# Why Google has used $\sqrt{X}$ and $\sqrt{Y}$ instead of $X$ and $Y$ in supremacy experiment?

In supremacy experiment Google has used $$\sqrt{X}$$ and $$\sqrt{Y}$$ as two of their single qubit gates (paper).

So My questions are:

• Is there any specific reason for choosing these gates and not $$X$$ and $$Y$$ instead?
• Is it because of technical issues like the natural choice for transmon qubits are these gates?
• Or is it about complexity of the algorithm that is needed to simulate these gates?
• Without the $\sqrt{X}$ and $\sqrt{Y}$ gates, there's nothing creating any superposition. – Craig Gidney May 7 '20 at 22:24

While Craig Gidney (from Google) is correct in his comment which says that $$X$$ and $$Y$$ do not create superpositions on states that are not in superposition, such as $$|0\rangle$$ and $$|1\rangle$$; even if we assume that the initial state must not be in superposition, it is still possible to create superpositions with the 2-qubit gates, even if the 1-qubit gates are in fact $$X$$ and $$Y$$ (without the square roots).

It would still be a perfectly fair question to ask your exact same question, but for 2-qubit gates instead of 1-qubit gates:

• Is there any specific reason for not using text-book gates such as $$\textrm{CZ}$$ (controlled-Z) ?
• Is it because of technical issues, such as the natural choice for transmon qubits not being these gates?
• Or is it because of the complexity of the algorithm that is needed to simulate these gates?

These are actually very, very good questions, and while the answer is subtle, the answer is in fact yes for all three questions.

I will start by pointing out that page 3 of the paper says:

The implementation of high-fidelity ‘textbook gates’ natively, such as CZ or iSWAP, is work in progress.

Therefore the chips cannot yet do the text-book gates CZ or iSWAP with the high-fidelities demonstrated in the experiment to claim quantum supremacy.

The details about the reason for this, are buried all the way down to pages 15-16 of the Supplementary Information document. They say:

For quantum supremacy, the two-qubit gate of choice is the iSWAP gate. For example, CZ is less computationally expensive to simulate on a classical computer by a factor of two [37, 49]. A dominant error-mechanism when trying to implement an iSWAP is a small conditional-phase that is generated by an interaction of the |11>-state with higher states of the transmons (|02> and |20>). For this reason, the fSim gate with swap-angle θ ~ 90◦ and conditional phase φ ~ 30◦ has become the gate of choice in our supremacy experiment. gates. These gates result from the natural evolution of two qubits making them easy to calibrate, high intrinsic fidelity gates for quantum supremacy.

## So now here are the answers to the three questions:

• Yes, there is a specific reason why they didn't implement the $$\textrm{CZ}$$ gate:
• Yes, one reason is because of the complexity of the algorithm needed to simulate these gates: The $$\textrm{iSWAP}$$ gate happens to be two times harder to simulate on a classical computer than the $$\textrm{CZ}$$ gate. They do not give the specific numbers for how long the classical simulation would take if $$\textrm{CZ}$$ were used, but you can imagine that they might have to say that the classical computer would have to take only 5000 years (rather than 10,000 years). If you believe IBM's estimate, that using not only the RAM but also the HDD space on the Summit supercomputer, would allow the classical calculation to finish in 2.5 days rather than 10,000 years, then this could be reduced down to maybe 1.25 days if using the CZ gate (still significantly more than the 200 seconds it took Sycamore, but the calculation would be exact rather than merely a quantum calculation sampled one million times). So yes, they deliberately wanted to use iSWAP instead of CZ for a complexity reason, which is that the classical algorithm against which they were comparing in order to claim quantum supremacy, is 2 times slower for iSWAP than it is for CZ.
• Yes, there is also a technical issue which caused them not to implement the iSWAP gate exactly, but instead what they call a "partial-iSWAP gate". The most natural way for them to implement an iSWAP gate is partially "polluted" by higher excited states which they label $$|02\rangle$$ and $$|20\rangle$$, so they instead implement a "partial-iSWAP" gate. This is in a sub-section of the Supplementary Information called "The natural two-qubit gate for transmon qubits" which is exactly what you suggested in your question, so you seem to have been very correct in your thinking!

Furthermore, I personally wouldn't call CZ or iSWAP "text-book" gates either. The 2-qubit gates most people are familiar with are the CNOT and the SWAP gates. They never mention why they can't do SWAP instead of iSWAP, but they do say that CNOT would require at least 3 of their "natural" gates instead of just 1. This would likely not affect the competing classical computer, as the bottleneck there is RAM and storage space, and a CNOT (containing only real elements) can be implemented far easier on a classical computer than iSWAP (which requires doing more complex/imaginry-number arithmetic), but it would badly affect the quantum computer's performance. They say the gate times were set to 12ns because any slower and there would be more decoherence and any faster and there would be leakage into the higher states, and who knows what consequences that would have on the whole experiment. If they have to do 3 times as many gates, they might have to have the gate times set to 4ns instead of 12ns to get similar levels of decoherence, and if the effect is linear this would mean 3 times as much leakage into excited states. No matter how you think of it (allow more decoherence, or cause more leakage into excited states) the performance of the quantum computer would be badly affected by switching from partial-iSWAP to CNOT.

In any case, the point of the paper still stands no matter what: A classical computer cannot simulate a 50-qubit system easily. We already knew this since the dawn of quantum mechanics in the 1920s, and we know that we also can't simulate a 50-spin Ising Hamiltonian or D-Wave's 2000-spin Ising Hamiltonian either. The difference is that Google's 53-qubit system can do 1-qubit and 2-qubit gates (for an extremely limited subset of the $${ 53 \choose{2}}$$ possible qubit pairs, but at least they can apply up to four 2-qubit gates for each of the qubits) which would be required for the most general quantum algorithms like Shor's algorithm. Google's quantum computer still can't do Shor's algorithm on any number faster than the computer in a watch could factor it, and we're at least several years away from that type of accomplishment, but they made a 53-qubit machine that can do fairly impressive (by today's standards) 2-qubit gates on each of them, and no one had done that before.

• The factor of two difference between the iswap and the CZ is a compounding factor. When you attempt to cut the computation in two, you gain a factor of 4 for each iSwap you cut and a factor of 2 for each CZ you cut. If CZ gates were used, the circuit would have had to be twice as deep to achieve the same difficulty. – Craig Gidney May 11 '20 at 11:06
• Yes, way more than twice as long. More like 2^(depth*ops_on_min_cut_per_layer) ~= 2^(20*7/4) times as long. Tensor network simulation strategies involve treating the circuit diagram as a tensor network diagram, and finding a minimal cost set of edges to cut that separate the tensor network into two pieces. iSwap edges are more expensive to cut than CZ edges. – Craig Gidney May 11 '20 at 17:37
• Thank you for the reply! So basically, the classical simulation of the supremacy experiment would have been 34359738368 times faster if CZ gates were used, meaning that 10,000 years would become 9 seconds, which is far faster than the 200 seconds it took for the 53 qubits? I appreciate that the QC was doing a partial-iSWAP not a true iSWAP, so the speed-up wouldn't necessarily be the same, but it was still much closer to iSWAP than CZ. – user1271772 May 11 '20 at 17:55
• On tensor network part: I didn't recall seeing tensor networks being mentioned in my brief glance at the paper and supplementary material (I was simply looking for the parts necessary, to answer this question, as I remembered from months ago that a different gate from CZ was preferred due to CZ being 2x easier to simulate on a classical computer, so thought I could at least answer that part of the question). I remember seeing in the paper last night, there was a Schroedinger solver up to N and then Sch-Feynman solver for >N. Maybe the cuts have to do with the pieces used for the path integral. – user1271772 May 11 '20 at 17:56
• Right. It may have been phrased in Sch-Feynman terms, but the same constraints apply. And yes, the fact it wasn't a pure iswap reduces the advantage somewhat. Also, since an iswap can be decomposed (roughly) into swap*cz, and those swaps can be replaced by relabeling in the classical simulation, it was important to check that the CZ-using tensor network you get when relabelling had a much larger minimum sized cut. – Craig Gidney May 11 '20 at 17:59