arXiv subjects

Hendra I. Nurdin

Publications and source records attributed to Hendra I. Nurdin.

At least 19 recordsLinked to original sources

arXiv2026

Quantum reservoir computing with repeated measurements on superconducting devices

Reservoir computing is a machine learning framework that uses artificial or physical dissipative dynamics to predict time-series data using nonlinearity and memory properties of dynamical systems. Quantum systems are considered promising physical reservoirs, and several quantum reservoir computing (QRC) schemes that exploit intrinsic quantum mechanical properties as computational resources have been proposed. This paper focuses on a particular QRC scheme that was theoretically proposed to use the repeated quantum non-demolition measurement to generate a time series, mainly for the purpose of reducing the total execution time compared to the conventional method without repeated measurement. We experimentally implemented this scheme on IBM superconducting devices and demonstrated that shorter execution time and higher accuracy can be achieved than with a specific conventional method. Then, we study the temporal information processing capacity to quantify the computational capability of the proposed QRC; in particular, we use this quantity to identify the measurement strength that finds the best balance between the amount of available information and the strength of dissipation. An experimental demonstration with a soft robot is also provided, where repeated measurement over 1,000 timesteps was effectively implemented. Finally, we implemented the repeated-measurement QRC scheme using 120 qubits, to test the potential computational power of a large-scale QRC.

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arXiv2026

A Single Atom in Front of a Mirror is a Universal Reservoir Computer

Universal approximation in reservoir computing is typically associated with a class of reservoirs. We show that universality can be associated with a single reservoir, considering a minimal setup of a single atom in front of a mirror. In its linear-transducer limit, our reservoir is a universal approximator of fading-memory maps under an operating class of checkable conditions, with a rate constant measured at the operating point. A given reservoir can reach arbitrary accuracy by changing measurement settings. The proof gives an explicit recipe: for a target accuracy, it specifies the required physical resources and resonator modes. Enlarging the number of accessible modes increases the matchable kernel span without reducing capability. Beyond the linear limit, the atom's saturation replaces high-order polynomial readouts, and the device operates on real-world tasks alongside classical baselines. Our results highlight an example of universality with a minimal quantum setup.

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arXiv2025

Minimalistic and Scalable Quantum Reservoir Computing Enhanced with Feedback

Quantum Reservoir Computing (QRC) leverages quantum systems to perform complex computational tasks with exceptional efficiency and reduced energy consumption. We introduce a minimalistic QRC framework utilizing as few as five atoms in a single-mode optical cavity, combined with continuous quantum measurement. The system is conveniently scalable, as newly added atoms naturally couple with existing ones via the shared cavity field. To achieve high computational expressivity with a minimal reservoir, we include two critical elements: reservoir feedback and polynomial regression. Reservoir feedback modifies the reservoir's dynamics without altering its internal quantum hardware, while polynomial regression nonlinearly enhances output resolution. We demonstrate significant QRC performance in memory retention and nonlinear data processing through two tasks: predicting chaotic time-series data via the Mackey-Glass task and classifying sine-square waveforms. This framework fulfills QRC's objectives to minimize hardware size and energy consumption, marking a significant advancement in integrating quantum physics with machine learning technology.

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arXiv2025

Practical Few-Atom Quantum Reservoir Computing

Quantum Reservoir Computing (QRC) harnesses quantum systems to tackle intricate computational problems with exceptional efficiency and minimized energy usage. This paper presents a QRC framework that utilizes a minimalistic quantum reservoir, consisting of only a few two-level atoms within an optical cavity. The system is inherently scalable, as newly added atoms automatically couple with the existing ones through the shared cavity field. We demonstrate that the quantum reservoir outperforms traditional classical reservoir computing in both memory retention and nonlinear data processing through two tasks, namely the prediction of time-series data using the Mackey-Glass task and the classification of sine-square waveforms. Our results show significant performance improvements with an increasing number of atoms, facilitated by non-destructive, continuous quantum measurements and polynomial regression techniques. These findings confirm the potential of QRC as a practical and efficient solution to addressing complex computational challenges in quantum machine learning.

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arXiv2025

Physical Reduced Stochastic Equations for Continuously Monitored Non-Markovian Quantum Systems with a Markovian Embedding

An effective approach to modeling non-Markovian quantum systems is to embed a principal (quantum) system of interest into a larger quantum system. A widely employed embedding is one that uses another quantum system, referred to as the auxiliary system, which is coupled to the principal system, and both the principal and auxiliary can be coupled to quantum white noise processes. The principal and auxiliary together form a quantum Markov system and the quantum white noises act as a bath (environment) for this system. Recently it was shown that the conditional evolution of the principal system in this embedding under continuous monitoring by a travelling quantum probe can be expressed as a system of coupled stochastic differential equations (SDEs) that involve only operators of the principal system. The reduced conditional state of the principal only (conditioned on the measurement outcomes) is determined by the ``diagonal" blocks of this coupled systems of SDEs. It is shown here that the ``off-diagonal" blocks can be exactly eliminated up to their initial conditions, leaving a reduced closed system of SDEs for the diagonal blocks only. Under additional conditions the off-diagonal initial conditions can be made to vanish. This new closed system of equations, which includes an integration term involving a two-time stochastic kernel, represents the non-Markovian stochastic dynamics of the principal system under continuous-measurement. The system of equations determine the reduced conditional state of the principal only and may be viewed as a stochastic Nakajima-Zwanzig type of equation for continuously monitored non-Markovian quantum systems.

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arXiv2025

Stochastic Reservoir Computers

Reservoir computing is a form of machine learning that utilizes nonlinear dynamical systems to perform complex tasks in a cost-effective manner when compared to typical neural networks. Many recent advancements in reservoir computing, in particular quantum reservoir computing, make use of reservoirs that are inherently stochastic. However, the theoretical justification for using these systems has not yet been well established. In this paper, we investigate the universality of stochastic reservoir computers, in which we use a stochastic system for reservoir computing using the probabilities of each reservoir state as the readout instead of the states themselves. In stochastic reservoir computing, the number of distinct states of the entire reservoir computer can potentially scale exponentially with the size of the reservoir hardware, offering the advantage of compact device size. We prove that classes of stochastic echo state networks, and therefore the class of all stochastic reservoir computers, are universal approximating classes. We also investigate the performance of two practical examples of stochastic reservoir computers in classification and chaotic time series prediction. While shot noise is a limiting factor in the performance of stochastic reservoir computing, we show significantly improved performance compared to a deterministic reservoir computer with similar hardware in cases where the effects of noise are small.

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arXiv2025

Improving the Performance of Echo State Networks Through State Feedback

Reservoir computing, using nonlinear dynamical systems, offers a cost-effective alternative to neural networks for complex tasks involving processing of sequential data, time series modeling, and system identification. Echo state networks (ESNs), a type of reservoir computer, mirror neural networks but simplify training. They apply fixed, random linear transformations to the internal state, followed by nonlinear changes. This process, guided by input signals and linear regression, adapts the system to match target characteristics, reducing computational demands. A potential drawback of ESNs is that the fixed reservoir may not offer the complexity needed for specific problems. While directly altering (training) the internal ESN would reintroduce the computational burden, an indirect modification can be achieved by redirecting some output as input. This feedback can influence the internal reservoir state, yielding ESNs with enhanced complexity suitable for broader challenges. In this paper, we demonstrate that by feeding some component of the reservoir state back into the network through the input, we can drastically improve upon the performance of a given ESN. We rigorously prove that, for any given ESN, feedback will almost always improve the accuracy of the output. For a set of three tasks, each representing different problem classes, we find that with feedback the average error measures are reduced by $30\%-60\%$. Remarkably, feedback provides at least an equivalent performance boost to doubling the initial number of computational nodes, a computationally expensive and technologically challenging alternative. These results demonstrate the broad applicability and substantial usefulness of this feedback scheme.

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arXiv2024

Saturation of the Multiparameter Quantum Cramér-Rao Bound at the Single-Copy Level with Projective Measurements

Quantum parameter estimation theory is an important component of quantum information theory and provides the statistical foundation that underpins important topics such as quantum system identification and quantum waveform estimation. When there is more than one parameter the ultimate precision in the mean square error given by the quantum Cramér-Rao bound is not necessarily achievable. For non-full rank quantum states, it was not known when this bound can be saturated (achieved) when only a single copy of the quantum state encoding the unknown parameters is available. This single-copy scenario is important because of its experimental/practical tractability. Recently, necessary and sufficient conditions for saturability of the quantum Cramér-Rao bound in the multiparameter single-copy scenario have been established in terms of i) the commutativity of a set of projected symmetric logarithmic derivatives and ii) the existence of a unitary solution to a system of coupled nonlinear partial differential equations. New sufficient conditions were also obtained that only depend on properties of the symmetric logarithmic derivatives. In this paper, key structural properties of optimal measurements that saturate the quantum Cramér-Rao bound are illuminated. These properties are exploited to i) show that the sufficient conditions are in fact necessary and sufficient for an optimal measurement to be projective, ii) give an alternative proof of previously established necessary conditions, and iii) describe general POVMs, not necessarily projective, that saturate the multiparameter QCRB. Examples are given where a unitary solution to the system of nonlinear partial differential equations can be explicitly calculated when the required conditions are fulfilled.

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arXiv2024

Saturability of the Quantum Cramér-Rao Bound in Multiparameter Quantum Estimation at the Single-Copy Level

The quantum Cramér-Rao bound (QCRB) as the ultimate lower bound for precision in quantum parameter estimation is only known to be saturable in the multiparameter setting in special cases and under conditions such as full or average commutavity of the symmetric logarithmic derivatives (SLDs) associated with the parameters. Moreover, for general mixed states, collective measurements over infinitely many identical copies of the quantum state are generally required to attain the QCRB. In the important and experimentally relevant single-copy scenario, a necessary condition for saturating the QCRB in the multiparameter setting for general mixed states is the so-called partial commutativity condition on the SLDs. However, it is not known if this condition is also sufficient. This paper establishes necessary and sufficient conditions for saturability of the multiparameter QCRB in the single-copy setting in terms of the commutativity of a set of projected SLDs and the existence of a unitary solution to a system of nonlinear partial differential equations. New necessary conditions that imply partial commutativity are also obtained, which together with another condition become sufficient. Moreover, when the sufficient conditions are satisfied an optimal measurement saturating the QCRB can be chosen to be projective and explicitly characterized. An example is developed to illustrate the case of a multiparameter quantum state where the conditions derived herein are satisfied and can be explicitly verified.

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arXiv2024

Markovian Embeddings of Non-Markovian Quantum Systems: Coupled Stochastic and Quantum Master Equations for Non-Markovian Quantum Systems

Quantum Markov models are employed ubiquitously in quantum physics and in quantum information theory due to their relative simplicity and analytical tractability. In particular, these models are known to give accurate approximations for a wide range of quantum optical and mesoscopic systems. However, in general, the validity of the Markov approximation entails assumptions regarding properties of the system of interest and its environment, which may not be satisfied or accurate in arbitrary physical systems. Therefore, developing useful modelling tools for general non-Markovian quantum systems for which the Markov approximation is inappropriate or deficient is an undertaking of significant importance. This work considers non-Markovian principal quantum systems that can be embedded in a larger Markovian quantum system with one or more compound baths consisting of an auxiliary quantum system and a quantum white noise field, and derives a set of coupled stochastic and quantum master equations for embedded non-Markovian quantum systems. The case of a purely Hamiltonian coupling between the principal and auxiliary systems as a closed system without coupling to white noises is included as a special case. The results are expected to be of interest for (open-loop and feedback) control of continuous-time non-Markovian systems and studying reduced models for numerical simulation of such systems. They may also shed more light on the general structure of continuous-time non-Markovian quantum systems.

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arXiv2022

Multi-Axis Control of a Qubit in the Presence of Unknown Non-Markovian Quantum Noise

In this paper, we consider the problem of open-loop control of a qubit that is coupled to an unknown fully quantum non-Markovian noise (either bosonic or fermionic). A graybox model that is empirically obtained from measurement data is employed to approximately represent the unknown quantum noise. The estimated model is then used to calculate the open-loop control pulses under constraints on the pulse amplitude and timing. For the control pulse optimization, we explore the use of gradient descent and genetic optimization methods. We consider the effect of finite sampling on estimating expectation values of observables and show results for single- and multi-axis control of a qubit.

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arXiv2022

Parameter estimation and system identification for continuously-observed quantum systems

This paper gives an overview of parameter estimation and system identification for quantum input-output systems by continuous observation of the output field. We present recent results on the quantum Fisher information of the output with respect to unknown dynamical parameters. We discuss the structure of continuous-time measurements as solutions of the quantum Zakai equation, and their relationship to parameter estimation methods. Proceeding beyond parameter estimation, the paper also gives an overview of the emerging topic of quantum system identification for black-box modeling of quantum systems by continuous observation of a traveling wave probe, for the case of ergodic quantum input-output systems and linear quantum systems. Empirical methods for such black-box modeling are also discussed.

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arXiv2021

From the Heisenberg to the Schrödinger Picture: Quantum Stochastic Processes and Process Tensors

A general theory of quantum stochastic processes was formulated by Accardi, Frigerio and Lewis in 1982 within the operator-algebraic framework of quantum probability theory, as a non-commutative extension of the Kolmogorovian classical stochastic processes. More recently, studies on non-Markovian quantum processes have led to the discrete-time process tensor formalism in the Schrödinger picture to describe the outcomes of sequential interventions on open quantum systems. However, there has been no treatment of the relationship of the process tensor formalism to the quantum probabilistic theory of quantum stochastic processes. This paper gives an exposition of quantum stochastic processes and the process tensor and the relationship between them. In particular, it is shown how the latter emerges from the former via extended correlation kernels incorporating ancillas.

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arXiv2021

A Small-Gain Theorem for Discrete-Time Convergent Systems and Its Applications

Convergent, contractive or incremental stability properties of nonlinear systems have attracted interest for control tasks such as observer design, output regulation and synchronization. The convergence property plays a central role in the neuromorphic (brain-inspired) computing of reservoir computing, which seeks to harness the information processing capability of nonlinear systems. This paper presents a small-gain theorem for discrete-time output-feedback interconnected systems to be uniformly input-to-output convergent (UIOC) with outputs converging to a bounded reference output uniquely determined by the input. A small-gain theorem for interconnected time-varying discrete-time uniform input-to-output stable systems that could be of separate interest is also presented as an intermediate result. Applications of the UIOC small-gain theorem are illustrated in the design of observer-based controllers and interconnected nonlinear classical and quantum dynamical systems (as reservoir computers) for black-box system identification.

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arXiv2020

Synthesizing Averaged Virtual Oscillator Dynamics to Control Inverters with an Output LCL Filter

In commercial inverters, an LCL filter is considered an integral part to filter out the switching harmonics and generate a sinusoidal output voltage. The existing literature on the averaged virtual oscillator controller (VOC) dynamics is for current feedback before the output LCL filter that contains the switching harmonics or for inductive filters ignoring the effect of filter capacitance. In this work, a new version of averaged VOC dynamics is presented for islanded inverters with current feedback after the LCL filter thus avoiding the switching harmonics going into the VOC. The embedded droop-characteristics within the averaged VOC dynamics are identified and a parameter design procedure is presented to regulate the output voltage magnitude and frequency according to the desired ac-performance specifications. Further, a power dispatch technique based on this newer version of averaged VOC dynamics is presented to simultaneously regulate both the active and reactive output power of two parallel-connected islanded inverters. The control laws are derived and a power security constraint is presented to determine the achievable power set-point. Simulation results for load transients and power dispatch validate the proposed version of averaged VOC dynamics.

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arXiv2020

Temporal Information Processing on Noisy Quantum Computers

The combination of machine learning and quantum computing has emerged as a promising approach for addressing previously untenable problems. Reservoir computing is an efficient learning paradigm that utilizes nonlinear dynamical systems for temporal information processing, i.e., processing of input sequences to produce output sequences. Here we propose quantum reservoir computing that harnesses complex dissipative quantum dynamics. Our class of quantum reservoirs is universal, in that any nonlinear fading memory map can be approximated arbitrarily closely and uniformly over all inputs by a quantum reservoir from this class. We describe a subclass of the universal class that is readily implementable using quantum gates native to current noisy gate-model quantum computers. Proof-of-principle experiments on remotely accessed cloud-based superconducting quantum computers demonstrate that small and noisy quantum reservoirs can tackle high-order nonlinear temporal tasks. Our theoretical and experimental results pave the path for attractive temporal processing applications of near-term gate-model quantum computers of increasing fidelity but without quantum error correction, signifying the potential of these devices for wider applications including neural modeling, speech recognition and natural language processing, going beyond static classification and regression tasks.

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arXiv2019

Mathematical Models of Markovian Dephasing

We develop a notion of dephasing under the action of a quantum Markov semigroup in terms of convergence of operators to a block-diagonal form determined by irreducible invariant subspaces. If the latter are all one-dimensional, we say the dephasing is maximal. With this definition, we show that a key necessary requirement on the Lindblad generator is bistochasticity, and focus on characterizing whether a maximally dephasing evolution may be described in terms of a unitary dilation with only classical noise, as opposed to a genuine non-commutative Hudson-Parthasarathy dilation. To this end, we make use of a seminal result of Kümmerer and Maassen on the class of commutative dilations of quantum Markov semigroups. In particular, we introduce an intrinsic quantity constructed from the generator, which vanishes if and only if the latter admits a self-adjoint representation and which quantifies the degree of obstruction to having a classical diffusive noise model.

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arXiv2019

Quantum Stochastic Processes and the Modelling of Quantum Noise

This brief article gives an overview of quantum mechanics as a {\em quantum probability theory}. It begins with a review of the basic operator-algebraic elements that connect probability theory with quantum probability theory. Then quantum stochastic processes is formulated as a generalization of stochastic processes within the framework of quantum probability theory. Quantum Markov models from quantum optics are used to explicitly illustrate the underlying abstract concepts and their connections to the quantum regression theorem from quantum optics.

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