Quantum Information, Probability and Computation
Hello, I need an urgent assistance for my final exam which is going to start at 1:30 pm EST. The duration of the exam is 2 hours and 15 minutes. The syllabus of the course:: Extended introduction and overview of the field; linear algebra fundamentals; postulates of quantum mechanics; quantum probability models, quantum circuits and gates; entanglement, teleportation, super-dense coding and Bell’s inequality; quantum computation and algorithms.I'm also adding previous year exams to the attachments. Looking for help, thank you!
State TRUE or FALSE by giving reasons [8 points each] You must state a correct reason to get credit. (a) Consider the quantum state |ψ〉 = 1√ 3 |00〉+ √ 2√ 3 |11〉. This is an entangled state. (b) Consider the quantum state |ψ〉 = 1√ 2 |+−〉 − 1√ 2 |−+〉, where |+〉 = 1√ 2 [|0〉+ |1〉] and |−〉 = 1√ 2 [|0〉 − |1〉]. A (observable) measurement given by H ⊗ Y is performed on this state, where H is Hadamard operator and Y is Pauli-Y operator. Let X denote the classical outcome. Then E(X) = 12 . (c) Consider the quantum state |ψ〉 = 1√ 3 |0〉+ √ 2√ 3 |1〉. Consider the measurement performed on this state given by observable Pauli-Z operator. Let X denote the classical output. Then Var(X) = 34 . (d) Consider the observable ~n · ~σ = n1σ1 + n2σ2 + n3σ3, where σi is the ith Pauli operator. This measurement is performed on the state |1〉. Then the probability of getting outcome equal to 1 is 14 . (e) Consider an operator A that acts on two qubits as follows: A |++〉 = |00〉, A |+−〉 = |01〉, A |−+〉 = |11〉 and A |−−〉 = |10〉. Then, in the computational basis, the operator A is given by 1√ 2 −1 0 −1 0 1 1 1 0 0 1 1 −1 1 0 −1 0 . (f) Consider the following quantum circuit shown in Figure 1, where f(x1, x2) = x1 ∧ x2. The input is initialized to |001〉. Then the output is given by Figure 1: 1 2 [|00〉+ |01〉+ |10〉 − |11〉]⊗ |−〉 . 2 (g) Consider Grover’s algorithm with n = 2. Consider the gate A stated in the quantum algo- rithm. The quantum circuit shown in Figure 2 implements this gate, where the input (ancilla) to the Z-gate is initialized to |1〉. Figure 2: (h) Consider a quantum state given by the density operator ρA, where ρA = 1 8 [ 4 −3 −3 4 ] . Then a purification of ρA is given by |ψ〉AB = 1 2 |++〉+ √ 3 2 |−−〉. (i) Consider the quantum state |ψ〉 = 1√ 2 |01〉 − 1√ 2 |10〉. The Schmidt number of this state is 1. (j) Consider a quantum state given by the density operator ρA given in Q.1(h). Consider two observables E and F each with two classical outcomes (eigen values) 0 and 1. Suppose the measurement E is performed first, and then measurement F is performed, where E = |0〉 〈0| and F = |+〉 〈+|. Then the probability of observing 1 on the first and 1 on the second measurements is 23 . (k) Consider a bit-flip channel where the probability of flipping is p = 0.1. A quantum state ρA is fed to the channel, where ρA is given in Q.1(h). The output density operator is given by 1 8 [ 3 −2 −2 5 ] . (l) Let α be a positive integer with partial fraction expansion [0, 2, 2, 3, 6, 5]. The third convergent of α is given by 13 . 3 State TRUE or FALSE by giving reasons [7 points each] You must state a correct reason to get credit. (a) Consider 3 operators Mi = √ 2 3 |vi〉 〈vi|, for i = 0, 1, 2, where |v0〉 = |0〉 , |v1〉 = − 1 2 |0〉+ √ 3 2 |1〉 , |v2〉 = − 1 2 |0〉 − √ 3 2 |1〉 . Then one can define a measurement based on these 3 operators, i.e., they satisfy the com- pleteness relation. (b) Consider a quantum state |ψ〉 = 14 |00〉+ √ 3 4 |01〉 − √ 3 4 |10〉 − 3 4 |11〉. |ψ〉 is an entangled state. (c) Alice and Bob share a pair of entangled qubits given by 1√ 2 |01〉 − 1√ 2 |10〉. They perform the measurement given by observable A = Y ⊗ Z, where Y and Z are Pauli operators. Let X denote the classical output. Then E(X) = 14 . (d) Let |ψ〉 = |0〉 be a quantum state. Consider two observables given by Y and Z, where Y and Z are Pauli operators. Let σ2X and σ 2 Y denote the variances of the classical outputs from the two observables, respectively. Then σXσY ≥ 1. (e) Consider a quantum circuit acting on two qubits with two gates: CNOT gate followed by H gate on the top line. Then the unitary transformation corresponding to this circuit is given by 1√ 2 1 0 1 0 0 1 0 1 0 1 0 −1 1 0 −1 0 . (f) Consider Grover’s algorithm with a toy example n = 2 and f(x) = x1 ∧ x2, i.e., the logical AND operation. Then the Uf gate used in Grover’s algorithm satisfies: Uf |101〉 = |111〉. (g) The set of gates given by (a) C(n−1)-NOT gates, (b) X-gates, and (c) C(n−1)-U gates (U is single-qubit unitary operation) is universal. (h) Consider a quantum state given by a density operator ρ. Consider a measurement given by {Mα1 , . . . ,MαK}. Let X denote the classical output. Suppose X = αi is observed. Then after the observation, the state of the quantum system is MαiρM † αi tr [ MαiρM † αi ] . 2 (i) Consider a quantum state given by a density operator ρ. Consider two observables A and B each with two classical outcomes (eigen values) 0 and 1. Suppose the measurement A is performed first, and then measurement B is performed. Then the probability of observing 1 on the first and 1 on the second measurements is tr(ρBAB). (j) Consider a quantum state with a density operator ρ. It is given that tr(ρ2) < 1. then the state is pure. (k) consider the quantum state of a pair of entangled particles ab given by |ψ〉ab = 1 2 |01〉 −√ 3 2 |10〉. let ρa denote the state of particle a. then ρa = [ 1 4 − √ 3 4 − √ 3 4 3 4 ] . (l) consider a bit-flip channel where the probability of flipping is p = 0.1. then the krause operators associated with this channel is given by e0 = (1− p)i, e1 = px where x is the pauli operator. (m) let x = 3 and n = 5. consider shor’s algorithm for finding the order of x mod n . consider the operator u given by u |y〉 = |xymodn〉 for y ∈ {0, 1, . . . , 4} and u |y〉 = |y〉 for y ∈ {5, 6, 7}. then the operator u has 4 distinct eigen values. 3 1.="" then="" the="" state="" is="" pure.="" (k)="" consider="" the="" quantum="" state="" of="" a="" pair="" of="" entangled="" particles="" ab="" given="" by="" |ψ〉ab="1" 2="" |01〉="" −√="" 3="" 2="" |10〉.="" let="" ρa="" denote="" the="" state="" of="" particle="" a.="" then="" ρa="[" 1="" 4="" −="" √="" 3="" 4="" −="" √="" 3="" 4="" 3="" 4="" ]="" .="" (l)="" consider="" a="" bit-flip="" channel="" where="" the="" probability="" of="" flipping="" is="" p="0.1." then="" the="" krause="" operators="" associated="" with="" this="" channel="" is="" given="" by="" e0="(1−" p)i,="" e1="pX" where="" x="" is="" the="" pauli="" operator.="" (m)="" let="" x="3" and="" n="5." consider="" shor’s="" algorithm="" for="" finding="" the="" order="" of="" x="" mod="" n="" .="" consider="" the="" operator="" u="" given="" by="" u="" |y〉="|xymodN〉" for="" y="" ∈="" {0,="" 1,="" .="" .="" .="" ,="" 4}="" and="" u="" |y〉="|y〉" for="" y="" ∈="" {5,="" 6,="" 7}.="" then="" the="" operator="" u="" has="" 4="" distinct="" eigen="" values.="">