# Efficient classical simulation of random shallow 2D quantum circuits

@article{Napp2020EfficientCS, title={Efficient classical simulation of random shallow 2D quantum circuits}, author={John Napp and Rolando L. La Placa and Alexander M. Dalzell and Fernando G. S. L. Brand{\~a}o and Aram Wettroth Harrow}, journal={ArXiv}, year={2020}, volume={abs/2001.00021} }

Random quantum circuits are commonly viewed as hard to simulate classically. In some regimes this has been formally conjectured, and there had been no evidence against the more general possibility that for circuits with uniformly random gates, approximate simulation of typical instances is almost as hard as exact simulation. We prove that this is not the case by exhibiting a shallow circuit family with uniformly random gates that cannot be efficiently classically simulated near-exactly under…

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## References

SHOWING 1-10 OF 90 REFERENCES

### Achieving quantum supremacy with sparse and noisy commuting quantum computations

- Computer Science
- 2016

It is shown that purely classical error-correction techniques can be used to design IQP circuits which remain hard to simulate classically, even in the presence of arbitrary amounts of noise of this form.

### On the complexity and verification of quantum random circuit sampling

- Computer ScienceNature Physics
- 2018

Evidence is provided that quantum random circuit sampling, a near-term quantum computational task, is classically hard but verifiable, making it a leading proposal for achieving quantum supremacy.

### Efficient classical simulation of noisy quantum computation

- Physics, Computer Science
- 2018

It is proved that under general conditions most of the quantum circuits at any constant level of noise per gate can be efficiently simulated classically with the cost increasing only polynomially with the size of the circuits.

### Unitary designs from statistical mechanics in random quantum circuits.

- Mathematics, Computer Science
- 2019

It is argued that random circuits form approximate unitary $k$-designs in O(nk) depth and are thus essentially optimal in both £n and $k, and can be shown in the limit of large local dimension.

### Polynomial simulations of decohered quantum computers

- Physics, Computer ScienceProceedings of 37th Conference on Foundations of Computer Science
- 1996

This work presents a simulation of decohered sequential quantum computers, on a classical probabilistic Turing machine, and proves that the expected slowdown of this simulation is polynomial in time and space of the quantum computation, for any non zero decoherence rate.

### Classical simulability, entanglement breaking, and quantum computation thresholds (11 pages)

- Physics
- 2005

We investigate the amount of noise required to turn a universal quantum gate set into one that can be efficiently modeled classically. This question is useful for providing upper bounds on…

### Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy

- Computer Science, MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2010

The class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection is introduced, and it is proved first that post- IQP equals the classical class PP, and that if the output distributions of uniform IQP circuit families could be classically efficiently sampled, then the infinite tower of classical complexity classes known as the polynomial hierarchy would collapse to its third level.

### Can Chaotic Quantum Circuits Maintain Quantum Supremacy under Noise

- Physics
- 2017

Although the emergence of a fully-functional quantum computer may still be far away from today, in the near future, it is possible to have medium-size, special-purpose, quantum devices that can…

### Classical algorithms for quantum mean values

- Physics, Computer ScienceNature Physics
- 2021

It is shown that a classical approximation is possible when the quantum circuits are limited to constant depth, and sub-exponential time classical algorithms are developed for solving the quantum mean value problem for general classes of quantum observables and constant-depth quantum circuits.

### Complexity-Theoretic Foundations of Quantum Supremacy Experiments

- Computer ScienceComputational Complexity Conference
- 2016

General theoretical foundations are laid for how to use special-purpose quantum computers with 40--50 high-quality qubits to demonstrate "quantum supremacy": that is, a clear quantum speedup for some task, motivated by the goal of overturning the Extended Church-Turing Thesis as confidently as possible.