Draft
Ex 01 - Tutorial
Status: Done
Ex 02 - Tutorial
Status: Done
Ex 03 - Tutorial
Status: Done
Ex 05 - Oracle decryption
[Doron]
Status: appears in form: https://myforum.bgu.ac.il/phpBB3/viewtopic.php?f=57&t=16649
There is a black box (so called Oracle) that flips bits of an \(x\) register in a deterministic way that looks random. This is the "encryption" operation. This black box can be used to encrypt/decrypt
messages.
Without loss of generality we can associate with the black box a SEED \(k\), such that the encryption function is written as
\( f(x) = (-1)^{k \cdot x } \)
The challenge is to find out what is \(k\). If we find \(k\), we can duplicate the
box, and use it to decrypt messages that were encrypted by the original black box. To find \(k\) is a classically hard problem.
The quantum way to find \(k\), assuming that we can integrate the original black box in a quantum computer, is to
use the Shor's algorithm with FT replaced by \(H\)...
Ex 13 - Teleportation
https://arxiv.org/pdf/1907.09415.pdf
סעיף 1.5 - טלפורטציה
Ex04 - Cat state preparation
[Doron]
Status: appears in form: https://myforum.bgu.ac.il/phpBB3/viewtopic.php?f=57&t=16646
Refs:
https://qiskit.org/textbook/ch-gates/more-circuit-identities.html
http://www.physics.udel.edu/~msafrono/650/Lecture%206.pdf
Our objective is to provide tools for preparations of e.g. cat states. We shall consider 8 site system with sites labeled as 0=000, 1=001,..., 7=111.
The cat state is the superposition 000+111.
(1) An R operation can be used to form 0+1 superposition in a two site system that is represented by a single qubit. Explain how to realize the same operation in a 4-site system for the sites 00 and 01, while the other sites are disconnected.
(2) Explain how to realize the same operation in an 8-site system for the sites 00 and 01, while the other sites are disconnected. You will have to explain how to construct CC-R (note that CC-X is the so-called Toffoli gate).
(3) Explain how CC-R operation can be used in order to perform R between two sites. Draw a diagram that show all the possible pairings that you can get with a single CC-R operation.
(4) Use Gray-code sequence (sequences of CC-X operations) in order to perform R between arbitrary sites, say 000 and 111. This can be used in order to prepare a cat state.
Ex 11 - Two site simulation
such that 0=first site, 1=second site.
The dynamics is generated by
H = c \sigma_x + (\epsilon/2) \sigma_z = T+V
Define
U(t) = exp(-iHt) = R(tau)
\tau = dimensionless time = ct
(1) Use a single rotation matrix in order to simulate the dynamics.
Plot P(\tau) = |<0|R(\tau)|0>|^2 for $\tau \in[0,6\pi]$.
The plot should contain the result of the simulation (stars),
compared with the analytical result (line).
(2) Choose $N$. Defined $d\tau=6\pi/N$. Plot
P(\tau) = |<0|R(d\tau)...R(d\tau)|0>|^2
Check what is the largest value of $N$ that still provides good results.
This will help you to choose the optimal $N$ (large but not too large).
(3) Trotter... (for the chosen $N$)
Ex 12 - two site Zeno effect
Ex 15,16 -coupled spins
[Daniel/Doron?????]
Ex06 - Tools for simulations
[Doron]
Status: appears in form: https://myforum.bgu.ac.il/phpBB3/viewtopic.php?f=57&t=16646
Our objective is to provide tools for simulations. We shall consider 8 site system with sites labeled as 0=000, 1=001,..., 7=111.Refs:
https://qiskit.org/textbook/ch-gates/more-circuit-identities.html
http://www.physics.udel.edu/~msafrono/650/Lecture%206.pdf
(1) Explain how \( e^{-i H dt} \) can be realized for a 2 site system that is represented by a single qubit.Distinguish between the dynamics that is generated by the "potential" and the dynamics that is generated the "hopping".
As for a demonstration, assume that "\( Hdt \)" generates \( \frac{\pi}{6} \) rotation.
(2) Explain how to realize the same dynamics in a 4-site system for the sites 00 and 01, while the other sites are disconnected. Use CX (CNOT) and Rotation matrices. You will have to explain how to construct C-Rotations.
(3) Explain how to realize the same dynamics in an 8-site system for the sites 00 and 01, while the other sites are disconnected. You will have to explain how to construct CC-Rotations (note that CCX is the so-called Toffoli gate).
(4) Explain how CC-phase operation can be used in order to simulate the effect of potential at a given site.
(5) Use Gray-code sequence in order to perform a jump between arbitrary sites, say 000 and 111.
(6) Explain the generalization for any \( e^{-i H dt} \) hopping between two given sites.