Improved upper bounds on the stabilizer rank of magic states

@article{Qassim2021ImprovedUB,
  title={Improved upper bounds on the stabilizer rank of magic states},
  author={Hammam Qassim and Hakop Pashayan and David Gosset},
  journal={Quantum},
  year={2021},
  volume={5},
  pages={606}
}
<jats:p>In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>m</mml:mi></mml:math> copies of the magic state <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo stretchy="false">|</mml:mo></mml:mrow><mml:mi>T… 
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References

SHOWING 1-10 OF 17 REFERENCES

Explicit Lower Bounds on Strong Quantum Simulation

It is shown that a universal simulator computing any amplitude to precision simulator must take at least at least time, which yields a conditional exponential lower bound on the growth of the stabilizer rank of magic states.

Lower Bounds on Stabilizer Rank

The stabilizer rank of a quantum state ψ is the minimal r such that |ψ〉 = ∑j=1 cj ∣ ∣ so that ψ ≥ 1 and r ≥ 1.

Simulation of quantum circuits by low-rank stabilizer decompositions

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.

Lower bounds on the non-Clifford resources for quantum computations

It is shown that catalysis is necessary for many conversions and introduced new catalytic conversions, some of which are optimal, which optimally lower bound the number of T or CCZ states needed to implement the ubiquitous modular adder and multiply-controlled-Z operations.

Improved Strong Simulation of Universal Quantum Circuits

We find a scaling reduction in the stabilizer rank of the twelve-qubit tensored T gate magic state. This lowers its asymptotic bound to 2 ∼ 0 . 463 t for multi-Pauli measurements on t magic states,…

Universal quantum computation with ideal Clifford gates and noisy ancillas (14 pages)

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.…

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

It is shown how each Clifford circuit can be reduced to an equivalent, manifestly simulatable circuit (normal form), which provides a simple proof of the Gottesman-Knill theorem without resorting to stabilizer techniques.

Improved Classical Simulation of Quantum Circuits Dominated by Clifford Gates.

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.

Fine-grained quantum computational supremacy

This paper studies "fine-grained" version of quantum supremacy that excludes some exponential-time classical simulations, and shows that output probability distributions of Clifford-$T quantum computing cannot be classically sampled within a constant multiplicative error.

Explicit lower bounds on strong simulation of quantum circuits in terms of $T$-gate count

Using the sparsification lemma, this work identifies time complexity lower bounds in terms of $T$-gate count below which a strong simulator would improve on the state-of-the-art $3$-SAT solving.