# Interactive shallow Clifford circuits: Quantum advantage against NC¹ and beyond

@article{Grier2020InteractiveSC,
title={Interactive shallow Clifford circuits: Quantum advantage against NC¹ and beyond},
author={Daniel Grier and Luke Schaeffer},
journal={Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing},
year={2020}
}
• Published 6 November 2019
• Computer Science
• Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
Recent work of Bravyi et al. and follow-up work by Bene Watts et al. demonstrates a quantum advantage for shallow circuits: constant-depth quantum circuits can perform a task which constant-depth classical (i.e., 0) circuits cannot. Their results have the advantage that the quantum circuit is fairly practical, and their proofs are free of hardness assumptions (e.g., factoring is classically hard, etc.). Unfortunately, constant-depth classical circuits are too weak to yield a convincing real…
11 Citations

## Figures and Tables from this paper

Interactive quantum advantage with noisy, shallow Clifford circuits
• Computer Science
ArXiv
• 2021
A key component of this reduction is showing average-case hardness for the classical simulation tasks—that is, showing that a classical simulation of the quantum interactive task is still powerful even if it is allowed to err with constant probability over a uniformly random input.
Possibilistic simulation of quantum circuits by classical circuits
This paper constructs classical circuits, using {NOT, AND, OR} gates of fan-in $\leq 2$, that can simulate any given quantum circuit with Clifford+$T$ gates, and implies that the constant-vs-log circuit depth separation achieved by BGK is the largest achievable for simulating quantum circuits, like theirs, with Clifford+.
Adaptive constant-depth circuits for manipulating non-abelian anyons
• Mathematics
• 2022
We consider Kitaev’s quantum double model based on a ﬁnite group G and describe quantum circuits for (a) preparation of the ground state, (b) creation of anyon pairs separated by an arbitrary
Quantum advantage for computations with limited space
• Computer Science, Physics
Nature Physics
• 2021
Space-restricted computations, where input is a read-only memory and only one (qu)bit can be computed on, are considered, and it is shown that n-bit symmetric Boolean functions can be implemented exactly through the use of quantum signal processing as restricted space quantum computations using O(n^2) gates.
Quantum computational advantage with string order parameters of 1D symmetry-protected topological order
• Mathematics
• 2021
Nonlocal games with advantageous quantum strategies give arguably the most fundamental demonstration of the power of quantum resources over their classical counterparts. Recently, certain multiplayer
Quantum Computational Advantage with String Order Parameters of One-Dimensional Symmetry-Protected Topological Order.
• Mathematics
Physical review letters
• 2021
This work shows advantageous strategies for these nonlocal games for generic ground states of one-dimensional symmetry-protected topological orders (SPTOs) when a discrete invariant of a SPTO known as a twist phase is nontrivial and -1.
3XOR Games with Perfect Commuting Operator Strategies Have Perfect Tensor Product Strategies and are Decidable in Polynomial Time
• Mathematics, Computer Science
ArXiv
• 2020
It is shown that for perfect 3XOR games the advantage of a quantum strategy over a classical strategy (defined by the quantum-classical bias ratio) is bounded, in contrast to the general3XOR case where the optimal quantum strategies can require high dimensional states and there is no bound on the quantum advantage.
Provable quantum computational advantage with the cyclic cluster state
• Physics
• 2021
Austin K. Daniel,1, ∗ Yingyue Zhu,2 C. Huerta Alderete,2 Vikas Buchemmavari,1 Alaina M. Green,2 Nhung H. Nguyen,2 Tyler G. Thurtell,1 Andrew Zhao,1 Norbert M. Linke,2, † and Akimasa Miyake1, ‡
Fast simulation of planar Clifford circuits
• Computer Science, Mathematics
ArXiv
• 2020
This work improves known classical algorithms with cubic runtime by providing a classical algorithm with runtime scaling asymptotically as n^{\omega/2}<n^{1.19}\$ which samples from the output distribution obtained by measuring all £n qubits of a planar graph state in given Pauli bases.
Test of Quantumness with Small-Depth Quantum Circuits
• Computer Science
MFCS
• 2021
This paper shows that this test of quantumness, and essentially all the above applications, can actually be implemented by a very weak class of quantum circuits: constant-depth quantum circuits combined with logarithmic-depth classical computation.

## References

SHOWING 1-10 OF 49 REFERENCES
Average-case quantum advantage with shallow circuits
• F. Gall
• Computer Science
Computational Complexity Conference
• 2019
This paper constructs a computational task that can be solved on all inputs by a quantum circuit of constant depth with bounded-fanin gates (a "shallow" quantum circuit) and shows that any classical circuit solving this problem on a non-negligible fraction of the inputs must have logarithmic depth.
Quantum Advantage with Noisy Shallow Circuits in 3D
• Computer Science
2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
• 2019
This work constructs a relation problem which can be solved with near certainty using a noisy constant-depth quantum circuit composed of geometrically local gates in three dimensions, provided the noise rate is below a certain constant threshold value.
Classical simulation complexity of extended Clifford circuits
• Mathematics
Quantum Inf. Comput.
• 2014
The results reveal a surprising proximity of classical to quantum computing power viz. a class of classically simulatable quantum circuits which yields universal quantum computation if extended by a purely classical additional ingredient that does not extend the class of quantum processes occurring.
Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits
• Mathematics
STOC
• 2019
The Parity Halving Problem is constructed by constructing a new problem in QNC^0, which is easier to work with, and it is proved that AC^0 lower bounds for this problem are proved, and that it reduces to the 2D HLF problem.
Trading Locality for Time: Certifiable Randomness from Low-Depth Circuits
• Computer Science
Communications in mathematical physics
• 2021
A protocol for exponential certified randomness expansion using a single quantum device and relies on the physical assumption that the adversarial device being tested implements a circuit of sub-logarithmic depth to be able to be easily verified in classical linear time.
Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
• Computer Science, Mathematics
Proceedings 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.
Quantum Fan-out is Powerful
• Mathematics
Theory Comput.
• 2005
It is demonstrated that the unbounded fan-out gate is very powerful and can approximate with polynomially small error the follow- ing gates: parity, mod(q), And, Or, majority, threshold (t), exact(t), and Counting.
Improved Simulation of Stabilizer Circuits
• Computer Science
ArXiv
• 2004
The Gottesman-Knill theorem, which says that a stabilizer circuit, a quantum circuit consisting solely of controlled-NOT, Hadamard, and phase gates can be simulated efficiently on a classical computer, is improved in several directions.
Forging quantum data: classically defeating an IQP-based quantum test
A classical algorithm is described that can not only convince the verifier that the (classical) prover is quantum, but can in fact can extract the secret key underlying a given protocol instance.
Parallel Quantum Computation and Quantum Codes
• Computer Science
SIAM J. Comput.
• 2001
While it is noted the exact quantum Fourier transform can be parallelized to linear depth, it is conjecture that neither it nor a simpler "staircase" circuit can be Parallelized to less than this.