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… 

Figures and Tables from this paper

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.

Simulating quantum circuits with ZX-calculus reduced stabiliser decompositions

We introduce an enhanced technique for strong classical simulation of quantum circuits which combines the ‘sum-of-stabilisers’ method with an automated simplification strategy based on the

Classical Simulation of Quantum Circuits with Partial and Graphical Stabiliser Decompositions

This work finds a new technique of partial stabiliser decompositions that allow us to trade magic states for stabiliser terms, and manages to reliably simulate 50-qubit 1400 T-count hidden shift circuits in a couple of minutes on a consumer laptop.

Quantum circuit compilation and hybrid computation using Pauli-based computation

Practical ways of implementing PBC as adaptive quantum circuits, and code to do the required classical side-processing are proposed, and the practical advantage of PBC techniques for circuit compilation and hybrid computation is demonstrated.

How to Simulate Quantum Measurement without Computing Marginals.

Algorithms for classically simulating measurement of an n-qubit quantum state in the standard basis, that is, sampling a bit string from the probability distribution determined by the Born rule are described and analyzed.

Magic hinders quantum certification

The resource theory of magic quantifies the hardness of quantum certification protocols, showing that the resources needed to certify the quality of the application of a given unitary U are governed by the magic in the Choi state associated with U, which is shown to possess a profound connection with out-of-time order correlators.

Faster Born probability estimation via gate merging and frame optimisation

Two classical sub-routines are proposed: circuit gate merging and frame optimisation, which optimise the circuit representation to reduce the sampling overhead and show that the runtimes of both sub- routines scale polynomially in circuit size and gate depth.

New techniques for bounding stabilizer rank

A number-theoretic theorem of Moulton is refined to exhibit an explicit sequence of product states with exponential stabilizer rank but constant approximate stabilizers rank, and alternate (and simplified) proofs of the best-known asymptotic lower bounds on stabilizerRank and approximate stabilizerrank are provided.

Magic-state resource theory for the ground state of the transverse-field Ising model

Ground states of quantum many-body systems are both entangled and possess a kind of quantum complexity as their preparation requires universal resources that go beyond the Clifford group and

Efficient classical simulation of cluster state quantum circuits with alternative inputs

Sahar Atallah , Michael Garn 1,∗, Sania Jevtic , Yukuan Tao 3,†, and Shashank Virmani1,‡ Department of Mathematics, Brunel University London, Kingston Ln, Uxbridge, UB8 3PH, United Kingdom, 2

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.