Speaker
Description
Quantum walks (QWs) play an important role in quantum computing. On the one hand, some algoritmical problems can be recast as a QW. On the other hand, many physical phenomena can be simulated with the help of a QW. Here we concentrate on discrete-time QWs, and we discuss quantum circuits that can implement discrete-time quantum walks having an arbitrary position-dependent coin operator [1]. The position of the walker is encoded in base 2: with n qubits, we encode $2^n$ position states. 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 “transfers” all the depth into ancillae, yielding a final depth that is linear in n at the cost of an exponential number of ancillae. The main idea of this linear-depth circuit is to implement in parallel all coin operators at the different positions. Finally, we extend the result of [2] from position-dependent unitaries which are diagonal in the position basis to position-dependent $2 \times 2$-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 $O(1/\epsilon)$. Applications of a coin-dependent QW range from the quantum simulation of a relativistic spin-1/2 particle on a lattice, coupled to a smooth external gauge field, to spatial noise on the coin operator.
[1] https://arxiv.org/abs/2211.05271.
[2] J. Welch, D. Greenbaum, S. Mostame, and A. AspuruGuzik, “Efficient quantum circuits for diagonal unitaries without ancillas,” New J. Phys. 16, 033040 (2014).
