Adjustable-depth quantum circuit for position-dependent coin operators of discrete-time quantum walks

TL;DR

This paper tackles the challenge of exactly implementing position-dependent coin operators in discrete-time quantum walks with circuits whose depth remains tractable. It introduces an adjustable-depth circuit parametrized by $m\in\mathbb{N}$ that partitions $2^n$ coin operators into $2^{n-m}$ packs of size $2^m$, enabling a trade-off between depth and ancilla resources by executing packs in parallel and sequencing them. A key advancement is the new ingredient in $Q_{1,i}^{(n,m)}$, which encodes ancilla only for current-stage position states using generalized $(n-m)$-Toffoli gates, yielding an exact implementation with depth scaling $d(U^{(n,m)})=2^{n-m}(20m+2\varepsilon_d(n-m)+8\delta_{m,0}-5)-2$. The approach is validated by implementing the circuit on IBM's QASM simulator, analyzing depth/width scaling, and discussing potential extensions to quantum cellular automata and field-theory simulations that preserve locality and symmetries, thereby broadening hardware-efficient quantum walk-based schemes.

Abstract

Discrete-time quantum walks with position-dependent coin operators have numerous applications. For a position dependence that is sufficiently smooth, it has been provided in Ref. [1] an approximate quantum-circuit implementation of the coin operator that is efficient. If we want the quantum-circuit implementation to be exact (e.g., either, in the case of a smooth position dependence, to have a perfect precision, or in order to treat a non-smooth position dependence), but the depth of the circuit not to scale exponentially, then we can use the linear-depth circuit of Ref. [1], which achieves a depth that is linear at the cost of introducing an exponential number of ancillas. In this paper, we provide an adjustable-depth quantum circuit for the exact implementation of the position-dependent coin operator. This adjustable-depth circuit consists in (i) applying in parallel, with a linear-depth circuit, only certain packs of coin operators (rather than all of them as in the original linear-depth circuit [1]), each pack contributing linearly to the depth, and in (ii) applying sequentially these packs, which contributes exponentially to the depth.

Figures (14)

• Figure 1: Registers necessary for the implementation of $U^{(n,m)}$.
• Figure 2: Decomposition of $U^{(n,m)}$ in $2^{n-m}$ packs $U_i^{(n,m)}$, as written in Eq. \ref{['eq:packs']}.
• Figure 3: Decomposition of each $U_i^{(n,m)}$, written in Eq. \ref{['eq:packsQ']}.
• Figure 4: Quantum circuits implementing $Q_{0,i}^{(n=3,m=1)}$ for $i=0,1,2,3$, that is, we implement, at each stage, $2^m=2$ coin operators (out of $2^n=8$) in parallel.
• Figure 5: We choose $n=3$ and $m=1$. On the far left, there is a circuit with only a NOT gate on $\mathinner{|{b'_0}\rangle}$, which is what would be used at the beginning of the controlled-SWAP operations of each $Q_{1,i}^{(n,m)}$, if there was no difference with $Q_1^{(n)}$ of $U_{\text{lin.}}^{(n)}$ in Ref. NZDPplus2022. But, here is precisely the main new ingredient of the present adjustable-depth circuit to implement the position-dependent coin operators: instead of this NOT gate, we have to apply, at the beginning of the controlled-SWAP operations of $Q_{1,i}^{(n,m)}$, a generalized $(n-m)$-Toffoli gate which activates the NOT gate if and only if $b_{n-1}...b_m = i_2$, where $i_2$ is the binary writing of $i$. From left to right starting from the second circuit, we have depicted this generalized $(n-m)$-Toffoli gate for $i=0,1,2,3$. The term "generalized" simply refers to the fact that the controls can be positive (black dot) or negative (white dot).