Introduction to Quantum Information Science Homework 7
Point total: 34
All homework assignments are weighted equally in the final grade. Point values are unique to each assignment.
For problems which ask you to explain or show your work, you don’t need to show every step of each calculation, but the answer should include an explanation written with words of what you did. Even when work is not required to be shown, it’s a good idea to include anyways so that you can earn partial credit.
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1. [10 points] Give two different decompositions of the 1-qubit mixed state cos2(π/8) 0
as a mixture of two pure states. Show your work.
Draw a 2D-projection of the Bloch sphere with your solutions.
Additionally, give a short interpretation of what this mixed state with these two different decomposi-
tions is — specifically explain the fact there are multiple decompositions (e.g. “it represents 2 different qubits, one in the first state, and one in the other”).
2. Quantum computation with real amplitudes “Real quantum mechanics” is a hypothetical theory that’s identical to standard quantum mechanics except that the amplitudes always need to be real, and instead of unitary matrices, we’re restricted to applying real orthogonal matrices.
a) [8 points] Prove that any standard quantum circuit acting on n qubits can be simulated exactly by a real quantum circuit acting on n + 1 qubits and containing exactly as many gates as the original circuit (the gates may act on slightly more qubits than the gates of the original circuit). The standard quantum circuit may include arbitrary n-qubit gates.
Your simulation should include a mapping from complex-valued states to real-valued states and from unitary matrices to real orthogonal matrices such that if A|ψ⟩ = |φ⟩ are arbitrary standard states and operations, and A′, |ψ′⟩ , |φ′⟩ are the “real” encodings of each, then A′ |ψ′⟩ = |φ′⟩.
Hint: Begin by encoding an arbitrary n-qubit, complex-valued state using n+1 “real qubits”, then create a mapping for the operations. Consider how many additional entries exist in a vector or matrix when an additional qubit is added.
Note: If you try finding the answer online, you’ll probably find a paper. But, that paper doesn’t explain each part of this problem, and you’ll likely find that if you just use their answer to the first part, you will struggle or be unable to make the other parts work.
ρ = 0 sin2(π/8)
Introduction to QIS The University of Texas
b) [4 points] Conclude the proof that “real quantum computing” is equivalent to standard quantum computing by explaining how measurements are performed, such that when you simulate a standard quantum circuit with your construction above, you are able to output the same measurement statistics.
Thus, you have proven that complex amplitudes are never actually needed for quantum computing speedups — positive and negative real amplitudes suffice.
c) [2 points] To illustrate your construction, show how the phase gate, defined by P |0⟩ = |0⟩ and P |1⟩ = i |1⟩, gets converted into a purely real gate in your simulation. Show that it correctly transforms an arbitrary single-qubit state.
3. Universal gate sets [10 points] Identify the following gate sets as either universal or not uni- versal. If it is not, briefly argue why (if it is, then you don’t have to give an argument). Note that when we say “universal gate set”, we are referring to approximate universality: the set is able to approximate any target unitary to any desired precision, without ancilla.
Recall: P = [ 1 0 ]. 0i
a) {All single qubit gates, CNOT} b) {Toffoli, Hadamard}
c) {Toffoli, P}
d) {Toffoli, P, Hadamard}
e) {Hadamard, P, Controlled Z}
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