On the role of entanglement in qudit-based circuit compression

Xiaoqin Gao1,2, Paul Appel2, Nicolai Friis3,2, Martin Ringbauer4, and Marcus Huber3,2

1Department of Physics, University of Ottawa, Advanced Research Complex, 25 Templeton Street, K1N 6N5, Ottawa, ON, Canada
2Institute for Quantum Optics and Quantum Information – IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
3Atominstitut, Technische Universität Wien, Stadionallee 2, 1020 Vienna, Austria
4Universität Innsbruck, Institut für Experimentalphysik, Technikerstrasse 25, 6020 Innsbruck, Austria

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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.

Quantum computing inherited the zeros and ones of binary information processing from its very successful classical counterpart. Yet, the quantum systems we use for information processing typically have many more than two states that can be used for storing information. An intriguing way of using these additional states is to encode more than one qubit per quantum system. Such an approach has the potential to get more out of existing quantum computers by reducing the number of quantum systems that need to be controlled and the number of entangling operations that need to be performed to realize certain quantum computations.
Here we show that this approach can be highly beneficial already with existing quantum computing hardware, but the usefulness depends greatly on the available forms of entangling gates. A central observation is that although multiple qubits can be combined into a smaller number of high-dimensional qudits, such an embedding changes the structure of the associated state space. Consequently, traditionally complicated multi-qubit gates can become particularly easy to implement with the right kind of high-dimensional entangling gate, while certain "simple" two-qubit gates can become more complicated due to the modified entanglement structure. This highlights the importance of taking into account the available set of entangling operations when considering embeddings of qubits into qudits.

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