (y) 1

1

x y0

011

0

11

x y z0

111 1

10

00

x

yz = (x) OR (y)OR gate

x y z0

111 1

100

0

x

yz = (x) NOR (y)NOR gate 0

00

x y z0

111 1

10

00

x

yz = (x) XOR (y)XOR gate

1

0

0


One qubit gatesOne qubit gates

Transform an input qubit into an output qubitTransform an input qubit into an output qubit Characterized by a 2 x 2 matrix with complex Characterized by a 2 x 2 matrix with complex

coefficientscoefficients


One-qubit gate

10 10

G

2221

1211

gg

ggG

1

0

2221

1211

1

0

gg

gg

10 '1

'0



I I identity gate; leaves a qubit identity gate; leaves a qubit unchanged.unchanged.

X or NOT gateX or NOT gate transposes the transposes the components of an input qubit.components of an input qubit.

Y gate.Y gate. Z gate Z gate flips the sign of a qubit. flips the sign of a qubit. H H the Hadamard gate. the Hadamard gate.



2221

1211

gg

ggG

2212

2111

gg

ggGT

*22

*12

*21

*11

gg

ggG

Igggggggg

ggggggggGG

22*2212

*1221

*2211

*12

22*2112

*1121

*2111

*11


10

010 I

10

013 Z

0

02 i

iY

01

101 X

11

11

2

1H

10 10

10 01

10 01 ii

10 10

2

10

2

1010

Identity transformation, Pauli matrices, Identity transformation, Pauli matrices, HadamardHadamard


Tensor products and ``outer’’ productsTensor products and ``outer’’ products

0000

0000

0000

0001

0001

0

0

0

1

0000

0

0

0

1

0

1

0

10|0|00


CNOT a two qubit gateCNOT a two qubit gate

Two inputsTwo inputs Control Control TargetTarget

The control qubit is transferred to the output as is.The control qubit is transferred to the output as is. The target qubitThe target qubit

Unaltered if the control qubit is 0Unaltered if the control qubit is 0 Flipped if the control qubit is 1.Flipped if the control qubit is 1.

addition modulo 2+O

Target input

Control input

O+


CNOT

CNOTV

0100

1000

0010

0001

CNOTG

CNOTCNOTCNOT VGW


The two input qubits of a two qubit The two input qubits of a two qubit gatesgates

10 10

10 10

11

01

10

00

1

0

1

0

CNOTV


Two Two qubitqubit gates gates

1011111001010000 CNOTG

0100

1000

0010

0001

CNOTG


Two qubit gatesTwo qubit gates

01

11

10

00

11

01

10

00

0100

1000

0010

0001

CNOTW

CNOTCNOTCNOT VGW

11100100 01111000 CNOTW

)10(1)10(0 011100 CNOTW


Final comments on the CNOT gateFinal comments on the CNOT gate CNOT preserves the control qubit (the first and CNOT preserves the control qubit (the first and

the second component of the input vector are the second component of the input vector are replicated in the output vector) and flips the replicated in the output vector) and flips the target qubit (the third and fourth component of target qubit (the third and fourth component of the input vector become the fourth and the input vector become the fourth and respectively the third component of the output respectively the third component of the output vector).vector).

The CNOT gate is reversible. The control qubit is The CNOT gate is reversible. The control qubit is replicated at the output and knowing it we can replicated at the output and knowing it we can reconstruct the target input qubit.reconstruct the target input qubit.

)10(1)010(0 011100 CNOTW


Fredkin gateFredkin gate

Three input and three output qubitsThree input and three output qubits One controlOne control Two targetTwo target

When the control qubit isWhen the control qubit is 0 0 the target qubits are replicated to the the target qubits are replicated to the

outputoutput 1 1 the target qubits are swapped the target qubits are swapped


a

b

c

a'

b'

c'

a

b

0

a

b

0

a

b

1

b

a

1

0

Input Outputa b c a' b' c'

0 0

0

0

0

0

0

0 0

0

0 0

0 0

01

1

1

1

1 1 1 1


1 1

1

1

1

1

1

0 0

0

0 1

0 0

11

1

0

0

1 1 1 1


0 0

0

0

0

0

0

0 0

0

0 0

0 0

01

1

1

1

1 1 1 1

0 1 0

1

1

1

1

1

1

1

1

0 0

0

0 0

0

1

1

1 1

1

1

11

0

0

(a) (b)

(c)

(d) (e)

0

b

c

bc

bc

c'

1

0

c

c

c

c'

a=0 a=1 & b=0 a’= bc & b’ = bc a’= c & b’ = c

Threequbitgate


Toffoli gateToffoli gate

Three input and three output qubitsThree input and three output qubits Two controlTwo control One targetOne target

When both control qubit When both control qubit are 1 are 1 the target qubit is flipped the target qubit is flipped otherwise the target qubit is not changed.otherwise the target qubit is not changed.


a

b

c

a

b

c O ab+

(a)

a a

(b) (c)

1 1

b b

1 NAND(ab) 0

b b

b

Toffoli gate is universal. It may Toffoli gate is universal. It may emulate an AND and a NOT gateemulate an AND and a NOT gate


Controlled H gateControlled H gate

H

H

(a) (b)


Generic one qubit controlled gateGeneric one qubit controlled gate

(b)

A CB

| c >

| t >

(a)

U

| c >

| t >


U

|c0>

|c1>

|c2>

|c3>

|c4>

|c5>

| w0>=| 0 >

| 0>

| t >

Control

Target

| 0>

| 0>

| 0>

| 0>

| w1>=| 0 >

| w2>=| 0 >

| w3>=| 0 >

| w4>=| 0 >

|c0>

|c1>

|c2>

|c3>

|c4>

|c5>


ContentsContents I. Computing and the Laws of PhysicsI. Computing and the Laws of Physics II. A Happy Marriage; Quantum Mechanics &II. A Happy Marriage; Quantum Mechanics & ComputersComputers III. Qubits and Quantum GatesIII. Qubits and Quantum Gates IV. IV. Quantum ParallelismQuantum Parallelism V. Deutsch’s AlgorithmV. Deutsch’s Algorithm VI. Bell States, Teleportation, and Dense VI. Bell States, Teleportation, and Dense



A quantum circuitA quantum circuit

Given a function f(x) we can construct a reversible Given a function f(x) we can construct a reversible quantum circuit consisting of Fredking gates only, quantum circuit consisting of Fredking gates only, capable of transforming two qubits as followscapable of transforming two qubits as follows

The function f(x) is hardwired in the circuitThe function f(x) is hardwired in the circuit

| x > | x >

| y >

Uf

| y o f(x )>+


A quantum circuit (cont’d)A quantum circuit (cont’d)

If the second input is zero then the If the second input is zero then the transformation done by the circuit istransformation done by the circuit is

| x > | x >

| 0 >

Uf

| f(x )>


A quantum circuit (cont’d)A quantum circuit (cont’d)

Now apply the first qubit through a Hadamad gate.Now apply the first qubit through a Hadamad gate.

The resulting sate of the circuit isThe resulting sate of the circuit is

The output state contains information about f(0) and The output state contains information about f(0) and f(1).f(1).

| 0 >

Uf

H| 0 > | 0 >

)2

10(0

f

2

)1(1)0(0 ff

2

10


O

(c)

Uf

H

| 0 >

+

| x >

f(x) >

| x >

| y| y >

(a)

| x >

| f(x) >

| x >

| 0 >

(b)

| 0 >

Uf Uf


Quantum parallelismQuantum parallelism

The output of the quantum circuit contains information The output of the quantum circuit contains information about both f(0) and f(1). This property of quantum about both f(0) and f(1). This property of quantum circuits is called circuits is called quantum parallelismquantum parallelism..

Quantum parallelism allows us to construct the entire Quantum parallelism allows us to construct the entire truth table of a quantum gate array having 2truth table of a quantum gate array having 2nn entries at entries at once. In a classical system we can compute the truth once. In a classical system we can compute the truth table in one time step with 2table in one time step with 2nn gate arrays running in gate arrays running in parallel, or we need 2parallel, or we need 2nn time steps with a single gate time steps with a single gate array.array.

We start with n qubits, each in state |0> and we apply a We start with n qubits, each in state |0> and we apply a Welsh-Hadamard transformation.Welsh-Hadamard transformation.


Uf

| x > | x >

| y > | y O f(x) >+

(m-dimensional)

(k-dimensional) (n=m+k)-dimensional)


2

100

H

)]10()10()10[(2

1

n

12

02

1n

xn

x

000)( HHH

12

0

12

0

12

0

))(,2

1)0,(

2

1)0,

2

1(

nnn

xn

xfn

xnf xfxxUxU


ContentsContents I. Computing and the Laws of PhysicsI. Computing and the Laws of Physics II. A Happy Marriage; Quantum Mechanics &II. A Happy Marriage; Quantum Mechanics & ComputersComputers III. Qubits and Quantum GatesIII. Qubits and Quantum Gates IV. Quantum ParallelismIV. Quantum Parallelism V. V. Deutsch’s AlgorithmDeutsch’s Algorithm VI. Bell States, Teleportation, and Dense VI. Bell States, Teleportation, and Dense



Deutsch’s problemDeutsch’s problem Consider a black box characterized by a Consider a black box characterized by a

transfer function that maps a single input bit x transfer function that maps a single input bit x into an output, f(x). It takes the same amount of into an output, f(x). It takes the same amount of time, T, to carry out each of the four possible time, T, to carry out each of the four possible mappings performed by the transfer function mappings performed by the transfer function f(x) of the black box:f(x) of the black box:

f(0) = 0 f(0) = 0 f(0) = 1f(0) = 1 f(1) = 0 f(1) = 0 f(1) = 1f(1) = 1 The problem posed is to distinguish if The problem posed is to distinguish if )1()0(

)1()0(

ff

ff


0 f(0) 1 f(1)

2T

0 f(0)

1 f(1)

T

(a) (b)

O+

| x >

| y > f(x) >

| x >

| y >

T(c)

Uf


| y > + f(x)

Uf

| x >

| y >

| x >

O

| 0 > H

H| 1 >

H

0 1 2 3

A quantum circuit to solve Deutsch’s A quantum circuit to solve Deutsch’s problemproblem


1111

1111

1111

1111

2

1

11

11

2

1

11

11

2

11 HHG

2

10

2

10)11100100(

2

11

1

1

1

1

2

1

0

0

1

0

1111

1111

1111

1111

2

1011 G

0

0

1

0

1

0

0

1100

2

10 x

2

10 y


2

)(1)(

2

)(1)(0 xfxfxfxf

)(2

10)( xfxfy

2

10)1()( )(

xfxfy

1)(

2

10

0)(2

10

)(

xfif

xfifxfy


1)1()0(

1

1

1

1

2

1

2

10

2

10

0)1()0(

1

1

1

1

2

1

2

10

2

10

))((2

ffif

ffif

xfyx

0)1(,1)0(

1

1

1

1

2

1

2

10

2

10

1)1(,0)0(

1

1

1

1

2

1

2

10

2

10

))((2

ffif

ffif

xfyx

2

10 x

2

10 y


)1()0(

1

1

1

1

2

1

)1()0(

1

1

1

1

2

1

))((2

ffif

ffif

xfyx


1010

0101

1010

0101

2

1

10

01

11

11

2

13 IHG

)1()0(2

101

1

1

0

0

2

1

1

1

1

1

2

1

1010

0101

1010

0101

2

1

)1()0(2

100

0

0

1

1

2

1

1

1

1

1

2

1

1010

0101

1010

0101

2

1

233

ffif

ffif

G


Evrika!!Evrika!!

By measuring the first output qubit qubit we are By measuring the first output qubit qubit we are able to determine performing a able to determine performing a single evaluation.single evaluation.

)1()0(1

)1()0(0)1()0(

ffif

ffifff

2

10)1()0(3

ff

)1()0( ff


ContentsContents I. Computing and the Laws of PhysicsI. Computing and the Laws of Physics II. A Happy Marriage; Quantum Mechanics &II. A Happy Marriage; Quantum Mechanics & ComputersComputers III. Qubits and Quantum GatesIII. Qubits and Quantum Gates IV. Quantum ParallelismIV. Quantum Parallelism V. Deutsch’s AlgorithmV. Deutsch’s Algorithm VI. VI. Bell States, Teleportation, and Dense Bell States, Teleportation, and Dense



H|a>

|b>

|a>|a’>

H

I|b>|b’>

(a)

|V> |W>

(b)

stage 1 stage 2

Quantum circuit to create Bell statesQuantum circuit to create Bell states


Caroll

iY

CNOTparticle 1 - target qubitparticle 3 - control qubit

particle1

Aliceparticle

2

The measurement onthe pair (1&3) changesthe state of particle 2 toone of four states: S1,

S2, S3, S4

particle1

particle2

ClassicalChannel

Bob

particle3

I ZX

Pair of entangled qubits

particle3

Measurementparticle 3 - measuredparticle 1 - unchanged

QuantumChannel

Send to Bob results ofmeasurement

00 01 10 11

Receive from Aliceresults of measurements

00 01 10 11

Z

Particle 2 is in the samestate as particle 3


Caroll

I Z X iY

Alice’s modifiedbut still entangled qubit

Alice’squbit

Alice

Bob’squbit

Qubit fromAlice

Alice’squbit

Bob’squbit

Pair of entangled qubits

CNOT

Quantumchannel

00 01 10

H

1

Bob

Secondqubit

Firstqubit

0

0 1

11


ContentsContents I. Computing and the Laws of PhysicsI. Computing and the Laws of Physics II. A Happy Marriage; Quantum Mechanics &II. A Happy Marriage; Quantum Mechanics & ComputersComputers III. Qubits and Quantum GatesIII. Qubits and Quantum Gates IV. Quantum ParallelismIV. Quantum Parallelism V. Deutsch’s AlgorithmV. Deutsch’s Algorithm VI. Bell States, Teleportation, and Dense VI. Bell States, Teleportation, and Dense

CodingCoding VII. VII. SummarySummary


Final remarksFinal remarks

A tremendous progress has been made in the area A tremendous progress has been made in the area of quantum computing and quantum information of quantum computing and quantum information theory during the past decade. Thousands of theory during the past decade. Thousands of research papers, a few solid reference books, and research papers, a few solid reference books, and many popular-science books have been published in many popular-science books have been published in recent years in this area. recent years in this area.

The growing interest in quantum computing and The growing interest in quantum computing and quantum information theory is motivated by the quantum information theory is motivated by the incredible impact this discipline could have on how incredible impact this discipline could have on how we store, process, and transmit data and knowledge we store, process, and transmit data and knowledge in this information age.in this information age.


Final remarks (cont’d)Final remarks (cont’d)

Computer and communication systems using Computer and communication systems using quantum effects have remarkable properties. quantum effects have remarkable properties. Quantum computers enable efficient simulation of the Quantum computers enable efficient simulation of the

most complex physical systems we can envision.most complex physical systems we can envision. Quantum algorithms allow efficient factoring of large Quantum algorithms allow efficient factoring of large

integers with applications to cryptography. integers with applications to cryptography. Quantum search algorithms speedup considerably the Quantum search algorithms speedup considerably the

process of identifying patterns in apparently random data. process of identifying patterns in apparently random data. We can guarantee the security of our quantum We can guarantee the security of our quantum

communication systems because eavesdropping on a communication systems because eavesdropping on a quantum communication channel can always be detected.quantum communication channel can always be detected.


Final remarks (cont’d)Final remarks (cont’d) It is true that we are years, possibly It is true that we are years, possibly

decades away from actually building a decades away from actually building a quantum computer quantum computer requiring little if any power at all, requiring little if any power at all, filling up the space of a grain of sand, and filling up the space of a grain of sand, and computing at speeds that are unattainable today computing at speeds that are unattainable today

even by covering tens of acres of floor space even by covering tens of acres of floor space with clusters made from tens of thousands of with clusters made from tens of thousands of the fastest processors built with current state of the fastest processors built with current state of the art solid state technology. the art solid state technology.


Final remarks (cont’d)Final remarks (cont’d)

All we have at the time of this writing is a seven All we have at the time of this writing is a seven qubit quantum computer able to compute the qubit quantum computer able to compute the prime factors of a small integer, 15. Building a prime factors of a small integer, 15. Building a quantum computer faces tremendous quantum computer faces tremendous technological and theoretical challenges. technological and theoretical challenges.

At the same time, we witness a faster rate of At the same time, we witness a faster rate of progress in quantum information theory where progress in quantum information theory where applications of quantum cryptography seem applications of quantum cryptography seem ready for commercialization. Recently, a ready for commercialization. Recently, a successful quantum key distribution experiment successful quantum key distribution experiment over a distance of some 100 km has been over a distance of some 100 km has been announced.announced.


SummarySummary

Quantum computing and quantum information Quantum computing and quantum information theory is truly an exciting field.theory is truly an exciting field.

It is too important to be left to the physicists It is too important to be left to the physicists alone….alone….

Introduction to Quantum Computing and Quantum Information Theory Dan C. Marinescu and Gabriela M....

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