Quantum Speed Limit Research Articles

Quantum speed limit for mixed states in a unitary system

Since the evolution of a mixed state in a unitary system is equivalent to the joint evolution of the eigenvectors contained in it, we could use the tool of instantaneous angular velocity for pure states to study the quantum speed limit (QSL) of a mixed state. We derive a lower bound for the evolution time of a mixed state to a target state in a unitary system, which automatically reduces to the quantum speed limit induced by the Fubini–Study metric for pure states. The computation of the QSL of a degenerate mixed state is more complicated than that of a non-degenerate mixed state, where we have to make a singular value decomposition (SVD) on the inner product between the two eigenvector matrices of the initial and target states. By combing these results, a lower bound for the evolution time of a general mixed state is presented. In order to compare the tightness among the lower bound proposed here and lower bounds reported in the references, two examples in a single-qubit system and in a single-qutrit system are studied analytically and numerically, respectively. All conclusions derived in this work are independent of the eigenvalues of the mixed state, which is in accord with the evolution properties of a quantum unitary system.

Quantum speed limits for information and coherence

The quantum speed limit indicates the maximal evolution speed of the quantum system. In this work, we determine speed limits on the informational measures, namely the von Neumann entropy, maximal information, and coherence of quantum systems evolving under dynamical processes. These speed limits ascertain the fundamental limitations on the evolution time required by the quantum systems for the changes in their informational measures. Erasing of quantum information to reset the memory for future use is crucial for quantum computing devices. We use the speed limit on the maximal information to obtain the minimum time required to erase the information of quantum systems via some quantum processes of interest.

Quantum lower and upper speed limits using reference evolutions

We derive generalized quantum speed limit inequalities that represent limitations on the time evolution of quantum states. They are extensions of the original inequality and are applied to the overlap between the time-evolved state and an arbitrary state. We can discuss the lower limit of the Bures angle, in addition to the upper limit as in the original inequality, which allows us to evaluate the lower and upper bounds of processing time for the evolution toward a target state. The inequalities are written by using an arbitrary reference state and are flexibly used to obtain a tight bound. We demonstrate these properties by using the twisted Landau–Zener model, the Grover Hamiltonian, and a periodically-oscillating Hamiltonian.

Resource speed limits: maximal rate of resource variation

Recent advances in quantum resource theories have been driven by the fact that many quantum information protocols make use of different facets of the same physical features, e.g. entanglement, coherence, etc. Resource theories formalise the role of these important physical features in a given protocol. One question that remains open until now is: how quickly can a resource be generated or degraded? Using the toolkit of quantum speed limits we construct bounds on the minimum time required for a given resource to change by a fixed increment, which might be thought of as the power of said resource, i.e., the rate of resource variation. We show that the derived bounds are tight by considering several examples. Finally, we discuss some applications of our results, which include generalisations of thermodynamic work, heat, and power to any given resource, as well as fundamental bounds on the computation rate in leading quantum processing architectures.

Quantum speed limit and stability of coherent states in quantum gravity

Utilizing the program of expectation values in coherent states and its recently developed algorithmic tools, this letter investigates the dynamical properties of cosmological coherent states for loop quantum gravity. To this end, the quantum speed limit (QSL) is adapted to quantum gravity, yielding necessary consistency checks for any proposal of stable families of states. To showcase the strength of the developed tools, they are applied to a prominent model: the Euclidean part of the quantum scalar constraint. We report the variance of this constraint evaluated on a family of coherent states showing that, for short times, this family passes the QSL test, allowing the transition from one coherent state to another one.

Shortcut to adiabaticity in a cavity with a moving mirror

Shortcuts to adiabaticity constitute a powerful alternative that speed up time-evolution while mimicking adiabatic dynamics. They are also relevant to clarify fundamental questions such as a precise quantification of the third principle of thermodynamics and quantum speed limits. In this letter we describe, for the first time, how to implement shortcuts to adiabaticity in quantum field theory, for the particular case of a massless scalar field inside a cavity with a moving wall, in 1 + 1 dimensions. The approach is based on the known solution to the problem that exploits the conformal symmetry, and the shortcuts take place whenever there is no dynamical Casimir effect. We obtain a fundamental limit for the efficiency of an Otto cycle with the quantum field as a working system, that depends on the maximum velocity that the mirror can attain. We describe possible experimental realizations of the shortcuts using superconducting circuits.

Optimal bounds on the speed of subspace evolution* *This work was partially supported by the Heisenberg–Landau Program.

By a quantum speed limit one usually understands an estimate on how fast a quantum system can evolve between two distinguishable states. The most known quantum speed limit is given in the form of the celebrated Mandelstam–Tamm inequality that bounds the speed of the evolution of a state in terms of its energy dispersion. In contrast to the basic Mandelstam–Tamm inequality, we are concerned not with a single state but with a (possibly infinite-dimensional) subspace which is subject to the Schrödinger evolution. By using the concept of maximal angle between subspaces we derive optimal bounds on the speed of such a subspace evolution. These bounds may be viewed as further generalizations of the Mandelstam–Tamm inequality. Our study includes the case of unbounded Hamiltonians.

Geometric quantum speed limits for Markovian dynamics in open quantum systems

We study theoretically the geometric quantum speed limits (QSLs) of open quantum systems under Markovian dynamical evolution. Three types of QSL time bounds are introduced based on the geometric inequality associated with the dynamical evolution from an initial state to a final state. By illustrating three types of QSL bounds at the cases of presence or absence of system driving, we demonstrate that the unitary part, dominated by system Hamiltonian, supplies the internal motivation for a Markovian evolution which deviates from its geodesic. Specifically, in the case of unsaturated QSL bounds, the parameters of the system Hamiltonian serve as the eigen-frequency of the oscillations of geodesic distance in the time domain and, on the other hand, drive a further evolution of an open quantum system in a given time period due to its significant contribution in dynamical speedup. We present physical pictures of both saturated and unsaturated QSLs of Markovian dynamics by means of the dynamical evolution trajectories in the Bloch sphere which demonstrates the significant role of system Hamiltonian even in the case of initial mixed states. It is further indicated that whether the QSL bound is saturated is ruled by the commutator between the Hamiltonian and reduced density matrix of the quantum system. Our study provides a detailed description of QSL times and reveals the effects of system Hamiltonian on the unsaturation of QSL bounds under Markovian evolution.

Extensions of the Mandelstam–Tamm quantum speed limit to systems in mixed states

The Mandelstam–Tamm quantum speed limit (QSL) puts a bound on how fast a closed system in a pure state can evolve. In this paper, we derive several extensions of this QSL to closed systems in mixed states. We also compare the strengths of these extensions and examine their tightness. The most widely used extension of the Mandelstam–Tamm QSL originates in Uhlmann’s energy dispersion estimate. We carefully analyze the underlying geometry of this estimate, an analysis that makes apparent that the Bures metric, or equivalently the quantum Fisher information, will rarely give rise to tight extensions. This observation leads us to address whether there is a tightest general extension of the Mandelstam–Tamm QSL. Using a geometric construction similar to that developed by Uhlmann, we prove that this is indeed the case. In addition, we show that tight evolutions of mixed states are typically generated by time-varying Hamiltonians, which contrasts with the case for systems in pure states.

From quantum speed limits to energy-efficient quantum gates

While recent breakthroughs in quantum computing promise the nascence of the quantum information age, quantum states remain delicate to control. Moreover, the required energy budget for large scale quantum applications has only sparely been considered. Addressing either of these issues necessitates a careful study of the most energetically efficient implementation of elementary quantum operations. In the present analysis, we show that this optimal control problem can be solved within the powerful framework of quantum speed limits. To this end, we derive state-independent lower bounds on the energetic cost, from which we find the universally optimal implementation of unitary quantum gates, for both single and N-qubit operations.

Speed Limits for Macroscopic Transitions

Speed of state transitions in macroscopic systems is a crucial concept for foundations of nonequilibrium statistical mechanics as well as various applications in quantum technology represented by optimal quantum control. While extensive studies have made efforts to obtain rigorous constraints on dynamical processes since Mandelstam and Tamm, speed limits that provide tight bounds for macroscopic transitions have remained elusive. Here, by employing the local conservation law of probability, the fundamental principle in physics, we develop a general framework for deriving qualitatively tighter speed limits for macroscopic systems than many conventional ones. We show for the first time that the speed of the expectation value of an observable defined on an arbitrary graph, which can describe general many-body systems, is bounded by the "gradient" of the observable, in contrast with conventional speed limits depending on the entire range of the observable. This framework enables us to derive novel quantum speed limits for macroscopic unitary dynamics. Unlike previous bounds, the speed limit decreases when the expectation value of the transition Hamiltonian increases; this intuitively describes a new tradeoff relation between time and quantum phase difference. Our bound is dependent on instantaneous quantum states and thus can achieve the equality condition, which is conceptually distinct from the Lieb-Robinson bound. Our work elucidates novel speed limits on the basis of local conservation law, providing fundamental limits to various types of nonequilibrium quantum macroscopic phenomena.

Reachable-set characterization of an open quantum system by the quantum speed limit

In recent years, the idea of reachable-set characterization based on the quantum speed limit (QSL) was proposed by Arenz et al. [New J. Phys. 19, 103015 (2017)]; that is, the reachable set of the target unitary gate in a closed qubit system can be characterized by considering the QSL as the necessary condition that the control setup must satisfy in order to achieve the goal. Inspired by this idea, in this paper we characterize a general Markovian open quantum system based on the QSL derived by Kobayashi and Yamamoto [Phys. Rev. A 102, 042606 (2020)]. Note that this bound is not only explicitly computable with respect to system parameters but also tighter than the other bounds. Some examples for demonstrating this analysis will be given.

Parametrically driven THz magnon-pairs: Predictions toward ultimately fast and minimally dissipative switching

Finding ways to achieve switching between magnetic states at the fastest possible timescale that simultaneously dissipates the least amount of energy is one of the main challenges in magnetism. Antiferromagnets exhibit intrinsic dynamics in the THz regime, the highest among all magnets, and are, therefore, ideal candidates to address this energy-time dilemma. Here, we study theoretically the THz-driven parametric excitation of antiferromagnetic magnon-pairs at the edge of the Brillouin zone and explore the potential for switching between two stable oscillation states. Using a semi-classical theory, we predict that switching can occur at the femtosecond timescale with an energy dissipation down to a few zepto Joule. This result touches the thermodynamical bound of the Landauer principle and approaches the quantum speed limit up to 5 orders of magnitude closer than demonstrated with magnetic systems so far.

Witnessing localization of a quantum state via quantum speed limits in a driven avoided-level crossing system.

In this work, we investigate the witnessing of the localization of quantum states through quantum speed limits (QSLs) in a two-level driven avoided-level crossing system. As the characteristic natures of the localized quantum states, the QSL presents the periodic oscillations and coherence. The coherence partition of QSL is much bigger than the population partition of QSL. Our study gives us the possibilities to manipulate dynamics of quantum states locally by employing the coherent destruction of tunneling, which is significant in quantum information process. In addition, we analyze the effects of the rotating-wave approximation and the generalized Van Vleck approach on QSL and show that they wipe out the quantum coherence.

Quantum distance to uncontrollability and quantum speed limits

Distance to Uncontrollability is a crucial concept in classical control theory. Here, we introduce Quantum Distance to Uncontrollability as a measure how close a universal quantum system is to a non-universal one. This allows us to provide a quantitative version of the Quantum Speed Limit, decomposing the bound into a geometric and dynamical component. We consider several physical examples including globally controlled solid state qubits and a cross-Kerr system, showing that the Quantum Distance to Uncontrollability provides a precise meaning to spectral crowding, weak interactions and other bottlenecks to universality. We suggest that this measure should be taken into consideration in the design of quantum technology.

Dynamics of Quantum Speed Limit Time for Correlated and Uncorrelated Noise Channels

Dynamics of Quantum Speed Limit Time for Correlated and Uncorrelated Noise Channels

How long does it take to implement a projective measurement?

According to the Schrödinger equation, a closed quantum system evolves continuously in time. If it is subject to a measurement however, its state changes randomly and discontinuously, which is mathematically described by the projection postulate. But how long does it take for this discontinuous change to occur? Based on simple estimates, whose validity rests solely on the fact that all fundamental forces in nature are finite-ranged, we show that the implementation of a quantum measurement requires a minimum time. This time scales proportionally with the diameter of the quantum mechanical object, on which the measured observable acts non-trivially, with the proportionality constant being around 10−5 s m−1. We confirm our bound by comparison with experimentally reported measurement times for different platforms. We give a pedagogical exposition of our argumentation introducing along the way modern concepts such as ancilla-based measurements, the quantum speed limit, and Lieb–Robinson velocity bounds.

Quantum Speed Limit Time of Topological Qubits Influenced by the Fermionic and Bosonic Environments

Quantum Speed Limit Time of Topological Qubits Influenced by the Fermionic and Bosonic Environments

Orthogonality catastrophe and quantum speed limit for spin chain at finite temperature

We present an interesting relationship between the orthogonality catastrophe (OC) and the quantum speed limit (QSL) for a spin chain with uniform nearest neighbour couplings perturbed by an impurity spin. We thoroughly study the catastrophic QSL that specifies a bound on the evolution time between the initial and final states and in this respect, link it to the emerging OC effect. It is found that the speed of state evolution subtle but fundamental, and the bound characterized by QSL shows the same behaviours as the OC effect in the thermodynamic limit. It allows us to reveal some universal properties, in particular finite temperature effects. Significantly, the threshold of temperature and system size is clearly demonstrated for the QSL under finite temperature.

Quantum speed limit and topological quantum phase transition in an extended [formula omitted] model

We explore quantum speed limit time τQSL of a qubit system coupling to a spin chain environment which is described by an extended Ising model associated with non-trivial topological characterization. For an initial pure state, we find that τQSL exhibits a behavior of local maximum around all critical points when the environment spin chain undergoes a topological quantum phase transition driven by external magnetic field λ or the anisotropy γ of nearest-neighbor interaction. And the value of local maximum would increase by strengthening the coupling between the system and environment around the critical points. However, the behavior of τQSL are both driving parameters and critical points dependent when the topological quantum phase transition is driven by the three-site interaction α or its anisotropy δ. Additionally, we also investigate τQSL for arbitrary time-evolution state in the whole dynamics process and find that τQSL exhibits a behavior of oscillation at non-critical points. However, τQSL will decay rapidly to zero around all critical points, and such trend accompany with oscillation behavior at some critical points.