# What is a gate-level circuit used in the 2022 Jafferis et al. experiment on Sycamore?

A recently published Nature paper of Jafferis et al. describes an experiment with a handful of qubits performed on Google's Sycamore processor to explore the SYK model in the context of AdS/CFT and quantum gravity, and states that the experiment "represents a step towards a program for studying quantum gravity in the laboratory".

The figures in the Nature article appear to present a high-level summary of the experiment but not at the level of gates acting on the particular topology of Sycamore. Figure 1c of Jafferis et al. appears to provide a standard wormhole-inspired teleportation circuit diagram, with time advancing upwards.

Figure 1 in an earlier counterpart paper of Shapoval et al., as below, also provides a high-level summary of a wormhole-inspired teleportation protocol - this appears to be almost the same as in Figure 1c of Jafferis et al. with time advancing to the right.

In step 0, the Left and Right qubits are created as Bell pairs and correspond to the mouths of a wormhole. In step 1, the Left black hole scrambles and evolves backwards in time until the target qubit $$|\psi\rangle$$ is introduced in step 2 while then evolving forward in time in step 3; in step 4 the wormhole gets injected with a negative-energy Hamiltonian $$V$$; while in step 5, by time-evolving the Right black hole the qubit $$|\psi\rangle$$ is teleported.

Additionally, Figure 4 in Shapoval et al. gets into the weeds of individual gates, and how they might be implemented on IBM's transmon devices and/or Quantinuum's trapped ion devices.

We know that the native 2-qubit gates used in earlier Sycamore experiments are related to iSWAP gates. The Jafferis et al. experiment reportedly was done on a 72-qubit version of Sycamore but only uses nine of these qubits and apparently involves several hundred 1- and 2-qubit gates.

What was the particular circuit used in this experiment, preferably at the level provided by the other Sycamore experiments or at the level of Shapoval et al.'s proposal?

For example, were the other qubits on Sycamore Idling, or were they doing any other supportive ancillary work (such as SWAPping info and out)? Did the circuit use these iSWAP gates, or something more subtle? How deep was the circuit? Apparently an important aspect of the experiment involved classical machine-learning to simplify and sparsify the Hamiltonian - did this machine learning characterize other qubits not used in the circuit, to know which ones not to use?

• If it helps, I can add that they use 164 CZ gates and 295 single-qubit gates. The sublattice they use is almost a 3x3 square lattice, but the bottom-right qubit is actually one unit to the right of the middle-right qubit -- I believe these are all the qubits used. Also, since they are simulating fermions they would need to use fswaps, not normal swaps. I don't see any number for total circuit depth but they say that N=10 SYK model is "prohibitively deep", so they sparsify it to a N=7 model. Dec 3, 2022 at 6:47
• @chrysaor4 that does help! Was this in the paper itself? Dec 3, 2022 at 16:20
• The paper can be accessed from this link. Also, rather than duplicate the circuit, what does it take to simulate the 7 fermion "learned" hamiltonian in the paper: $H_{L,R}=-0.36\psi^1\psi^2\psi^4\psi^5+0.19\psi^1\psi^3\psi^4\psi^7-0.71\psi^1\psi^3\psi^5\psi^6+0.22\psi^2\psi^3\psi^4\psi^6+0.49\psi^2\psi^3\psi^5\psi^7?$ Dec 4, 2022 at 23:01
• This doesn't answer your question, but you actually can contact the authors through their email listed on the paper and ask this question. If you want them to respond, I suggest you write no more than two sentences and be very clear in your writing Dec 7, 2022 at 4:00
• Reference 9 in the Nature paper is Gao and Jafferis 2021. It also talks about the gates at some level of detail: arxiv.org/abs/1911.07416 Dec 8, 2022 at 9:03