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.

The Parameterized Complexity of Quantum Verification

It is shown that for the problem of parameterized quantum circuit satisfiability, there exists a classical algorithm solving the problem with a runtime scaling exponentially in the number of non-Clifford gates but only polynomially with the system size.

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.

Thrifty shadow estimation: re-using quantum circuits and bounding tails

This paper proposes and analyzes a more practically-implementable variant of the protocol, thrifty shadow estimation, in which quantum circuits are reused many times instead of having to be freshly generated for each measurement.

Measuring magic on a quantum processor

This work proposes and experimentally demonstrate a protocol for measuring magic based on randomized measurements that can provide a characterization of the effectiveness of a quantum hardware in producing states that cannot be effectively simulated on a classical computer.

Magic determines the hardness of direct fidelity estimation

This work shows how the resource theory of magic quantifies the hardness of direct fidelity estimation protocols, and extends the results to quantum evolutions, showing that the resources needed to certify the quality of the implementation of a given unitary U are governed by the magic in the Choi state associated with U.

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.

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.

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.

Hybrid Techniques for Simulating Quantum Circuits using the Heisenberg Representation.

This work designs new, more efficient data structures and algorithms for beyond-stabilizer simulation using superpositions of stabilizer states, and demonstrates a parallel version of Quipu that achieves a computational speedup.

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.