RT @Nahuel_L_Diaz: New work on Quantum resource theories! . A big thanks to my collaborators @QuAntonioMele @Berm….

RT @MvsCerezo: 🚨Applications for LANL's 2025 Quantum Computing Summer School are open!. Please apply here.👇. Repost….

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RT @BermeQP: Huge thanks to @TheImPolster @QuantumManuel @qZoeHolmes @LCincio @MvsCerezo!!! . 👇👇Our work at a glance 👇👇.

However, while some of their techniques are similar to the ones we use, the extension to local orthogonal or free-fermionic gates introduced in our work is, to our knowledge, completely novel.

Lastly let me acknowledge which appeared on the ArXiv a few days before our manuscript. There the authors introduce a method for treating the P-net as a TN.similar to ours.

Let me sincerely thank @RobertHuangHY, @MartinLaroo, @DiegoGM_quantum, and Francesco Caravelli for useful discussions 🙏.

The MPS representation of the back-propagated measurement operator also opens up a new dimension of analysis, namely we can access entropic entanglement properties of the random circuits!.(Here are some of these properties for the HEA, should we cite Tolkien?)

For comparison, the same calculations with MC methods converge to our exact result in supra-exponential (in the accepted Kullback-Leibler divergence) number of shots! .(QCNN on top, HEA at the bottom)

And in case you wondered about what happens for deep HEAs, somebody's favorite ansatz, here you can watch it converge to a design after a number of layers roughly equal to n/2. (n=200, hence at the end we are dealing with almost 40k gates)

We call k-purities the sum of the squared overlaps between the back-propagated measurement operator and all the (normalized) Paulis with bodyness k. This is what their average over a random 1264-qubit QCNN looks like (yes, there are quite a few Paulis contributing to each bar)

Btw, don't try computing the following by sampling the gates and running the resulting circuit instance to gather statistics, you would only get an approximate result and would probably melt your machine. (I almost did that to Chicoma, our HPC cluster)

Let me show you some cool results we can obtain when the gates are sampled from the fundamental rep of U(4) (standard VQA ansatze with two qubits gates fall in this category). Ah, did I say this computations are classically efficient?.

We study the P-gates dimension, showing that they are well behaved for gates sampled from the fundamental reps of the standard U, O and Sp groups, and even for the free-fermionic rep of O. We also provide bounds for the bond dimension of the MPSs arising from deep circuits.

The measurement operator and the input state are conveniently described as MPSs whose physical legs (as many as the qubits the circuit acts on) take values in the "local" bases of the t-th order gates' commutants (assuming you want to compute the t-th moment of the circuit).

While standard MC methods require sampling a Markov chain driven by process matrices associated to each Haar random gate in the circuit, we slap those matrices (P-gates in our paper) in a TN and use the latter to back-propagate the measurement operator at the end of the circuit.

New work out! .Huge thanks to P. Bermejo, @LCincio, @MvsCerezo. Do you need to compute exact moments of quantum circuits composed of local random gates? Come join the Tensor Networks side!. 🧵 ⬇️.