Tommaso Caneva, Michael Murphy, Tommaso Calarco, Rosario Fazio, Simone Montangero, Vittorio Giovannetti, Giuseppe E. Santoro
Optimal control theory is a promising candidate for a drastic improvement of the performance of quantum information tasks. We explore its ultimate limit in paradigmatic cases, and demonstrate that it coincides with the maximum speed limit allowed by quantum evolution (the quantum speed limit).
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We present a simple derivation of the formula for the Hamiltonian operator(s) that achieve the fastest possible unitary evolution between given initial and final states. We discuss how this formula is modified in pseudo-Hermitian quantum mechanics and provide an explicit expression for the most general optimal-speed quasi-Hermitian Hamiltonian. Our approach allows for an explicit description of the metric- (inner product-) dependence of the lower bound on the travel time and the universality (metric-independence) of the upper bound on the speed of unitary evolutions.
We numerically investigate the performance of atomic transport in optical microtraps via the so called spatial adiabatic passage technique. Our analysis is carried out by means of optimal control methods, which enable us to determine suitable transport control pulses. We investigate the ultimate limits of the optimal control in speeding up the transport process in a triple well configuration for both a single atomic wave packet and a Bose-Einstein condensate within a regime of experimental parameters achievable with current optical technology.
We propose a method of optimally controlling state transfer through a noisy quantum channel (spin-chain). This process is treated as qubit state-transfer through a fermionic bath. We show that dynamical modulation of the boundary-qubits levels suffices to ensure fast and high-fidelity state transfer. This is achievable by dynamically optimizing the transmission spectrum of the channel. The resulting optimal control is robust against both static and fluctuating noise.
A remarkably simple result is derived for the minimal time $T_{\rm min}$ required to drive a general initial state to a final target state by a Landau-Zener type Hamiltonian or, equivalently, by time-dependent laser driving. The associated protocol is also derived. A surprise arises for some states when the interaction strength is assumed to be bounded by a constant $c$. Then, for large $c$, the optimal driving is of type bang-off-bang and for increasing $c$ one recovers the unconstrained result. However, for smaller $c$ the optimal driving can suddenly switch to bang-bang type. We discuss the…
The solution of the quantum Zermelo navigation problem is applied to the non-Markovian open system dynamics of a set of quantum systems interacting with a common environment. We consider a case allowing an exact time-optimal realization of environment-mediated non-local system unitaries. For a linear coupling to a harmonic bosonic bath, we derive a speed limit for the implementation time in terms of the fundamental frequency of the bath modes. As a product of two exponentials of the local free wind and the pairwise system-coupling, the Zermelo unitary forms a natural building block for reachin…
The minimal time a system requires to transform from an initial state to target state is defined as the quantum speed limit time. quantum speed limit time can be applied to quantify the maximum speed of the evolution of a quantum system. Quantum speed limit time is inversely related to the speed of evolution of a quantum system. That is, shorter quantum speed limit time means higher speed of quantum evolution. In this work, we study the quantum speed limit time of a two-level atom under Homodyne-based feedback control. The results show that the quantum speed limit time is decreased by increasi…
This paper focuses on accelerating quantum optimal control design for complex quantum systems. Based on our previous work [{arXiv:1607.04054}], we combine Pulse Width Modulation (PWM) and gradient descent algorithm into solving quantum optimal control problems, which shows distinct improvement of computational efficiency in various cases. To further apply this algorithm to potential experiments, we also propose the smooth realization of the optimized control solution, e.g. using Gaussian pulse train to replace rectangular pulses. Based on the experimental data of the D-Norleucine molecule, we …
A gradient ascent method for optimal quantum control synthesis is presented that employs a gradient derived with respect to the coefficients of a functional basis expansion of the control. Restricting the space of allowable controls to weighted sums of the Slepian sequences efficiently parameterizes the control in terms of bandwidth, resolution and pulse duration. A bound showing minimum time evolutions scaling with the inverse of the control bandwidth [S. Lloyd and S. Montangero, PRL, 113, 010502, (2014)] is recovered and the method is shown numerically to achieve the bound on entangling two-…
Efficient implementation of quantum algorithms requires single- or multi-qubit gates with high fidelity. In this report, we report that the fidelity of single-qubit gate operations on open quantum systems has a maximum value corresponding to an optimum value of the drive-amplitude in the presence of drive-induced decoherence. To show this, we use a previously reported fluctuation-regulated quantum master equation [Phys. Rev. A 97, 063837 (2018)]. The fidelity is found to be a function of the drive-induced dissipative terms as well as the relaxation terms arising from the qubit-environment coup…
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 Schroedinger evolution. By using the concept of maximal angle between subspaces we derive optimal bounds on…
Optimal quantum speed for mixed states
该研究构建框架求解d维系统在不含时哈密顿量下混合态的最优量子演化速度，明确最优态为副对角对称的X态，揭示副对角非对角元相干性及纠缠对加快演化、压缩速度极限的作用。
The speed of quantum computation is investigated through the time evolution of the speed of the orthogonality. The external field components for classical treatment beside the detuning and the coupling parameters for quantum treatment play important roles on the computational speed. It has been shown that the number of photons has no significant effect on the speed of computation. However, it is very sensitive to the variation in both detuning and the interaction coupling parameters.
Inequalities of Mandelstam-Tamm and Margolus-Levitin type provide lower bounds on the time it takes for a quantum system to evolve from one state into another. Knowledge of such bounds, called quantum speed limits, is of utmost importance in virtually all areas of physics, where determination of the minimum time required for a quantum process is of interest. Most Mandelstam-Tamm and Margolus-Levitin inequalities found in the literature have been derived from growth estimates for the Bures length, which is a statistical distance measure. In this paper we derive such inequalities by differential…
The quantum speed limit sets the minimum time required to transfer a quantum system completely into a given target state. At shorter times the higher operation speed has to be paid with a loss of fidelity. Here we quantify the trade-off between the fidelity and the duration in a system driven by a time-varying control. The problem is addressed in the framework of Hilbert space geometry offering an intuitive interpretation of optimal control algorithms. This approach is applied to non-uniform time variations which leads to a necessary criterion for control optimality applicable as a measure of …
It is recognized that Grover arrived at his original quantum search algorithm inspired by his comprehension of the interference of classical waves originating from an array of antennas. It is also known that quantum-mechanical characterization of electromagnetic radiation is isomorphic to the treatment of the orientation of a spin-1/2 particle. In this paper, motivated by Grover's original intuition and starting from this mathematical equivalence, we present a quantitative link between the geometry of time-independent optimal-speed Hamiltonian evolutions on the Bloch sphere and the geometry of…
Quantum mechanics establishes a fundamental bound for the minimum evolution time between two states of a given system. Known as the quantum speed limit (QSL), it is a useful tool in the context of quantum control, where the speed of some control protocol is usually intended to be as large as possible. While QSL expressions for time-independent hamiltonians have been well studied, the time-dependent regime has remained somewhat unexplored, albeit being usually the relevant problem to be compared with when studying systems controlled by external fields. In this paper we explore the relation betw…
We study controlled phasegates for ultracold atoms in an optical potential. A shaped laser pulse drives transitions between the ground and electronically excited states where the atoms are subject to a long-range 1/R^3 interaction. We fully account for this interaction and use optimal control theory to calculate the pulse shapes. This allows us to determine the minimum pulse duration, respectively, gate time T that is required to obtain high fidelity. We accurately analyze the speed limiting factors, and we find the gate time to be limited either by the interaction strength in the excited stat…
Limitations to the speed of evolution of quantum systems, typically referred to as quantum speed limits (QSLs), have important consequences for quantum control problems. However, in its standard formulation, is not straightforward to obtain meaningful QSL bounds for time-dependent Hamiltonians with unknown control parameters. In this paper we present a short introductory overview of quantum speed limit for unitary dynamics and its connection to quantum control. We then analyze potential methods for obtaining new bounds on control times inspired by the QSL. We finally extend the work in [Poggi,…
The minimum time a system needs to change from an initial state to a final orthogonal state is called quantum speed limit time. Quantum speed limit time can be used to quantify the speed of the quantum evolution. The speed of the quantum evolution will increase, if the quantum speed limit time decreases. In this work we will use relative purity based bound for quantum speed limit time. It is applicable for any arbitrary initial state. Here, we investigate the effects of filtering operation on quantum speed limit time. It will be observed that for some intervals of filtering operation parameter…
We propose a scheme to implement arbitrary-speed quantum entangling gates on two trapped ions immersed in a large linear crystal of ions, with minimal control of laser beams. For gate speeds slower than the oscillation frequencies in the trap, a single appropriately-detuned laser pulse is sufficient for high-fidelity gates. For gate speeds comparable to or faster than the local ion oscillation frequency, we discover a five-pulse protocol that exploits only the local phonon modes. This points to a method for efficiently scaling the ion trap quantum computer without shuttling ions.
Optimal Quantum Speedups for Repeatedly Nested Expectation Estimation
本文提出了一种量子算法，实现对多次嵌套期望估计的最优加速，扩展了单层嵌套的量子加速方法，并解决了多级蒙特卡洛算法的去随机化问题。
We consider finite-dimensional Markovian open quantum systems, and characterize the extent to which time-independent Hamiltonian control may allow to stabilize a target quantum state or subspace and optimize the resulting convergence speed. For a generic Lindblad master equation, we introduce a dissipation-induced decomposition of the associated Hilbert space, and show how it serves both as a tool to analyze global stability properties for given control resources and as the starting point to synthesize controls that ensure rapid convergence. The resulting design principles are illustrated in r…
We study optimization of fidelity for ultrafast transformation of a spin chain via external control of a local exchange coupling. We show that infidelity of such a process can be dramatically decreased by choosing a proper control profile in nonadiabatic time domain, predict main features of this profile analytically, corroborate them numerically with a gradient search algorithm, and discuss the corresponding quantum speed limit. For ultrafast transformations, the qualitative features of the obtained optimal control are system-independent. Moreover, the main restrictions on its shape do not de…
We investigate the influence of environment noise on the control cost and the quantum speed limit time (QSLT) in the process of almost exact state transmission (AEST) through a spin chain under pulse control. The chain is immersed in its surrounding non-Markovian, finite temperature heat baths. We find that AEST can be realized in weak system-bath coupling, low temperature, and strong non-Markovian baths under effective external control. Correspondingly, the control cost and QSLT increases with increasing bath temperature and coupling strength. It is noticeable that non-Markovianity from the b…
We implement a quantum optimal control algorithm based on automatic differentiation and harness the acceleration afforded by graphics processing units (GPUs). Automatic differentiation allows us to specify advanced optimization criteria and incorporate them in the optimization process with ease. We show that the use of GPUs can speed up calculations by more than an order of magnitude. Our strategy facilitates efficient numerical simulations on affordable desktop computers, and exploration of a host of optimization constraints and system parameters relevant to real-life experiments. We demonstr…
Constructions of Optimal-Speed Quantum Evolutions: A Comparative Study
本文比较了Bloch球面上两种最优速度量子Hamiltonian演化的构造方法，显式计算了最优Hamiltonian、酉演化算符和磁场配置，并证明在特定条件下两种方法等价且最优算符为绕特定正交轴的旋转。
We extend the concept of quantum speed limit -- the minimal time needed to perform a driven evolution -- to complex interacting many-body systems. We investigate a prototypical many-body system, a bosonic Josephson junction, at increasing levels of complexity: (a) within the two-mode approximation {corresponding to} a nonlinear two-level system, (b) at the mean-field level by solving the nonlinear Gross-Pitaevskii equation in a double well potential, and (c) at an exact many-body level by solving the time-dependent many-body Schrödinger equation. We propose a control protocol to transfer atoms…
Characterizing Optimal-speed unitary time evolution of pure and quasi-pure quantum states
本研究通过构造纯态流形与旗流形的等距关系，表征了产生纯态与准纯量子态最优速度幺正时间演化的哈密顿量，并将结果从纯态推广到准纯量子态。
Quantum Speed Limits from Symmetries in Quantum Control
本文从对称性角度探讨量子控制中的速度限制，通过李代数方法提供定量界限，应用于耦合量子比特等系统，帮助设计更快的量子控制方案。
We study the decoherence speed limit (DSL) of a single impurity atom immersed in a Bose-Einsteincondensed (BEC) reservoir when the impurity atom is in a double-well potential. We demonstrate how the DSL of the impurity atom can be manipulated by engineering the BEC reservoir and the impurity potential within experimentally realistic limits. We show that the DSL can be controlled by changing key parameters such as the condensate scattering length, the effective dimension of the BEC reservoir, and the spatial configuration of the double-well potential imposed on the impurity. We uncover the phys…
Approaching the Quantum Speed Limit in Quantum Gates with Geometric Control
提出基于几何原理的最优控制策略，通过控制哈密顿量参数实现量子门，逼近Mandelstam-Tamm量子速度极限，在双量子比特门上显著降低偏差并保持高保真度。
We study quantum information processing by means of optimal control theory. To this end, we analyze the damped Jaynes-Cummings model, and derive optimal control protocols that minimize the heating or energy dispersion rates, and controls that drive the system at the quantum speed limit. Special emphasis is put on analyzing the subtleties of optimal control theory for our system. In particular, it is shown how two fundamentally different approaches to the quantum speed limit can be reconciled by carefully formulating the problem.
We consider a two-level quantum system prepared in an arbitrary initial state and relaxing to a steady state due to the action of a Markovian dissipative channel. We study how optimal control can be used for speeding up or slowing down the relaxation towards the fixed point of the dynamics. We analytically derive the optimal relaxation times for different quantum channels in the ideal ansatz of unconstrained quantum control (a magnetic field of infinite strength). We also analyze the situation in which the control Hamiltonian is bounded by a finite threshold. As byproducts of our analysis we f…
Supersymmetry allows one to build a hierarchy of Hamiltonians that share the same spectral properties and which are pairwise connected through common superpotentials. The iso-spectral properties of these Hamiltonians imply that the dynamics and therefore control of different eigenstates are connected through supersymmetric intertwining relations. In this work we explore how this enables one to study general dynamics, shortcuts to adiabaticity (STA) and quantum speed limits for distinct states of different supersymmetric partner potentials by using the infinite box as an example.
A scheme for controlling light speed from slower-than-c to faster-than-c in an atomic system is presented in this paper. The scheme is based on far detuning Raman effect. Two far detuning coupling fields with small frequency difference will produce two absorptive peaks for the probe field in a $Λ$ structure, and an optical pump between the two ground states can change the absorptive peaks into enhanced peaks, which makes the normal dispersion between the two peaks change into anomalous dispersion, so the probe field can change from slow light to superluminal propagation.
Optimal light cone for macroscopic particle transport in long-range systems: A quantum speed limit approach
本文利用量子速度极限和最优输运理论，严格证明了长程Bose-Hubbard模型中宏观粒子输运的最短时间受源与目标区域距离约束，并给出目标区域粒子数概率上界，解决了通用玻色子系统粒子输运的开放问题。
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