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Quantum Computing and
Qbit Cryptography
Patrick Lii
5 May 2009
Physics 138
Outline
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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:
𝑛 = log 2 𝑁
▫ the factorization time of a
classical comp is:
𝑡𝐶𝐶 = 𝐴 ∗ 2
𝑛
▫ For a QC
𝑡𝑄𝐶 = 𝐵 ∗ 𝑛3
ORNL’s Jaguar Supercomputer
Speed Comparison
• Assume CC and QC can factor a 78 digit number (n = 256) in 1 hour
Classical Computer
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n = 256 (AES)
n = 512
n = 1024
n = 2048 (RSA)
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1 hr
4.11 days
7.47 years
~73000 years
• QC easily defeats RSA encryption!
Quantum Computer
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1 hr
8 hrs
2.76 days
21.3 days
Outline
•
•
•
•
•
Motivation for Quantum Computing
A Review of Classical Computers
Qbits and Quantum Algorithms
Quantum Cryptography
Conclusion
The Classical Computer
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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:
𝜓⟩ = 𝛼0 0⟩ + 𝛼1 |1⟩
• w/ normalization condition:
𝛼0 2 + 𝛼1 2 = 1
• In matrix form:
0⟩ =
1
0
𝑎𝑛𝑑 1⟩ =
0
1
• n qbits are in superposition of
2n states
• qbits can be any two-level
quantum system
The Quantum Bit
• In general, for n bits:
Ψ⟩ =
𝛼𝑥 𝑥 ⟩𝑛
0≤𝑥<2𝑛
• w/ normalization:
𝛼𝑥
0≤𝑥<2𝑛
2
=1
Qbit Entanglement
• Purely quantum property of
qbit
• Two qbits are entangled if
wavefunction cannot be
written as product of 1 qbit
states
Ψ⟩ = 𝛼00 00⟩ + 𝛼01 01⟩ + 𝛼10 10⟩ + 𝛼11 11⟩
Quantum Logic Gates
• All quantum operations are unitary
▫ UU† = U†U = 1
• Gate can be any unitary quantum operator
• Ex: quantum NOT gate
𝐗 0⟩ =
0 1 1
0
=
= |1⟩
1 0 0
1
Quantum logic gate using lasers
• 2-bit gates can operate on entangled pairs
Important Quantum Gates
CNOT Gate
• Conditional Not
• Hadamard Transformation
1 1 1
𝐇=
2 1 −1
𝐇|0⟩ =
1
2
(|0⟩ + |1⟩)
𝐇|1⟩ =
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
• π/8 Phase Gate
𝐓=
1
2
(|0⟩ − |1⟩)
𝐓 0⟩ = |0⟩
1
0
0 𝑒
𝑖
𝜋
4
π
𝐓 1⟩ = ei 4 |1⟩
These 3 gates form a universal set
The Measurement Gate
• Born’s Law
• M gate Most important gate in
QC
▫ Given qbit:
▫ Collapses qbit wavefunction
𝜓⟩ = 𝛼0 0⟩ + 𝛼1 |1⟩
▫ Probability of measuring
state = amplitude squared
p(0) = 𝛼0
2
p(1) = 𝛼1
2
• Result based on probability
▫ We may not always get
“correct” answer
• Irreversible!
Quantum Algorithms
• Similarly to classical algorithms, quantum
algorithms are sequences of quantum gates
• In general, QCs have a simple processing
structure:
Ψ⟩ = 𝛼0 0⟩ + 𝛼1 1⟩ → 𝐔 𝐆𝐚𝐭𝐞𝐬 → 𝛽0 0⟩ + 𝛽1 1⟩ → 𝐌 𝐆𝐚𝐭𝐞 → 𝑟𝑒𝑠𝑢𝑙𝑡
• Complex processing lies in the U Gates
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
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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
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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:
 +
 O
Linear
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?)
Conclusion
1. Quantum computers are based on enacting quantum
operations on qbits
2. Quantum operations are simply unitary operators in
the Hilbert space of the system
3. 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
References
Quantum 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 Computing
http://joshblog.net/projects/logic-gate-simulator/Logicly.html [cool logic gate simulator]
Quantum Cryptography
C. 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 Algorithm
Shor, 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:
𝐔(|ϕμ ⟩ ⊗ |Φ⟩) = |𝜙𝜇 ⟩ ⊗ |Ψ𝜇 ⟩
• Eve must find a U that yields four distinct |ψμ>
𝜙𝜈 𝜙𝜇 Φ Φ⟩ = 𝜙𝜈 𝜙𝜇 Ψ𝜈 Ψ𝜇