# Simulation of quantum circuits by low-rank stabilizer decompositions

@article{Bravyi2019SimulationOQ,
title={Simulation of quantum circuits by low-rank stabilizer decompositions},
author={Sergey Bravyi and Dan E. Browne and Padraic Calpin and Earl T. Campbell and David Gosset and Mark Howard},
journal={Quantum},
year={2019}
}
• Published 1 August 2018
• Computer Science
• Quantum
Recent work has explored using the stabilizer formalism to classically simulate quantum circuits containing a few non-Clifford gates. The computational cost of such methods is directly related to the notion of stabilizerrank, which for a pure state ψ is defined to be the smallest integer χ such that ψ is a superposition of χ stabilizer states. Here we develop a comprehensive mathematical theory of the stabilizer rank and the related approximate stabilizer rank. We also present a suite of…
117 Citations

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## References

SHOWING 1-10 OF 107 REFERENCES
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.
Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates.
• Physics, Computer Science
Physical 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.
Efficient Inner-product Algorithm for Stabilizer States
• Computer Science
ArXiv
• 2012
It is proved that each n-qubit stabilizer state has exactly 4(2^n - 1) nearest-neighbor stabilizer states, and is generalized to compute the inner product between two such frames.
On the geometry of stabilizer states
• Computer Science
Quantum Inf. Comput.
• 2014
This work characterize and count nearest-neighbor stabilizer states, quantify the distribution of angles between pairs of stabilizerStates, study succinct stabilizer superpositions and stabilizer bivectors, explore the approximation of non-stabilizer states by single stabilizers states and short linear combinations of stabiliser states, and develop an improved inner-product computation for stabilizer States via synthesis of compact canonical stabilizer circuits.
Universal quantum computation with ideal Clifford gates and noisy ancillas (14 pages)
• Physics
• 2005
We consider a model of quantum computation in which the set of elementary operations is limited to Clifford unitaries, the creation of the state |0>, and qubit measurement in the computational basis.
Pareto-Efficient Quantum Circuit Simulation Using Tensor Contraction Deferral
• Physics, Computer Science
• 2017
It is shown that deferring tensor contractions can extend the boundaries of what can be computed on classical systems and can be used to simulate $7 \times 7$-qubit random circuits to arbitrary depth by leveraging secondary storage.
Simulation of low-depth quantum circuits as complex undirected graphical models
• Physics
• 2017
Near term quantum computers with a high quantity (around 50) and quality (around 0.995 fidelity for two-qubit gates) of qubits will approximately sample from certain probability distributions beyond
Simulating quantum computers with probabilistic methods
It is shown that the exponential speed-ups of Simon's and Shor's algorithms crucially depend on the very last stage in these algorithms, dealing with the classical postprocessing of the measurement outcomes, and it is proved that both algorithms would be classically simulatable if the function classically computed in this step had a sufficiently peaked Fourier spectrum.
Fast simulation of stabilizer circuits using a graph-state representation
• Computer Science
• 2006
An algorithm for this task, which is based on the graph-state formalism, shows significant improvement in comparison to an existing algorithm, given by Gottesman and Aaronson, in terms of speed and of the number of qubits the simulator can handle.
Shorter Stabilizer Circuits via Bruhat Decomposition and Quantum Circuit Transformations
• Computer Science
IEEE Transactions on Information Theory
• 2018
This paper improves the layered implementation of arbitrary stabilizer circuits over the gate library, and develops a two-qubit gate depth circuit executable in the Linear Nearest Neighbor (LNN) architecture and shows that a circuit in this normal form is optimal in the number of Hadamard gates used.