LIMDD: A Decision Diagram For Simulation Of Quantum Computing Including Stabilizer States

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Lieuwe Vinkhuijzen¹, Tim Coopmans^1,2, David Elkouss^2,3, Vedran Dunjko¹, and Alfons Laarman¹

¹Leiden University, The Netherlands
²Delft University of Technology, The Netherlands
³Networked Quantum Devices Unit, Okinawa Institute of Science and Technology Graduate University, Okinawa, Japan

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Abstract

Efficient methods for the representation and simulation of quantum states and quantum operations are crucial for the optimization of quantum circuits. Decision diagrams (DDs), a well-studied data structure originally used to represent Boolean functions, have proven capable of capturing relevant aspects of quantum systems, but their limits are not well understood. In this work, we investigate and bridge the gap between existing DD-based structures and the stabilizer formalism, an important tool for simulating quantum circuits in the tractable regime. We first show that although DDs were suggested to succinctly represent important quantum states, they actually require exponential space for certain stabilizer states. To remedy this, we introduce a more powerful decision diagram variant, called Local Invertible Map-DD (LIMDD). We prove that the set of quantum states represented by poly-sized LIMDDs strictly contains the union of stabilizer states and other decision diagram variants. Finally, there exist circuits which LIMDDs can efficiently simulate, while their output states cannot be succinctly represented by two state-of-the-art simulation paradigms: the stabilizer decomposition techniques for Clifford + $T$ circuits and Matrix-Product States. By uniting two successful approaches, LIMDDs thus pave the way for fundamentally more powerful solutions for simulation and analysis of quantum computing.

Featured image: Left: Venn diagram of the set of quantum states efficiently represented by LIMDDs and several other formalisms. Right: An example of a LIMDD, which compactly represents a $3$-qubit quantum state.

Popular summary

Classical simulation of a quantum circuit is a computationally difficult task. In a straightforward approach, the memory requirements for storing a description of a quantum state grow as $2^n$ for an $n$-qubit circuit. Decision diagrams address this problem by providing a compressed representation of a quantum state. However, the limits of DD-based methods were not well understood. In this work, we investigate and bridge the gap between existing DD-based structures and the stabilizer formalism, another important tool for simulating quantum circuits. We first show that although DDs were suggested to succinctly represent important quantum states, they actually require exponential space for certain stabilizer states. To remedy this, we introduce a more powerful decision diagram variant, called Local Invertible Map-DD (LIMDD). We prove that there are quantum circuits which can be efficiently analyzed by LIMDDs, but not by existing DD-based methods, nor stabilizer decomposition techniques, nor matrix product states. By leveraging the strengths of both DD and the stabilizer formalism in a strictly more succinct data structure, LIMDDs thus pave the way for fundamentally more powerful simulation and analysis of quantum computing.

► BibTeX data

@article{Vinkhuijzen2023limdddecision, doi = {10.22331/q-2023-09-11-1108}, url = {https://doi.org/10.22331/q-2023-09-11-1108}, title = {{LIMDD}: {A} {D}ecision {D}iagram for {S}imulation of {Q}uantum {C}omputing {I}ncluding {S}tabilizer {S}tates}, author = {Vinkhuijzen, Lieuwe and Coopmans, Tim and Elkouss, David and Dunjko, Vedran and Laarman, Alfons}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1108}, month = sep, year = {2023}<br /> }

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Cited by

[1] Dimitrios Thanos, Tim Coopmans, and Alfons Laarman, “Fast equivalence checking of quantum circuits of Clifford gates”, arXiv:2308.01206, (2023).

[2] Robert Wille, Stefan Hillmich, and Lukas Burgholzer, “Tools for Quantum Computing Based on Decision Diagrams”, arXiv:2108.07027, (2021).

The above citations are from SAO/NASA ADS (last updated successfully 2023-09-13 02:58:16). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2023-09-13 02:58:15).

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LIMDD: A Decision Diagram for Simulation of Quantum Computing Including Stabilizer States

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