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You will now show that we can always convert quantum circuits with internal measurements into quantum circuits where all measurements occur at the end. This is nice because, as we know, a circuit without measurements just corresponds to multiplication by a unitary matrix. So suppose you have an n-qubit quantum circuit that does some internal measurements, and suppose the first measurement is at step t. It measures the state of the kth qubit in the standard basis and outputs a classical bit b = {0, 1} and the corresponding state collapses to the outcome. Now make two changes: 1. Introduce an (n + 1)st qubit to the circuit 2. Replace the measurement on qubit k by a CNOT gate whose control is the kth bit and the target bit is the new (n+1)st qubit. Solve the following: (i) Show that this exactly simulates the original circuit's operation. (ii) Explain how this gives a way of taking any quantum circuit and converting it into one where all measurements occur only at the final step. (The idea is that you want to be able to reproduce all the values output by the internal measurements in the circuit, using measurement gates that only happen at the end.)

You will now show that we can always convert quantum circuits with internal measurements into quantum circuits where all measurements occur at the end. This is nice because, as we know, a circuit without measurements just corresponds to multiplication by a unitary matrix. So suppose you have an n-qubit quantum circuit that does some internal measurements, and suppose the first measurement is at step t. It measures the state of the kth qubit in the standard basis and outputs a classical bit b = {0, 1} and the corresponding state collapses to the outcome. Now make two changes: 1. Introduce an (n + 1)st qubit to the circuit 2. Replace the measurement on qubit k by a CNOT gate whose control is the kth bit and the target bit is the new (n+1)st qubit. Solve the following: (i) Show that this exactly simulates the original circuit's operation. (ii) Explain how this gives a way of taking any quantum circuit and converting it into one where all measurements occur only at the final step. (The idea is that you want to be able to reproduce all the values output by the internal measurements in the circuit, using measurement gates that only happen at the end.)

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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You will now show that we can always convert quantum circuits with internal measurements into quantum circuits where all measurements occur at the end. This is nice because, as we know, a circuit without measurements just corresponds to multiplication by a unitary matrix. So suppose you have an n-qubit quantum circuit that does some internal measurements, and suppose the first measurement is at step t. It measures the state of the kth qubit in the standard basis and outputs a classical bit 6 € {0, 1} and the corresponding state collapses to the outcome. Now make two changes: 1. Introduce an (n + 1)st qubit to the circuit 2. Replace the measurement on qubit k by a CNOT gate whose control is the kth bit and the target bit is the new (n + 1)st qubit. Solve the following: (i) Show that this exactly simulates the original circuit's operation. (ii) Explain how this gives a way of taking any quantum circuit and converting it into one where all measurements occur only at the final step. (The idea is that you want to be able to reproduce all the values output by the internal measurements in the circuit, using measurement gates that only happen at the end.)
Transcribed Image Text:You will now show that we can always convert quantum circuits with internal measurements into quantum circuits where all measurements occur at the end. This is nice because, as we know, a circuit without measurements just corresponds to multiplication by a unitary matrix. So suppose you have an n-qubit quantum circuit that does some internal measurements, and suppose the first measurement is at step t. It measures the state of the kth qubit in the standard basis and outputs a classical bit 6 € {0, 1} and the corresponding state collapses to the outcome. Now make two changes: 1. Introduce an (n + 1)st qubit to the circuit 2. Replace the measurement on qubit k by a CNOT gate whose control is the kth bit and the target bit is the new (n + 1)st qubit. Solve the following: (i) Show that this exactly simulates the original circuit's operation. (ii) Explain how this gives a way of taking any quantum circuit and converting it into one where all measurements occur only at the final step. (The idea is that you want to be able to reproduce all the values output by the internal measurements in the circuit, using measurement gates that only happen at the end.)
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