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

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 −− = 1 2 𝑎+𝑏 2 Add diagonals Probability 𝑆 𝑥 spin down eigenvalue = 𝑓 −+ + 𝑓 + − = 1 2 𝑎−𝑏 2 Write the probabilities as dependent on 2 indices: 𝑓 + - 1 2 𝑎 2 + 1 4 𝑎𝑏 1 2 𝑎 2 − 1 4 𝑎𝑏 1 2 𝑏 2 − 1 4 𝑎𝑏 1 2 𝑏 2 + 1 4 𝑎𝑏 Only sums of 𝑓 terms can be probabilities! Any 𝑓 term can be negative!

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 = 1 2 cos 2 ( 𝜃 1 − 𝜃 2 )

Quantum Probability vs. Classical Probability My detector: 𝜃 1 Your detector: 𝜃 2 = 𝜃 1 + 30 0 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 2 30 0 = 3 4

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⟩

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

References Feynman, Richard P. “Simulating Physics with Computers.” International Journal of Theoretical Physics, vol. 21, no. 6-7, 1982, pp. 467–488., doi:10.1007/bf02650179. 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:10.1145/237814.237866.