# Questions tagged [gate-synthesis]

For questions about finding (short) gate sequences to implement a specific unitary operation, for example decomposing a complicated multi-qubit gate into a sequence of basic gates. It might apply to optimizing circuits with respect to length or depth or finding gate sequences to implement an algorithm.

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### Optimal decompositions of some standard multi-qubit gates

To have a concrete example in mind: 3-qubit Toffoli gate can be decomposed into 6 $CNOT$s as shown here I believe this is the most economic decomposition in terms of the number of $CNOT$s used. My ...
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### How is a Toffoli gate built without using T gates?

Can someone tell me how to make a Toffoli gate without using T gates? Can we use $R_x$ and $R_y$. If yes, then how? I tried many circuits but I was unable to create the CCNOT gate out of $R_x$, $R_y$ ...
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### How to derive the CNOT matrix for a 3-qubit system where the control & target qubits are not adjacent?

In a three-qubit system, it's easy to derive the CNOT operator when the control & target qubits are adjacent in significance - you just tensor the 2-bit CNOT operator with the identity matrix in ...
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### What is Qiskit's Transpiler method for unitary synthesis?

As I could found in here how the transpile works in qiskit, I understood that transpile gets arbitrary Unitary gate $U$ and some set of basis gates as input, and produce some quantum circuit of $U$ ...
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### How to construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift?

How can I construct a controlled-Hadamard gate using single qubit gates and controlled phase-shift? I am stuck in this and any help would be appreciated.
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### Sampling random circuits vs Solovay-Kitaev compiler

Suppose I want to obtain a gate sequence representing a particular 1 qubit unitary matrix. The gate set is represented by a discrete universal set, e.g. Clifford+T gates or $\{T,H\}$ gates. A well ...
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### Compiling Quantum Circuits using the Palindrome Transform

This paper shows a way to produce optimal circuits. I haver verified most of them and they are correct except this procedure: procedure ProduceArray(n) I cannot ...
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### Representation of multiple qubit matrices in Dirac notation

Imagine one wants to represent the and function for any number of qubits in Dirac notation. The and gate flips the target qubit if all the control qubits are in state 1. This is its matrix ...
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