1. Plamen Kamenov Physics 502 Advanced Quantum Mechanics
Quantum Computing
Plamen Kamenov
Physics 502 Advanced Quantum Mechanics

2. Why Do We Need Quantum Computers?
Take a spin-1/2 system:
Probability 𝑆 𝑧 spin up eigenvalue = 𝑓 ++ + 𝑓 + − = 𝑎 2
Add columns
Probability 𝑆 𝑧 spin down eigenvalue = 𝑓 −+ + 𝑓 − − = 𝑏 2
|𝜓 = 𝑎 𝑏
Probability 𝑆 𝑦 spin up eigenvalue = 𝑓 ++ + 𝑓 − + = 1 2 ( 𝑎 2 + 𝑏 2 )
Add rows
Probability 𝑆 𝑦 spin down eigenvalue = 𝑓 −− + 𝑓 + − = 1 2 ( 𝑎 2 +𝑏 2 )
Normalized state with
𝑎 2 + 𝑏 2 =1
𝑎,𝑏 𝜖 ℝ
Probability 𝑆 𝑥 spin up eigenvalue = 𝑓 ++ + 𝑓 −− = 𝑎+𝑏 2
Add diagonals
Probability 𝑆 𝑥 spin down eigenvalue = 𝑓 −+ + 𝑓 + − = 𝑎−𝑏 2
Write the probabilities as dependent on 2 indices: 𝑓 + -
1 2 𝑎 𝑎𝑏
1 2 𝑎 2 − 1 4 𝑎𝑏
1 2 𝑏 2 − 1 4 𝑎𝑏
1 2 𝑏 𝑎𝑏
Only sums of 𝑓 terms can
be probabilities!
Any 𝑓 term can be negative!

3. Can We Observe This Discrepancy?
Imagine a simple optics experiment:
𝜃 1
𝜃 2
Detector 1
Detector 2
2-photon emission
Ordinary Ray
Extraordinary Ray
Excited atom
Crystal
Crystal
Polarized light is analogous to the spin-1/2 system
By changing the crystal angles, we ask about the probability of mixing spin components (i.e. 𝑆 𝑧 , 𝑆 𝑥 eigenstates)
Quantum version of Malus’ Law: 𝑝 𝐸𝐸 = 𝑝 𝑂𝑂 = cos 2 ( 𝜃 1 − 𝜃 2 )

4. Quantum Probability vs. Classical Probability
My detector:
𝜃 1
Your detector:
𝜃 2 = 𝜃
Experiment 1:
90°
90°
60°
120°
60°
120°
30°
150°
30°
150° 0° 0°
Coincidence Probability: 2 3
Experiment 2:
90°
90°
60°
120°
60°
120°
30°
150°
30°
150° 0° 0°
Coincidence Probability: 0
Quantum Probability: p EE + p OO = cos 2 𝜃 2 − 𝜃 1 = cos = 3 4

5. What Could Quantum Computers Do?
Grover’s Algorithm:
How quickly can you find a name in a phonebook?
Classically, you need to search N/2 names to find a name with probability ½
Set up state: 𝜓 = 1 𝑁 𝑥=0 𝑁−1 |𝑥⟩
⟨0|
⟨1|
⟨2|
⟨𝑁−1|
⟨𝜔| …
⟨𝑥|𝜓⟩
Apply the “quantum oracle” operator: 𝑈 𝜔 =𝐼−2|𝜔⟩⟨𝜔| …
⟨𝑥| 𝑈 𝜔 |𝜓⟩
Rotates desired state in
complex plane by phase 𝑒 𝑖𝜋
Apply the Grover diffusion operator: 𝑈 𝜓 =2|𝜓⟩⟨𝜓|−𝐼 …
⟨𝑥| 𝑈 𝜓 𝑈 𝜔 |𝜓⟩
Repeat or measure the probability you’re in state |𝜔⟩: p 𝜔 =⟨𝜔 𝑈 𝜓 𝑈 𝜔 𝜓⟩

6. Is Grover’s Algorithm Quantum Mechanical?
Both operators are unitary and Hermitian:
𝑈 𝜔 𝑈 𝜔 † =𝑈 𝜔 † 𝑈 𝜔 =(𝐼−2|𝜔⟩⟨𝜔|)(𝐼−2|𝜔⟩⟨𝜔|) = 𝐼−2|𝜔⟩⟨𝜔 −2 𝜔⟩⟨𝜔 +4 𝜔⟩⟨𝜔|=𝐼
𝑈 𝜓 𝑈 𝜓 † = 𝑈 𝜓 † 𝑈 𝜓 =(2|𝜓⟩⟨𝜓|−𝐼) 2 𝜓 𝜓 −𝐼 =4|𝜓⟩⟨𝜓|−2|𝜓⟩⟨𝜓|−2|𝜓⟩⟨𝜓|+𝐼=𝐼
Following the algorithm, all amplitudes are positive:
You only need to search 𝒪( 𝑁 ) names to find a name with probability ~1

7. References
Feynman, Richard P. “Simulating Physics with Computers.” International Journal of Theoretical Physics,
vol. 21, no. 6-7, 1982, pp. 467–488., doi: /bf
Grover, Lov K. “A Fast Quantum Mechanical Algorithm for Database Search.” Proceedings of the
Twenty-Eighth Annual ACM Symposium on Theory of Computing - STOC '96, 1996, doi: /