arXiv subjects

Francesco Ticozzi

Publications and source records attributed to Francesco Ticozzi.

At least 37 records · Page 2Linked to original sources

arXiv2024

Dissipative Feedback Switching for Quantum Stabilization

Switching controlled dynamics allows for fast, flexible control design methods for quantum stabilization of pure states and subspaces, which naturally include both Hamiltonian and dissipative control actions. A novel approach to measurement-based, dissipative feedback design is introduced, and extends the applicability of switching techniques with respect to previously proposed ones, as it does not need stringent invariance assumptions, while it still avoids undesired chattering or Zeno effects by modulating the control intensity. When the switching dynamics do leave the target invariant, on the other hand, we show that exponential convergence to the target can be enforced without modulation, and switching times that can be either fixed or stochastic with hysteresis to avoid chattering. The effectiveness of the proposed methods is illustrated via numerical simulations of simple yet paradigmatic examples, demonstrating how switching strategies converge faster than open-loop engineered dissipation.

quant-ph↗

arXiv2024

Improving robustness of quantum feedback control with reinforcement learning

Obtaining reliable state preparation protocols is a key step towards practical implementation of many quantum technologies, and one of the main tasks in quantum control. In this work, different reinforcement learning approaches are used to derive a feedback law for state preparation of a desired state in a target system. In particular, we focus on the robustness of the obtained strategies with respect to different types and amount of noise. Comparing the results indicates that the learned controls are more robust to unmodeled perturbations with respect to simple feedback strategy based on optimized population transfer, and that training on simulated nominal model retain the same advantages displayed by controllers trained on real data. The possibility of effective off-line training of robust controllers promises significant advantages towards practical implementation.

quant-ph↗

arXiv2023

Optimizing Measurements Sequences for Quantum State Verification

We consider the problem of deciding whether a given state preparation, i.e., a source of quantum states, is accurate, namely produces states close to a target one within a prescribed threshold. We show that, when multiple measurements need to be used, the order of measurements is critical for quickly assessing accuracy. We propose and compare different strategies to compute optimal or suboptimal measurement sequences either relying solely on a priori information, i.e., the target state for state preparation, or actively adapting the sequence to the previously obtained measurements. Numerical simulations show that the proposed algorithms reduce significantly the number of measurements needed for verification, and indicate an advantage for the adaptive protocol especially assessing faulty preparations.

quant-ph↗

arXiv2023

Algebraic Reduction of Hidden Markov Models

The problem of reducing a Hidden Markov Model (HMM) to one of smaller dimension that exactly reproduces the same marginals is tackled by using a system-theoretic approach. Realization theory tools are extended to HMMs by leveraging suitable algebraic representations of probability spaces. We propose two algorithms that return coarse-grained equivalent HMMs obtained by stochastic projection operators: the first returns models that exactly reproduce the single-time distribution of a given output process, while in the second the full (multi-time) distribution is preserved. The reduction method exploits not only the structure of the observed output, but also its initial condition, whenever the latter is known or belongs to a given subclass. Optimal algorithms are derived for a class of HMM, namely observable ones.

cs.LG↗

arXiv2021

Stabilization via feedback switching for quantum stochastic dynamics

We propose a new method for pure-state and subspace preparation in quantum systems, which employs the output of a continuous measurement process and switching dissipative control to improve convergence speed, as well as robustness with respect to the initial conditions. In particular, we prove that the proposed closed-loop strategy makes the desired target globally asymptotically stable both in mean and almost surely, and we show it compares favorably against a time-based and a state-based switching control law, with significant improvements in the case of faulty initialization.

quant-ph↗

arXiv2021

Dissipative Encoding of Quantum Information

We formalize the problem of dissipative quantum encoding, and explore the advantages of using Markovian evolution to prepare a quantum code in the desired logical space, with emphasis on discrete-time dynamics and the possibility of exact finite-time convergence. In particular, we investigate robustness of the encoding dynamics and their ability to tolerate initialization errors, thanks to the existence of non-trivial basins of attraction. As a key application, we show that for stabilizer quantum codes on qubits, a finite-time dissipative encoder may always be constructed, by using at most a number of quantum maps determined by the number of stabilizer generators. We find that even in situations where the target code lacks gauge degrees of freedom in its subsystem form, dissipative encoders afford nontrivial robustness against initialization errors, thus overcoming a limitation of purely unitary encoding procedures. Our general results are illustrated in a number of relevant examples, including Kitaev's toric code.

quant-ph↗

arXiv2019

Uniquely determined pure quantum states need not be unique ground states of quasi-local Hamiltonians

We consider the problem of characterizing states of a multipartite quantum system from restricted, quasi-local information, with emphasis on uniquely determined pure states. By leveraging tools from dissipative quantum control theory, we show how the search for states consistent with an assigned list of reduced density matrices may be restricted to a proper subspace, which is determined solely by their supports. The existence of a quasi-local observable which attains its unique minimum over such a subspace further provides a sufficient criterion for a pure state to be uniquely determined by its reduced states. While the condition that a pure state is uniquely determined is necessary for it to arise as a non-degenerate ground state of a quasi-local Hamiltonian, we prove the opposite implication to be false in general, by exhibiting an explicit analytic counterexample. We show how the problem of determining whether a quasi-local parent Hamiltonian admitting a given pure state as its unique ground state is dual, in the sense of semidefinite programming, to the one of determining whether such a state is uniquely determined by the quasi-local information. Failure of this dual program to attain its optimal value is what prevents these two classes of states to coincide.

quant-ph↗

arXiv2017

Simulation of Quantum Walks and Fast Mixing with Classical Processes

We compare discrete-time quantum walks on graphs to their natural classical equivalents, which we argue are lifted Markov chains, that is, classical Markov chains with added memory. We show that these can simulate quantum walks, allowing us to answer an open question on how the graph topology ultimately bounds their mixing performance, and that of any stochastic local evolution. The results highlight that speedups in mixing and transport phenomena are not necessarily diagnostic of quantum effects, although superdiffusive spreading is more prominent with quantum walks.

quant-ph↗

arXiv2017

Generic pure quantum states as steady states of quasi-local dissipative dynamics

We investigate whether a generic multipartite pure state can be the unique asymptotic steady state of locality-constrained purely dissipative Markovian dynamics. In the simplest tripartite setting, we show that the problem is equivalent to characterizing the solution space of a set of linear equations and establish that the set of pure states obeying the above property has either measure zero or measure one, solely depending on the subsystems' dimension. A complete analytical characterization is given when the central subsystem is a qubit. In the N-partite case, we provide conditions on the subsystems' size and the nature of the locality constraint, under which random pure states cannot be quasi-locally stabilized generically. Beside allowing for the possibility to approximately stabilize entangled pure states that cannot be exact steady states in settings where stabilizability is generic, our results offer insights into the extent to which random pure states may arise as unique ground states of frustration free parent Hamiltonians. We further argue that, to high probability, pure quantum states sampled from a t-design enjoy the same stabilizability properties of Haar-random ones as long as suitable dimension constraints are obeyed and t is sufficiently large. Lastly, we demonstrate a connection between the tasks of quasi-local state stabilization and unique state reconstruction from local tomographic information, and provide a constructive procedure for determining a generic N-partite pure state based only on knowledge of the support of any two of the reduced density matrices of about half the parties, improving over existing results.

quant-ph↗

arXiv2017

Bounding the convergence time of local probabilistic evolution

Isoperimetric inequalities form a very intuitive yet powerful characterization of the connectedness of a state space, that has proven successful in obtaining convergence bounds. Since the seventies they form an essential tool in differential geometry, graph theory and Markov chain analysis. In this paper we use isoperimetric inequalities to construct a bound on the convergence time of any local probabilistic evolution that leaves its limit distribution invariant. We illustrate how this general result leads to new bounds on convergence times beyond the explicit Markovian setting, among others on quantum dynamics.

cs.DM↗

arXiv2017

Lifting Markov Chains To Mix Faster: Limits and Opportunities

Lifted Markov chains are Markov chains on graphs with added local "memory" and can be used to mix towards a target distribution faster than their memoryless counterparts. Upper and lower bounds on the achievable performance have been provided under specific assumptions. In this paper, we analyze which assumptions and constraints are relevant for mixing, and how changing these assumptions affects those bounds. Explicitly, we find that requesting mixing on either the original or the full lifted graph, and allowing for reducible lifted chains or not, have no essential influence on mixing time bounds. On the other hand, allowing for suitable initialization of the lifted dynamics and/or imposing invariance of the target distribution for any initialization do significantly affect the convergence performance. The achievable convergence speed for a lifted chain goes from diameter-time to no acceleration over a standard Markov chain, with conductance bounds limiting the effectiveness of the intermediate cases. In addition, we show that the relevance of imposing ergodic flows depends on the other criteria. The presented analysis allows us to clarify in which scenarios designing lifted dynamics can lead to better mixing, and provide a flexible framework to compare lifted walks with other acceleration methods.

math.OC↗

arXiv2017

Quantum and classical resources for unitary design of open-system evolutions

A variety of tasks in quantum control, ranging from purification and cooling, to quantum stabilization and open-system simulation, rely on the ability to implement a target quantum channel over a specified time interval within prescribed accuracy. This can be achieved by engineering a suitable unitary dynamics of the system of interest along with its environment -- which, depending on the available level of control, is fully or partly exploited as a coherent quantum controller. After formalizing a controllability framework for completely positive trace-preserving quantum dynamics, we provide sufficient conditions on the environment state and dimension that allow for the realization of relevant classes of quantum channels -- including extreme channels, stochastic unitaries, or simply any channel. The results hinge on generalizations of Stinespring's dilation via a subsystem principle. In the process, we show that a conjecture by Lloyd on the minimal dimension of the environment required for arbitrary open-system simulation, albeit formally disproved, can in fact be salvaged -- provided that classical randomization is included among the available resources. Existing measurement-based feedback protocols for universal simulation, dynamical decoupling, and dissipative state preparation are recast within the proposed coherent framework as concrete applications, and the resources they employ discussed in the light of the general results.

quant-ph↗

arXiv2017

Distributed finite-time stabilization of entangled quantum states on tree-like hypergraphs

Preparation of pure states on networks of quantum systems by controlled dissipative dynamics offers important advantages with respect to circuit-based schemes. Unlike in continuous-time scenarios, when discrete-time dynamics are considered, dead-beat stabilization becomes possible in principle. Here, we focus on pure states that can be stabilized by distributed, unsupervised dynamics in finite time on a network of quantum systems subject to realistic quasi-locality constraints. In particular, we define a class of quasi-locality notions, that we name "tree-like hypergraphs," and show that the states that are robustly stabilizable in finite time are then unique ground states of a frustration-free, commuting quasi-local Hamiltonian. A structural characterization of such states is also provided, building on a simple yet relevant example.

quant-ph↗

arXiv2017

Exact stabilization of entangled states in finite time by dissipative quantum circuits

Open quantum systems evolving according to discrete-time dynamics are capable, unlike continuous-time counterparts, to converge to a stable equilibrium in finite time with zero error. We consider dissipative quantum circuits consisting of sequences of quantum channels subject to specified quasi-locality constraints, and determine conditions under which stabilization of a pure multipartite entangled state of interest may be exactly achieved in finite time. Special emphasis is devoted to characterizing scenarios where finite-time stabilization may be achieved robustly with respect to the order of the applied quantum maps, as suitable for unsupervised control architectures. We show that if a decomposition of the physical Hilbert space into virtual subsystems is found, which is compatible with the locality constraint and relative to which the target state factorizes, then robust stabilization may be achieved by independently cooling each component. We further show that if the same condition holds for a scalable class of pure states, a continuous-time quasi-local Markov semigroup ensuring rapid mixing can be obtained. Somewhat surprisingly, we find that the commutativity of the canonical parent Hamiltonian one may associate to the target state does not directly relate to its finite-time stabilizability properties, although in all cases where we can guarantee robust stabilization, a (possibly non-canonical) commuting parent Hamiltonian may be found. Beside graph states, quantum states amenable to finite-time robust stabilization include a class of universal resource states displaying two-dimensional symmetry-protected topological order, along with tensor network states obtained by generalizing a construction due to Bravyi and Vyalyi. Extensions to representative classes of mixed graph-product and thermal states are also discussed.

quant-ph↗

arXiv2017

Exponential Stability of Subspaces for Quantum Stochastic Master Equations

We study the stability of quantum pure states and, more generally, subspaces for stochastic dynamics that describe continuously--monitored systems. We show that the target subspace is almost surely invariant if and only if it is invariant for the average evolution, and that the same equivalence holds for the global asymptotic stability. Moreover, we prove that a strict linear Lyapunov function for the average evolution always exists, and latter can be used to derive sharp bounds on the Lyapunov exponents of the associated semigroup. Nonetheless, we also show that taking into account the measurements can lead to an improved bound on stability rate for the stochastic, non-averaged dynamics. We discuss explicit examples where the almost sure stability rate can be made arbitrary large while the average one stays constant.

math-ph↗

arXiv2016

Alternating Projections Methods for Discrete-time Stabilization of Quantum States

We study sequences (both cyclic and randomized) of idempotent completely-positive trace-preserving quantum maps, and show how they asymptotically converge to the intersection of their fixed point sets via alternating projection methods. We characterize the robustness features of the protocol against randomization and provide basic bounds on its convergence speed. The general results are then specialized to stabilizing en- tangled states in finite-dimensional multipartite quantum systems subject to a resource constraint, a problem of key interest for quantum information applications. We conclude by suggesting further developments, including techniques to enlarge the set of stabilizable states and ensure efficient, finite-time preparation.

quant-ph↗

arXiv2016

General fixed points of quasi-local frustration-free quantum semigroups: from invariance to stabilization

We investigate under which conditions a mixed state on a finite-dimensional multipartite quantum system may be the unique, globally stable fixed point of frustration-free semigroup dynamics subject to specified quasi-locality constraints. Our central result is a linear-algebraic necessary and sufficient condition for a generic (full-rank) target state to be frustration-free quasi-locally stabilizable, along with an explicit procedure for constructing Markovian dynamics that achieve stabilization. If the target state is not full-rank, we establish sufficiency under an additional condition, which is naturally motivated by consistency with pure-state stabilization results yet provably not necessary in general. Several applications are discussed, of relevance to both dissipative quantum engineering and information processing, and non-equilibrium quantum statistical mechanics. In particular, we show that a large class of graph product states (including arbitrary thermal graph states) as well as Gibbs states of commuting Hamiltonians are frustration-free stabilizable relative to natural quasi-locality constraints. Likewise, we provide explicit examples of non-commuting Gibbs states and non-trivially entangled mixed states that are stabilizable despite the lack of an underlying commuting structure, albeit scalability to arbitrary system size remains in this case an open question.

quant-ph↗

arXiv2016

Symmetrizing quantum dynamics beyond gossip-type algorithms

Recently, consensus-type problems have been formulated in the quantum domain. Obtaining average quantum consensus consists in the dynamical symmetrization of a multipartite quantum system while preserving the expectation of a given global observable. In this paper, two improved ways of obtaining consensus via dissipative engineering are introduced, which employ on quasi local preparation of mixtures of symmetric pure states, and show better performance in terms of purity dynamics with respect to existing algorithms. In addition, the first method can be used in combination with simple control resources in order to engineer pure Dicke states, while the second method guarantees a stronger type of consensus, namely single-measurement consensus. This implies that outcomes of local measurements on different subsystems are perfectly correlated when consensus is achieved. Both dynamics can be randomized and are suitable for feedback implementation.

quant-ph↗