arXiv subjects

Paul Appel

Publications and source records attributed to Paul Appel.

3 recordsLinked to original sources

arXiv2022

On the role of entanglement in qudit-based circuit compression

Gate-based universal quantum computation is formulated in terms of two types of operations: local single-qubit gates, which are typically easily implementable, and two-qubit entangling gates, whose faithful implementation remains one of the major experimental challenges since it requires controlled interactions between individual systems. To make the most of quantum hardware it is crucial to process information in the most efficient way. One promising avenue is to use higher-dimensional systems, qudits, as the fundamental units of quantum information, in order to replace a fraction of the qubit-entangling gates with qudit-local gates. Here, we show how the complexity of multi-qubit circuits can be lowered significantly by employing qudit encodings, which we quantify by considering exemplary circuits with exactly known (multi-qubit) gate complexity. We discuss general principles for circuit compression, derive upper and lower bounds on the achievable advantage, and highlight the key role played by entanglement and the available gate set. Explicit experimental schemes for photonic as well as for trapped-ion implementations are provided and demonstrate a significant expected gain in circuit performance for both platforms.

quant-ph↗

arXiv2020

Finite-Function-Encoding Quantum States

We introduce finite-function-encoding (FFE) states which encode arbitrary $d$-valued logic functions, i.e., multivariate functions over the ring of integers modulo $d$, and investigate some of their structural properties. We also point out some differences between polynomial and non-polynomial function encoding states: The former can be associated to graphical objects, that we dub tensor-edge hypergraphs (TEH), which are a generalization of hypergraphs with a tensor attached to each hyperedge encoding the coefficients of the different monomials. To complete the framework, we also introduce a notion of finite-function-encoding Pauli (FP) operators, which correspond to elements of what is known as the generalized symmetric group in mathematics. First, using this machinery, we study the stabilizer group associated to FFE states and observe how qudit hypergraph states introduced in arXiv:1612.06418v2 admit stabilizers of a particularly simpler form. Afterwards, we investigate the classification of FFE states under local unitaries (LU), and, after showing the complexity of this problem, we focus on the case of bipartite states and especially on the classification under local FP operations (LFP). We find all LU and LFP classes for two qutrits and two ququarts and study several other special classes, pointing out the relation between maximally entangled FFE states and complex Butson-type Hadamard matrices. Our investigation showcases also the relation between the properties of FFE states, especially their LU classification, and the theory of finite rings over the integers.

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arXiv2018

Monogamy of correlations and entropy inequalities in the Bloch picture

We investigate monogamy of correlations and entropy inequalities in the Bloch representation. Here, both can be understood as direct relations between different correlation tensor elements and thus appear intimately related. To that end we introduce the split Bloch basis, that is particularly useful for representing quantum states with low dimensional support and thus amenable to purification arguments. Furthermore, we find dimension dependent entropy inequalities for the Tsallis 2-entropy. In particular, we present an analogue of the strong subadditivity and a quadratic entropy inequality. These relations are shown to be stronger than subadditivity for finite dimensional cases.

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