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arXiv:2003.13257 · Physics Letters A

Quantum Stochastic Walk models for quantum state discrimination

Nicola Dalla Pozza, Filippo Caruso

Published Dec 11, 2019• 3 citations
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Abstract

Quantum Stochastic Walks (QSW) allow for a generalization of both quantum and classical random walks by describing the dynamic evolution of an open quantum system on a network, with nodes corresponding to quantum states of a fixed basis. We consider the problem of quantum state discrimination on such a system, and we solve it by optimizing the network topology weights. Finally, we test it on different quantum network topologies and compare it with optimal theoretical bounds.

DOI
10.1016/j.physleta.2019.126195

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Quantum Stochastic Walks (QSW) allow for a generalization of both quantum and classical random walks by describing the dynamic evolution of an open quantum system on a network, with nodes corresponding to quantum states of a fixed basis. We consider the problem of quantum state discrimination on such a system, and we solve it by optimizing the network topology weights. Finally, we test it on different quantum network topologies and compare it with optimal theoretical bounds.
A fundamental problem in quantum information processing is the discrimination among 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.
We introduce the quantum stochastic walk (QSW), which determines the evolution of a generalized quantum-mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all possible quantum, classical, and quantum-stochastic transitions from a vertex as defined by its connectivity. We show how the family of possible QSWs encompasses both the classical random walk (CRW) and the quantum walk (QW) as special cases but also includes more general probability distributions. As an example, we study the QSW on a line and the glued tree of depth three to observe the behavior of the QW-to-CRW transition.
Abstract Quantum walks have been shown to have a wide range of applications, from artificial intelligence, to photosynthesis, and quantum transport. Quantum stochastic walks (QSWs) generalize this concept to additional non-unitary evolution. In this paper, we propose a trajectory-based quantum simulation protocol to effectively implement a family of discrete-time QSWs in a quantum device. After deriving the protocol for a 2-vertex graph with a single edge, we show how our protocol generalizes to a graph with arbitrary topology and connectivity. The straightforward generalization leads to simple scaling of the protocol to complex graphs. Finally, we show how to simulate a restricted class of continuous-time QSWs by a discrete-time QSW, and how this is amenable to our simulation protocol for discrete-time QSWs.
Abstract The study of quantum walks has been shown to have a wide range of applications in areas such as artificial intelligence, the study of biological processes, and quantum transport. The quantum stochastic walk (QSW), which allows for incoherent movement of the walker, and therefore, directionality, is a generalization on the fully coherent quantum walk. While a QSW can always be described in Lindblad formalism, this does not mean that it can be microscopically derived in the standard weak-coupling limit under the Born–Markov approximation. This restricts the class of QSWs that can be experimentally realized in a simple manner. To circumvent this restriction, we introduce a technique to simulate open system evolution on a fully coherent quantum computer, using a quantum trajectories style approach. We apply this technique to a broad class of QSWs, and show that they can be simulated with minimal experimental resources. Our work opens the path towards the experimental realization of QSWs on large graphs with existing quantum technologies.
Abstract Financial markets are noisy yet contain a latent graph–theoretic structure that can be exploited for superior risk-adjusted returns. We propose a \emph{quantum-stochastic-walk} (QSW) optimizer that embeds assets in a weighted graph—nodes are securities, edges encode the return–covariance kernel—and derives portfolio weights from the stationary distribution of the walk. Three empirical studies support the method. (i)~On the \textbf{Top-100 S\&P constituents} (2016–2024) six scenario portfolios fitted on 1- and 2-year windows lift the out-of-sample Sharpe ratio by up to \textbf{+27\,\%} while slashing annual turnover from 480%480\%480% (mean–variance) to 222–90%90\%90%. (ii)~A \mbox{625625625} grid search isolates a robust sweet-spot—α,λ ⁣≲ ⁣0.5\alpha,\lambda\!\lesssim\!0.5α,λ≲0.5, ω ⁣∈[0.2,0.4]\omega\!\in[0.2,0.4]ω∈[0.2,0.4]—that delivers Sharpe ≈0.97\approx0.97≈0.97 at ≤5%\le5\%≤5% turnover and Herfindahl–Hirschman index(HHI) ∼0.01{\sim}0.01∼0.01. (iii)~Repeating the full grid on \textbf{50 random 100-stock subsets} of the S\&P\,500 generates 31,350 additional back-tests: the best-per-draw QSW beats re-optimized mean–variance on Sharpe in 54\,\% of samples and \emph{always} wins on trading efficiency, with median turnover \textbf{36\,\%} versus \textbf{351\,\%}. Overall, QSW raises the annualized Sharpe ratio by \textbf{15\,\%} and cuts average turnover by \textbf{90\,\%} relative to classical optimization, all while remaining comfortably within UCITS 5/10/40 rule. The findings demonstrate that hybrid quantum–classical dynamics can uncover non-linear dependencies overlooked by quadratic models, offering a practical low-cost weighting engine for themed ETFs and other systematic mandates.
Classical mean-variance optimization is powerful in theory but fragile in practice, often producing highly concentrated, high-turnover portfolios. Naive equal-weight (1/N) portfolios are more robust but largely ignore cross-sectional information. We propose a quantum stochastic walk (QSW) framework that embeds assets in a weighted graph and derives portfolio weights from the stationary distribution of a hybrid quantum-classical walk. The resulting allocations behave as a “smart 1/N” portfolio: structurally close to equal-weight, but with small, data-driven tilts and a controllable level of trading. On recent S&P 500 universes, QSW portfolios match the diversification and stability of 1/N while delivering higher risk-adjusted returns than both mean-variance and naive benchmarks. A comprehensive hyper-parameter grid search shows that this behavior is structural rather than the result of fine-tuning and yields simple design rules for practitioners. A 34-year, multi-universe robustness study with rolling re-optimization further demonstrates that the QSW optimizer preserves these advantages across market regimes. Overall, the QSW framework improves risk-adjusted performance while maintaining strong diversification and moderate turnover.