Quantum circuits for discrete-time quantum walks with position-dependent coin operator

TL;DR

This work addresses the challenge of implementing discrete-time quantum walks with arbitrary position-dependent coin operators on a quantum computer. It introduces three circuit paradigms: a naive circuit with exponential depth, a linear-depth circuit that parallelizes coin-operator applications at the cost of an exponential ancilla budget, and an efficient approach for smooth position dependence based on Walsh decompositions. The paper rigorously analyzes exact and approximate constructions, proving linear-depth performance and $O(1/\epsilon)$ scaling under suitable smoothness, and demonstrates practical implementations on a classical IBM simulator for small system sizes. These developments enable more faithful quantum simulations of coupled gauge fields and relativistic spin-1/2 dynamics on lattices, with potential applicability to spatial search and disorder/ noise models in DQWs.

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 $n$. 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 Ref. [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 $ε$ (in terms of the spectral norm), the depth and size of which scale as $O(1/ε)$. 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 [3, 4]. 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).

Figures (18)

• Figure 1: Circuits implementing the position-dependent coin operator $U^{(n)}={C}^{(n)}$ for $n=1$ (top left figure), $n=2$ (top right figure), $n=3$ (middle figure), and $n=4$ (bottom figure).
• Figure 2: Circuit corresponding to Formula \ref{['eq:the_circuit']} for $n=3$, implementing the position-dependent coin operator $U^{(n)}={C}^{(n)}$, see Eq. \ref{['eq:The_equality']}.
• Figure 3: Registers necessary for the implementation of the linear-depth quantum circuit for the position-dependent coin operator. From top to bottom, there is (i) the positions register $\mathcal{H}_{\text{pos.}}$, (ii) the ancillary-coins register $\mathcal{H}'_{\text{coins}}$, (iii) the principal coin register $\mathcal{H}_0$, and (iv) the ancillary-positions register $\mathcal{H}'_{\text{pos.}}$.
• Figure 4: Main operation, called $Q_0$, of the linear-depth quantum circuit implementing the position-dependent coin operator defined in Eq. \ref{['eq:non-uniform_coin_operator']}, for $n=2$. We see that all controlled-$C_k$ operations can by applied simultaneously. The explicit definition of $Q_0$ is given in Eq. \ref{['eq:Q0']}.
• Figure 5: Linear-depth quantum circuit $U^{(n)}_{\text{lin.}}$, defined in Eq. \ref{['eq:U_linear']}, implementing the position-dependent coin operator ${C}^{(n)}$, defined in Eq. \ref{['eq:non-uniform_coin_operator']}, that is, we have that $U^{(n)}_{\text{lin.}} = {C}^{(n)} \otimes I_{2^{(2^n-1)}} \otimes I_{2^{(2^n)}}$, where ${C}^{(n)}$ acts on the tensor product $\mathcal{H}$ of the position space and the principal-coin space, $I_{2^{(2^n-1)}}$ acts on the ancillary coins $\mathcal{H}'_{\text{coins}}$, and $I_{2^{(2^n)}}$ on the ancillary positions $\mathcal{H}'_{\text{pos.}}$.