# Possibilistic simulation of quantum circuits by classical circuits

@article{Wang2019PossibilisticSO, title={Possibilistic simulation of quantum circuits by classical circuits}, author={Daochen Wang}, journal={Physical Review A}, year={2019} }

In a recent breakthrough, Bravyi, Gosset and Konig (BGK) [Science, 2018] proved that a family of shallow quantum circuits cannot be simulated by shallow classical circuits. In our paper, we first formalise their notion of simulation, which we call possibilistic simulation. We then construct classical circuits, using {NOT, AND, OR} gates of fan-in $\leq 2$, that can simulate any given quantum circuit with Clifford+$T$ gates. Our constructions give log-depth classical circuits that solve the…

## One Citation

### Single-qubit gate teleportation provides a quantum advantage

- Computer Science
- 2022

It is shown that even for single-qubit Cliﬀord-gate-teleportation circuits this simulation problem cannot be solved by constant-depth classical circuits with bounded fan-in gates, and a reduction to the problem of computing the parity is obtained, a well-studied problem in classical circuit complexity.

## References

SHOWING 1-10 OF 34 REFERENCES

### Average-case quantum advantage with shallow circuits

- Computer ScienceComputational 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.

### Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

- MathematicsSTOC
- 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.

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

- Computer ScienceSTOC
- 2020

This work gives a two-round interactive task which is solved by a constant-depth quantum circuit, but such that any classical solution would necessarily solve -hard problems, and proves hardness results for weaker complexity classes under more restrictive circuit topologies.

### Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates.

- Physics, Computer SciencePhysical review letters
- 2016

The algorithm may serve as a verification tool for near-term quantum computers which cannot in practice be simulated by other means and can be used in practice to simulate medium-sized quantum circuits dominated by Clifford gates.

### Improved Simulation of Stabilizer Circuits

- Computer ScienceArXiv
- 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.

### Trading Locality for Time: Certifiable Randomness from Low-Depth Circuits

- Computer ScienceCommunications 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.

### Quantum advantage with noisy shallow circuits

- Computer Science
- 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.

### Simulation of quantum circuits by low-rank stabilizer decompositions

- Computer ScienceQuantum
- 2019

A comprehensive mathematical theory of the stabilizerRank and the related approximate stabilizer rank is developed and a suite of classical simulation algorithms with broader applicability and significantly improved performance over the previous state-of-the-art are presented.

### Quantum Fan-out is Powerful

- MathematicsTheory 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.

### Quantum advantage with shallow circuits

- Computer Science, PhysicsScience
- 2018

It is shown that parallel quantum algorithms running in a constant time period are strictly more powerful than their classical counterparts; they are provably better at solving certain linear algebra problems associated with binary quadratic forms.