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 neuromorph
ic (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.
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 i
nformation 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.
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.
In this paper, we develop a theory of learning nonlinear input-output maps with fading memory by dissipative quantum systems, as a quantum counterpart of the theory of approximating such maps using classical dynamical systems. The theory identifies t
he properties required for a class of dissipative quantum systems to be {em universal}, in that any input-output map with fading memory can be approximated arbitrarily closely by an element of this class. We then introduce an example class of dissipative quantum systems that is provably universal. Numerical experiments illustrate that with a small number of qubits, this class can achieve comparable performance to classical learning schemes with a large number of tunable parameters. Further numerical analysis suggests that the exponentially increasing Hilbert space presents a potential resource for dissipative quantum systems to surpass classical learning schemes for input-output maps.
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 i
s 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 K{u}mmerer 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.
Kerr nonlinear cavities displaying optical thresholding have been proposed for the realization of ultra-low power photonic logic gates. In the ultra-low photon number regime, corresponding to energy levels in the attojoule scale, quantum input-output
models become important to study the effect of unavoidable quantum fluctuations on the performance of such logic gates. However, being a quantum anharmonic oscillator, a Kerr-cavity has an infinite dimensional Hilbert space spanned by the Fock states of the oscillator. This poses a challenge to simulate and analyze photonic logic gates and circuits composed of multiple Kerr nonlinearities. For simulation, the Hilbert of the oscillator is typically truncated to the span of only a finite number of Fock states. This paper develops a quasi-principal components approach to identify important subspaces of a Kerr-cavity Hilbert space and exploits it to construct an approximate reduced model of the Kerr-cavity on a smaller Hilbert space. Using this approach, we find a reduced dimension model with a Hilbert space dimension of 15 that can closely match the magnitudes of the mean transmitted and reflected output fields of a conventional truncated Fock state model of dimension 75, when driven by an input coherent field that switches between two levels. For the same input, the reduced model also closely matches the magnitudes of the mean output fields of Kerr-cavity-based AND and NOT gates and a NAND latch obtained from simulation of the full 75 dimension model.
We derive the explicit analytical form of the time-dependent coupling parameter to an external field for perfect absorption of traveling single photon fields with arbitrary temporal profiles by a tunable single input-output open quantum system, which
can be realized as either a single qubit or single resonator system. However, the time-dependent coupling parameter for perfect absorption has a singularity at $t=0$ and constraints on real systems prohibit a faithful physical realization of the perfect absorber. A numerical example is included to illustrate the absorbers performance under practical limitations on the coupling strength.
Quantum error correction (QEC) is fundamental for quantum information processing but entails a substantial overhead of classically-controlled quantum operations, which can be architecturally cumbersome to accommodate. Here we discuss a novel approach
to designing elementary QEC memory cells, in which all control operations are performed autonomously by an embedded optical feedback loop. Our approach is natural for nanophotonic implementations in which each qubit can be coupled to its own optical resonator, and our design for a memory cell based on the quantum bit-flip or phase-flip code requires only five qubit-cavities (three for the register and two for the controller) connected by wave-guides. The photonic QEC circuit is entirely on-chip, requiring no external clocking or control, and during steady-state operation would only need to be powered by the injection of constant-amplitude coherent fields.
In this paper, we consider a linear quantum network composed of two distantly separated cavities that are connected via a one-way optical field. When one of the cavity is damped and the other is undamped, the overall cavity state obtains a large amou
nt of entanglement in its quadratures. This entanglement however immediately decays and vanishes in a finite time. That is, entanglement sudden-death occurs. We show that the direct measurement feedback method proposed by Wiseman can avoid this entanglement sudden-death, and further, enhance the entanglement. It is also shown that the entangled state under feedback control is robust against signal loss in a realistic detector, indicating the reliability of the proposed direct feedback method in practical situations.
