Quantum and random walks as universal generators of probability distributions

Quantum walks and random walks bear similarities and divergences. One of the most remarkable disparities affects the probability of finding the particle at a given location: typically, almost a flat function in the first case and a bell-shaped one in the second case. Here I show how one can impose any desired stochastic behavior (compatible with the continuity equation for the probability function) on both systems by the appropriate choice of time- and site-dependent coins. This implies, in particular, that one can devise quantum walks that show diffusive spreading without loosing coherence, as well as random walks that exhibit the characteristic fast propagation of a quantum particle driven by a Hadamard coin.

Quantum walks [1] and random walks [2] have a long list of affinities and disparities. One can found the (now mostly deprecated) mixed expression "quantum random walks" in the first references exploring these new processes [3][4][5][6], because they were developed as the quantum variants of the discrete random walk in one dimension: the Markov process in which, at every time step, a particle moves (either leftward or rightward) to one of the two neighboring sites as a result of the random outcome of a coin toss. The quantum particle however, like in the renowned case of the double-slit experiment, moves to both directions simultaneously, and this propagation takes place in a deterministic way: the wave function describing the system evolves unambiguously according to the value of some inner binary property -as, e.g., the spin or the chirality-whose state is locally updated by the action a unitary operator, known as the coin operator. Therefore, in this case, the location of the particle at a given instant of time is a probabilistic magnitude due to the intrinsic uncertainty inherent in every quantum phenomenon.
One of the first coin operators considered in the quantum-walk literature is the Hadamard coin [7], a real-valued unitary operator that performs a Hadamard transformation on the chirality of the particle. Since all the probabilities associated with this transformation are identical, the Hadamard walk can be considered as the quantum counterpart of a random walk with a fair coin. In both cases, the occupation probabilities of the more distant (although accessible) locations are exponentially small. But, while the central part of the distribution of the unbiased random walk quickly converges to a Gaussian, the location of a particle doing a Hadamard walk after t steps is almost uniformly distributed in the range [−t/ √ 2, t/ √ 2], centered around the initial position of the particle, and therefore the quantum walker connects this point with any site within this interval after a lapse of time that is thus proportional to their relative distance. To perform the same operation, the unbiased random walker needs an amount of time that grows quadratically with the separation between the sites. These two diverging statistical traits are sometimes seen as paradigms of the two processes. The truth, however, is that these properties depend strongly on how the coin (operator) is chosen, and correspond to the homogeneous, time-independent (Markovian) case. Researchers have relaxed these conditions in the past and detected the emergence of new features in the system as, e.g., Anderson localization. Thus, among the publications on quantum walks, one can find examples of processes whose evolution is driven by site-dependent coins [8][9][10][11][12], timedependent coins [13][14][15][16][17], history-dependent coins [18,19], and even random coins, unitary operators which are randomly chosen [20][21][22][23][24]. The lack of homogeneity is also a recurrent topic in the random-walk literature [25][26][27][28].
My goal in this letter is, in a sense, just the opposite: starting from a given probability function, I want to deduce what is the proper coin selection to retrieve this distribution. With this aim, I consider here the discretetime evolution of a particle moving on the integers as a result of the interaction with a set of site-and timedependent (either quantum or random) coins. In a previous work [29], I examined a particular instance of this problem, the design of a quantum walk that shown a binomial probability function, the distribution of a random walk with a fair coin. Here, I am going to generalize these results in both directions: I will find quantum walks with classical distributions, as well as random walks with quantum-like properties, provided that the comparison is limited to their common probabilistic aspects.
Let us begin with the fundamentals of the quantummechanical side of the problem. As I announced previously, along this letter we will identify particle positions through integer numbers, so let us call H p the associated Hilbert space, with the usual span {|n : n ∈ Z}. H c will represent the Hilbert space of coin states and {|+ , |− } its orthogonal basis. The mathematical representation of the state of our discrete-time, discrete-space quantum walk resides in the tensor-product space H ≡ H c ⊗ H p and changes as a result of the action of the evolution operator T t on it: T t ≡ S U t , where the coin U t is a timeand site-dependent, real-valued unitary operator of the form: with 0 ≤ θ n,t ≤ π, and S is the operator that shifts the walker position according to the coin component of the state vector: (2) In the discrete-time version of quantum (and random) walks, time increases in regular ticks, so one can adjust time units so that t becomes an integer variable: the state of the system at a later time, |ψ t+1 , is recovered after applying T t to |ψ t : Equation (3) leads to the following set of recursive equations that fully characterizes the dynamics of the system, ψ + (n + 1, t + 1) = cos θ n,t ψ + (n, t) + sin θ n,t ψ − (n, t), expressed in terms of the wave-function components, ψ ± (n, t), the projections of the state of the walker into the elements of the basis of the Hilbert space: We will assume that the particle is initially located at the origin, ψ ± (n, 0) = 0 if n = 0, implying this that ψ ± (n, t) = 0 for |n| > t, in general. We also assume that the wave function is real. The reason behind considering real-valued magnitudes is to clearly ensure that quantum walks and random walks to be introduced here share the same number of degrees of freedom. The viability of approaches to this same issue based on complex-valued operators and wave functions are not discarded, however. My first aim is to show how a quantum experiment can be designed with custom probabilistic properties -as long as the null sets are kept unchanged. So, let us introduce ρ(n, t), the likelihood of finding the particle in a particular position n at a given time t, the probability function. In the case of a quantum walker, this probability is recovered through the wave-function components: The free parameters that determine the features of the coin operators are in this case the angular variables θ n,t .
The inhomogeneous, time-dependent random walk, X t , is a non-Markovian process whose one-step evolution can be expressed as follows: If at time t the walker is at a given location, X t = n, then at time t + 1 one has X t+1 = n + 1, with probability p n,t , n − 1, with probability (1 − p n,t ).
The corresponding recursive equation for the probability function reads: where we have expressed p n,t as p n,t = cos 2θ n,t for comparison purposes. From Eq. (19) one can easily conclude the validity of expression (16) also in this case, since now J(n, t) ≡ (2p n,t − 1) ρ(n, t) = cos 2θ n,t ρ(n, t).
The general solution of the classical problem for arbitrary ρ(n, t) can be attained, in this case, with the help of the z transform,
Provided with this information, we can solve the classical problem without passing through Eq. (21) in this case: 3 recall that J(n, t) is the same in both flavors of the walk, so we can substitute (23) and (28) in Eq. (22) to find cos 2θ n,t = J(n, t) ρ(n, t) 3 The explicit functional forms of ρ(z, t) and J(z, t) for this case are: The surprisingly disparity in the complexity of these formulas when compared to Eqs. (23) and (28) is in great measure due to the fact that the last expressions only apply for alternating sites, i.e., n ∈ {−t, −t + 2, · · · , t − 2, t}, being zero otherwise. that implies p n,t = cos 2θ n,t = 1 2 1 + n t + 2 .
Finally, I want to explore the possibility of role interchange. In a recent work [29], I considered the case in which the ρ(n, t) corresponding to a time-and sitedependent quantum walk matched the probability function of a site-homogeneous Markovian random walk, i.e., the binomial distribution: for n ∈ {−t, −t + 2, · · · , t − 2, t}. There it was shown that the solution for this problem reads two expressions whose suitability can be checked by direct insertion in Eqs. (8) and (9). Alternatively, it is very elucidative the computation of J(n, t), since in this case what corresponds to the flux of probability of a random walk with a constant jump likelihood, cf. Eq. (20).
Here, I will examine the opposite situation: how a time-and site-dependent random walk can mimic the characteristic properties of a standard quantum walk. In particular, we are going to focus our attention on the celebrated Hadamard walk, for which θ n,t = π/4. This means that, on the one side, see Eq. (17), and, on the other side, see Eq. (20), that is Closed expressions for the wave-function components of plain quantum walks (including Hadamard walks) are unwieldy but available [30]. In Fig. 1 we can observe the almost perfect correspondence between the probability function of the random walk with inhomogeneous probabilities, and the one of the Hadamard walk with initial state: This apparently capricious choice was made to get a quasi-symmetrical ρ(n, t). Full symmetry in Hadamard walks necessarily involves the use of complex coefficients for describing the initial coin state. In this letter, I have shown how a time-and sitedependent coin is an extremely useful and versatile tool for the design of both quantum and random walks on the line. Such approach entails enough generality to give rise to any desired probabilistic fingerprint either through quantum or classical randomness: I have deduced the rules that must be employed for unambiguously assessing the values of the parameters that fully determine the evolution of the two kind of systems.
This means, in particular, that the extra degree of freedom of the quantum walker associated with its chirality does not introduce further arbitrariness into the problem. This fact is not the consequence of the restriction that I have considered along the text by demanding that the Hilbert space of the quantum particle is defined on the reals rather than on the complex plane: Since a quantum walk with a time-and site-dependent coin operator taking values on the reals can mimic any desired probability function, it is also capable of reproducing the probabilistic behavior of general, complex-valued quantum walks.
The author acknowledges support from the Spanish