# Classical simulation of quantum computation, the Gottesman-Knill theorem, and slightly beyond

@article{Nes2010ClassicalSO, title={Classical simulation of quantum computation, the Gottesman-Knill theorem, and slightly beyond}, author={Maarten Van Den Nes}, journal={Quantum Information \& Computation}, year={2010}, volume={10}, pages={258-271} }

We study classical simulation of quantum computation, taking the Gottesman-Knilltheorem as a starting point. We show how each Clifford circuit can be reduced to anequivalent, manifestly simulatable circuit (normal form). This provides a simple proofof the Gottesman-Knill theorem without resorting to stabilizer techniques. The normalform highlights why Clifford circuits have such limited computational power in spiteof their high entangling power. At the same time, the normal form shows how…

## 39 Citations

### Classical simulation complexity of extended Clifford circuits

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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.

### Simulating quantum computers with probabilistic methods

- Computer ScienceQuantum Inf. Comput.
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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.

### Commuting quantum circuits with few outputs are unlikely to be classically simulatable

- Computer Science, PhysicsQuantum Inf. Comput.
- 2016

This work shows for the first formal evidence that a commuting quantum circuit is not classically simulatable even when the number of output qubits is exponentially small, and applies the argument for the above evidence to Clifford circuits in a similar setting and provides evidence that such a circuit augmented by a depth-1 non-Clifford layer is notClassical simulatable.

### From estimation of quantum probabilities to simulation of quantum circuits

- Computer ScienceQuantum
- 2020

It is argued that a notion of classical simulation, which is called EPSILON-simulation (or ϵ-simulations for short), captures the essence of possessing ``equivalent computational power'' as the quantum system it simulates: it is statistically impossible to distinguish an agent with access to an ϵ -simulator from one possessing the simulated quantum system.

### 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.

### Simulating Quantum Circuits with Sparse Output Distributions

- Computer ScienceElectron. Colloquium Comput. Complex.
- 2013

We show that several quantum circuit families can be simulated efficiently classically if it is promised that their output distribution is approximately sparse i.e. the distribution is close to one…

### Computing quopit Clifford circuit amplitudes by the sum-over-paths technique

- Computer Science, MathematicsQuantum Inf. Comput.
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It is shown that the sum over paths takes a special form: it can be expressed as a product of Weil sums with quadratic polynomials, which can be computed efficiently, and provides a method for computing the outcome probabilities and amplitudes of such circuits efficiently.

### Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus

- Computer ScienceQuantum
- 2020

A simplification strategy for ZX-diagrams is given based on the two graph transformations of local complementation and pivoting and it is shown that the resulting reduced diagram can be transformed back into a quantum circuit.

### Commuting quantum circuits and complexity of Ising partition functions

- MathematicsArXiv
- 2013

Instantaneous quantum polynomial-time (IQP) computation is a class of quantum computation consisting only of commuting two-qubit gates and is not universal. Nevertheless, it has been shown that if…

### Hardness of classically simulating quantum circuits with unbounded Toffoli and fan-out gates

- Computer ScienceQuantum Inf. Comput.
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This work shows that there exists a constant-depth quantum circuit with only two unbounded fan-out gates that is not strongly simulatable, unless P = PP, which is in contrast to the fact that any constant- depth quantum circuit without additional gates on an unbounded number of qubits is strongly and weakly simulatable.

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