Diagrammatic Differentiation for Quantum Machine Learning

@article{Toumi2021DiagrammaticDF,
  title={Diagrammatic Differentiation for Quantum Machine Learning},
  author={Alexis Toumi and Richie Yeung and Giovanni de Felice},
  journal={ArXiv},
  year={2021},
  volume={abs/2103.07960}
}
We introduce diagrammatic differentiation for tensor calculus by generalising the dual number construction from rigs to monoidal categories. Applying this to ZX diagrams, we show how to calculate diagrammatically the gradient of a linear map with respect to a phase parameter. For diagrams of parametrised quantum circuits, we get the well-known parameter-shift rule at the basis of many variational quantum algorithms. We then extend our method to the automatic differentation of hybrid classical… 

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References

SHOWING 1-10 OF 45 REFERENCES

Diagrammatic Design and Study of Ansätze for Quantum Machine Learning

This thesis pioneers the use of diagrammatic techniques to reason with QML ansatze, taking commonly used QML Ansatze circuits and converting them to diagrammatic form and giving a full description of how these gates commute, making the circuits much easier to analyse and simplify.

Picturing Quantum Processes: A First Course in Quantum Theory and Diagrammatic Reasoning

This entirely diagrammatic presentation of quantum theory represents the culmination of ten years of research, uniting classical techniques in linear algebra and Hilbert spaces with cutting-edge developments in quantum computation and foundations.

Analyzing the barren plateau phenomenon in training quantum neural network with the ZX-calculus

The barren plateaus theorem is extended from unitary 2-design circuits to any parameterized quantum circuits under certain reasonable assumptions and is shown that, for the hardware efficient ansatz and the MPS-inspired ansatz, there exist barren Plateaus.

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

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.

Evaluating analytic gradients on quantum hardware

This paper shows how gradients of expectation values of quantum measurements can be estimated using the same, or almost the same the architecture that executes the original circuit, and proposes recipes for the computation of gradients for continuous-variable circuits.

A Complete Axiomatisation of the ZX-Calculus for Clifford+T Quantum Mechanics

The ZX-Calculus is made complete for the so-called Clifford+T quantum mechanics by adding two new axioms to the language, and it is proved that the π/4-fragment of the ZX -Calculus represents exactly all the matrices over some finite dimensional extension of the ring of dyadic rationals.

Two complete axiomatisations of pure-state qubit quantum computing

Extended versions of ZW and ZX calculus are presented, and their completeness for pure-state qubit theory is proved by a strategy that rewrites all diagrams into a normal form, thus solving two major open problems in categorical quantum mechanics.

Supervised learning with quantum-enhanced feature spaces

Two classification algorithms that use the quantum state space to produce feature maps are demonstrated on a superconducting processor, enabling the solution of problems when the feature space is large and the kernel functions are computationally expensive to estimate.

Coherence for compact closed categories