# Is there a quantum processor with physically implemented Toffoli gate?

Recently, I came across the article Realization of efficient quantum gates with a superconducting qubit-qutrit circuit where its authors proposed a physical implementation of three qubits quantum gates with two physical qubits and one physical qutrit controlled by microwave pulses.

As a result, they were able to implement physically CCZ gate which when accompanied by two Hadamards implements Toffoli gate.

They also proposed configuration implementing Fredking gate, another three qubits gate which together with Hadamard gate form universal set.

Finally, they showed that their design can be used for implementation of $$CC-U$$ gate, where $$U= \begin{pmatrix} \cos(\theta) & \mathrm{e}^{i\varphi}\sin(\theta)\\ \mathrm{e}^{-i\varphi}\sin(\theta) & -\cos(\theta)\\ \end{pmatrix},$$ which is more or less similar to $$U3$$ gate on IBM Q (to be precise $$U(\theta, \varphi) = U3(2\theta,-\varphi,\pi+\varphi)$$).

As it seems that this proposal is able to cut lenght of circuit, particularly in case of Toffoli gates (implemented by physical CC-Z gate and two Hadamards only) and therefore enables us to implement more complex cirucits, I am curious if such design is being implemented or at least considered by companies like IBM or Google for their next generation of quantum processors. I highly appreciate any links.

• As far as I know, ion trap platforms can natively implement Toffoli gates ... However, I'm not an experimentalist and cannot give you good references on this. Maybe look in the direction of the Innsbruck, Oxford, Siegen, ... groups ? Nov 28, 2020 at 14:00
• I don't have a reference, but I remember talking with someone who worked with neutral atoms and they claimed that one of the benefits of neutral atoms was that there was a native CC..CCZ operation. That's effectively a Toffoli, modulo some trivial single qubit operations. Edit: arxiv.org/abs/2006.12326 mentions this. Dec 7, 2020 at 18:01

Although I can't find an online resource currently, I'm pretty certain that people in QuTech (Tim Taminiau and his group, specifically) are trying to physically implement a Toffoli gate in their NV-center systems. It's definitely on-going research though.

Both Google and IBM use transmons as their systems - entangling gates are implemented here using physical 'coupling' resonance cavities - as far as I known, it would in principle be possible to connect multiple (i.e. $$3$$) qubits, so that one would get a coupled Hamiltonian between $$3$$ qubits simultaneously. I, however, am not knowledgeable enough to judge the feasibility of this.

Not quite what you're asking, but...

With something like the IBM processors, if you have the right permissions, you can get "pulse level" control of the system. This means you can directly manipulate the Hamiltonian. In principle, this can give you multi-qubit gates directly, if only you can design your pulses correctly.

• I was curious if design proposed in the artile is implemented or considered to be implemented. Thanks for the answer, but I think that there is no qutrit in IBM Q processors. Nov 27, 2020 at 8:28
• OK, I was taking your question more generally - there's a variety of mechanisms by which you might imagine implementing a 3-qubit gate, and I assumed you were asking if there were plans for implementing any of them rather than the specific design you mentioned initially. Nov 27, 2020 at 9:14
• Wouldn't that just be a efficient compilation for the Toffoli into the gates that the IBM device can actually physically do? For a physical Toffoli implementation (or something 'LU'-equivalent to it) one needs to have intrinsic coupling between 3 different qubits - the transmons of the IBM devices only offer (tunable) coupling between pairs of two qubits.
– JSdJ
Nov 27, 2020 at 9:19
• @JSdJ No, in principle you can do things somewhat quicker than decomposing them into gates. Effectively, this is because you can use multi-qubit interference rather than two-at-a-time interference for the creation of gates. Nov 27, 2020 at 9:54
• @DaftWullie Ah, okay. I always thought that the transmon systems of IBM, Rigetti had only physical ('tunable') coupling between pairs of qubits, so that the pulse control only gave you control over these couplings (be it more detailed than the standard gates that are offered).
– JSdJ
Nov 27, 2020 at 10:04