- #1

Haorong Wu

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- Homework Statement
- Verify that the circuit implement the operation ##exp \left ( -i \left | \psi \right > \left < \psi \right | \Delta t \right )## with ##\left | \psi \right >=\frac {\sum _{x=0}^{N-1} \left | x \right >} {\sqrt N}##, and ##N## is the number of elements in a search space.

- Relevant Equations
- For small ##\Delta t##, ##exp \left ( -i \left | \psi \right > \left < \psi \right | \Delta t \right ) = I -i \Delta t \left | \psi \right > \left < \psi \right |##.

This is an exercise from "Quantum search as a quantum simulation " in Chapter "Quantum search algorithms".

The circuit is shown as the following picture.

For small time interval, the effect of the operation in the problem statement could be written as ## exp \left ( -i \left | \psi \right > \left < \psi \right | \Delta t \right ) \left | \psi \right >=\left ( I -i \Delta t \left | \psi \right > \left < \psi \right | \right ) \left | \psi \right >=\left ( 1-i \Delta t \right ) \left | \psi \right >##.

In the circuit, let the input qubits ##\left | y \right > = \left | \psi \right >##, since ##\left | \psi \right >## is defined to be the initial state and I can generate other states from it if the following reasoning is correct.

The qubits change as follow:

##\begin{align} & \left | \psi \right > \left | 0 \right > \nonumber \\ \rightarrow & \left | 000 \right > \left | 0 \right > \text{, (Hadamard Gates)} \nonumber \\ \rightarrow & \left | 000 \right > \left | 1 \right > \text{, (CNOT Gate)} \nonumber \\ \rightarrow & \left | 000 \right > e^{i \Delta t} \left | 1 \right > \text{, (U on the fourth qubit)} \nonumber \\ \rightarrow & \left | 000 \right > e^{i \Delta t} \left | 0 \right > \text{, (CNOT)} \nonumber \\ \rightarrow & \left | \psi \right > e^{i \Delta t} \left | 0 \right > \text{, (Hadamard Gates)} \nonumber \end{align} ##

Then the first three qubits would be in state ## e^{i \Delta t} \left | \psi \right > =\left ( I + i \Delta t \right ) \left | \psi \right >##, which does not match the operation in the problem statement.

Should the ##e^{i \Delta t}## in the circuit be ##e^ {-i \Delta t}##? But I think the circuit should be correct since the book has been in ##10^{th}## anniversary edition. So where did I make a mistake?

Thanks!

The circuit is shown as the following picture.

For small time interval, the effect of the operation in the problem statement could be written as ## exp \left ( -i \left | \psi \right > \left < \psi \right | \Delta t \right ) \left | \psi \right >=\left ( I -i \Delta t \left | \psi \right > \left < \psi \right | \right ) \left | \psi \right >=\left ( 1-i \Delta t \right ) \left | \psi \right >##.

In the circuit, let the input qubits ##\left | y \right > = \left | \psi \right >##, since ##\left | \psi \right >## is defined to be the initial state and I can generate other states from it if the following reasoning is correct.

The qubits change as follow:

##\begin{align} & \left | \psi \right > \left | 0 \right > \nonumber \\ \rightarrow & \left | 000 \right > \left | 0 \right > \text{, (Hadamard Gates)} \nonumber \\ \rightarrow & \left | 000 \right > \left | 1 \right > \text{, (CNOT Gate)} \nonumber \\ \rightarrow & \left | 000 \right > e^{i \Delta t} \left | 1 \right > \text{, (U on the fourth qubit)} \nonumber \\ \rightarrow & \left | 000 \right > e^{i \Delta t} \left | 0 \right > \text{, (CNOT)} \nonumber \\ \rightarrow & \left | \psi \right > e^{i \Delta t} \left | 0 \right > \text{, (Hadamard Gates)} \nonumber \end{align} ##

Then the first three qubits would be in state ## e^{i \Delta t} \left | \psi \right > =\left ( I + i \Delta t \right ) \left | \psi \right >##, which does not match the operation in the problem statement.

Should the ##e^{i \Delta t}## in the circuit be ##e^ {-i \Delta t}##? But I think the circuit should be correct since the book has been in ##10^{th}## anniversary edition. So where did I make a mistake?

Thanks!

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