Quantum Dynamics With an Ensemble of Hamiltonians

Rahmani, Armin

doi:10.1142/s0217984913300196

Cited by 11 publications

(11 citation statements)

References 76 publications

(120 reference statements)

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“…Here W characterizes the strength of the noise (note that γ is dimensionless and W 2 has units of time). White noise is a good approximation to ubiquitous colored noise with exponentially decaying correlations such as the Ornstein-Uhlenbeck process [33,34]. While our quantitative predictions may change for power-law correlated noise, we expect the slow decay of the noise spectrum (in the frequency domain) to allow for absorption of energy from the noisy drive and the aforementioned competition between adiabaticity and heating, qualitatively leading to the same anti-KZ behavior.…”

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confidence: 74%

Anti-Kibble-Zurek Behavior in Crossing the Quantum Critical Point of a Thermally Isolated System Driven by a Noisy Control Field

2016

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We show that a thermally isolated system driven across a quantum phase transition by a noisy control field exhibits anti-Kibble-Zurek behavior, whereby slower driving results in higher excitations. We characterize the density of excitations as a function of the ramping rate and the noise strength. The optimal driving time to minimize excitations is shown to scale as a universal power law of the noise strength. Our findings reveal the limitations of adiabatic protocols such as quantum annealing and demonstrate the universality of the optimal ramping rate.Understanding adiabatic dynamics and its breakdown in many-body systems is fundamental to the progress of quantum technologies [1]. Adiabatic evolution is the cornerstone of the quantum annealing scheme for state preparation and quantum computation [2, 3]. The adiabatic theorem states that the dynamics of a physical system is free from diabatic transitions under slow driving [4]. The suppression of excitations becomes challenging in the absence of an energy gap, e.g., when crossing a quantum critical point (QCP) [5][6][7]. The density of excitation follows a universal power aw as a function of the rate of change of the control field driving the system through the QCP [8-11] and can be reduced by resorting to slow ramps. This universal scaling is the key prediction of the Kibble-Zurek mechanism (KZM), initially developed for classical continuous phase transitions [12,13].While its experimental verification still calls for further studies [7], KZM is believed to be broadly applicable. Yet, a conflicting observation has been reported in the study of mutiferroic systems: approaching the adiabatic limit, slower ramps generate more excitations [14]. This counterintuitive phenomenon was termed anti-Kibble-Zurek (anti-KZ) dynamics. While tests of KZM in the quantum regime are scarce, the data in one of them hint at a possible anti-KZ behavior [15]. Here we show that in a thermally isolated quantum system, the presence of noisy fluctuations in the control field naturally provides an explanation for anti-KZ behavior.We start by considering a linear passage through the QCP. A control field g is turned on from zero value to unity as in standard quantum annealing schemes, crossing a QCP at g c = 1 2 . When the transition is crossed at a rate 1/τ fixed by the ramp duration τ, KZM predicts universal power-law for the density of excitations n 0 ∝ τ −β , with β = dν/(1 + zν), where ν and z are the correlation length and dynamic critical exponents, and d is the dimensionality of the system. The subindex in n 0 is introduced to denote noise-free driving. The density of excitations monotonically decreases with τ and vanishes in the limit of τ → ∞.The control over the system, however, is never perfect. In particular, the modulation in time of the control field might be subject to noise [16][17][18][19][20]. In our study, we consider a thermally isolated system with no coupling to a thermal environment or heat bath, discussed in [21][22][23][24][25][26][27][28]. While the dynamics is ...

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confidence: 74%

Anti-Kibble-Zurek Behavior in Crossing the Quantum Critical Point of a Thermally Isolated System Driven by a Noisy Control Field

2016

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“…This method relies on starting from an easy-to-prepare initial state and evolving into the desired ground state of a target Hamiltonian by variationally modifying the parameters in the time-dependent Hamiltonian of the device. The idea has been explored for state preparation [9][10][11], and has showed remarkable theoretical [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] and experimental [27][28][29][30][31][32][33][34] success, particularly, in quantum chemistry simulations. It is also closely related to the Quantum Approximate Optimization Algorithm [35,36].…”

Section: Introductionmentioning

confidence: 99%

Topological and geometric patterns in optimal bang-bang protocols for variational quantum algorithms: application to the $XXZ$ model on the square lattice

Scoggins,

Rahmani

2020

Preprint

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In this work, we address the challenge of uncovering patterns in variational optimal protocols for taking the system to ground states of many-body Hamiltonians, using variational quantum algorithms. We develop highly optimized classical Monte Carlo (MC) algorithms to find the optimal protocols for transformations between the ground states of the square-lattice XXZ model for finite systems sizes. The MC method obtains optimal bangbang protocols, as predicted by Pontryagin's minimum principle. We identify the minimum time needed for reaching an acceptable error for different system sizes as a function of the initial and target states and uncover correlations between the total time and the wave-function overlap. We determine a dynamical phase diagram for the optimal protocols, with different phases characterized by a topological number, namely the number of onpulses. Bifurcation transitions as a function of initial and final states, associated with new jumps in the optimal protocols, demarcate these different phases. The number of pulses correlates with the total evolution time. In addition to identifying the topological characteristic above, i.e., the number of pulses, we introduce a correlation function to characterize bang-bang protocols' quantitative geometric similarities. We find that protocols within one phase are indeed geometrically correlated. Identifying and extrapolating patterns in these protocols may inform efficient large-scale simulations on quantum devices.

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“…Will the antiadiabatic effect still occur when using noise-canceling optimal protocols? To answer these questions, we focus on a widely used minimal effective model of MZM braiding [60][61][62], whose evolution is governed by a Lindblad master equation [59,[63][64][65]. We find optimal protocols by numerical minimization (through simulated-annealing Monte Carlo simulations) of a cost function based on the trace distance of the noise-averaged final density matrix from the perfectly adiabatic target density matrix.…”

Section: Introductionmentioning

confidence: 99%

Optimal noise-canceling shortcuts to adiabaticity: application to noisy Majorana-based gates

Ritland

Rahmani

2018

New J. Phys.

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Adiabatic braiding of Majorana zero modes can be used for topologically protected quantum information processing. While extremely robust to many environmental perturbations, these systems are vulnerable to noise with high-frequency components. Ironically, slower processes needed for adiabaticity allow more noise-induced excitations to accumulate, resulting in an antiadiabatic behavior that limits the precision of Majorana gates if some noise is present. In a recent publication (2017 Phys. Rev. B 96 075158), fast optimal protocols were proposed as a shortcut for implementing the same unitary operation as the adiabatic braiding in a low-energy effective model. These shortcuts sacrifice topological protection in the absence of noise but provide performance gains and remarkable robustness to noise due to the shorter evolution time. Nevertheless, gates optimized for vanishing noise are suboptimal in the presence of noise. If we know the noise strength beforehand, can we design protocols optimized for the existing unavoidable noise, which will effectively correct the noiseinduced errors? We address this question in the present paper, focusing on the same low-energy effective model. We find such optimal protocols using simulated-annealing Monte Carlo simulations. The numerically found pulse shapes, which we fully characterize, are in agreement with Pontryagin's minimum principle, which allows us to arbitrarily improve the approximate numerical results (due to discretization and imperfect convergence) and obtain numerically exact optimal protocols. The protocols are bang-bang (sequence of sudden quenches) for vanishing noise, but contain continuous segments in the presence of multiplicative noise due to the nonlinearity of the master equation governing the evolution. We find that such noise-optimized protocols completely eliminate the above-mentioned antiadiabatic behavior. The final error corresponding to these optimal protocols monotonically decreases with the total time (in three different regimes). A liner fit to 1/τ indicates extrapolation of the cost function to finite value in the t  ¥ limit. However, quadratic and cubic fits are more suggestive of the cost function extrapolating to zero in the limit of infinite time. Our results set the precision limit of the device as a function of the noise strength and total time.

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Quantum Dynamics With an Ensemble of Hamiltonians

Cited by 11 publications

References 76 publications

Anti-Kibble-Zurek Behavior in Crossing the Quantum Critical Point of a Thermally Isolated System Driven by a Noisy Control Field

Anti-Kibble-Zurek Behavior in Crossing the Quantum Critical Point of a Thermally Isolated System Driven by a Noisy Control Field

Topological and geometric patterns in optimal bang-bang protocols for variational quantum algorithms: application to the $XXZ$ model on the square lattice

Optimal noise-canceling shortcuts to adiabaticity: application to noisy Majorana-based gates

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