# Optimal compression of quantum many-body time evolution operators into brickwall circuits

### Submission summary

 As Contributors: Maurits Tepaske Preprint link: scipost_202205_00013v1 Date submitted: 2022-05-16 10:17 Submitted by: Tepaske, Maurits Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Condensed Matter Physics - Computational Quantum Physics Approaches: Theoretical, Computational

### Abstract

Near term quantum computers suffer from a degree of decoherence which is prohibitive for high fidelity simulations with deep circuits. An economic use of circuit depth is therefore paramount. For digital quantum simulation of quantum many-body systems, real time evolution is typically achieved by a Trotter decomposition of the time evolution operator into circuits consisting only of two qubit gates. To match the geometry of the physical system and the CNOT connectivity of the quantum processor, additional SWAP gates are needed. We show that optimal fidelity, beyond what is achievable by simple Trotter decompositions for a fixed gate count, can be obtained by compiling the evolution operator into optimal brickwall circuits for the $S=1/2$ quantum Heisenberg model on chains and ladders, when mapped to one dimensional quantum processors without the need of additional SWAP gates.

###### Current status:
Editor-in-charge assigned

### Submission & Refereeing History

Submission scipost_202205_00013v1 on 16 May 2022

## Reports on this Submission

### Report

In this article, Tepaske et al study methods to obtain optimal brickwork circuit for time evolution operators generated by 1-d local spin Hamiltonians. They optimize the brickwork circuit with respect to an objective function defined as the Frobenius distance between the exact time evolution operator and the brickwork unitary, and also compare the gate count and fidelity with respect to the various orders of Trotter evolution. Apart from the distance from the exact unitary, they also compute OTOCs of local operators, and show that the OTOCs are efficiently captured by the optimized circuit.

The results and analysis reported in the article are sound, and overall appear to be significant, especially in the context of simulating time evolution on quantum simulation platforms which are limited by geometric locality. I do have a few questions and suggestions for the authors, and I think the paper warrants a publication in this journal following their response and modifications.

1. The authors don't provide much detail in the manuscript about the optimization process, especially how much resource and optimization time is required for achieving the quoted results. This is relevant, since that will determine the scalability and accuracy of the optimization procedure. The authors comment that they encounter the 'barren plateau' in optimization, but do not provide any quantitative details. It would be useful if the authors could provide more details of the parameters and the performance of the optimization procedure, and also comment on the scalability of the optimization to larger systems, at least in an appendix. It would also be useful to visualize the `slow' convergence that they comment upon. It would also help the reader to briefly explain the differential programming method that they employ.

2. In the discussions, the authors comment upon the scope of using their methods with translational invariance. It would be interesting to check and present details about whether the optimized gates show any emergent translational invariance, since the underlying dynamics that they consider is translationally invariant. Do the results change significantly if the gates are enforced to be uniform (when periodic boundary conditions are considered)?

3. Regarding the OTOCs, one suggestion is to improve the presentation of the data in the main text, by showing the error in the optimized/Trotter circuits with respect to the exact computation in the space-time diagrams, since it is difficult to compare the different panels in Fig. 7. It would probably be better to show Fig. 14 in the main-text as that is a better comparison of the method with the exact computation. Furthermore, it would be interesting to compare the intermediate-time OTOCs for an optimized circuit, to explore the refocusing phenomenon that the authors found in some numerical experiments (in the discussion around Fig. 9 and 10). Do the OTOC data show an elevated error at intermediate times, which becomes low again at late times?

4. On a minor note, there are some comments (second paragraph, pg 10) about how the performance of the compressed optimized circuits has similar asymptotic slopes with the best Trotter circuits in Fig. 4. However, the data in the figures show that the 4th order Trotter circuits have consistently more favorable slopes for the parameters considered in the plot. It would be better to qualify or clarify that comment.

• validity: high
• significance: good
• originality: good
• clarity: good
• formatting: excellent
• grammar: excellent