How to synthesize quantum circuits?

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SUMMARY

This discussion focuses on synthesizing quantum circuits, specifically converting matrix representations into gate lists. The user seeks to build a circuit for the exclusive OR operation on three qubits, utilizing controlled NOT gates to achieve the desired output. The conversation highlights the importance of understanding quantum circuit synthesis, referencing quantum computing textbooks and the Quirk quantum circuit simulator as valuable resources for further exploration.

PREREQUISITES
  • Understanding of quantum circuits and qubits
  • Familiarity with controlled NOT gates and their applications
  • Knowledge of matrix representations in quantum computing
  • Basic concepts of truth tables in quantum operations
NEXT STEPS
  • Study quantum circuit synthesis techniques
  • Learn about the Quirk quantum circuit simulator
  • Explore methods for converting multi-control NOT gates into single-control operations
  • Read quantum computing textbooks focusing on circuit design and synthesis
USEFUL FOR

Quantum computing enthusiasts, researchers, and developers interested in circuit design and synthesis, particularly those working with qubits and gate operations.

nashed
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I wonder if anyone can explain how to actually synthesis a quantum circuit, what I mean by this is that given some requirements, I am able to write the matrix form of the circuit needed, but I have no idea how to turn that into a list of gates.

For example:
Build a circuit that performs the exclusive or operation on three qubits $$ |abc\rangle \rightarrow |ab\rangle\otimes|a\oplus b\oplus c\rangle $$ looking at a truth table for the circuit I can see that when a and b are the same then c is conserved, otherwise c is flipped so I can deduce that if I connect c to a controlled not where a is the control and another controlled not where b is the control I get the desired operation.
That said if I try and write the matrix form of the circuit I have no idea how to manipulate it so it looks like the repeated application of controlled not gates.

BTW this is not a homework question but I am not sure this is the correct forum either so feel free to tell me and I"ll delete the post.
 
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This kind of information is covered in quantum computing textbooks, and those might be your best bet.

Other than that, you could try reading this blog post about cutting down a NOT with many controls into single-control NOTs and single-qubit operations.

You will probably also find my drag-and-drop quantum circuit simulator Quirk very useful. For example, I bet the truth table you have looks quite a lot like the grid on the right:

Screenshot from 2017-02-27 15:32:56.png
 

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