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Days on Diffraction

Quantum reservoir computing based on decoherence-free subspaces

Mikhail Altaisky, Natalia Kaputkina

Published Jun 16, 2025• 0 citations
Publisher

Abstract

The idea of coding the information in decoherence-free subspaces (DFS) to avoid decoherence in quantum computations has been receiving a lot of attention in circuit-based quantum computations, since first proposed by Zanardi & Rasetti [1] (Phys. Rev. Lett., 1997). Following [2] (Altaisky & Kaputkina, JISST, 2025), we study the possibility to harness DFS to quantum machine learning, where no explicit algorithm is present, but the quantum system learns to classify data. We have shown that a simple classification problem of distinguishing between 2-qubit product states and 2-qubit entangled states can be solved by 4-qubit quantum reservoir [3] (Ghosh, et al., npj Quantum Inf., 2019), the output of which we fed to decoherence-free subspaces.

DOI
10.1109/dd66835.2025.11263475

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•May 27, 2026•0 citations
The idea of coding the information in decoherence-free subspaces (DFS) to avoid decoherence in quantum computations has been receiving a lot of attention in circuit-based quantum computations, since first proposed by Zanardi & Rasetti [1] (Phys. Rev. Lett., 1997). Following [2] (Altaisky & Kaputkina, JISST, 2025), we study the possibility to harness DFS to quantum machine learning, where no explicit algorithm is present, but the quantum system learns to classify data. We have shown that a simple classification problem of distinguishing between 2-qubit product states and 2-qubit entangled states can be solved by 4-qubit quantum reservoir [3] (Ghosh, et al., npj Quantum Inf., 2019), the output of which we fed to decoherence-free subspaces.
We present numerical simulation of a six-qubit quantum reservoir network with an output implemented on a 5-dimensional decoherence-free subspace (DFS), working as a classifier between entangled and product states of the input quantum system, fed to the reservoir during a finite learning time. Since the dynamics of DFS is not affected by external fluctuations, no cooling is required, and the proposed model seems a promising candidate for future quantum artificial intelligence systems working at room temperatures and free of huge energy consumption.
In a quantum computer, qubits are often stored in identical two-level systems separated by a distance shorter than the characteristic wavelength of the reservoirs that are responsible for decoherence. In this case the collective qubit-reservoir interaction, rather than the individual qubit-reservoir interaction, may determine the decoherence properties. We study the collective decoherence behavior in between each step in certain quantum algorithms and propose a simple alternative of implementing quantum algorithms using a quantum trajectory that is close to a decoherence-free subspace that avoids unstable Dicke's superradiant states and Schrödinger's cat state.
It was shown recently [D. A. Lidar et al., Phys. Rev. Lett. 81, 2594 (1998)] that within the framework of the semigroup Markovian master equation, decoherence-free (DF) subspaces exist which are stable to first order in time to a perturbation. Here this result is extended to the non-Markovian regime and generalized. In particular, it is shown that within both the semigroup and the non-Markovian operator sum representation, DF subspaces are stable to all orders in time to a symmetry-breaking perturbation. DF subspaces are thus ideal for quantum memory applications. For quantum computation, however, the stability result does not extend beyond the first order. Thus, to perform robust quantum computation in DF subspaces, they must be supplemented with quantum error correcting codes.
A general scheme to perform universal, fault-tolerant quantum computation within decoherence-free subspaces (DFSs) is presented. At most two-qubit interactions are required, and the system remains within the DFS throughout the entire implementation of a quantum gate. We show explicitly how to perform universal computation on clusters of the four-qubit DFS encoding one logical qubit each under spatially symmetric (collective) decoherence. Our results have immediate relevance to quantum computer implementations in which quantum logic is implemented through exchange interactions, such as the recently proposed spin-spin coupled quantum dot arrays and donor-atom arrays.
A basic idea of quantum computing is surprisingly similar to that of kernel methods in machine learning, namely, to efficiently perform computations in an intractably large Hilbert space. In this Letter we explore some theoretical foundations of this link and show how it opens up a new avenue for the design of quantum machine learning algorithms. We interpret the process of encoding inputs in a quantum state as a nonlinear feature map that maps data to quantum Hilbert space. A quantum computer can now analyze the input data in this feature space. Based on this link, we discuss two approaches for building a quantum model for classification. In the first approach, the quantum device estimates inner products of quantum states to compute a classically intractable kernel. The kernel can be fed into any classical kernel method such as a support vector machine. In the second approach, we use a variational quantum circuit as a linear model that classifies data explicitly in Hilbert space. We illustrate these ideas with a feature map based on squeezing in a continuous-variable system, and visualize the working principle with two-dimensional minibenchmark datasets.
Reservoir computing is a neural network framework capable of nonlinear information processing. The ambiguous nature of the information processing unit provides a fascinating opportunity of employing physical systems as reservoirs. This has also spiked an interest in the research and development of quantum systems. As quantum computing still faces many problems such as scalability and the loss of coherence, quantum reservoir computing offers an alternative way of conducting intricate quantum tasks. These results are not limited to quantum tasks, as the evidence in quantum reservoir computing suggest a possible advantage over classical algorithms in classical nonlinear tasks. This work provides a comprehensive inspection of the theory behind quantum reservoir computing and an entanglement classification task conducted on the reservoir. This includes the review of tools for the measurement of entanglement, Gaussian and non-Gaussian nature of systems, machine learning and quantum computing. The theoretical review is accompanied by results of classical reservoir computing and a greater research on the capabilities of different quantum substrates used in quantum reservoir computing. The main part of this work is to build a working quantum reservoir computing system containing a fermionic reservoir. The task of the reservoir is to learn to classify quantum entanglement of two-mode bosonic inputs. The bosonic inputs used in this work are squeezed thermal states, photon added squeezed vacuum states, photon subtracted squeezed vacuum states and number states. This way Gaussian and non-Gaussian entanglement recognition is tested. Reservoirs of sizes 2, 3 and 4 modes are used and different connectivities inside the reservoir networks are studied. Even though the entanglement is assessed for different classes of states, the training is conducted with only squeezed thermal states