Abstract
Quantum walks exhibit many unique characteristics compared to classical random walks. In the classical setting, selfavoiding random walks have been studied as a variation on the usual classical random walk. Here the walker has memory of its previous locations and preferentially avoids stepping back to locations where it has previously resided. Classical selfavoiding random walks have found numerous algorithmic applications, most notably in the modelling of protein folding. We consider the analogous problem in the quantum setting – a quantum walk in one dimension with tunable levels of selfavoidance. We complement a quantum walk with a memory register that records where the walker has previously resided. The walker is then able to avoid returning back to previously visited sites or apply more general memory conditioned operations to control the walk. We characterise this walk by examining the variance of the walker's distribution against time, the standard metric for quantifying how quantum or classical a walk is. We parameterise the strength of the memory recording and the strength of the memory backaction on the walker and investigate their effect on the dynamics of the walk. We find that by manipulating these parameters, which dictate the degree of selfavoidance, the walk can be made to reproduce ideal quantum or classical random walk statistics, or a plethora of more elaborate diffusive phenomena. In some parameter regimes we observe a close correspondence between classical selfavoiding random walks and the quantum selfavoiding walk.
Introduction
Quantum walks^{1,2,3,4}, the quantum equivalent of classical random walks, have been studied extensively for their applications in quantum information processing^{5}. Here a walker (a particle such as a photon) resides at a vertex (e.g. an optical mode) in a graph and is allowed to ‘hop’ along the edges in the graph to reach other vertices. In the classical random walk this process takes place randomly, whereas in the quantum case the walker coherently enters a superposition across different vertices. Numerous elementary optical demonstrations of quantum walks have been performed^{6,7,8,9,10,11,12,13,14}, experimentally confirming the unique behaviour of quantum walks compared to classical random walks and providing an alternate route towards optical quantum information processing^{15,16}.
In this paper we consider onedimensional quantum walks, where the walker has memory of its previous location history and the coin operator is a function of the memory. This can be used, for example, to preferentially avoid previously visited vertices or implement more elaborate conditional coins, allowing for tunable degrees of selfavoidance in the evolution of the walk. Selfavoiding walks have been studied extensively in the classical case^{17,18,19,20,21,22}, having been applied to applications such as protein folding and percolation theory.
Previous authors have considered quantum walks complemented by memory, where the memory is of previous coin values^{23,24,25,26}. Here, however, we consider the case where the memory is of previous positions rather than of previous coins.
Selfavoiding quantum walks may find applications in situations where ordinary quantum walks have been applied, but where we explicitly wish to avoid reexploring previously visited sites. While classical selfavoiding walks have been applied to modelling protein folding^{27}, longchain polymers^{28} and molecular conformation, perhaps quantum selfavoiding walks may be applied to modelling the analogous problems in the quantum context.
In our model each vertex in the graph is complimented by a qubit, which marks whether the site has previously been visited. As the walker walks it leaves a record on the site at which it is presently located. The coin, which determines the subsequent step direction, is biased to avoid walking onto previously marked sites. We will see that due to the unitarity of our model, we cannot maintain a permanent record of where the walker has been and the walker has some chance of revisiting previously visited sites.
We find that selfavoiding quantum walks exhibit diverse diffusive characteristics, ranging from ideal classical to ideal quantum walk dynamics and a plethora of more elaborate behaviours. In particular, we find that there is good agreement between the partially avoiding quantum walk and a corresponding partially avoiding classical walk in onedimension. In the limit of maximum selfavoidance, the walk can exhibit significantly faster rate of spread than an ordinary quantum walk.
Classical selfavoiding random walks
Selfavoiding classical random walks^{17,18,19,20} are a variation on the standard classical walk, where the walker is unable to revisit any sites it has previously visited. A one dimensional, classical, completely selfavoiding walk is trivial; once the first step has been taken by the walker, every step thereafter has only one possibility. The variance of the walker will always be zero, as after the initial step there is only one possible location for the walker to be at any given step. Selfavoiding random walks become less trivial and more useful in two and three dimensions, but they also become exceedingly difficult to analyse in higher dimensions.
Selfavoiding walks have the property that they explore a graph more rapidly, since evolution time is not wasted reexploring already visited sites. Selfavoiding classical random walks are particularly useful in two and three dimensions as they find applications in modelling solvents and polymers^{28}, protein folding^{27}, molecular conformation, percolation lattices and drug discovery. When using classical random walks to model the proliferation of forest fires or the propagation of liquids through a porous material – two archetypal applications for classical random walks – we wish to determine whether a route through a graph exists. Thus, there is no benefit in exploring vertices that have previously been explored and a selfavoiding walk will be far more efficient.
Partially selfavoiding classical walks in one dimension have been studied previously and we review a standard model for one of these in Section below and find good comparison with the partial avoiding quantum walk.
Quantum walks
A (discretetime) quantum walk is a bipartite system comprising position and coin degrees of freedom, with Hilbert space . The position parameter specifies the location (vertex) of the walker in the graph, whilst the coin parameter specifies the direction the walker is heading. On a onedimensional graph the state of the quantum walker at time t is of the form ψ(t)〉 = Σ_{x}_{, c} α_{x}_{, c}(t)x, c〉, where is the discrete position of the walker and c = ±1 is the direction of the walker (left or right respectively).
The evolution of the walk is specified by unitary and operators, which have the actions and respectively, where U is the unitary coin matrix. Thus the coin operator coherently manipulates the direction of the walker, leaving the position unchanged, whilst the step operator uses the coin parameter to update the location of the walker, leaving the coin unchanged. The evolution of the walk over t time steps proceeds as .
One of the interesting features of the quantum walk is that it exhibits a quadratically faster rate of spread (measured using variance) compared to the classical random walk. The variance is defined as, σ^{2}(t) = Σ_{x}p_{x}(t) · (x − µ(t))^{2}, µ(t) = Σ_{x}p_{x}(t) · x, where p_{x}(t) is the total probability of finding the walker at location x at time t, given by summing over the coin degree of freedom, p_{x}(t) = Σ_{c}α_{x}_{, c}(t)^{2}, effectively tracing out the coin.
Quantum selfavoiding walks
We fashion our selfavoiding quantum walk with three components: (1) we record the current position of the walker into the local quantum memory. This is performed using a ‘memory update operator’; (2) we perform the usual coin operator, where the bias of the coin is determined by the state of the memory registers at the current location of the walker; (3) the usual step operator is applied, which results in the walker's movement being a function of whether the site has previously been visited. This allows, for example, the walker to reverse direction upon reaching a previously visited site. An illustration of the components in the selfavoiding quantum walk are shown in Fig. 1.
Recording the current position — To model the selfavoiding quantum walk whereby the walker has memory of its previous location history and avoids positions where it has previously resided, we complement the walker by a sequence of qubits – one for each site in the lattice – where the qubits indicate whether the walker has previously visited the respective site. In this case the state of the system at time t is of the form,
where q_{i} = {0, 1} is the qubit associated with site i and there are N sites. Initially q_{i} = 0 µ i, indicating that none of the sites have been visited. Evidently, the size of the Hilbert space grows exponentially with the size of the lattice (), in contrast to linear growth for an ordinary quantum walk (). This limits our numerical investigations to relatively small evolution times.
We introduce a memory update operator, which updates the memory qubits as a function of the walker's position. We have chosen to implement the memory update operator unitarily, which implies that the memory is reversible and temporary and that the record will evaporate after a second visit to the respective site. A permanent memory recording would require a Lindblad model, which would introduce decoherence into the system. Taking our memory recording to be unitary we fix it to be of the form,
a Pauli X rotation applied to the xth qubit. Now θ_{M} determines the strength of the action of the position on the memory. In the extreme case where θ_{M} = π/2, if the walker has not previously visited site x, the memory update operator flips the respective qubit to mark that this position has now been visited. Of course, a second visit to the site will flip the qubit again, marking that the site has not been visited. Note that this latter property is unavoidable if the evolution is to remain unitary. If θ_{M} = 0 the position has no effect on the memory and the memory only couples with position via the coin operator.
Biasing the coin according to neighbouring memory registers — The coin operator is defined analogously to the usual quantum walk coin operator,
where is a 2 × 2 coin matrix that depends on the state of the qubit at location x, q_{x}. This operator coherently manipulates the coin degree of freedom locally, whilst leaving the position and memory qubits unchanged. The choice of coin matrix is a function of the local memory qubit – when q_{x} = 0 we let (a usual balanced walk with equal amplitudes of stepping in either direction) and when q_{x} = 1 we let . Here, R(θ) is a general Pauli X rotation,
Thus, θ = 0 implements an identity gate, θ = π/2 implements a full bitflip and θ = π/4 implements a balanced step. Our memory will have the effect that if the xth qubit is set (indicating that we previously visited location x) then the coin operator implements a partially reflecting mirror on the walker's dynamics, with strength determined by θ_{B}. This mirror is located at spatial location x and the reflectivity (r) is determined by θ_{B} (when θ_{B} = 0, r = 0 and when θ_{B} = π/2, r = 1). Thus, the parameter θ_{B} controls how strong the effect of memory on the coin is. The probability of being reflected at a site where the memory is set is given by r = sin θ_{B}^{2}, otherwise it is given by r = sin θ_{C}^{2}.
Propagating the walker — The step operator is defined in the usual way – it updates the position as a function of the coin parameter,
The step operator affects only the position, leaving the coin and memory systems unchanged.
Combined evolution — The total evolution now progresses as,
where a single step begins with memory recording, followed by the coin operator and finally the step operator.
Our formalism differs from that presented in Ref. 29, where no memory was required to ensure selfavoidance. Rather, the coin matrix was chosen so as to guarantee there is always zero amplitude in the reverse direction. This is the socalled ‘nonreversing quantum walk’. However, such a formalism is constrained by the fact that one is unable to tune between different memory strengths, making our formalism much more versatile, as we will see in the next section.
Whilst the presented formalism is in the context of a onedimensional walk, it could be trivially generalised to higher dimensions, by expanding the position space. However, we will not perform simulations in this context as the complexity of simulations grows exponentially with the number of discrete position states.
Results
To characterise the selfavoiding walk we will again consider the variance of the position probability distribution against time. We compute the variance, where now p_{x}(t) is obtained by summing over the coin and memory degrees of freedom,
effectively tracing out the coin and memory subsystems.
In the purely classical case σ^{2}(t) ∝ t and in the quantum case σ^{2}(t) ∝ t^{2}. In our quantum walk with memory we anticipate more elaborate dynamics, which we will now explore and characterise.
In Fig. 2(centre) we plot the variance of the selfavoiding quantum walk against varying memory recording (θ_{M}) and coin backaction (θ_{B}) when evolved 7 steps. In all cases the regular coin is fixed as a balanced coin, θ_{C} = π/4. Here we have employed the symmetrised input state,
which guarantees symmetric position probability distributions in the case of the standard unbiased quantum walk.
Each point in the plot in Fig. 2(centre) corresponds to the final variance of an entire quantum walk with given choices of coin backaction and memory recording. To understand these dynamics, the insets illustrate the complete quantum walk evolution dynamics corresponding to some individual points in the plot.
We observe that the variance is maximal when the memory has maximum recording strength (θ_{M} = π/2) and the coin has identity backaction (θ_{B} = 0), as shown in Fig. 2(a). Because the memory recording strength is maximal, the memory qubit corresponding to the initial location is set. Then, because the qubit is set, the identity coin will be applied, which propagates the walker outwards. At the next step the same process applies, marking the current site and propagating the walker outwards. Thus, the two terms in the symmetrised input state shoot off in straight line paths in opposite directions, yielding maximum quadratic variance against time. Fitting to a second order polynomial,
the second order coefficient of the aforementioned maximum variance walk is k_{2} = 1, compared to k_{2} = 0.27 for the ideal quantum walk, reflecting the significantly faster rate of spread of the θ_{M} = π/2, θ_{B} = 0 selfavoiding quantum walk compared to the normal ideal quantum walk.
Intuitively, the maximum variance in this case appears to be a result of the choice of symmetrised input state. We expect that if we instead adopt an unsymmetrised input state,
there will be only a single straight line trajectory in one direction, yielding zero variance. This is confirmed in Fig. 3. Thus, we see that the variance of the selfavoiding quantum walk is highly dependent on the initial choice of input state and can in fact swap us between maximal and zero variance. Fig. 4 illustrates the variance of an unsymmeterised input state across all parameter regimes.
In the intermediate case of 0 < θ_{B} < π/2, the coin creates a superposition, yielding lower variance than Fig. 2(a).
In the case of θ_{M} = π/4 and θ_{B} = π/4 we observe statistics that are qualitatively very classical (Fig. 2(b)). This is because now the coin is a π/4 rotation irrespective of the memory (recall that θ_{C} = π/4 also), yielding a balanced walk, regardless of history. But simultaneously the memory register is highly entangled with the position degree of freedom, which, upon being traced out, yields decoherence, which can cause a transition from a quantum to a classical walk. This transition was nicely experimentally demonstrated by Broome et al.^{8} in the optical context.
If we increase the memory recording strength to θ_{M} = π/2, whilst retaining θ_{B} = π/4, the walker is maximally entangling with the memory, yielding complete decoherence upon tracing out the memory. Thus, in this instance we observe the binomial distribution and linear variance associated with the ideal classical walk (Fig. 2(e)).
When θ_{M} = 0 there is no memory recording and so no entanglement between the memory and the walker. The memory stays permanently fixed at its initial value and thus the coin is always the θ_{C} = π/4 coin, yielding a balanced quantum walk, irrespective of θ_{B} (Fig. 2(c)). Thus, when θ_{B} = π/4, the memory recording strength can tune us between a perfect classical walk (θ_{M} = π/2) and a perfect quantum walk (θ_{M} = 0).
In the limiting case of maximum coin backaction (θ_{B} = π/2) and maximum memory recording strength (θ_{M} = π/2), the walker resists outward diffusion, yielding a distribution more localised around the origin (Fig. 2(d)). This is because as the walker visits sites the respective memory qubit is completely flipped, toggling between marking the site as visited or not visited. When visited, the walker will incur a full bitflip, sending the walker back whence it came. Otherwise a balanced coin is implemented and the walker spreads equally in both directions. Thus, on alternate steps the walker either completely flips direction or spreads evenly in both directions, resulting in reduced outward diffusion compared to a normal balanced walk.
An interesting feature of Fig. 2(centre) is to note that σ^{2}(θ_{B}, θ_{C}, θ_{M}) is not monotonically related to θ_{B} and θ_{M}. For low values of θ_{B} we see that σ^{2} increases monotonically with θ_{M}, whereas for large θ_{B} it decreases with θ_{M}. Similarly, for large θ_{M}, σ^{2} decreases monotonically with θ_{B}, whereas for small θ_{M} it increases with θ_{M}. However, the nonmonotonic nature of σ^{2} may be due to the small values of t that we are able to simulate. In the case of unsymmetrised input states (Fig. 4) the dynamics are even more complex.
Evidently, different parameter regimes for θ_{B}, θ_{C} and θ_{M} yield different couplings between the different subsystems when acted upon by the coin, step and memory operators. The couplings are illustrated in Fig. 5. The coin operator will couple all three subsystems, provided that θ_{C} ≠ θ_{B}. The step operator will always couple the coin and position subsystems, irrespective of θ_{B}, θ_{C} and θ_{M}. And the memory operator will couple the position and memory subsystems provided that θ_{M} ≠ 0.
Comparison with classical selfavoiding random walks
In the classical case, selfavoiding random walks can be constructed in a variety of ways. For example, a classical random walk could be completely selfavoiding, in the sense that it may never return to a visited site. More generally, a metric could be defined, which specifies the probability with which a walker may return to a previously visited site. In Refs. 17,18 such a metric is defined as,
where q_{i} is the probability of stepping to site i, n_{i} is the number of times site i has been previously visited, g ≥ 0 is the strength of the selfavoidance (g = 0 is an ordinary classical random walk, g = ∞ is a completely selfavoiding walk and 0 < g < ∞ is a weakly selfavoiding walk) and j sums over the nearest neighbours of the current vertex. Thus, the probability of returning to a given site is weighted exponentially with the number of times the respective site has been visited. Using this formalism one can tune between a regular nonselfavoiding classical random walk, a weakly selfavoiding random walk, or a completely selfavoiding random walk. A generic characteristic of such a selfavoiding classical random walk is that as g increases the walker is more likely to be found further away from the origin.
The selfavoidance metric given in Eq. 11 biases how likely the walker is to revisit a site given its prior visitation history and the strength of the selfavoidance, g. However, this bias requires the total number of previous visits to each site. To compare with our quantum selfavoiding walk we restrict to the case where θ_{M} = π/2. Then the memory is marked and then unmarked at subsequent visits, i.e. the bias effectively depends on the number of visits modulo 2. Therefore, we modify Eq. 11 to count modulo 2, such that,
Alternately, one could achieve a situation more analogous to Eq. 11 by replacing the qubits in the selfavoiding quantum walk with qudits and letting the memory recording implement q_{i} → (q_{i} + 1) mod d, thus giving us the ability to record up to d instances of previous visits. However we will not explore this option given the computational intractability.
As discussed earlier, in the quantum case when θ_{M} = π/2 we have maximum coupling between the walker and the memory, yielding a decoherence effect. Thus, we expect the comparison between the classical selfavoiding random walk and the quantum selfavoiding walk to be closest in the regime where θ_{M} = π/2.
Recall that in the classical limit σ^{2}(t) ∝ t, whereas in the quantum limit θ^{2}(t) ∝ t^{2}. Thus, to characterize the classical selfavoiding random walk we fit the variance to σ^{2}(t) = t^{β}. In Fig. 6 (top) we plot the exponent β against the selfavoidance strength g for a classical selfavoiding random walk. Contrast this with Fig. 6 (bottom), where we plot the analogous situation for the quantum selfavoiding walk with θ_{M} = π/2 while varying θ_{B}, corresponding to the classical regime of maximum coupling with the memory. Now θ_{B} takes the role of the selfavoidance strength. In the limit of θ_{B} = π/4 we observe an ideal classical random walk with β = 1, which corresponds to Fig. 2(e). In the opposing limit of θ_{B} = 0 we observe maximum dispersion and β = 2, corresponding to Fig. 2(a). Thus, within this region we observe a correspondence between g (in the classical case) and θ_{B} (in the quantum case). Specifically, g = 0 corresponds to θ_{B} = π/4 and corresponds to θ_{B} = 0. In these two limits the selfavoiding classical random walk exhibits dynamics close to the respective selfavoiding quantum walk (Figs. 2(e) and (a) respectively). In both cases β exhibits a monotonic relationship to the selfavoidance strength, ranging between β = 1 (linear spread) and β = 2 (ballistic, quadratic spread).
Conclusion
We have considered a onedimensional quantum walk with tunable degrees of selfavoidance. Here the walker is complemented by a memory register that records the previous position history and the coin operator is a function of the memory, allowing the walker to preferentially avoid previously visited sites.
Similar behaviour could be modelled using the formalism of Ref. 26, where the walker has memory of past coin values rather than past position values. In that formalism the walker's movement may be an arbitrary unitary function of the walker's history, allowing arbitrary dynamics to be implemented.
Whilst we have focused on onedimensional walks, the formalism may be easily generalized to higher dimensions. But in that instance numerical simulation faces the obstacle that computational complexity grows exponentially with the number of positions.
We have defined our selfavoiding walk such that the walker's direction is made a function of whether the present site has been previously visited. A possibility we have not explored is where the coin is a function of whether the neighbour in the direction of movement has been visited, but such rules often are difficult to implement in a unitary fashion. A variation where the walker's movements depends on both neighbours, though not consistent with our formalism, can be made unitary and constitutes a type of quantum cellular automata.
With appropriate choices of conditional coins and recording strengths, the walk can be made to exhibit zero dispersion or maximal dispersion, which is highly dependent on the symmetrisation of the input state. The walker can be made to reproduce ideal quantum or classical statistics, or to exhibit richer, more diverse diffusive characteristics, which can be manipulated with the strength of the memory recording (θ_{M}) and backaction (θ_{B}). In the regime of maximum memory recording strength, we observe a close correspondence between the quantum and classical selfavoiding walks, where the coin backaction takes the role of the selfavoidance strength.
The rich diffusive characteristics of the quantum selfavoiding walk may be applicable to modelling a broader range of diffusive phenomenon than ordinary quantum walks and may lend themselves to novel quantum information processing applications such as applications in quantum simulation.
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Acknowledgements
This research was conducted by the Australian Research Council Centre of Excellence for Engineered Quantum Systems (Project number CE110001013).
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J.T. provided the central idea, which was further developed by all authors. E.C. and P.P.R. developed the numerical codes and performed the analysis. All authors contributed to interpreting the data and writing the manuscript.
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Camilleri, E., Rohde, P. & Twamley, J. Quantum walks with tuneable selfavoidance in one dimension. Sci Rep 4, 4791 (2014). https://doi.org/10.1038/srep04791
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DOI: https://doi.org/10.1038/srep04791
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