@Article{Nzongani_etal2023,
author="Nzongani, U.
and Zylberman, J.
and Doncecchi, C. E.
and Perez, A.
and Debbasch, F.
and Arnault, P.",
title="Quantum circuits for discrete-time quantum walks with position-dependent coin operator",
journal="Quantum Information Processing",
year="2023",
publisher="Springer",
volume="22",
number="7",
pages="270--46pp",
optkeywords="Quantum walks; Quantum circuits; Quantum simulation",
abstract="The aim of this paper is to build quantum circuits that implement discrete-time quantum walks having an arbitrary position-dependent coin operator. The position of the walker is encoded in base 2: with n wires, each corresponding to one qubit, we encode 2(n) position states. The data necessary to define an arbitrary position-dependent coin operator is therefore exponential in n. Hence, the exponentiality will necessarily appear somewhere in our circuits. We first propose a circuit implementing the position-dependent coin operator, that is naive, in the sense that it has exponential depth and implements sequentially all appropriate position-dependent coin operators. We then propose a circuit that {\textquoteleft}{\textquoteleft}transfers{\textquoteright}{\textquoteright} all the depth into ancillae, yielding a final depth that is linear in n at the cost of an exponential number of ancillae. Themain idea of this linear-depth circuit is to implement in parallel all coin operators at the different positions. Reducing the depth exponentially at the cost of having an exponential number of ancillae is a goal which has already been achieved for the problem of loading classical data on a quantum circuit (Araujo in Sci Rep 11:6329, 2021) (notice that such a circuit can be used to load the initial state of the walker). Here, we achieve this goal for the problem of applying a position-dependent coin operator in a discrete-time quantum walk. Finally, we extend the result of Welch (New J Phys 16:033040, 2014) from position-dependent unitaries which are diagonal in the position basis to position-dependent 2 x 2-block-diagonal unitaries: indeed, we show that for a position dependence of the coin operator (the block-diagonal unitary) which is smooth enough, one can find an efficient quantum-circuit implementation approximating the coin operator up to an error epsilon (in terms of the spectral norm), the depth and size of which scale as O(1/epsilon). A typical application of the efficient implementation would be the quantum simulation of a relativistic spin-1/2 particle on a lattice, coupled to a smooth external gauge field; notice that recently, quantum spatial-search schemes have been developed which use gauge fields as the oracle, to mark the vertex to be found (Zylberman in Entropy 23:1441, 2021), (Fredon arXiv:2210.13920). A typical application of the linear-depth circuit would be when there is spatial noise on the coin operator (and hence a non-smooth dependence in the position).",
optnote="WOS:001022408900002",
optnote="exported from refbase (https://references.ific.uv.es/refbase/show.php?record=5587), last updated on Thu, 03 Aug 2023 07:45:10 +0000",
issn="1570-0755",
doi="10.1007/s11128-023-03957-8",
opturl="https://arxiv.org/abs/2211.05271",
opturl="https://doi.org/10.1007/s11128-023-03957-8",
language="English"
}