Quantum Computing and Qbit CryptographyPatrick Lii5 May 2009Physics 138

Outline

•Motivation for Quantum Computing•A Review of Classical Computers•Qbits and Quantum Algorithms•Quantum Cryptography•Conclusion

What is a Quantum Computer?• A quantum computer (QC)

is a computational device which operates on data using quantum algorithms

• QC in proof-of-concept stage

• Current motivations:▫ Cryptography▫ Factorization▫ Database searching http://www.acceleratingfuture.com/michael/blog/category/random/

page/2/

Classical versus Quantum Computers

• Example: Large number factorization

• QCs ->advantage of parallelism▫ qbits are in superpositions

of states

• ‘backwards compatible’ w/ classical algorithms

Performance Advantage of QCs• classical: ~1-10 gflops of

computing power• quantum: ~10 tflops

• Factorization speed:▫ for an integer N with size:

▫ the factorization time of a classical comp is:

▫ For a QC

𝑛 = log2 𝑁

𝑡𝐶𝐶 = 𝐴∗2ξ𝑛

𝑡𝑄𝐶 = 𝐵∗𝑛3

ORNL’s Jaguar Supercomputer

Speed Comparison

Classical Computer

Quantum Computer

• 1 hr• 4.11 days• 7.47 years• ~73000 years

• 1 hr• 8 hrs• 2.76 days• 21.3 days

• Assume CC and QC can factor a 78 digit number (n = 256) in 1 hour

• QC easily defeats RSA encryption!

• n = 256 (AES)• n = 512• n = 1024• n = 2048 (RSA)

Outline

•Motivation for Quantum Computing•A Review of Classical Computers•Qbits and Quantum Algorithms•Quantum Cryptography•Conclusion

The Classical Computer• Classical bits (cbits): 0 or 1• 2 cbits 4 states, • 3 cbits 8 states• n cbits 2n states

• data is represented in binary▫ 138 10001010▫ q 01110001

Classical Operations• Based on logic gates

• Example: 1-bit gate ▫ NOT gate▫ X:0 1▫ X:1 0

• 2-bit gates:▫ AND/NAND▫ OR/NOR▫ XOR/XNOR

AND Gate

Bit 1 Bit 2 Output

1 1 1

1 0 0

0 1 0

0 0 0

XOR Gate

Bit 1 Bit 2 Output

1 1 0

1 0 1

0 1 1

0 0 0

Classical Algorithms• All 1, 2, 3-cbit gates together

form universal set

• classical algorithm: a complex operation that uses a sequence of classical gates

http://www.inetdaemon.com/img/gates.gif

Outline

•Motivation for Quantum Computing•A Review of Classical Computers•Qbits and Quantum Algorithms•Quantum Cryptography•Conclusion

The Quantum Bit (Qbit)• Unlike cbits, state of a qbit

is a superposition of 1 and 0:

• w/ normalization condition:

• In matrix form:

• n qbits are in superposition of 2n states

• qbits can be any two-level quantum system

ȁ�𝜓⟩ = 𝛼0ȁ�0⟩ + 𝛼1|1⟩ ȁ�𝛼0ȁ�2 + ȁ�𝛼1ȁ�2 = 1

ȁ�0ۧ = ൬10൰ 𝑎𝑛𝑑 ȁ�1ۧ = ൬01൰

The Quantum Bit

• In general, for n bits:

•w/ normalization:

ȁ�Ψۧ = 𝛼𝑥ȁ��ۧ� 𝑛0≤𝑥<2𝑛

ȁ�𝛼𝑥ȁ�20≤𝑥<2𝑛 = 1

•Purely quantum property of qbit

•Two qbits are entangled if wavefunction cannot be written as product of 1 qbit states

Qbit Entanglement

ȁ�Ψۧ = 𝛼00ȁ�00ۧ + 𝛼01ȁ�01ۧ + 𝛼10ȁ�10ۧ + 𝛼11ȁ�11ۧ

Quantum Logic Gates• All quantum operations are unitary

▫ UU† = U†U = 1

• Gate can be any unitary quantum operator

• Ex: quantum NOT gate

• 2-bit gates can operate on entangled pairs

𝐗ȁ�0ۧ = ቂ0 11 0ቃቂ10ቃ= ቂ

01ቃ= |1⟩ Quantum logic gate using lasers

Important Quantum Gates• Conditional Not

• Hadamard Transformation

CNOT Gate

Bit 1 In Bit 2 In Bit 1 Out

Bit 2 Out

1 1 1 0

1 0 1 1

0 1 0 1

0 0 0 0

𝐇= 1ξ2ቂ1 11 −1ቃ 𝐇|0⟩ = 1ξ2(|0⟩+ |1⟩) 𝐇|1⟩ = 1ξ2(|0⟩− |1⟩)

• π/8 Phase Gate

𝐓= 1 00 𝑒𝑖𝜋4൨ 𝐓ȁ�1ۧ = eiπ4|1⟩ 𝐓ȁ�0ۧ = |0⟩ These 3 gates form a universal set

p(1) = ȁ�𝛼1ȁ�2

The Measurement Gate• Born’s Law

▫ Given qbit:

▫ Probability of measuring state = amplitude squared

• M gate Most important gate in QC▫ Collapses qbit

wavefunction

• Result based on probability▫ We may not always get

“correct” answer• Irreversible!

ȁ�𝜓⟩ = 𝛼0ȁ�0⟩ + 𝛼1|1⟩

p(0) = ȁ�𝛼0ȁ�2

Quantum Algorithms

•Similarly to classical algorithms, quantum algorithms are sequences of quantum gates

• In general, QCs have a simple processing structure:

•Complex processing lies in the U Gates

ȁ�Ψۧ = 𝛼0ȁ�0ۧ + 𝛼1ȁ�1ۧ →ሾ𝐔 𝐆𝐚𝐭𝐞𝐬ሿ→ 𝛽0ȁ�0ۧ + 𝛽1ȁ�1ۧ →ሾ𝐌 𝐆𝐚𝐭𝐞ሿ→𝑟𝑒𝑠𝑢𝑙𝑡

Shor’s Algorithm• Developed by Peter Shor in 1994• Efficient factorization of large

numbers

• RSA Encryption▫ Based on multiplying 2 very

large prime numbers (~200 digits each)

• CCs cannot factor this in a reasonable time

• However, using Shor’s algorithm, a QC can▫ Lots of interest from

government

Physical Implementations of QCs• In 2001, a group at IBM led

by Vandersypen created a 7-qbit QC▫ NMR implementation▫ Used it to demonstrate

Shor’s algorithm by factoring 15 into 3 and 5

• Other possibilities▫ Optical lattices▫ Polarized light▫ Diamond based▫ Superconductor (SQUIDs)▫ Trapped ion▫ any two level system w/

orthogonal bases

Biggest problem in implementation of QC: controlling decoherence of qbits

Outline

•Motivation for Quantum Computing•A Review of Classical Computers•Qbits and Quantum Algorithms•Quantum Cryptography•Conclusion

Quantum Cryptography (BB84)• Called BB84: Bennett and

Brassard 1984• Method of secure key

distribution▫ Created using only 1-qbit

gates▫ Can be implemented

using current tech (transmission w/ polarized light)

▫ interception can be detected

Message Security• Say we want to transmit the number 83▫ In binary: 1010011 (7-bits)

• We securely (and randomly) generate a key w/ equal bit-length▫ take: 1011011

• We then use this key to encode the message▫ “flip” message bits everywhere the key equals 1▫Message becomes 0001000

• impossible for someone w/o a key to unencrypt this• Cryptography comes down to:▫ Random key generation▫ Secure key distribution

Key Generation I• Alice sends Bob a long stream of photons (qbits)

▫ She randomly assigns each a type: circular or linear pol▫ Then, randomly assigns a polarization sub-state based on

the type LH or RH for circ X or Y for linear

• Example: Alice sends 8 qbits

▫−|<−<|<−▫ Legend:

— X, 0 bit | Y, 1 bit > RHCP, 0 bit < LHCP, 1 bit

Key Generation II• Bob randomly decides on a linear or circular measurement

of each incoming photon

• For measurement, Bob chooses:

▫O+O++OOO▫ Legend:

+ Linear O Circular

• And he measures:▫<|<−−><>▫ for reference, Alice sent: −|<−<|<−

Key Generation III•Bob calls Alice and tells her his choice of

measurement (circ or lin) for each photon▫Alice then tells Bob which of his types agree

with her transmission types NYYYNNYN

•They then use the agreeing values as a key▫In example, A&B have 4 agreeing qbits: |<-<▫Their key is: 1101

Eavesdropping• for ‘Eve’ to eavesdrop on

A&B’s transmission, she must also randomly make circ or lin measurements of each photon▫ This changes polarization

of about half the qbits

• 1/4th of Bob’s result will not agree w/ Alice’s prep▫ A&B can compare some

‘check bits’ over the phone to see if anyone is eavesdropping

Qbit Interception• Suppose Eve uses a more sophisticated attack:▫ intercepts the transmission▫processes it in a QC ▫ restores it to original state and sends it back off

to Bob

• This is defeated by the no-cloning theorem▫Forbids creation of identical copy of an arbitrary

state• Eve gets no useful information from her

interception

Outline

•Motivation for Quantum Computing•A Review of Classical Computers•Qbits and Quantum Algorithms•Quantum Cryptography•Conclusion

Future Developments in QC

•Largely in proof-of-concept stage▫formidable technological obstacles

•Still need to:▫discover more algorithms▫overcome decoherence of qbits

•Deeper understanding of QM may make it easier to do this

•We are decades away from truly powerful QC (~2050?)

Conclusion1. Quantum computers are based on enacting quantum

operations on qbits2. Quantum operations are simply unitary operators in

the Hilbert space of the system3. QCs have the potential to vastly outperform classical

computers because of the QM nature of their operations

4. QCs are still many years off; however, they will fundamentally change computation as we know it

5. Qbits can also be employed in generating an undefeatable cryptography scheme which may prove useful once RSA encryption is defeated by QCs

ReferencesQuantum Computing (General)Kaye, Phillip, Raymond Laflamme, and Michele Mosca. An Introduction to Quantum Computing. 1st ed.

Oxford: Oxford University Press, 2007. Print. Lieven M.K. Vandersypen et al. (1999). "Separability of Very Noisy Mixed States and Implications for NMR

Quantum Computing". Phys. Rev. Lett 83: 1054–1057.Mermin, David. Quantum Computer Science. 1st Ed. Cambridge: Cambridge University Press, 2007. Print.

[great introductory resource for Quantum Computers from a professor at Cornell, not rigorous however]

Classical Computinghttp://joshblog.net/projects/logic-gate-simulator/Logicly.html [cool logic gate simulator]

Quantum CryptographyC. H. Bennet and G. Brassard, “Quantum Cryptography: Public key distribution and coin tossing”, in

Proceedings of the IEEE International Conference on Computers, Systems, and Signal Processing, Bangalore, p. 175 (1984)

http://fredhenle.net/bb84/ [BB84 transmission simulator]

Shor’s AlgorithmShor, P. (1994) Algorithms for Quantum Computation: Discrete Logarithms and Factoring.

Proceedings of the 35th Annual IEEE Symposium on Foundations of Computer Science, Santa Fe, NM, Nov. 20-22, 1994.

•Let:▫ |Фμ>, μ = 0, …, 3 = four states of Alice’s

qbits (X, Y, RH, LH)▫|ψ> = initial state of qbits on Eve’s QC

•Since qbit must emerge in original state:

•Eve must find a U that yields four distinct |ψμ>

𝐔(|ϕμ⟩⊗ |Φ⟩) = |𝜙𝜇⟩ ⊗ |Ψ𝜇⟩ ൻ𝜙𝜈ห𝜙𝜇ൿۦΦȁ�Φۧ =ൻ𝜙𝜈ห𝜙𝜇ൿൻΨ𝜈หΨ𝜇ൿ