Discrete-time Quantum Walks in Qudit Systems

• URL: http://arxiv.org/abs/2207.04319v3
• Date: Thu, 03 Oct 2024 14:29:31 GMT
• Title: Discrete-time Quantum Walks in Qudit Systems
• Authors: Amit Saha, Debasri Saha, Amlan Chakrabarti,
• Abstract summary: We introduce a first of its kind one-dimensional quantum walk in the $d$-dimensional quantum domain. We show its equivalence for circuit realization in an arbitrary finite-dimensional quantum logic.
• Score: 3.452050192629253
• License:
• Abstract: Quantum walks contribute significantly to developing quantum algorithms and quantum simulations. Here, we introduce a first of its kind one-dimensional quantum walk in the $d$-dimensional quantum domain, where $d>2$, and show its equivalence for circuit realization in an arbitrary finite-dimensional quantum logic for utilizing the advantage of larger state space, which helps to reduce the run-time of the quantum walks as compared to the conventional binary quantum systems. We provide efficient quantum circuits for the implementation of discrete-time quantum walks (DTQW) in one-dimensional position space in any finite-dimensional quantum system when the dimension is odd using an appropriate logical mapping of the position space on which a walker evolves onto the multi-qudit states. With example circuits for various qudit state spaces, we also explore scalability in terms of $n$-qudit $d$-ary quantum systems. Further, the extension of one-dimensional DTQW to $d$-dimensional DTQW using $2d$-dimensional coin space on $d$-dimensional lattice has been studied, where $d>=2$. Thereafter, the circuit design for the implementation of scalable $d$-dimensional DTQW in $d$-ary quantum systems has been portrayed. Lastly, we exhibit the circuit design for the implementation of DTQW using different coins on various search spaces.

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