Search Talks

In this lecture, we provide an introduction to quantum trajectory theory. We present various mathematical problems that arise within this context. In particular, we introduce approaches for analyzing the asymptotic behavior, convergence speed, and stabilization of quantum trajectories toward different states or subspaces through feedback control strategies. Our study includes both quantum non-demolition (QND) measurements and generic (non-QND) measurements in discrete-time and continuous-time settings.

Recent developments in the architecture of quantum computers have enabled their use in applications for various information-processing tasks. This information becomes unreliable primarily due to the erroneous implementation of control methods for state preparation and measurements and the qubit’s inability to store information for long periods in the presence of uncontrollable noise sources. Conventional noise spectroscopy protocols can characterize and model environmental noise but are usually resource-intensive and lengthy. Moreover, the underlying noise can vary over time, making noise profile extraction futile as this new information cannot be harnessed to improve quantum error correction or dynamical decoupling protocols. In this work, we address this challenge using a machine learning-based methodology to swiftly extract noise spectra of multiple qubits and demonstrate a possible noise mitigation strategy. The procedure involves implementing undemanding dynamical decoupling sequences to record coherence decays of the investigated qubits and then predict the underlying noise spectra with the help of a convolution neural network pre-trained on a synthetic dataset. The protocol is virtually hardware-agnostic. However, we validated its effectiveness using IBM’s superconducting qubits. We used these rapidly obtained yet accurate noise spectra to design bespoke dynamic decoupling sequences and perform time-dependent noise spectroscopy.

A variable-range interacting Ising model with spin-$s$ particles exhibits distinct behavior depending on the fall-off rates in the range of interactions, notably non-local (NL), quasi-local (QL), and local, which are based on the equilibrium properties.  It is unknown if such a transition is respected in the dynamical framework. We use an analytically solvable model in arbitrary spatial dimension ($D$), to establish a dynamical non-local (dNL) behavior, which does not agree with the known result of equilibrium NL behavior. We analyze the profiles of topological entanglement entropy (TEE), mutual information, Lieb-Robinson bound (LRB) and genuine multipartite entanglement (GME) of the weighted graph state (WGS), prepared when the multi-level maximally coherent state at each site evolves according to the long-range spin-$s$ Ising Hamiltonian. Specifically, we demonstrate that the connection between the LRB profile and the divergence in the first derivative of GME with respect to the fall-off rate in the WGS can indicate the transition point from dNL to a dynamical local/quasi-local (dQL) regimes.

After introducing the basic notions about tensors, I will discuss different aspects of quantum entanglement in the framework of tensor norms. I will show how this point of view can bring new insights to this fundamental notion of quantum theory and how new entanglement criteria can be naturally obtained in this way.

Theoretical tools used in processing continuous measurement records from real experiments to obtain quantum trajectories can easily lead to numerical errors due to a non-infinitesimal time resolution. In this work, we propose a systematic assessment of the accuracy of a map. We perform error analyses for diffusive quantum trajectories, based on single-time-step Kraus operators proposed in the literature, and find the orders in time increment to which such operators satisfy the conditions for valid average quantum evolution (completely positive, convex-linear, and trace-preserving), and the orders to which they match the Lindblad solutions. Given these error analyses, we propose a Kraus operator that satisfies the valid average quantum evolution conditions and agrees with the Lindblad master equation, to second order in the time increment, thus surpassing all other existing approaches. In order to test how well our proposed operator reproduces exact quantum trajectories, we analyze two examples of qubit measurement, where exact maps can be derived: a qubit subjected to a dispersive (z-basis) measurement and a fluorescence (dissipative) measurement. We show analytically that our proposed operator gives the smallest average trace distance to the exact quantum trajectories, compared to existing approaches.

In this talk, I will consider a finite-level quantum system linearly coupled to a bosonic reservoir, that is the prototypical example of an open quantum system. I will present recent results on the reduced dynamics of the finite system when the coupling constant tends to infinity, i.e. in the ultrastrong coupling limit. In particular, I will show that the dynamics corresponds to a nonselective projective measurement followed by a unitary evolution with an effective (Zeno) Hamiltonian. I will also discuss the connection with the usual setting for the quantum Zeno effect, based on repeated measurements.
The rigorous proof of the limit is quite simple and can be generalized to the case of a small system interacting with two reservoirs when one of the couplings is finite and the other one tends to infinity. In this second scenario the reduced dynamics is richer and possibly non-Markovian.
Joint work with Marco Merkli, arXiv:2411.06817.    

Entanglement of pure and mixed quantum states.

After introducing the basic notions about tensors, I will discuss different aspects of quantum entanglement in the framework of tensor norms. I will show how this point of view can bring new insights to this fundamental notion of quantum theory and how new entanglement criteria can be naturally obtained in this way.

We consider weak unitary symmetries of Markovian open quantum systems at the level of the joint dynamics of the system and its environment described by a continuous matrix product state, as well as for stochastic quantum trajectories of the system, obtained by conditioning on counting measurements of the environment. We derive necessary and sufficient conditions under which the dynamics of these different descriptions exhibit a weak symmetry, in turn characterising the resulting symmetries of their generators. In particular, this depends on whether the counting measurement satisfies the conditions we derive. In doing so we also consider the possible gauge transformations for generators of quantum trajectories, i.e. when two representations of the master operator produce equivalent trajectory ensembles.

We consider the estimation of parameters encoded in the measurement record of a continuously monitored quantum system in the jump unraveling. This unraveling picture corresponds to a single-shot scenario, where information is continuously gathered. Here, it is generally difficult to assess the precision of the estimation procedure via the Fisher Information due to intricate temporal correlations and memory effects. In this paper we provide a full set of solutions to this problem. First, for multi-channel renewal processes we relate the Fisher Information to an underlying Markov chain and derive a easily computable expression for it. For non-renewal processes, we introduce a new algorithm that combines two methods: the monitoring operator method for metrology and the Gillespie algorithm which allows for efficient sampling of a stochastic form of the Fisher Information along individual quantum trajectories. We show that this stochastic Fisher Information satisfies useful properties related to estimation in the single-shot scenario. Finally, we consider the case where some information is lost in data compression/post-selection, and provide tools for computing the Fisher Information in this case. All scenarios are illustrated with instructive examples from quantum optics and condensed matter.

Quantum experiments are performed in noisy platforms. In NISQ devices, realistic setups can be described by open systems or noisy Hamiltonians. Using this setup, we explore a number of dynamical schemes and control techniques. First, starting from a generic noisy Hamiltonian, I will show how noise can help simulate long-range and many-body interaction in a quantum platform [1]. Second, in the setup of shortcut to adiabaticity extended to open quantum systems, we adapt our noisy Hamiltonian to control the thermalization of a harmonic oscillator [2] and generate a squeezed thermal state [3] in arbitrary time. 

Third, adding non-Hermiticity in the picture [3], I will show how noise allows for a rich control of the dynamics, and induced a new phase in which the lossy state becomes stable. More generally, we characterize the quantum dynamics generated by a non-Hermitian Hamiltonian subject to stochastic perturbations in its anti-Hermitian part, describing fluctuating gains and losses. 

Finally, I will briefly show results where we do not look at the noise-averaged density matrix but at an observable introduced as the stochastic operator variance (SOV), which characterizes the deviations of any operator from the noise-averaged operator in a stochastic evolution governed by the Hamiltonian.  Surprisingly, we find that the evolution of the noise-averaged variance relates to an out-of-time-order correlator (OTOC), which connects fluctuations of the system with scrambling, and thus allows computing the Lyapunov exponent.

[1] A. Chenu, M. Beau, J. Cao, and A. del Campo. Phys. Rev. Lett. 118:140403 (2017)
[2] L. Dupays, I. L. Egusquiza, A. del Campo,  and A. Chenu. Superadiabatic thermalization of a quantum oscillator by engineered dephasing, Phys. Rev. Res. 2:033178 (2020)
[3] L. Dupays and A. Chenu. Dynamical engineering of squeezed thermal state, Quantum 5:449 (2021)
[4] P. Martinez-Azcona, A.Kundu, A. Saxena, A. del Campo, and A. Chenu, ArXiv 2407.07746 
[5] P. Martinez-Azcona, A.Kundu, A. del Campo, and A. Chenu, Phys. Rev. Lett. 131:16202 (2023). 

State smoothing is a technique to estimate a state at a particular time, conditioned on information obtained both before (past) and after (future) that time. For a classical system, the smoothed state is a normalized product of the filtered state (a state conditioned only on the past measurement information and the initial preparation) and the retrofiltered effect (depending only on the future measurement information). For the quantum case, whilst there are well-established analogues of the filtered state (ρ) and retrofiltered effect (E), their product does not, in general, provide a valid quantum state for smoothing. However, this procedure does seem to work when ρ and E are mutually diagonalizable. This fact has been used to obtain smoothed quantum states — more pure than the filtered states — in a number of experiments on continuously monitored quantum systems, in cavity QED and atomic systems. In this paper we show that there is an implicit assumption underlying this technique: that if all the information were known to the observer, the true system state would be one of the diagonal basis states. This assumption does not necessarily hold, as the missing information is quantum information. It could be known to the observer only if it were turned into a classical measurement record, but then its nature depends on the choice of measurement. We show by a simple model that, depending on that measurement choice, the smoothed quantum state can: agree with that from the classical method; disagree with it but still be co-diagonal with it; or not even be co-diagonal with it. That is, just because filtering and retrofiltering appear classical does not mean classical smoothing theory is applicable in quantum experiments.
Kiarn T. Laverick, Prahlad Warszawski, Areeya Chantasri, and Howard M. Wiseman

Error mitigation (EM) methods are crucial for obtaining reliable results in the realm of noisy intermediate-scale quantum (NISQ) computers, where noise significantly impacts output accuracy. Some EM protocols are particularly efficient for specific types of noise. Yet the noise in the actual hardware may not align with that.

I will introduce "noise-tailoring" (NT) — an innovative strategy designed to modify the structure of the noise associated with two-qubit gates through statistical sampling. I will discuss its application for IBM's quantum computers and show evidence that NT+EM result can be up to an order of magnitude more accurate than the result of EM alone.

I will also discuss the overheads that accompany the NT method and that may limit its practical applicability.