On the equivalence between quantum and random walks on finite graphs

Abstract

Quantum walks on graphs are ubiquitous in quantum computing finding a myriad of applications. Likewise, random walks on graphs are a fundamental building block for a large number of algorithms with diverse applications. While the relationship between quantum and random walks has been recently discussed in specific scenarios, this work establishes a formal equivalence between the processes on arbitrary finite graphs and general conditions for shift and coin operators. It requires empowering random walks with time heterogeneity, where the transition probability of the walker is non-uniform and time dependent. The equivalence is obtained by equating the probability of measuring the quantum walk on a given vertex of the graph and the probability that the random walk is at that same vertex , for all vertices and time steps. The result is given by the construction procedure of a matrix sequence for the random walk that yields the exact same vertex probability distribution sequence of any given quantum walk, including the scenario with multiple interfering walkers. Interestingly, these matrices allow for a different simulation approach for quantum walks where vertex samples respect neighbor locality, and convergence is guaranteed by the law of large numbers, enabling efficient (polynomial) sampling of quantum graph trajectories (paths). Furthermore, the complexity of constructing this sequence of matrices is discussed in the general case.

Appendices

Appendix A Limitations of the rejection method for QWT

The question in hand is to generate samples of quantum walk trajectories. As defined in Sect. 5, quantum walk trajectories are random paths of a graph G for which the empirical distribution ${\hat{p}}(v, t)$ (fraction of instances that the walker is found in vertex v after t hops) approaches $\upsilon (v, t)$, the given vertex distribution of a quantum walk defined on G, for all $v \in V$, $t \in \{0,\ldots ,L - 1\}$. Essentially, a path of the graph, or a graph trajectory, of length L is a sequence of vertices $\{v_1,\ldots ,v_L\}$ such that $(v_i, v_{i+1}) \in E$ for $i \in \{1,\ldots ,L-1\}$.

The usual simulation method for a quantum walk consists in drawing independent samples from distinct time instants. Effectively, assume that one wants to draw a sequence of L samples from a quantum walker system such that the $i\text {-th}$ sample is drawn according with the distribution $\upsilon$ at instant i. Note that $\upsilon$ depends on the initial condition. Denoting $V_i = \{v \in V | \upsilon (v, i) \ne 0\}$ as the set of all vertices that can be the outcome of a measurement of the system at instant i, define ${\mathscr {X}}_L = V_0 \varvec{\times }... \varvec{\times }V_{L-1}$ as the set of all possible sequences of measurements of length L that can be obtained from the quantum walker system. Simply, the set ${\mathscr {X}}_L$ is the sample space for the usual simulation procedure of a quantum walk.

Analyzing the set ${\mathscr {X}}_L$ and considering the general case of initial conditions, it is clear that not every sequence $\tau \in {\mathscr {X}}_L$ is a valid path of the graph. However, every path of the graph that can be obtained as a quantum walk trajectory belongs to ${\mathscr {X}}_L$. Thus, this method cannot be directly used to sample quantum walk trajectories as some measured sequences would not be proper paths of the graph.

To overcome this limitation, rejection sampling could be applied where a sample is accepted as a quantum walk trajectory only if it is a valid path on G. In order to address this matter formally, let ${\mathscr {T}}_L \subset {\mathscr {X}}_L$ be the set of all sequences of measurements that are paths of length L of G. Let $p(\tau )$ be the probability of obtaining a sequence $\tau = \{\tau _0,\ldots ,\tau _{L-1}\} \in {\mathscr {X}}$ from the independent sampling procedure. Let ${\mathscr {X}}_L^{v,t} \subset {\mathscr {X}}_{L}$ and ${\mathscr {T}}_{L}^{v, t} \subset {\mathscr {T}}_{L}$ denote the set of all sequences and trajectories of length L in which v appears in position t. It follows trivially from independence that

$$\begin{aligned}&p(\tau ) = \prod _{t=1}^{L}\upsilon (\tau _t, t), \end{aligned}$$

(35)

where $\tau _t$ denotes the vertex measured at t. In this case, the vertex probability at t is simply,

$$\begin{aligned}&\upsilon (v, t) = \sum _{\tau \in {\mathscr {X}}_{L}^{v,t}} p(\tau ) . \end{aligned}$$

(36)

However, rejecting non-trajectory samples yields the following estimator ${\hat{p}}(v, t)$ for the probability of finding vertex v at instant t:

$$\begin{aligned}&{\hat{p}}(v, t) = \frac{\sum _{\tau \in {\mathscr {T}}_{L}^{v, t}} p(\tau )}{\sum _{\tau ^{'} \in {\mathscr {T}}_{L}} p(\tau ^{'})}. \end{aligned}$$

(37)

It is not clear whether Eqs. 36 and 37 are equal for every possible quantum walk, since the ratio between the probability of generating trajectories with vertex v at position t and the probability of generating any trajectory would have to be $\upsilon (v, t)$, for all $v \in V$ and $t \in \{0,\ldots , L - 1\}$.

Additionally, even for the cases where Eqs. 36 and 37 are equal, the expected time to accept a sample in the rejection procedure is precisely the inverse of the probability of generating a trajectory. Although this probability depends on a myriad of factors such as the graph structure, the number of trajectories of a given length L within a graph can be exponentially smaller than the number of possible sequences of measurements. As an example, a $D\text {-dimensional}$ torus with V vertices would have $VD^{L - 1}$ paths of length L and $V^{L}$ possible sequences of vertices, which for values of $D \in O(1)$ is exponentially larger than the number of trajectories, suggesting that the expected time to generate a trajectory sample would be unfeasible.

In precise terms, Theorem 1 offers a polynomial time procedure to sample trajectories for any quantum walk. Given the transition matrix P(t), a quantum trajectory of length L can be sampled in time $O(Ld_{\mathrm{max}})$ where $d_{\mathrm{max}}$ is the maximum degree of the graph. Moreover, if multiple trajectories are to be generated, then the alias method could be used to decide the next neighbor at time instant in which case the amortized time complexity for each trajectory is O(L), and no longer depends on vertex degrees.

Appendix B Dynamic programming for Grover walk on torus

The Grover-coined (Eq.8) walk on a $D\text {-dimensional}$ torus with moving-shift operator (Eq.15) and purely real initial conditions serves as an example where the probability distribution can be computed by a dynamic programming algorithm that is more efficient than direct matrix multiplication. The number of degrees of freedom within the $D\text {-dimensional}$ torus is 2D. Analyzing the action of the total walk operator MG on a given state

$$\begin{aligned} \varPsi (u, t) = \sum _{c \in C_u} \varPsi (u, c, t) \left| (u,c)\right\rangle \end{aligned}$$

and having $\eta (u, c) = v$, the probability $\rho (v, c, t+1)$ is described as

$$\begin{aligned}&\rho (v, c, t + 1) = \left| \sum _{c' \in C_u} \varPsi (u, c', t) - \varPsi (u, c, t)\right| . \end{aligned}$$

(38)

Assuming that $\psi (u,c,t) = \sqrt{\rho (u,c,t)}e^{i\cos {\theta _{uct}}}$ and noting that $\theta _{uct} = 0$ for every u, c and t whenever purely real initial conditions are considered yields

$$\begin{aligned}&\rho (v, c, t + 1) = \left| \sum _{c' \in C_u} \sqrt{\rho (u, c', t)} - \sqrt{\rho (u, c, t)}\right| . \end{aligned}$$

(39)

From Theorem 1, the entries of the random walk matrices for which $\rho (v, c, t) > 0$ are given by

$$\begin{aligned}&\frac{\rho (v, c, t + 1)}{\upsilon (u , t)} = \frac{1}{\upsilon (u , t)} \left| \sum _{c' \in C_u} \sqrt{\rho (u, c', t)} - \sqrt{\rho (u, c, t)}\right| . \end{aligned}$$

(40)

Equation 40 can be solved through a dynamic programming algorithm in which each time instant has complexity $O(|V|^2D)$, implying on an overall procedure of complexity $O(t|V|^2D)$ for all matrices up to time T. For $D \in O(1)$, the algorithm has complexity $O(t|V|^2)$, showing a quadratic improvement over the generic procedure, since $O(t|E|^2) \in O(t|V|^4)$.¹