CMSC 657 FALL 2020 HOMEWORK 12
DUE: 5PM ET FRIDAY, NOVEMBER 20TH. SUBMIT ON GRADESCOPE.
GORJAN ALAGIC
syllabus before starting the homework. Note that late homeworks are not accepted; no exceptions.
(1) KLM: Sections 10.5.4 and 10.6
(2) NC (optional): Sections 10.4.2, 10.5, 10.6
(3) Daniel Gottesman’s Intro to QEC and FT (optional): https://arxiv.org/abs/0904.2557
(4) Dave Bacon’s notes (optional):
• https://courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes18.pdf
• https://courses.cs.washington.edu/courses/cse599d/06wi/lecturenotes19.pdf
Video lectures. Watch lecture 12A (QEC II) and 12B (QEC III) on ELMS.
Problems.
(1) Analyze the code whose stabilizer is generated by the below.
X Z Z X I
I X Z Z X
X I X Z Z
Z X I X Z
Z Z X I X
(a) Use the stabilizer formalism to compute n, k, and d, and prove that this is a [[n, k, d]]
code for the values you found.
(b) How many arbitrary single-qubit errors can this code detect? How many can it correct?
(2) Analyze the code whose stabilizer is generated by the below.
X X I I
I I X X
Z Z Z Z
1
2 GORJAN ALAGIC
(a) Write down explicitly the codewords for logical |0〉 and logical |1〉. Verify that they are
stabilized by the stabilizer defined above.
(b) Use the stabilizer formalism to compute n, k, and d, and prove that this is a [[n, k, d]]
code for the values you found.
(c) How many arbitrary single-qubit errors can this code detect? How many can it correct?
(3) Give a fault-tolerant circuit for implementing logical X and logical Z gates in the 9-qubit
Shor code. Show that your circuits satisfy the definition of fault tolerance (Definition 10.6.1)
as stated in the KLM textbook. 