Quantum State Discrimination on Reconfigurable Noise-Robust Quantum Networks

A fundamental problem in Quantum Information Processing is the discrimination amongst a set of quantum states of a system. In this paper, we address this problem on an open quantum system described by a graph, whose evolution is defined by a Quantum Stochastic Walk. In particular, the structure of the graph mimics those of neural networks, with the quantum states to discriminate encoded on input nodes and with the discrimination obtained on the output nodes. We optimize the parameters of the network to obtain the highest probability of correct discrimination. Numerical simulations show that after a transient time the probability of correct decision approaches the theoretical optimal quantum limit. These results are confirmed analytically for small graphs. Finally, we analyze the robustness and reconfigurability of the network for different set of quantum states, and show that this architecture can pave the way to experimental realizations of our protocol as well as novel quantum generalizations of deep learning.


I. INTRODUCTION
In the last decade, Quantum Stochastic Walks (QSW) have been proposed as a model to generalize both quantum walks and classical random walks [1].Their formulation arises from the need to extend quantum walks to open quantum systems, with the aim of incorporating decoherence effects that are inevitably present in a real physical system.In fact, the formerly proposed quantum version of random walks, i.e.Quantum Walk (QW) [2][3][4][5][6], has been defined by a unitary evolution of the state, without taking into account incoherent effects.This allows the walker's position on a graph to be in a superposition of states, a property that has been exploited to show that QWs are universal for quantum computation [7][8][9] and that they allow to design quantum algorithms with computational advantages over classical algorithms [10][11][12][13][14][15][16][17][18][19].
In parallel to these results, there have been works showing the beneficial impact of decoherence for dephasing-enhanced transport in a variety of systems, in particular in light-harvesting complexes [20][21][22][23][24][25][26].This has motivated the study of quantum walks with an environmental interaction, as it is described by QSW.The framework of QSW has been investigated in the context of relaxing property [27][28][29] and propagation speed [30,31], showing advantages for speed-up in learning algorithms [32], and enhancement of excitation transport [33][34][35][36][37].
The evolution of QSW is defined by a Gorini-Kossakowski-Sudarshan-Lindblad master equation [38][39][40], where one assumes to work in units with = 1, H is the Hamiltonian, {L k } are the Lindblad operators, and both are defined from the adjacency matrix describing the network of nodes involved in the random walk.The smoothing parameter p accounts for the amount of coherent evolution given from H with respect to the irreversible evolution given by the Lindblad operators, and it allows to interpolate between a quantum walk (p = 0) and a classical random walk (p = 1).On the other side, quantum state discrimination has been one of the first problems faced in quantum information theory [41][42][43][44][45][46][47][48], but it is still a flourishing research field as demonstrated from recent theoretical [49][50][51][52][53][54] and experimental works [55][56][57][58][59][60], also considered in relation to machine learning approaches [61].In its most general formulation, an observer wants to guess the quantum state of a system that is prepared in one of a set of feasible states, possibly by optimizing the measurement operators to apply on the system.The performance measure is the probability of correct detection where {ρ (n) } is the set of quantum states to discriminate, n = 1 . . .M, {p n } their a priori probabilities, and {Π n } the measurement operators to estimate them.
In this work we consider the discrimination of quantum states as a result of their time evolution.Our structure is inspired by neural networks, with its evolution described by a quantum stochastic walk.The connection between the implementation of measurement operators (possibly to solve the discrimination problem) and quantum walks has been investigated in recent works [62,63], but in these papers an alternative formulation of quantum walks is used, and we explicitly refer to a neural network structure for the quantum system.We test different sets of quantum states and several networks in order to understand the best topologies for the discrimination problem.
The paper is organized as follows.In Section II we review the formalism of quantum stochastic walks and we introduce the network model that describes the quantum system.We present different topologies for the binary discrimination and for the discrimination between M > 2 quantum states.In Section III we formalize the discrimination problem, discussing in Subsection III A the binary case and in Subsection III B the M-ary case.In Section IV we report the conclusions and final discussions.

II. QUANTUM STOCHASTIC WALKS
In this section we introduce the QSW model that we here apply to the discrimination problem for the first time.
Classical random walks, quantum walks and quantum stochastic walks are usually defined on an graph G, which is defined by a pair G = (N , E), with N being a set of elements called nodes (or vertices) and E being a set of pairs of nodes (N i , N j ) representing arcs from N i to N j .The pairs in E can be summarized in the adjacency matrix A, with As a generalization, weighted graphs can have any real values A j,i assigned to an arc.Also, in the case the adjacency matrix is a symmetric matrix, i.e., A i,j = A j,i , the graph is called undirected, otherwise the graph is said to be directed.Undirected graphs have pairs (N i , N j ) and (N j , N i ) with the same weight on them, and in this case the arcs are also called edges or links.
The name random walk comes from the fact that a walker, starting from an initial node and moving around randomly according to the link connections, assumes a time dependent probability distribution that can be predicted with this framework.In particular, in the case of an undirected graph with equal weights on the edges, we can define the transition-probability matrix T of the possible node transitions as T = AD −1 , where D is the diagonal degree matrix, with D i,i = j A j,i representing the number of nodes connected to i.The probability distribution of the node occupation, written as a column vector q(t), is evaluated for a discrete time (t) random walk as and for a continuous time random walk as d q dt = (T − I) q . (5) In the quantum scenario, the nodes are associated with the elements of the site basis [34] (see subsection II A for an extensive description of the network).The evolution of the system can be given by the Gorini-Kossakowski-Sudarshan-Lindblad master equation (1), with both the Hamiltonian H and the Lindblad operators {L i,j } depending on the adjacency matrix defined on the graph.
In some models of QSW [1,34], the Hamiltonian operator H is defined from the adjacency matrix, i.e., H = A, and with the Lindblad operators depending on the transition matrix defined on the graph as L i,j = T i,j |i j|.With this approach, optimizing the coefficients of a weighted undirected adjacency matrix A fixes the Hamiltonian H and the Lindblad operators via T = AD −1 .Here, we relax this assumption and, once the adjacency matrix defines the topology of the network, we optimize H and T independently.More precisely, the coefficients A i,j are used to decide whether the corresponding H i,j and T i,j will be optimized (independently) or are set to zero.This approach physically corresponds to optimize independently the hopping rates in H and the noise rates in L. Of course, to define proper transition probabilities the matrix T must satisfy a set of constraints, while for simplicity we take H to be any real symmetric matrix, H i,j = H j,i , with zero entries on the diagonal.Moreover, in the graph some nodes may have a particular role.There is usually a starting node that identifies the initial position of the walker.There might be also sink nodes, i.e., nodes that can irreversibly trap the received population.The latter are connected to the rest of the network only through an arc connecting a sinker node in the network to the sink, preventing a transition in the reverse direction.In the QSW, this is obtained with a Lindblad operator L n = |n s n | for each sink, which is added on the right-side of Eq. ( 1), with s n being the sinker node connected to the n-th sink |n , 1 ≤ n ≤ M. Overall, the master equation for the density operator ρ describing the system reads and the population at the n-th sink at time t = τ (corresponding to the total evolution time) can be evaluated as From here on, we assume Γ = 1 since this parameter is just a factor defining the time scale (Γτ is dimensionless).

A. Network model
To define the topology of the graph we mimic the structure of neural networks [64][65][66].The latter are described by complex graphs where the nodes (resembling neurons) are grouped into input, hidden or output layers.Input nodes are those where the data to be processed are set.Output nodes are those containing the results of the desired task.Hidden nodes represent intermediate steps in the elaboration.
In the quantum case a similar network could be physically realized with an ensemble of two-level systems, one for each node, and with a walker realized by a single quantum exciton moving around.Each node is then associated to the state |i = |0 . . . 1 . . .0 , corresponding to have one excitation (|1 ) at the i-th node and |0 elsewhere.
Then, we classify the nodes of the quantum network into input, intermediate and output layers.For a M − N − O network we mean that there are M input nodes, N intermediate and O output ones (see for instance Fig. 1).Multiple intermediate layers may also be present, for instance a 2 − 6 − 5 − 4 network has 2 nodes in the input layer, 6 nodes in the first intermediate layer, 5 nodes in the second intermediate one, and 4 output nodes.The input nodes are associated to a subset of the Hilbert space , where we prepare the initial quantum (pure or mixed) state of the system, initially in the ground state (no excitons).The network will then evolve in time according to Eq. (8).Note that in general the number of input nodes M is not related to the number M of quantum states to discriminate.By default, each node in a layer is fully connected with all the nodes of the same layer and with all the nodes of the following one.Only in the output layer each sink is connected only from its sinker.We also consider different topologies by reducing the connections between nodes within the same layer.When we want to refer to a topology that is not the default one, we use 'r' to indicate that the connectivity is reduced, i.e. some links are removed.The output nodes are sink nodes where the population gets trapped.After the time evolution of the network dynamic, we measure the sink population to estimate the initial quantum state in the discrimination problem (see Section III).

Models for binary discrimination
In the case of binary discrimination, we first consider the 2r − 2r − 2 model (see Fig. 1a).This is probably the simplest model one can imagine, with 2 input nodes connected to 2 sinker nodes and 2 sinks, but with no links between nodes of the same layer.Then, we consider some of its variants obtained adding some links, for instance the 2−2r−2 model, where the input nodes are connected among themselves, the 2r − 2 − 2 model, with an additional link between the sinkers, the 2 − 2 − 2 model, with both these links added, and a 2r − 4 − 2 model, which has 4 intermediate nodes.By comparing the performances of these models we analyze the role of the added edges.In addition, we investigate the role of the intermediate layers optimizing the 2r − 2r − . . .− 2 model for increasing number of intermediate fully connected layers.This latter model is represented in Fig. 1b.

Models for M-ary discrimination
In the case of M-ary discrimination, we consider a setup with the same number of quantum states to discriminate as the input nodes, M = M , and one with a larger number of quantum state, i.e. with M > M .For instance, we consider the 2 − M − M, 2 − Mr − M and 2r − M − M models (see Fig. 1c) for M = 4 and M = 8.As in the binary case, the reduction in the connectivity indicates the absence of links between nodes of the same layer.We also consider Mr − Mr − M, Mr − M − M (see Fig. 1d) and M − M − M models for M = 4.

III. QUANTUM STATE DISCRIMINATION
In this section we introduce the problem of quantum state discrimination (for more details see reviews [45][46][47][48]).
Assume that a quantum system is prepared in a quantum state drawn from a set of given states, represented by density operators {ρ (m) , 1 ≤ m ≤ M} in the Hilbert Space H.The a-priori probabilities p m by which the quantum states are prepared are also known, i.e. {p m , 1 ≤ m ≤ M}, m p m = 1.In the discrimination problem, we search for the Positive Operator-Valued Measurements operators (POVM) {Π n , 1 ≤ n ≤ M} satisfying that allow to estimate the prepared state with the highest probability of correct detection P c , or equivalently, the lowest probability of error P e = 1 − P c .In (10) the term I H represents the identity operator acting on the space of the density matrices of the quantum system H.
If the quantum states {ρ (m) } span orthogonal subspaces, a perfect discrimination is possible by appropriate measurement operators and P c = 1.If this is not the case, the outcome corresponding to the measurement operator Π n may be correctly recorded when the prepared quantum state is ρ (n) , or it may be wrongly recorded when the quantum state is ρ (m) , m = n, leading to the probability of correct detection in Eq. ( 2).
The conditions for the optimal solution have been derived by Holevo [42] and by Yuen, Kennedy and Lax [43].In the most general scenario the problem can be solved numerically via semidefinite programming [67], but, in the binary case or if the set of quantum states exhibits symmetry features, the optimization can be further carried on analytically to better understand the structure of the measurement operators [68][69][70], and possibly to find a closed form for the probability of correct decision.In the case of only two quantum states, the problem of binary discrimination has been solved by Helstrom [41], and the optimal probability of correct decision is known as Helstrom bound.If the quantum states are pure, ρ (1) = |ψ (1) ψ (1) | and ρ (2) = |ψ (2) ψ (2) |, the Helstrom bound reads If instead ρ (1) and ρ (2) are mixed, the Helstrom bound can be evaluated numerically [41].When the discrimination is set among M > 2 quantum states, the theoretical optimal probability of correct decision P c * is evaluated numerically, and it is used as a reference for the performance of the network.
Note that in our setup we actually consider an equivalent formulation of the problem, where instead of optimizing the measurement operators we fix the measurement projectors (on the population of the output nodes) and optimize the evolution of the network.The optimization of the evolution of the system concerns the coefficients of the Hamiltonian and the Lindblad operators to obtain the best evolution from the subspace of the input nodes, where the quantum states to discriminate are prepared, to the subspace of the output nodes, where the measurement is performed.The two problems are equivalent, and one can interpret the optimized evolution with the measurement on the output nodes as realizing the Naimark extension of the POVM defined on the input nodes for the original discrimination problem.To be more precise, to have a resolution of the identity as in Eq. (10) we formally need to include a projection on the subspace outside the output nodes.This is necessary since part of the population can be trapped in the network [24,34].The outcome associated to this extra projector is considered inconclusive for the discrimination.
We fix the measurement operators to be {Π n = |n n|}, with n identifying the n-th sink, 1 ≤ n ≤ M, which is associated with the estimation of the input quantum state ρ (n) .The projector associated with the inconclusive output is Π The probability of correct decision can then be written as where we have defined ρ The optimization of the network coefficients has been performed numerically using standard routines employing an interior-point algorithm [71][72][73][74] to maximize the probability of correct decision P c .
In the case of the simple network 2r − 2r − 2 for the binary discrimination, we further carry on the optimization analytically and solve the problem for p = 0 and p = 1.This has given us some insights on how to interpret the behaviour of the performance as a function of p and τ .More details are reported in the Appendices A-D.

A. Binary discrimination
We set up the discrimination problem with different pairs of states, for different values of p ∈ [0, 1] and of the total evolution time τ in Eqs ( 8), (9).For each pair (p, τ ) we optimize H, T in Eqs ( 6), ( 8) assuming equal a-priori preparation probabilities of the states to get discriminated.
We have further investigated this behaviour solving the optimization problem analytically for p = 0 and p = 1.We provide the expression for the optimal H, T , in the Appendices A, B and C, and we prove that for p = 0 asymptotically we can reach the Helstrom bound while for p = 1 we reach the theoretical classical bound, that is, the Helstrom bound evaluated on the quantum states with the coherences set to zero.Now we give a sketch of the solution in the particular case p = 0 and the discrimination between the pure quantum states (14), θ ∈ [0, π/2], on a 2r − 2r − 2 model.Assuming the Hamiltonian in the form we can evaluate the time evolution of the node population (more details in Appendix C) and the probability of correct decision results with z = √ 1 − 8h 2 .The maximization of P c (τ ) for a finite τ requires the minimization of the term f (z) = (z sinh (zτ ) + cosh (zτ ) − 1)/z 2 , which can be accomplished numerically.In the asymptotic limit of τ → ∞ the term in the brackets vanishes, and P c (∞) equals the Helstrom bound evaluated on |φ 1 , |φ 2 .
While in the case of the quantum states ( 14) the optimized network approaches the optimal performance for p = 0 in the asymptotic limit, this is not the case for any pair of quantum states.In fact, the 2r − 2r − 2 model has an invariant subspace [24] not connected to any sink.An invariant subspace of a quantum system dynamics is an Hilbert subspace where the dynamics is confined, i.e., span of the eigenstates of H that are orthogonal to the output nodes.In the case of the 2r − 2r − 2 model, the invariant subspace is present for any p and includes the y-component of the state in the Bloch sphere, that is, the r y component of the quantum state starts in this invariant subspace and its evolution remains trapped there.This means that the problem is equivalent to discern the quantum states after setting r y to zero, effectively projecting the quantum states in the (σ x , σ z ) plane of the Bloch sphere (see Fig. 2

right panel).
Secondly, we investigate the role of some links in the network performance, by evaluating the performance of some variants of the 2r − 2r − 2 model.For instance, we add a link in the input layer (  performances in Fig. 3, where we optimize the discrimination between a pure state and a mixed state, both with some r y coordinates, ρ (1) = |ψ (1) ψ (1) |, |ψ (1) = cos θ|1 + sin θe −iξ |2 , We believe we cannot observe the beneficial impact of the noise since the graphs we consider are too small and simple where noise assisted transport is not present since interference effects are neutralized from static disorder in the Hamiltonian coefficients (see [24,33] and the references therein).

Robustness
Here, we analyze the robustness of our discrimination scheme in the case of noisy preparation of the quantum states, a noisy configuration of the network and also an increasing number of intermediate layers.
In the former case, we optimize the 2 − 2 − 2 model assuming to discriminate the quantum states in (18) while only a noisy preparation is actually available, for instance due to experimental imperfections in the preparation stage.We prepare the network with the optimal setup, i.e., with the optimal coefficients in H, L, but we input two random quantum states ρ (1) , ρ (2) by uniformly sampling θ, ξ, r around their nominal value with a maximum of 5% percent error.We run 10 4 simulations with this setup as a function of p for τ = 1, 10, sampling new pairs of quantum states at each run.In Fig. 4 we show that even with a noisy preparation of the quantum states, for any p the performances remain close to the theoretical values.Additionally, we focus on how the performance varies as a function of the preparation error ranging from 0% to 100% for p = 0 -see Fig. 5.We find that up to around 5% of error the correct decision probability remains very close to the Helstrom bound and anyway decreases linearly with the preparation error up to around 25%, before exponentially dropping down below the random guess case (i.e.P c = 0.5).As a second robustness test, we consider a perfect preparation of the quantum states of Eq. ( 18) but a noisy setup of the network coefficients.In this case, we focus on p = 0, 0.1 and τ = 1, 10, and in each simulation we sample the coefficients of the Hamiltonian H uniformly around the optimal values with a given maximum percent error corresponding to a sort of network static disorder.Figure 6 shows that indeed the discriminatory network is robust against noisy preparations of the network dynamics, due for instance to experimental imperfections.It is also interesting to compare the two plots corresponding to the cases with p = 0 and p = 0.1.The former case achieves asymptotically a better performance with small network static disorder.The probability of correct decision slowly decreases as a function of this error and only at around 50% of error it approaches the random guess limit.However, the p = 0.1 case has a more robust performance with respect to this disorder, with the performance range crossing the threshold of random guess at around 80%. Towards an experimental implementation of our protocol, it could be beneficial to consider p = 0.1, slightly sacrificing the performances in favor of a more robust discrimination.
As a third robustness test we study whether the probability of correct decision improves adding or removing more intermediate layers.We consider p = 0 and the 2r − 2r − 2 model with 1, 2, 4, 8 and 16 intermediate layers, and we plot the performance for different evolution times in Fig. 7.It turns out that the increased amount of layers does not allow to close the gap with the Helstrom bound.As the number of intermediate layers increases, the performance lowers due to the fact that a deeper network, i.e., a network with more layers, requires more time to move the quantum state from the input nodes to the output ones.Comparing Figs 3 and 7, we find that to increase the discrimination performance it is more convenient to add links inside an intermediate layer rather than increasing the depth of the network.

B. M-ary discrimination
We now consider the generalization of our scheme to the discrimination of M quantum states.In particular, we investigate whether the number of nodes M in the input layer poses limitations in the distinguishability of the quantum states.In general, M and M are not related, meaning that it could be M < M, M = M or M > M.
As a first case, we consider the discrimination of the M pure qubits (M = 2) with M = 4, 8.In the Bloch sphere representation, these quantum states are equally spaced along the equator defined by the Tr [ρσ z ] = 0 plane, and because of this symmetry they are often used to test discrimination protocols [41,75].We consider the models 2r − Mr − M, 2r − M − M and 2 − M − M, whose performance are reported in Fig. 8, where we plot only the behaviour for p = 0 since the trend with respect to p and τ is similar to the binary case.In this figure we just focus on understanding whether the topology asymptotically closes the gap with the optimal bound P c * = 1 − 2/M, which has been reported for instance in [41].
There is however a fundamental difference here with respect to the binary case.While in the latter the optimal measurement operators are projectors, here the optimal ones are given by POVM.We can realize these POVM via projectors in an extended Hilbert space using the Naimark theorem [41,[75][76][77], meaning that the optimal network will try to implement such extended projectors via its dynamics and the measurement on the sink nodes.We find that 2 − M − M is the most general model and has the highest performance, approaching asymptotically the optimal P c * .Interestingly, the model 2r − M − M share the same behaviour, while for 2r − Mr − M the performances are clearly lower.
As a second case, we consider the discrimination with M = M .We define the initial quantum states as a linear combination of pure states and the completely mixed state, i.e.
with |ϕ m being the m-th state in the mutually unbiased basis of the input nodes, i.e.where |k is the quantum state associated with the kth input node.For α = 1 it leads to discrimination of the pure orthogonal states |k .On the other hand, α = 0 means that ρ (m) = I/M for all m, resulting in a completely random estimation with P c * = 1/M.With an intermediate value of α we want to simulate a noisy preparation of the states (21).We consider α = 0.3 and α = 0.7, and we show the performance for different variants of the model 4 − 4 − 4 in Fig. 9.The behaviour of the probability of correct detection as a function of p and τ is similar to the binary case.In the case of the model 4−4−4 we approach asymptotically the optimal theoretical bound for p = 0, while for the models 4r − 4 − 4 and 4r−4r−4 there is a gap which is more or less emphasized depending on the value of α.

IV. CONCLUSIONS
In this work we have applied the formalism of quantum stochastic walks on configurable networks to the problem of quantum state discrimination, inspired by the neural network approach for deep learning of classical information as images In particular, the input nodes encode the quantum states to discriminate, while the output nodes are used to guess the right answer.We test the discrimination of binary and M-ary set of quantum states with multiple topologies, optimizing the coefficients of the Hamiltonian and the Lindblad operators to obtain the maximum probability of correct detection.The reconfigurability of the network architecture allows to optimally discriminate numerous sets of quantum states.We observe that the general trend of the performances is to increase with the total evolution time τ , while for a fixed τ the best performances are obtained lowering p.Notice that we are not observing any beneficial noise effects probably because the considered networks are very small and not homogeneous (not equal Hamiltonian and noise terms).In many cases with a pure quantum walk (p = 0) we can asymptotically approach the optimal theoretical performance.When this happens, the optimized dynamics realize the Naimark extension (on the whole quantum system) of the optimal POVM for the discrimination.In some cases there is a gap between the theoretical and the asymptotic performance for two reasons, i.e., the lack of node connectivity, which prevents the realization of the optimal POVM, or the presence of an invariant subspace trapping a portion of the quantum states to discriminate, which prevents this component to reach the output nodes.
We have also analyzed the robustness of the optimized network with respect to the preparation of quantum states and the setup of the optimal coefficients of the Hamiltonian and Lindblad operators.Indeed, the architecture is very robust with respect to noise on both stages, as shown in Figs. 4 and 5.This analysis is promising for an experimental realization of the protocol, where imperfections in the preparation apparatus or in the network would be mitigated by the robustness of the architecture.
Therefore, we believe our results may represent further steps towards quantum implementations of machine learning protocols, for instance to solve classification problem.Further studies will address the model applicability beyond discrimination problems, with larger networks, and by experimental benchmark on photonicsbased architectures and in cold atoms platforms.the literature, this phenomenon is explained in terms of invariant subspaces [24,25], that is, a subspace that prevents the dynamics to escape from the network.When this invariant subspace does not contain any sink and it is initialized by the quantum states, it reduces the probability of correct detection since its time-evolution is irrelevant for the discrimination.
Note that the presence of the invariant subspace is due to the topology of the 2r − 2r − 2 model.The model 2−2−2 instead shows a greater connectivity between the nodes, and does not exhibit the separation of the ordinary differential equation system into two sub-systems that generate the invariant subspace.In this Appendix we solve the discrimination problem with p = 0 in the case of equal probable pure states.Since p = 0, there are no Lindblad operators in the master equation except the sink terms, i.e. vanishing terms (A4) and (A5), and the solution can be obtained by finding the optimal value of h 1 , h 2 , h 3 , h 4 .
It is convenient to apply a rotation to the quantum states in order to highlight the symmetry of the problem.By applying the unitary we obtain |φ (1) = U |ψ (1) = cos(θ) sin(θ) , (C3) We can then proceed to solve the discrimination problem on |φ (1) , |φ (2) .From its solution H * (θ) we can recover the solution of the original problem as H * (α, β) = U † H * (θ)U .
We also assume that the optimal solution H * verifies We will shortly see that under this ansatz we can optimize h such that for τ going to infinity we reach the Helstrom bound.Under these assumptions, the system of differential equation (B3) separates into two subsystems with disjoint variables.For instance, ρ 3,3 , s = b 1,3 + b 2,3 , p = ρ 1,1 +ρ 2,2 +2a 1,2 form the following system of differential equations The same system holds for the variables ρ A fundamental set of solutions for the system (C7) can be arranged in a matrix as The initial conditions for (C7) are with x = 0, 1 denoting the initial quantum state.Similarly, we have In particular, defining ρ To solve the discrimination problem, we need to maximize the probability of correct decision.This can be written as It is clear that in order to maximize P (p=0) c (τ ) we need to minimize the term f (z) = (z sinh (zτ )+cosh (zτ )−1)/z 2 , z = √ 1 − 8h 2 .Unfortunately, this transcendental real function in the complex variable z cannot be minimized analytically, and we need to resort to numerical methods.Graphically, we can see that the global minimum is located in the region where 1 − 8h 2 < 0, meaning that z is a pure imaginary number.Writing z = iξ/τ , the function to minimize becomes f (ξ) = τ sin ξ ξ + τ 1−cos ξ ξ 2 .
Both f 1 (ξ) = sin(ξ)/ξ and f 2 (ξ) = (1 − cos(ξ))/ξ 2 are oscillating functions with the global point of minimum corresponding to the first local minimum, and the same holds for f (ξ), with the point of minimum located close to those of f 1 (ξ) or f 2 (ξ) depending on τ (see Fig. 10).The asymptotic probability of correct decision is obtained from (C13) for τ → ∞.In this case the term in brackets vanished, and we obtain the Helstrom bound (11) independently of the value h, as long as the optimal Hamiltonian operator verifies (C5).In this Appendix we solve the discrimination problem with p = 1.In this case, the Hamiltonian operator no more contributes to the evolution of the quantum system since the term (A3) vanishes.We obtain hence a classical random walk, which can be optimize for the entries of T , With p = 1, not only the system of differential equations separates into the subsystems (B3) and (B4), but also the coherence components a i,j , b i,j evolves independently while the diagonal terms form the coupled system with v = [ρ 1,1 , ρ 2,2 , ρ 3,3 , ρ 4,4 ] T .As in the case p = 0, we can evaluate the fundamental set of solutions and the Wronskian of the system.The initial conditions for the system (D3) are defined from the diagonal entries of the initial states ρ 1 and ρ 2 .We then obtain the solutions ρ (1) 3,3 (t), ρ 4,4 (t) reported in Eqs.(D6), (D7), with the corresponding probability of correct decision evaluated in Eq. (D8).As we can see, this probability depends on ∆ρ 2 − ∆ρ 1 , ∆ρ x = ρ (D4) Asymptotically, for τ → ∞ the optimal probability of correct decision becomes which coincides with the Helstrom bound for the discrimination of the quantum states ρ (1,class.)and ρ (2,class.), obtained from ρ (1) , ρ (2) by removing the coherences, effectively turning a quantum state in a classical one.

FIG. 1 .
FIG.1.Examples of M −N −O network models.In each subfigure, the left layer (blue) collects the input nodes, the right layer (purple) the output nodes while in between (orange) the intermediate nodes, which can be organized in multiple layers as in Fig.1b.Directed edges refer to irreversible transfer of population, while plain links indicate both coherent and incoherent transport.By default the nodes of a layer are fully connected within themselves.We indicate with r when the connectivity is reduced, i.e. some links are removed.

FIG. 2 .
FIG.2.Optimized probability of correct decision with respect to (p, τ ) for the 2r − 2r − 2 model in the case of binary discrimination with pure states (1 st row), a pure state and a mixed state (2 nd row) and mixed states (3 rd row).The quantum states, reported in each subcaption, on the left column (a-c-e) have rx = 0, ry = 0, while those in the right column (b-d-f) have been rotated to have the ry coordinates in place of the rx ones.The Helstrom bound is reported in red dashed line, and in dash-dotted orange line we evaluate the Helstrom bound on the same quantum states after setting ry = 0.The color map is linear between blue and yellow and normalized between the minimum and the maximum point in each plot.

1 − 2 ,FIG. 3 .
FIG. 3. Probability of correct detection Pc for different variants of the 2r − 2r − 2 model as a function of τ for p = 0.The red dashed line shows the Helstrom bound for mixed quantum states in Eq. (18)

FIG. 4 .
FIG. 4. Probability of correct decision Pc (blue region) around the predicted performance (solid lines) with a noisy preparation of the quantum states to discriminate (5% of percent error).The red dashed line shows the Helstrom bound.For each p and τ = 1, 10 we run 10 4 evaluations of Pc, each with a different pair of quantum states.

FIG. 5 .
FIG. 5. Probability of correct decisionPc (blue region) as a function of the maximum percent error on the preparation of quantum states, for p = 0 and τ = 1, 10.The continuous line shows the average performance over 10 4 simulations.The black dash-dotted line shows the threshold corresponding to a random guess (0.5).The red dashed line shows the Helstrom bound.

1 FIG. 6 .
FIG.6.Probability of correct decision Pc (blue region) for a network prepared in a noisy configuration of its Hamiltonian coefficients Hi,j, as a function of its network static disorder percentage, for p = 0 (a), p = 0.1 (b) with τ = 1, τ = 10.The solid lines show the average performance over 10 4 runs.The black dash-dotted line shows the threshold corresponding to a random guess (0.5).The red dashed line shows the Helstrom bound.

= 8 FIG. 8 .
FIG. 8. Probability of correct detection Pc of pure qubits (19) for M = 4 (a) and M = 8 (b) as a function of time τ for different variants of a 2 − M − M model.

7 FIG. 9 .
FIG. 9. Probability of correct detection Pc for M = 4 as a function of time τ for different variants from a 4−4−4 model.The quantum states to discriminate are a mixture of a pure state and the completely mixed state with a factor α, 1 − α as in Eq. (20).
Appendix C: Solution of the master equation for p = 0

2 FIG. 10 . 1 .
FIG.10.Plot of the term f (ξ)/τ to minimize (blue line, circle markers), along with the terms f1(ξ) (orange line, triangular markers) and f2(ξ) (green line, square markers).Both f1(ξ) and f2(ξ) are oscillating functions with the point of minimum corresponding to the first local minimum.The global minimum of f (ξ) is between these two minimum, close to the minimum of f1(ξ) for τ ≈ 0 or the minimum of f2(ξ) for τ 1.In the plot the total evolution time is τ = 5.