Homework 2

Important: Write Names and Unity IDs on every submission file. Name the submission file correctly as asked for.

1. (20 points) Let's create the 1-bit full adder using D-Wave Ocean.
   • First form the QUBO for the full-adder.
   • Obtain the weights and strengths of all qubits and their connections using the 5/6 qubit Quantum Apprentice spreadsheet (5 qubit tab).
   • Let the default embedding provide an implicit mapping onto the D-Wave architecture in a first attempt. Report the results as comments.
   • Then create a version with explicit embedding onto the bi-partite graph in Quantum Apprentice (6-qubit tab) coded up and include results as comments.

   Turn in files fadd.xlsm, fadd-impl.py, fadd-expl.py

2. (40 points) Combine a half adder and two full adders to create an n-bit adder.
   • Look at the half-adder program in the slides.
   • Consider your full adder from the previous problem.
   • Combine the two by creating chains to connect corresponding qubits (for carry flags). Try it of for a 2-bit adder.
   • Re-factor your code to make it n-bit (as a parameter) by repeatedly chaining full adders to each other. See the factoring circuit as an example for hwo this is done in python. But remember, you need to do this with your own half and full adders.
   • Try it out as a 3-bit adder, include results as comments.

   Turn in file nadd.py

3. (15 points, 3 points each) There are several types of products used in quantum computing math. Show the result of each product below. The answer will either be a scalar, a vector, or a matrix. Do not use bra-ket notation in your answers.
   • \( <0|1>\)
   • \( |0><1| \)
   • \( |1> \otimes \ \ |1> \)
   • \( H \otimes X \)
   • \( <1|H|1> \)

   Submit your answers as \(\textbf{hw2math.pdf}\).

   Note: If you are a LaTeX/Overleaf user, the \(\textbf{braket}\) package defines \(\verb#\bra{}#, \verb#\ket{}#\), and \(\verb#\braket{}#\) macros. You should also check out the \(\textbf{qcircuit}\) package for drawing quantum circuits.

4. (10 points) Extend the one-bit adder, shown on slide 30 of Dr. Byrd's lecture slides, to create the 1-bit full adder on Quirk with inputs c0, a0, b0 and outputs s0 and c1.
   Hints:
   • Watch the tutorial.
   • Create the adder circuit by just dropping down X gates (marked as "cross", right below the Y gate) and adding one or two control probes (solid dot) for C-Not and Toffoli gates, respectively.
   • Put a Sample box on the inputs (left).
   • Test your circuit by verifying that inputs/outputs are correct. You can change inputs by putting one or more X gates on Qbit input lines (very left, left of the sample box), and then watch the output.
   • Label inputs and outputs. You can do this by creating a labeled gate for each symbol, i.e., click: make gate, from matrix "{{1,0},{0,1}}" circuit symbol "c0", create. Place it on the c0 input line, etc. for a0/b0/s0/c1.
   • Export your solution as a URL (copy clipboard) in the file adder.txt.

   Turn in file adder.txt.

5. (15 points) Combine two adders to create a 2-bit adder in Quirk with inputs c0, a0, a1, b0, b1 and outputs s0, s1, and c2. Make sure to label inputs/outputs.

   Turn in file doubleadder.txt.