Operators in Quantum State and Implications Assignment

Assignment Task

1. Consider the n-qubit Cat state (a.k.a generalized GHZ state) |GHZn) = 10)+11) √2 Apply a Hadamard gate to each qubit and then measure in the computational basis  

• Show that only strings with even Hamming weight (i.e. number of ones in the string) can show up after the measurement.
• Repeat the exercise for the state Z₁ GHZn), where 1 ≤ i ≤ n. What type of strings can show up after the measurement. (Compare it to the result (a).)
• Repeat the exercise for the state X, GHZn). What type of strings can show up after the measurement.

3. Some circuit identities. Let C denote a CNOT gate, with qubit 1 the control qubit and qubit 2 the target qubit. (Note that C = Ct, and H = H+, where H is the Hadamard gate).
a) Prove HZH = X and HXH = Z, and use them to find HYH. (Hint: you can multiply the two identities and then simplify.)
b) Prove that CX₁C+ = X1X2 and CZ₁C = Z1. Use these two identities to find CYC.
(Observation: this means CX1 = X1X2C. We can interpret it as: an X "error" on the control line of a CNOT spreads to two X errors on both control and target lines!)
c) Prove that CX2C+ = X2 and CZ2C+ = Z1 Z2. Use these two identities to find CY2Ct.
(Note how CNOT spreads Z operator from the target line to both control and target lines!)
d) Use the previous identities to find CZ1X2Ct.

5. In this exercise we try to come up with a simple strategy to use a biased coin to simulate a perfectly unbiased coin. We will use two rounds of (biased) coin flips and post-selection to probabilistically simulate an unbiased coin.

a) Let + be the probability of getting heads H (and probability of 7'), where €≤0.5. After two coin flips, what is the probability of getting HH, HT, TH, and TT. (Assume coin tosses are independent.)

b) Let's discard the result of the two coin flips if the outcomes are the same (i.e. discarding if the result is HH or TT). Conditioned on this post-selection rule, what is the probability of getting HT or TH.

c) Use the result of section (b) to develop a strategy to simulate a fair coin (a coin with equal probability of showing heads and tails).

d) What is the expected number of rounds we need to repeat the experiment to be able to simulate one flip of a fair coin (in terms of e).

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