# 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 $$R_x$$, $$R_y$$ and $$R_z$$.

The answer by Bertrand Einstein IV is the correct answer to the question as asked - if you only have single-qubit rotations and no entangling gate, you cannot create an entangling gate.

However, we can use the single qubit gates $$R_X$$ and $$R_Y$$ to create a $$T$$ gate and a Hadamard. These, combined with controlled-not give a standard construction for Toffoli. For example, the sequence $$R_Y(-\pi/2)R_X(\pi/4)R_Y(\pi/2)=e^{i\pi/4}T$$ while $$R_X(\pi)R_Y(\pi/2)=-i H.$$ Remember that global phases are irrelevant.

The $$T$$ gate as well as all possible single qubit rotations are non-entangling operations. That means if we have a circuit composed of single bit rotations, any non-entangled $$n$$-bit input, it will result in a $$n$$-bit non-entangled output.

The $$CCNOT$$ gate however is of course entangling, really all controlled gates are entangling, which means there exist non-entangled inputs to the gate which will result in entangled outputs. This is quite easy to show for the $$CNOT$$ and by extension the $$CCNOT$$.

You should now be able to deduce the answer to your question. The Toffoli Gate needs to be entangling, and rotations can never do that; hence we cannot build a Toffoli gate using the gate set proposed in the question.

• Thank you. But what gate set can we choose to reduce the gate complexity other than the actutal decompostion of the Toffoli gate using T gate. This question was asked in a seminar and they asked us to create the toffoli gate with or without T gate but the circuit should be different than actual decomposition. I tried but i am not able to reach to the solution. Can you please guide me in this. Sep 25, 2021 at 7:13

$$T$$ gate is defined as $$T= \begin{pmatrix} 1 & 0 \\ 0 & \mathrm{e}^{i\frac{\pi}{4}} \end{pmatrix},$$ and $$Rz(\theta)$$ is $$Rz(\theta)= \begin{pmatrix} \mathrm{e}^{-i\frac{\theta}{2}} & 0 \\ 0 & \mathrm{e}^{i\frac{\theta}{2}} \end{pmatrix}$$ If we factor out $$\mathrm{e}^{-i\frac{\theta}{2}}$$ we get $$\mathrm{e}^{-i\frac{\theta}{2}} \begin{pmatrix} 1 & 0 \\ 0 & \mathrm{e}^{i\theta} \end{pmatrix}.$$ So, setting $$\theta = \pi/4$$ we get $$T$$ gate up to a global phase. This means that with $$Rz$$ gate we can implement $$T$$. Note that set of $$Rx$$, $$Ry$$ and $$Rz$$ gates is universal for single-qubit gates, i.e. it allows to costruct any single-qubit gate including $$T$$ gate.

To construct Toffoli gate we also need CNOT gate. CNOT together with the rotations above allows to implement any unitary operation on a quantum computer.

• Thank you. But what gate set can we choose to reduce the gate complexity other than the actual decomposition of the Toffoli gate using T gate. This question was asked in a seminar and they asked us to create the toffoli gate with or without T gate but the circuit should be different than actual decomposition. I tried but i am not able to reach to the solution. Can you please guide me in this. Sep 25, 2021 at 12:18
• @MinhPham: Did you mean to react on my comment or Tarun Kumar's? If the second option is right, please start your comment with @ followed by name of the user you want to communicate with. Sep 29, 2021 at 15:57
• Well what is the "actual" decomposition you are referring to, because there are a few decompositions that I know off. @TarunKumar Sep 29, 2021 at 17:03

You can decompose the T gates themselves to create a Toffoli Gate. Here is one way of doing this:-

You can refer to this Qiskit chapter if you are interested and want to understand gate decomposition: https://qiskit.org/textbook/ch-gates/more-circuit-identities.html