11_exercise_compilation_verification.pdf

Problem 4 Compilation of Quantum Circuits (31 credits) Consider the following quantum computer architecture A : Q4 Q0 Q3...

Problem 4 Compilation of Quantum Circuits (31 credits) Consider the following quantum computer architecture A : Q4 Q0 Q3 Q5 Q1 Q2 0 a) Which of the following pairs of qubits can natively interact on this architecture? 1 2 3 (1, 3) (2, 5) (3, 4) A(1, 2) (0, 5) (0, 2) b) Which of the following circuits can be mapped to the architecture A without adding any SWAP gates? 0 1 q3 q2 q3 q3 2 q2 q1 q2 q2 3 q1 q0 q1 q1 4 q0 q0 q0 5 6 To this end, build an interaction graph out of all gates for each circuit. Then try to fit that graph to the 7 8 architecture. I IQ III 0 II I 0 0 0 0 LEE.name Suberchitecture 011 8 0 0 59 and 89 0 0 059 and 0 0 0 0 – Page 6 / 12 – c) Consider the circuit(s) from Problem b) that could not be perfectly mapped to the architecture. Determine 0 a suitable mapping for these circuits by choosing an initial layout and inserting SWAP gates to satisfy the 1 restricted connectivity. 2 3 4 5 IE mad 6 IE EEEE iEI e a 91 c Gaas L ÉÉ go E 91 an co qX 93 Q3 9227 Cz 90400 II 14T FI I Qg d) Can the mapping of any of these circuits be improved by considering more than four qubits on the 0 architecture? If yes, show the improved mapping. If no, give an argument why. 1 2 3 273 4 5 6 1 f 9 3 If 0.01 2 get Qr It Cz 9144 a II 9H00 of co 11641 garbage PG titi – Page 7 / 12 – 0 e) Given the following quantum circuit G : 1 2 q1 H H H 3 4 q0 X H Z Z H X 5 6 Your goal is to simplify the circuit as much as possible based on the following quantum circuit identities: 7 8 I H X Z H X X H Y Y H Y Y H Z X H © Z Z S X © X S† © H H Z Z H H S Y Y S† © H H S S† S Z S† S† S S† Z S H H X X Assume that a CNOT gate has cost 10, while all single-qubit gates have cost 1. What is the minimal cost for realizing G ? Make sure to document every step in your simplification. 22 11 31 13 I EIEIEIIE.EE IIEE E EIEIT.IN II E Et.E I EEE7d El EATE EKITI II n To EEI.iq td – Page 8 / 12 – LET To – Page 12 / 12 –  Problem 5 Verification of Quantum Circuits (14 credits) a) Verify the equivalence of the following two circuits G and G Õ by showing that both circuits act the same on 0 all possible computational basis states |i Í = |000Í ,... , |111Í. 1 2 q2 q2 3 q1 q1 4 q0 q0 5 6 7 Remember that the CNOT gate flips the state of the target qubit whenever the control qubit is in state |1Í. 8 Input G GÕ |q2 q1 q0 Í g0 g0Õ g1Õ g2Õ g3Õ |000Í 10007 10007 10007 10007 10007 |001Í 10077 10017 10017 10017 10017 |010Í 10107 1011710117 10 1 0710 107 |011Í 1011 10107 10107 1011710117 |100Í 11017 11007 11107 |101Í 1111711 017 1100 1107 11171110711 007 |110Í 11117 11117 10171101711 117 |111Í 1110711107 110071100711 107 000 on one on no ton the inn 8 0 0 0 0 0 0 0 a o o 0 0 1 0 0 0 0 o 1000 1 11 0 0 0 0 0 0 u ii ns 0 0 0 0 1 0 0 0 IN 0 0 0 0 0 0 0 0 0 A 0 – Page 9 / 12 – – Page 12 / 12 –