Homework 2

Important: Make one submission per group. Write Names and Unity IDs on every submission file of all group members. Name the submission files correctly as asked for.

1. (20 points) Let's create the 1-bit full adder in QMI.
   • First form the QUBO for the full-adder.
   • Obtain the weights and strengths of all qubits and their connections using the Quantum Apprentice spreadsheet.
   • Perform the embedding on to the bi-partite graph and then encode it fadd.q (QUBO terms) and fadd.e (embedding).

   Turn in file fadd.bash, fadd.q, fadd.e

2. (20 points) Combine the half and full adders to create a 2-bit adder QUBO.
   • Look at the half-adder program from HW1
   • Fuse both embeddings into one to create a 2 bit adder (see embedding slide mentioned in HW2)

   Turn in file add2.bash, add2.q, add2.e

3. (10 points) Let's create the 1-bit full adder from slide 41 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 Hadamard 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

4. (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

5. (15 points) Create a Toffoli gate in Quirk from basic gates by modifying this solution. We now try construct Toffoli variation. Start with the given circuit but substitute gates as follows:
   • T is Z^1/8
   • T-dagger is Z^-1/8
   • S is Z
   • add a Sample on the inputs
   • try to add one more gate, X^1/2, in the right place with the right wiring (controls)
   • try out all input combinations to verify output

   Turn in file toffoli2.txt

6. (20 points) Test drive the IBM Quantum Experience and create a Toffoli equivalent circuit.
   Hints:
   • Sign up for an account (free).
   • Open the composer, choose some machine, e.g., ibmqx4 (5 qubits).
   • Create a Toffoli circuit from this solution, i.e., by directly using the indicated gates on IBM Q.
   • Set both controls to 1 (using X gates).
   • Simulate a run, print out the result and capture the qasm code. Your output should be 111 on the 3 Qubits.
   • Run on the real machine. After you hit the run button, you will have to wait for an email (several minutes in my case) reporting the outcome. Your output should be 111 on the 3 Qubits with a higher probability (0.572 for me). This is your first program running on a real quantum computer!!!
   • Remove your X gates and instead place Hadamards as the first gate for each of the 3 qubits. Simulate and explain the output (in the txt file). Repeat on the real machine.

   Turn in file toffoli.pdf, toffoli.qasm and toffoli.txt.