Undergrad Help in understanding a quantum circuit

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The discussion centers on simulating quantum circuits, specifically focusing on the Hadamard gate's effect on the "C" entrance in a referenced paper. It is clarified that when the state is either off or on, the Hadamard gate functions like a classical coin flip, as it does not create superposition in this scenario. One participant finds the second state, ##S2##, to be irrelevant. For practical understanding, using the Quirk simulator is suggested to visualize specific cases of the circuit. Overall, the conversation aims to clarify the role of the Hadamard gate in quantum circuit simulation.
victor94
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I'm in a proyect to simulate quantum circuits in robots like in this paper
( http://ieeexplore.ieee.org/document/4215941/ ) ,the first thing that i need to do is to simulated the circuit that is in that paper:

upload_2017-3-29_15-15-14.png

But I'm having trouble understanding how the hadamard gate affects the "C" entrance, any help will be appreciated.

Thanks
 
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In this case, assuming ##S1## is off or on (i.e. not in a superposition w.r.t. the computational basis), the Hadamard gate just acts like a classical coin flip because its output is only ever measured or used to control other operations.

##S2## seems completely pointless as far as I can tell.

You can simulate the circuit in Quirk to see the specific cases:

circuit-cycle.gif
 
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Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?

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