The notebook describes some applications of scalar chain. The examples with discrete quantum walk on a scalar chain are discussed further:
Scalar chain with $N$ nodes is a quantum system with basic states $|k\rangle$, $k=1,\dots,N$. Let us denote $R$ and $L$ operators of right and left shift respectively. For an infinite chain without boundary:
$$R : |k\rangle \to |k+1\rangle,\qquad L : |k\rangle \to |k-1\rangle.$$For a cyclic boundary condition (motion on a ring) arithmetic modulo $N$ should be used instead and an alternative model with possibility of reflections is considered further.
The model of scalar chain is convenient for description of quantum walk. There are two broad classes of quantum walks: discrete and continuous (aka state transport). The examples below use discrete time models and 'quantum bots' together with coined and staggered discrete quantum walk – are simple illustrations of such approach.
Let us consider so-called coined quantum walk. The model uses (together with a chain $|k\rangle$, $k=1,\dots,N$) the special coin $|c\rangle$ with two states $|0\rangle$ and $|1\rangle$ to control direction of walk. So, basic states of the model are defined as $|c\rangle|k\rangle$.
The coin defines direction of motion using controlled shift operator:
$$S = |0\rangle\langle 0| \otimes R + |1\rangle\langle 1| \otimes L$$Such a model (aka quantum bot) is an example of conditional quantum dynamics. The operator $S$ for basic states of coins $|0\rangle$ and $|1\rangle$ produces right and left shifts respectively.
The coined quantum walk suggests modification of coin after each step $S$ using some operator $C$ on a two-dimensional "coin space"
$$W = S \cdot (C \otimes {\bf 1})$$to look for some analogue of classical random walk. Simplest choise is Hadamard matrix:
$$C_H = H = \frac{\sqrt 2}{2}\begin{pmatrix}1&1\\1&-1\end{pmatrix}.$$A balanced coin also may be used
$$C_b = \frac{\sqrt 2}{2}\begin{pmatrix}1&i\\i&1\end{pmatrix}.$$The model also permits to define reflection on boundaries. In such a case operator $S$ should "loop" both chains with opposite directions of motion:
The composite quantum system with coin and chain has $2\times N = 2N$ basic states $|c,k\rangle$. An equivalent model of staggered quantum walk is defined on a chain with $2N$ nodes and special partitions. Let us consider for $k =1,2,\ldots$ first partition with pairs of states $(2k-1,2k)$ and the second one with $(2k,2k+1)$.
Let us denote $S_1$ and $S_2$ swaps in all pairs for first and second partition, respectively. Without taking into account boundary, the composition of alternating swaps $S = S_1 S_2$ would produce double-step shifts of elements with even and odd indexes in opposite directions $2k \to 2k+2$ and $2k+1 \to 2k-1$.
See animated staggered classical walk in 3D for illustration of such a motion with boundary effects.
For a quantum model states $|c,k\rangle$ are mapped into $|2k+1\rangle$. The trivial motion of basic states would be represented as $S = S_1 S_2$, where $S_1 = s(1,2)\,s(3,4)\dots$ and $S_2 = s(2,3)\,s(4,5)\dots$ are two alternating sequences of swaps defined for given link simply as
$$s = \begin{pmatrix}0&1\\1&0\end{pmatrix}.$$Coin operator $C$ acts on pairs in first partition $C = c(1,2)\,c(3,4)\dots$ and an operator $S^\prime_1 = C S_1$ is used in quantum staggered walk
$$W_S = S^\prime_1\,S_2 = (C S_1) S_2.$$So, for single link a swap operator multiplied on Hadamard coin is
$$s_c = c \, s = \frac{\sqrt 2}{2}\begin{pmatrix}1&-1\\1&1\end{pmatrix}.$$Models discussed further use one-to-one correspondence $n = N$ between qubits and nodes of chain (or $n = 2N$ for staggered walk). More difficult method could "expand" $n$ qubits into a scalar chain with $N = 2^n$ states, but it should be discussed elsewhere. Thus, the scalar quantum chain model may use standard packages such as NumPy to compare with Qiskit application for qubit chain.
See more in notebook for scalar chain modeling.