## 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

[Doron]

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:

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

[Doron]
Status: discussed with Doron.

Consider a 2 site system that is represented by a qubit,
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

[Daniel/Doron?????]

### Ex 15,16 -coupled spins

[Daniel/Doron?????]
Status: unknown, sent links from past exams 4852-4858

Triangular AFM ring of 3 spins. Turn off adiabatically a magnetic field to get a frustrated (cat) ground state...   Optionally consider FM chain with AFM coupling between the first and the last spins.

A 3 qubits version of Hubbard model (each qubit is empty/full site).

### 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:

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.

### Ex08 - Particle in a ring, Spreading

[Daniel]

Status: Daniel Spoke with the group, still in draft mode.

As for a zero iteration, use regular computational methods in order to simulate the survival probability of a particle initially located at the first site, i.e, $$\left| \psi(t=0)\right\rangle=\left|1 \right\rangle$$, and the system contains 4 or 8 sites with a ring like geometry.

Do you wnat me to rephrase the question?
In any case I suggest that they measure the state and plot the probability distribution (i.e. |psi(x)|^2) at a sequence of time instants. Best is to use MATLAB image where each row is the image of the wavefunction at a different time instants. The survivavl probability is merely the "first column" of this plot.

Plot the survival probability as a function of time and via means of FFT evaluate the frequencies, check your results analytically.

The Hamiltonian is $$H=-\frac{c}{2} (D+D^\dagger)$$ and can be written as H=A+B, where A (B) contains only the odd (even) bonds, note that A and B do not commute.

(a) For a 4 site problem using 2 qubits, find the right combination of gates that implement $$U_A(t)=e^{-iAt}$$ and  $$U_B(t)=e^{-iBt}$$

(b) Use the Suzuki–Trotter expansion in order to simulate the evolution of a ring to the second order, i.e, for a small time step  $$U_{H}(dt)=e^{-iHdt} \approx U_A\left(\frac{dt}{2}\right)\cdot U_B(dt) \cdot U_A\left(\frac{dt}{2}\right)+O(dt)^3$$, plot the survival probability as function in time, compare your results with the analytical ones (exact and 2nd order) versus the real qubits on IBM, use FFT in order to find the frequencies.

(c) Can you find the correct combination of gate for an exact evolution for a 4 site problem? (Hint: use only CX and Ry gates). Show this equality by comparing the two matrices.

(d) Repeat (a) and (b) for an 8 site problem.

[Danile]
Status: unknown

[Daniel]
Status: unknown

[Daniel]
Status: unknown

### Ex 09 - STIRAP 3 sites

[Daniel?]
Status: unknown, sent example with 5

[Daniel?]
Status: unknown

### Ex 14 - Repeated measurement - "launch" and "target"

[Daniel?]
Status: Daniel Spoke with the group, still in draft mode.

האם נדרש ממני משהו?

אתם תדונו על שרשרת בת 4 אתרים שמיוצגת על ידי 2 קיוביטים. המצב 00 מיצג את האתר שבו החלקיק מתחיל את תנועתו. אתר המטרה 11 הוא המקום שבו מבוצעת מדידה למציאת החלקיק. המדידה מתבצעת על ידי צימוד לקיוביט נוסף.
הקיוביט הזה משמש כפוינטר: הוא אומר האם יש או אין חלקיק באתר המטרה. תצטרכו בשביל זה להבין איך CCX עובד, איך משתמשים ב if בשביל לסיים את הניסוי [[מציע לא להכניס את הסיבוך הזה. אם תתעורר בעיה נתמודד]] אם מצאתם את הפוינטר במצב "1" שמעיד על המצאות חלקיק באתר 11 (ייתכן ותצטרכו להסתדר ללא פקודת הif, ראו לינק מטה, כלומר למצוא דרך לעקוב "ידנית" לאחר התוצאות ואלו שהגיעו למטרה לא לעקוב אחריהם לאחר מכן).

שלבים נוספים:
* איך בונים מערכת 4 אתרים קבוצות 04 06 08 עובדות על ה"טכנולוגיה" הרלוונטית ליישום טבעת.
* לאחר מכן יש "להוריד" את הקפיצה המעגלית בשביל שרשרת של 4.
* לבסוף להשתמש בקיוביט נוסף בכדי לבצע מדידה.