Quantum Computer Architecture Prof Eric Rotenberg ECE 792

Quantum Computer Architecture Prof. Eric Rotenberg ECE 792 (046) Fall 2019 Topic 1: Quantum Circuits Part B: Elementary Quantum Gates

Single-qubit gates ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 2

Hadamard H https: //www. mathsisfun. com/algebra/trig-sin-cos-tan-graphs. html ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 3

Hadamard (cont. ) H z ϴ y x ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 4

Hadamard (cont. ) H z ϴ ϕ x ECE 792 (046), Quantum Comp. Arch. , Fall 2019 y (c) Eric Rotenberg 5

Hadamard (cont. ) • ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 6

Hadamard (cont. ) H constructive interference H H destructive interference ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 7

Hadamard (cont. ) H destructive interference H H constructive interference ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 8

Light beam-splitting experiment • https: //medium. com/@jonathan_hui/qc-control-quantum-computing -with-unitary-operators-interference-entanglement-7790 c 69 f 6 e 98 ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 9

“In the first experiment, we prepare all the inbound photons to have a polarization state |0�. This stream of polarized photons is split evenly by the beam splitter BA positioned at 45°, i. e. , it will split the beam into two orthogonally polarized light beams and exit in separate paths. Then we use mirrors to reflect the photons to two separate detectors and measure the intensity. From the perspective of classical mechanics, photons split into two separate paths and hit the detectors evenly. ” 1: all photons “In the second experiment, we put another beam splitter BB before the detectors. By intuition, the beam splitters operate independent of each other and split a light beam into two 0: no photons halves. Both detectors should detect half of. . . the photons. …. But that does not match with the experimental result! Only D₀ detects light. ” ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 10

“If we put another detector Dx after the first splitter, the experiment shows both detectors will detect half of the photons now. … when we add a measurement after the first splitter, we force a collapse in the superposition. The final result will be different than one without the additional detector. Nature tells us that if you know which path the photon takes, both detectors will detect half of the photons. In fact, we can achieve that with just one detector in one of the paths only. If no measurement is done before both detectors, all photons end up in detector D₀ if the photon is prepared to be |0�. ” ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 11

Pauli-X X X Rotation around the x-axis by 180° (π radians). ECE 792 (046), Quantum Comp. Arch. , Fall 2019 These differ by a global phase factor: statistics of measurement predicted for these two states is the same. Also, this factor forward-propagates as-is in future steps (global phase factor can be dropped even before measurement). (c) Eric Rotenberg 12

Rotation around the x-axis. ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 13

Pauli-Z Z Z These differ by a global phase factor: statistics of measurement predicted for these two states is the same. Also, this factor forward-propagates as-is in future steps (global phase factor can be dropped even before measurement). ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 14

ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 15

Pauli-Y Y like X Y Y ECE 792 (046), Quantum Comp. Arch. , Fall 2019 like Z Y (c) Eric Rotenberg 16

ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 17

Slides in progress • Phase (S) gate • pi/8 (T) gate ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 18

Two-qubit gates ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 19

controlled-NOT ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 20

Slides in progress • Two-qubit gates • swap gate • controlled-Z gate • controlled-phase gate • Three-qubit gates • Toffoli • Fredkin (controlled-swap) • Other • • measurement qubit classical bit n qubits ECE 792 (046), Quantum Comp. Arch. , Fall 2019 (c) Eric Rotenberg 21