ECS 189A: Assignment 6 辅导

ECS 189A: Assignment 6
2/24/2023
Due: 3/3/2023
1. Multi-qubit circuit identities.
(a) (1 Point) Reduce the presented circuit to Pauli, rotation, or their products.
X =?
(b) (1 Point) What is U? (Here |Y+? is the +1 eigenstate of Y .)
|Y+? |Y+?
= U
|Y+? |Y+?
.
(c) (1 Point) Reduce the presented circuit to Pauli, rotation, or their products.
Y
2. (4 Points) In Project 1, you were asked to find a quantum circuit that computes the parity of bits.
Checking a parity can be useful, because we can use those operations to check if an error has occurred. We
will go through an example that exemplifies this point.
(a) (1 Point) Consider two classical bit-strings 000 and 111, representing a binary information. Consider
the following three-step process.
1. You are given a bit-string 000 (or equivalently, with 111).
2. For each bit, flip the bit with probability p.
3. Perform a majority vote. If the majority is 1, return 111. If the majority is 0, return 000.
If the bit-string obtained is different from the original bit-string. We say that a logical error has occurred.
What is the probability of logical error occurring?
(b) (1 Point) Now consider two quantum states |000? and |111?. Suppose one of the qubits is flipped
(by applying X). No matter which qubit is flipped, it turns out that we can always revert the state back
1
to the original state by measuring the parity of the first two bits and the parity of the last two bits.
Draw a quantum circuit that includes two measurements, such that the parities can be obtained from the
measurement outcomes. (Hint: You will need two ancillary qubits for this purpose.)
(c) (1 Point) Suppose you are only given the parity information (without knowing the state). For each
parity measurement outcome, explain what unitary you will apply to get back the original state to perform
the majority vote.
(d) (1 Points) Suppose now we are in a superposition state α|000?+β|111?. Suppose we applied X1 and
then applied the protocol in (c). What do we get? (Optional Problem: What about the case of X2, X3?)
3. (3 Points) In the lecture, we learned about teleportation. (See Page 258-260 of the textbook for the
explanation of the protocol.) In that discussion, we used the EPR state |EPR? = 1√
2
(|00?+ |11?). (In the
textbook, this is denoted as |Φ+?.) Here we use a modified EPR state:
|EPR′? = 1√
2
(|00? ? |11?).
(Everything else about the protocol will be exactly the same.) In this problem, we will learn how this
change modifies the state that Alice gets. (Note: Please take a look at the teleportation circuit in the slides
for Lecture 12.)
(a) (1 Point) There is a single-qubit unitary operator U acting on the first qubit of |EPR?, such that
(U ? I)|EPR? = |EPR′?. What is this unitary?
(b) (1 Point) Using (a), we can see that our new teleportation protocol is exactly the same as our old
protocol except for the fact that we now have additional gates between CNOT and measurements. What
are these gates? (Hint 1. You may find it helpful to draw a circuit diagram. Hint 2. You may find the
cheat sheet for circuit identities helpful.)
(c) (1 Point) Suppose the sender in the teleportation protocol prepares her/his state as |ψ?. After
running through our new teleportation protocol, what state does the sendee get?