Quasi-polynomial Time Approximation of Output Probabilities of Geometrically-local, Shallow Quantum Circuits

  title={Quasi-polynomial Time Approximation of Output Probabilities of Geometrically-local, Shallow Quantum Circuits},
  author={Nolan J. Coble and Matthew Coudron},
  journal={2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)},
  • Nolan J. Coble, Matthew Coudron
  • Published 10 December 2020
  • Computer Science
  • 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
We present a classical algorithm that, for any 3D geometrically-local, polylogarithmic-depth quantum circuit $C$, and any bit string $x\in\{0,1\}^{n}$, can compute the quantity $\vert \langle x\vert C\vert 0^{\otimes n}\rangle\vert ^{2}$ to within any inverse-polynomial additive error in quasi-polynomial time. It is known that it is $\# P$-hard to compute this same quantity to within $2^{-n^{2}}$ additive error [1], [2], and worst-case hardness results for this task date back to [3]. The… 
1 Citations

Figures and Tables from this paper

Approximating Output Probabilities of Shallow Quantum Circuits which are Geometrically-local in any Fixed Dimension
This work here shows that, while handling each new dimension has historically required a new insight, and fixed algorithmic primitive, based on known techniques for D ≤ 3, this work can now handle any fixed dimension D > 3.


Improved robustness of quantum supremacy for random circuit sampling
It is proved under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to within exp(−Ω(m logm)) additive error is hard for any classical computer, where m is the number of gates in the quantum computation.
Quantum supremacy and random circuits.
It is proved that estimating the output probabilities of random quantum circuits is formidably hard for any classical computer, implying that there is an exponential hardness barrier for the classical simulation of most quantum circuits.
Classical Simulation of Quantum Supremacy Circuits
It is shown that achieving quantum supremacy may require a period of continuing quantum hardware developments without an unequivocal first demonstration, and an orders-of-magnitude reduction in classical simulation time is indicated.
Efficient classical simulation of random shallow 2D quantum circuits
It is proved by exhibiting a shallow circuit family with uniformly random gates that cannot be efficiently classically simulated near-exactly under standard hardness assumptions, but can be simulated approximately for all but a superpolynomially small fraction of circuit instances in time linear in the number of qubits and gates.
Classical algorithms for quantum mean values
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.
On the complexity and verification of quantum random circuit sampling
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.
Classical boson sampling algorithms with superior performance to near-term experiments
A classical algorithm solves the boson sampling problem for 30 bosons with standard computing hardware, suggesting that a much larger experimental effort will be needed to reach a regime where
How many qubits are needed for quantum computational supremacy?
A quantum computational supremacy argument is refined and it is concluded that Instantaneous Quantum Polynomial-Time circuits with 208 qubits and 500 gates, Quantum Approximate Optimization Algorithm circuits with 420 qu bits and 500 constraints and boson sampling circuits are large enough for the task of producing samples from their output distributions up to constant multiplicative error to be intractable on current technology.
Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
A new “Quantum singular value transformation” algorithm is developed that can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator.
Improvements in Quantum SDP-Solving with Applications
This paper improves on all previous quantum SDP-solvers and applies their results to the problem of shadow tomography to simultaneously improve the best known upper bounds on sample complexity due to Aaronson and complexity due Brandao et al.