A Gentle Introduction to Quantum Information Scienceckli/Courses/m162/m162-26.pdfQuantum Information Science (QIS) Chi-Kwong Li Information Science QC,QC,QC Schr¨odinger cat Quantum

QuantumInformation

Science (QIS)

Chi-Kwong Li

InformationScience

QC,QC,QCSchrodinger cat

QuantumcomputingA brief history

The computing model

Computing by physicalsystems

Mathematicalformulation

QuantumComplexityQuantum algorithms

QuantumcryptologyQuantum teleportation

Conclusion

Thank you

A Gentle Introduction toQuantum Information Science

Chi-Kwong Li

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What is Information Science?

Information Science concerns the study of data.

1. Computing Storage and processing of information.

Input −→ Computing Unit −→ Output

2. Communication

Transmission of information.

3. Complexity Efficiency of the computing process.

Polynomial time vs. Exponential time vs. NP hard

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2. Communication




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2. Communication




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2. Communication




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Why Quantum?

Quantum properties are useful in the study of information science.

Let Dr. Quantum (Fred-Alan Wolf) explain.

• Superposition.Useful in quantum computing/complexity; one can apply anoperation simultaneously to many different physical states.

• Measurement effect, no cloning theorem.Useful for the security of quantum communication (QKD);no eavesdropping and faking information.

• Quantum entanglement.Useful for quantum computing, quantum communicationsuch as teleportation, superdense coding, etc.

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Why Quantum?






QuantumInformation

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Why Quantum?






QuantumInformation

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Why Quantum?






QuantumInformation

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Why Quantum?






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Why Quantum?






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Challenges

• How to control the (evolution of) quantum states?

Construct quantum operations transforming quantum states

ρ1, . . . , ρk −→ ρ1, . . . , ρk .

• How to do measurement to obtain useful information?Construct measurement operators to extract desired outputstates.

• Quantum mechanical systems always try to interact with theenvironment leading to the decoherence problem.One needs to develop quantum error correction (QEC)schemes to deal with the problem.

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Challenges

• How to control the (evolution of) quantum states?Construct quantum operations transforming quantum states

ρ1, . . . , ρk −→ ρ1, . . . , ρk .



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Challenges


ρ1, . . . , ρk −→ ρ1, . . . , ρk .

• How to do measurement to obtain useful information?

Construct measurement operators to extract desired outputstates.


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Challenges


ρ1, . . . , ρk −→ ρ1, . . . , ρk .



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Challenges


ρ1, . . . , ρk −→ ρ1, . . . , ρk .


• Quantum mechanical systems always try to interact with theenvironment leading to the decoherence problem.

One needs to develop quantum error correction (QEC)schemes to deal with the problem.

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Challenges


ρ1, . . . , ρk −→ ρ1, . . . , ρk .



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Schrodinger cat interpretation

• In 1935, the Austrian physicist ErwinSchrodinger suggested a thoughtexperiment showing that there wasa problem with the Copenhageninterpretation of quantum mechanics(applied to everyday objects).

• The thought experiment presentsa cat that might be simultaneouslyalive and dead with nonzeroprobability.

• Note that there is no-cloning ofthe cat, and measurement willcollapse the states.

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• In the course of developing hiscat experiment, Schrodinger coinedthe term Verschrankung – literally,entanglement (mixing many cats!).

• If one has to handle/examine 10 cats each could be alive ordead, there are 210 = 1024 cases to study in classicalinformation theory.

• In quantum information theory one can do it in one step!

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Quantum computingMotivations / a brief history

• 1980’s R. Feynman suggested the useof quantum systems to do computing, andsimulate more complex quantum systems.

• 1990’s Quantum algorithms by Duestsch,Jozsa, Shor, etc. were proposed.Shor’s quantum algorithm will havesignificant impact on the current (RSA) cryptology systems.

• 2000’s (Small scale) Quantum computers were/are built totest quantum algorithms.

• A lot of advance in theory and physical implementation.• Still a long way to go, and many research needed to be done!• Additional motivation. Transistors in digital computers are

getting so small that quantum effects take place.

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• 1990’s Quantum algorithms by Duestsch,Jozsa, Shor, etc. were proposed.

Shor’s quantum algorithm will havesignificant impact on the current (RSA) cryptology systems.




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• A lot of advance in theory and physical implementation.

• Still a long way to go, and many research needed to be done!• Additional motivation. Transistors in digital computers are


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• A lot of advance in theory and physical implementation.• Still a long way to go, and many research needed to be done!

• Additional motivation. Transistors in digital computers aregetting so small that quantum effects take place.

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The computing model

The general model


Classical computing. Modern Computing

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The computing model

The general model


Classical computing. Modern Computing

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Quantum computing model

−→ Quantum Computing UnitOptical lattices, NMR

−→

• Input - suitable quantum states as Quantum bits (Qubits).• A computing unit which can provide a suitable environment

for the quantum algorithm to run (quantum system of qubitsto evolve).

• Output - measure the resulting quantum states (in a suitableway) to get the useful information.

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−→

• Input - suitable quantum states as Quantum bits (Qubits).

• A computing unit which can provide a suitable environmentfor the quantum algorithm to run (quantum system of qubitsto evolve).


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−→




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−→




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Computing by physical systems

Basic procedures

• Step 1. Set up the apparatus.

• Step 2. Let the system run.

• Step 3. Do a suitable measurement to find the useful quantity.

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Basic procedures




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Basic procedures




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Examples

• Use magnetic needle to findthe north-south direction.

• Set the length L of the arm of thependulum so that 2

√L/g = 1.

Then the period T of a full swing is π.

• Put $1 in the bank in 1960.Pay 2% annual interest rate deposited daily.In 2010 (50 years later), you gete = 2.718281828459.... (the Euler constant).

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Examples



√L/g = 1.



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Examples



√L/g = 1.



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Examples



√L/g = 1.



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Mathematical formulationProposed by von Neumann• Consider a quantum system with

two physical states, say, up spinand down spin of a particlerepresented by

|0〉 =(

10

)and |1〉 =

(01

).

• Before measurement, the vector state may be insuperposition state represented by a complex vector

v = |ψ〉 = α|0〉+ β|1〉 =(αβ

)∈ C2, |α|2 + |β|2 = 1.

• Thus the the famous Schrodinger cat has probability |α|2being alive and |β|2 being dead!

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|0〉 =(

10

)and |1〉 =

(01

).


v = |ψ〉 = α|0〉+ β|1〉 =(αβ

)∈ C2, |α|2 + |β|2 = 1.


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|0〉 =(

10

)and |1〉 =

(01

).


v = |ψ〉 = α|0〉+ β|1〉 =(αβ

)∈ C2, |α|2 + |β|2 = 1.


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|0〉 =(

10

)and |1〉 =

(01

).


v = |ψ〉 = α|0〉+ β|1〉 =(αβ

)∈ C2, |α|2 + |β|2 = 1.


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Matrix and Bloch sphere

• It is convenient to represent thequantum state |ψ〉 as a rank-oneorthogonal projection:

Q = |ψ〉〈ψ| = 12

(1 + z x + iyx− iy 1− z

)with x, y, z ∈ R, x2 + y2 + z2 = 1.

Bloch sphere

• The state of k qubits are convex sum of 2k × 2k matrices ofthe form |ψ〉〈ψ| with |ψ〉 = |x1 · · · xk〉.

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Matrix and Bloch sphere

• It is convenient to represent thequantum state |ψ〉 as a rank-oneorthogonal projection:

Q = |ψ〉〈ψ| = 12

(1 + z x + iyx− iy 1− z

)with x, y, z ∈ R, x2 + y2 + z2 = 1.

Bloch sphere

• The state of k qubits are convex sum of 2k × 2k matrices ofthe form |ψ〉〈ψ| with |ψ〉 = |x1 · · · xk〉.

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Mathematical tools

• As a consequence of the Schrodinger equation

All quantum gates and quantum evolutions (for a closedsystem) are unitary similarity transforms of the densitymatrices representing the states, i.e.,

A(t) 7→ U(t)A(0)U(t)∗ for some unitaries U(t).

• By the results of Choi in 70’s and Kraus in 80’sQuantum channels, quantum operations, quantummeasurement operators, etc. aretrace preserving completely positivelinear maps of the form

A 7→∑r

j=1 FjAF∗j .

• The theory was discovered waybefore the applications! Picture of Choi

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Mathematical tools

• As a consequence of the Schrodinger equationAll quantum gates and quantum evolutions (for a closedsystem) are

unitary similarity transforms of the densitymatrices representing the states, i.e.,



A 7→∑r

j=1 FjAF∗j .


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Mathematical tools

• As a consequence of the Schrodinger equationAll quantum gates and quantum evolutions (for a closedsystem) are unitary similarity transforms of the densitymatrices representing the states, i.e.,



A 7→∑r

j=1 FjAF∗j .


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Mathematical tools



• By the results of Choi in 70’s and Kraus in 80’sQuantum channels, quantum operations, quantummeasurement operators, etc. are

trace preserving completely positivelinear maps of the form

A 7→∑r

j=1 FjAF∗j .


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Mathematical tools




A 7→∑r

j=1 FjAF∗j .


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Mathematical tools




A 7→∑r

j=1 FjAF∗j .


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Quantum complexity

• For k = 100, we have a state represented as a convex sum ofmatrices corresponding to |ψ〉 = |x1 · · · x100〉 of size

2100 = (210)10 ≈ 1030.

• If a high speed computer can do 1015 operations per second,to do one operation on each of the states corresponding to|ψ〉〈ψ|, one needs

1030/1015 = 1015 seconds,

which is more than 300, 000 centuries!

• That is why Feyman in 1980’s suggested that one could notsimulate quantum systems using digital computer.

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Quantum complexity


2100 = (210)10 ≈ 1030.

• If a high speed computer can do 1015 operations per second,to do one operation on each of the states corresponding to|ψ〉〈ψ|,

one needs

1030/1015 = 1015 seconds,



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Quantum complexity


2100 = (210)10 ≈ 1030.


1030/1015 = 1015 seconds,



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Quantum complexity


2100 = (210)10 ≈ 1030.


1030/1015 = 1015 seconds,



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Quantum algorithms

• Step 1. Create a maximal entangled state:

|ψ〉 = 2−k/2∑|x1 · · · xk〉, xj ∈ {0, 1}.

• Step 2. Apply an (unitary) operation f to |ψ〉 to get

f (|ψ〉) = 2−k/2∑

f (|x1 . . . xk〉).

• Step 3. Choose a suitable unitary operator g so that

g(f (|ψ〉)) = |y〉 ⊗ |z〉,

where |y〉 will carry some useful information.

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Quantum algorithms


|ψ〉 = 2−k/2∑|x1 · · · xk〉, xj ∈ {0, 1}.


f (|ψ〉) = 2−k/2∑

f (|x1 . . . xk〉).


g(f (|ψ〉)) = |y〉 ⊗ |z〉,


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Quantum algorithms


|ψ〉 = 2−k/2∑|x1 · · · xk〉, xj ∈ {0, 1}.


f (|ψ〉) = 2−k/2∑

f (|x1 . . . xk〉).


g(f (|ψ〉)) = |y〉 ⊗ |z〉,


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Deutsch-Jozsa algorithm

• In 1985, David Deutsch used thesuperposition idea to design analgorithm to determine whether a function

f : {|0〉, |1〉} → {|0〉, |1〉} satisfies:

f (|0〉) = f (|1〉) or f (|0〉) 6= f (|1〉).

• In classical computing, one must compute f (|0〉) and f (|1〉)to answer the above question.

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Deutsch-Jozsa algorithm

• In 1985, David Deutsch used thesuperposition idea to design analgorithm to determine whether a function

f : {|0〉, |1〉} → {|0〉, |1〉} satisfies:

f (|0〉) = f (|1〉) or f (|0〉) 6= f (|1〉).

• In classical computing, one must compute f (|0〉) and f (|1〉)to answer the above question.

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• In 1992, David Deutsch and Richard Jozsaextended the algorithm to determinewhether a function

f : {|x1 · · · xn〉 : xi = 0 or 1} → {| ↑〉, | ↓〉}

is constant of balanced.

• The Deutsch-Jozsa algorithm has been implemented:

• Zhi-Ming Zhan, Implementation of Deutsch-Jozsa Algorithmwith Superconducting Quantum-Interference Devices viaRaman Transition, Commun. Theor. Phys. 51 (2009),135-138.

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f : {|x1 · · · xn〉 : xi = 0 or 1} → {| ↑〉, | ↓〉}




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f : {|x1 · · · xn〉 : xi = 0 or 1} → {| ↑〉, | ↓〉}




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Shor’s algorithm

• In 1994, Shor designed a quantum algorithmto factor an integer N = pq for two large primenumbers p and q in polynomial time of log N.

• This is exponentially faster than the mostefficient known classical factoring algorithm:O(

e(log N)1/3(log log N)2/3)

.

• Shor’s algorithm is important because it can be used to“break” the widely used public-key cryptology schemeknown as RSA.

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Shor’s algorithm




.


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Shor’s algorithm




.


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Implementation

• In 2001, Shor’s algorithm was demonstrated by a group atIBM, who factored 15 into 3 ∗ 5, using an NMR quantumcomputer with 7 qubits.

• However, some doubts have been raised as to whether IBM’sexperiment was a true demonstration of quantumcomputation, since no entanglement was observed.

• Since IBM’s implementation, several other groups haveimplemented Shor’s algorithm using photonic qubits,emphasizing that entanglement was observed in 2007.

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Implementation




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Implementation




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Quantum cryptology• Quantum cryptology provides a

scheme for Alice and Bob to producea shared random bit string known onlyto them, which can be used as a keyto encrypt and decrypt messages.

• For example, the key is k = (k1k2 · · · , kN). Every messagem = (x1x2 · · · xN) is encrypted and decrypted as follows:

[encrypt]→ [transmit/correct]→ [decrypt]

m −→ m⊕ k −→ m⊕ k −→ (m⊕ k)⊕ k = m,

where ⊕ is the entry-wise Z2-addition.• There will be no eavesdropping (observing will change the

quantum states) and no way to fake information (no cloning).

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m −→ m⊕ k −→ m⊕ k −→ (m⊕ k)⊕ k = m,

where ⊕ is the entry-wise Z2-addition.

• There will be no eavesdropping (observing will change thequantum states) and no way to fake information (no cloning).

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m −→ m⊕ k −→ m⊕ k −→ (m⊕ k)⊕ k = m,

where ⊕ is the entry-wise Z2-addition.• There will be no eavesdropping (observing will change the

quantum states) and no way to fake information (no cloning).

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BB84 Protocol

By Charles H. Bennett and Gilles Brassard of IBM (1984)suggested using photon polarization states to send an N-bit key.

• Alice sends Bob 4N photons.• Alice (Bob) randomly chooses

one of the two bases to send(measure) each photon.

• Exchange the information of the 4N channels by classicalchannel (unsecured) to identify roughly 2N of the photonswere sent and received by the same bases.

• Check the errors in N of the remaining 2N photons byclassical channel (unsecured) to ensure no eavesdropper,and the bit strings in the other N photons can be used.Else, repeat the process.

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BB84 Protocol


• Alice sends Bob 4N photons.

• Alice (Bob) randomly choosesone of the two bases to send(measure) each photon.



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BB84 Protocol






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BB84 Protocol






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BB84 Protocol






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Implementation

• There are different research groups developing thetechnology.

• There are currently at least four companies offeringcommercial quantum cryptography systems.

• In Vienna, Austria, bank transfer using quantum cryptologywas first done in 2004, and a security computer network wasset up in 2008.

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Implementation




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Implementation




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Quantum teleportation

It is a technique using quantum entanglement to transferinformation on a quantum level.

(See the Youtube clip.)

Controversies about entanglement• Einstein, Podolsky, and Rosen, who

introduced the thought experiment in a1935 paper to argue that quantum mechanicsis not a complete physical theory.

• Most physicists today regard the EPR paradox as anillustration of how quantum mechanics violates classicalintuitions.

• Einstein never accepted quantum mechanics as a “real” andcomplete theory, struggling to the end of his life.

• As he once said: “God does not play dice.”

http://www.youtube.com/watch?v=_qmSdC7aQpY

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It is a technique using quantum entanglement to transferinformation on a quantum level. (See the Youtube clip.)







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Controversies about entanglement

• Einstein, Podolsky, and Rosen, whointroduced the thought experiment in a1935 paper to argue that quantum mechanicsis not a complete physical theory.





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Conclusion

• Quantum Information Science concerns the study of QC,QC, QC.

• It was a dream in 1980’s, and there has been muchcontroversies.

• Because of the efforts of many researchers from differentareas, much progress has been made.

• Quantum cryptology will be used widely in the near future.• Quantum computing is in its infancy, but has a promising

prospect.• Much theoretical work has been done on quantum

complexity theory.• Stay tuned for the transformation, or join the workforce to

explore these new frontiers!

QuantumInformation

Science (QIS)

Chi-Kwong Li

InformationScience



The computing model





Conclusion

Thank you

Conclusion








QuantumInformation

Science (QIS)

Chi-Kwong Li

InformationScience



The computing model





Conclusion

Thank you

Conclusion








QuantumInformation

Science (QIS)

Chi-Kwong Li

InformationScience



The computing model





Conclusion

Thank you

Conclusion




• Quantum cryptology will be used widely in the near future.

• Quantum computing is in its infancy, but has a promisingprospect.

• Much theoretical work has been done on quantumcomplexity theory.

• Stay tuned for the transformation, or join the workforce toexplore these new frontiers!

QuantumInformation

Science (QIS)

Chi-Kwong Li

InformationScience



The computing model





Conclusion

Thank you

Conclusion





prospect.

• Much theoretical work has been done on quantumcomplexity theory.


QuantumInformation

Science (QIS)

Chi-Kwong Li

InformationScience



The computing model





Conclusion

Thank you

Conclusion






complexity theory.


QuantumInformation

Science (QIS)

Chi-Kwong Li

InformationScience



The computing model





Conclusion

Thank you

Conclusion








QuantumInformation

Science (QIS)

Chi-Kwong Li

InformationScience



The computing model





Conclusion

Thank you

Thank you for your attention!

Documents

A Gentle Introduction to Quantum Information Scienceckli/Courses/m162/m162-26.pdfQuantum Information Science (QIS) Chi-Kwong Li Information Science QC,QC,QC Schr¨odinger cat Quantum