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I have understood the concept of "universal quantum gates" in real quantum computer, and the quantum gates and states can be emulated as matrix in classical computer, and the matrix operation can be regarded as gate operation. In real quantum system, we can realize a set of universal quantum gates to decompose any unitary quantum gates to reduce the complexity.

So I want to know is if in a quantum emulator there is the same procedure of decomposition, or just matrix operations without any decomposition? Because from my perspective, if it can be emulated as a matrix operation, it seems there is no need to do the procedure of decomposition.

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In simulation, it is normally not needed to perform the process of decomposition - so your perspective is right, in principle.

Of course, it can be actually useful to perform the decomposition and synthesis, for various reasons, including but not limited to:

  • The gate decomposition algorithm itself needs to be simulated (Rather than the algorithm/computation).
  • Gate decomposition and synthesis might offer easier methods of circuit compilation, limiting or bringing down the number of operations that need to be simulated.
  • Simulations with noise often try to resemble some specific physical implementation; here the noise characteristics are only known/applicable to the gates from the specific gateset.
  • Simulations can sometimes be sped up/the memory usage can be brought down if all the entangling gates in the circuit are bundled together and/or pushed as far back in the circuit. Performing decompositions (of the multi-qubit gates, specifically) can help in determining the use-cases of this.
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