Similar to its classical version, quantum Markovian evolution can be either time-discrete or time-continuous. Discrete quantum Markovian evolution is usually modeled with completely positive trace-preserving maps, while time-continuous evolution is often specified with superoperators referred to as "Lindbladians." Here, we address the following question: Being given a quantum map, can we find a Lindbladian that generates an evolution identical-when monitored at discrete instances of time-to the one induced by the map? It was demonstrated that the problem of getting the answer to this question can be reduced to an NP-complete (in the dimension N of the Hilbert space, the evolution takes place in) problem. We approach this question from a different perspective by considering a variety of machine learning (ML) methods and trying to estimate their potential ability to give the correct answer. Complimentarily, we use the performance of different ML methods as a tool to validate a hypothesis that the answer to the question is encoded in spectral properties of the so-called Choi matrix, which can be constructed from the given quantum map. As a test bed, we use two single-qubit models for which the answer can be obtained using the reduction procedure. The outcome of our experiment is that, for a given map, the property of being generated by a time-independent Lindbladian is encoded both in the eigenvalues and the eigenstates of the corresponding Choi matrix.
Similar to its classical version, quantum Markovian evolution can be either time-discrete or time-continuous. Discrete quantum Markovian evolution is usually modeled with completely positive trace-preserving maps, while time-continuous evolution is often specified with superoperators referred to as "Lindbladians." Here, we address the following question: Being given a quantum map, can we find a Lindbladian that generates an evolution identical-when monitored at discrete instances of time-to the one induced by the map? It was demonstrated that the problem of getting the answer to this question can be reduced to an NP-complete (in the dimension N of the Hilbert space, the evolution takes place in) problem. We approach this question from a different perspective by considering a variety of machine learning (ML) methods and trying to estimate their potential ability to give the correct answer. Complimentarily, we use the performance of different ML methods as a tool to validate a hypothesis that the answer to the question is encoded in spectral properties of the so-called Choi matrix, which can be constructed from the given quantum map. As a test bed, we use two single-qubit models for which the answer can be obtained using the reduction procedure. The outcome of our experiment is that, for a given map, the property of being generated by a time-independent Lindbladian is encoded both in the eigenvalues and the eigenstates of the corresponding Choi matrix.
A mathematical description of the reduced dynamics of an open quantum system can often be given in terms of a completely positive and trace-preserving map, also known as a quantum channel. In a seminal work by Wolf [], it was shown that deciding whether a given quantum channel was generated from an underlying effective Markovian dynamics, with time-independent generator of Lindblad form, is generally an NP-hard problem. The difficulty is related to the fact that one has to search through all possible branches of the operator logarithm of the map in order to identify if any of the resulting effective generators is of Lindblad form. In this work we show that in cases where one has access to the full reduced dynamics at all previous times (the dynamical map) one can significantly facilitate the search for an effective generator by making use of Floquet theory. By performing a spectral unwinding such that the effective micromotion is minimized, the effective Floquet generator is often an excellent candidate for an effective generator of Lindblad form. This significantly reduces the complexity of the search for an effective Lindbladian in many (though not all) cases. Our results are relevant for engineering Floquet Lindbladians in complex many-body systems.
We study the problem of learning an unknown Markovian open-system generator from access to its physical time evolution. This generator, called a Lindbladian, contains Hamiltonian and dissipative coefficients indexed by an exponentially large family of possible Pauli terms. We propose an efficient algorithm that learns arbitrary Lindbladians from time evolution under minimal assumptions. For a Lindbladian of dynamical strength at most Λ, the algorithm estimates every coefficient to error ε using O(Λ2/ε2) experiments and O(Λ/ε2) total evolution time, together with polynomial classical running time. The algorithm consists of two nonadaptive, ancilla-free, and control-free stages: 1. The support-learning stage outputs a candidate support of size poly(Λ/η) that contains every Hamiltonian and dissipative coordinate of magnitude at least η, using O(Λ2/η2) experiments with preparations of product Pauli eigenstates and single-qubit Pauli measurements. 2.The coefficient-learning stage estimates all coefficients in any candidate support of size M to error ε, using O(Λ2logM/ε2) experiments with preparations of random stabilizer states and measurements in random Clifford bases. Composing the two stages identifies and estimates every coefficient of an arbitrary Lindbladian in polynomial time. The experiment-count and total-evolution-time scalings match the lower bounds up to logarithmic factors, so the algorithm is nearly optimal for learning arbitrary Lindbladians.
Learning dynamics from repeated observation of the time evolution of an open quantum system, namely, the problem of quantum process tomography is an important task. This task is difficult in general, but, with some additional constraints could be tractable. This motivates us to look at the problem of Lindblad operator discovery from observations. We point out that for moderate size Hilbert spaces, low Kraus rank of the channel, and short time steps, the eigenvalues of the Choi matrix corresponding to the channel have a special structure. We use the least-square method for the estimation of a channel where, for fixed inputs, we estimate the outputs by classical shadows. The resultant noisy estimate of the channel can then be denoised by diagonalizing the nominal Choi matrix, truncating some eigenvalues, and altering it to a genuine Choi matrix. This processed Choi matrix is then compared to the original one. We see that as the number of samples increases, our reconstruction becomes more accurate. We also use tools from random matrix theory to understand the effect of estimation noise in the eigenspectrum of the estimated Choi matrix.
Characterizing the memory properties of the environment has become critical for the high-fidelity control of qubits and other advanced quantum systems. However, current non-Markovian tomography techniques are either limited to discrete superoperators, or they employ machine learning methods, neither of which provide physical insight into the dynamics of the quantum system. To circumvent this limitation, we design learning architectures that explicitly encode physical constraints like the properties of completely-positive trace-preserving maps in a differential form. This method preserves the versatility of the machine learning approach without sacrificing the efficiency and fidelity of traditional parameter estimation methods. Our approach provides the physical interpretability that machine learning and opaque superoperators lack. Moreover, it is aware of the underlying continuous dynamics typically disregarded by superoperator-based tomography. This paradigm paves the way to noise-aware optimal quantum control and opens a path to exploiting the bath as a control and error mitigation resource.
Abstract The spectral properties of operators generating Markov evolution provide essential information about processes induced by these operators. The celebrated theorem of Perron and Frobenius implies that spectra of classical discrete-time Markov operators, represented by stochastic matrices, are restricted to the unit disk. This property holds also for spectra of quantum stochastic maps (quantum channels), which describe quantum Markovian evolution in discrete time. Moreover, the spectra of stochastic N × N matrices are additionally restricted to a subset of the unit disk, called Karpelevi c ˘ region, the shape of which depends onN. We pose the question of whether the spectra of generators that induce continuous-time Markovian evolution can be corralled in a similar way. We propose renormalization procedures that allow us to answer the question affirmatively. The eigenvalues of the renormalized classical Markov generators are confined to modified Karpelevi c ˘ regions, whereas the eigenvalues of the renormalized quantum Markov generators are confined to the unit disk. We discuss spectral corralling for different symmetry classes of the generators and also make a few steps beyond the conventional assumptions of complete positivity and strict Markovianity. Our approach enables direct comparisons of the spectra of Markov generators arising from diverse fields, spanning multiple timescales, parameter ranges, and dimensionalities.
This work focuses on quantum machine learning and analyzes the power of one single qubit to solve classical machine learning problems. We explore possible strategies to address traditional supervised, unsupervised, and reinforcement learning tasks. In particular, we study binary and multinomial classification, regression problems, time series forecasting, clustering, and quantum reinforcement learning. Our results suggest that the same methodology could be used to address all three types of learning with different measurement strategies and, despite the strong limitation of reduced data dimensionality of the candidate problems, a single qubit can achieve similar or even improved performance with respect to state-of-the-art classic machine learning methods in many cases. As simulating the evolution of one qubit state is computationally efficient, our study enables the possibility to use the qubit model as a candidate solution to implement simple decision-making machine learning models in hardware with extremely low memory resources such as embedded systems or edge devices. The outcomes provided could be also of interest in academia and for the construction of demonstrators using low-scale contemporary quantum hardware.