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arXiv:2204.03012 · Physical Review A

Classical dissipative cost of quantum control

Anthony Kiely, Steve Campbell, Gabriel Teixeira Landi

Published Jul 5, 2022• 0 citations
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Abstract

Protocols for nonadiabatic quantum control often require the use of classical time-varying fields. Assessing the thermodynamic cost of such protocols, however, is far from trivial. Here, we study the irreversible entropy produced by the classical apparatus generating the control fields, thus providing a direct link between the cost of a control protocol and dissipation. We focus, in particular, on the case of time-dependent magnetic fields and shortcuts to adiabaticity. Our results are showcased with two experimentally realizable case studies: the Landau-Zener model of a spin-$\frac{1}{2}$ particle in a magnetic field and an ion confined in a Penning trap.

DOI
10.1103/physreva.106.012202

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Protocols for nonadiabatic quantum control often require the use of classical time-varying fields. Assessing the thermodynamic cost of such protocols, however, is far from trivial. Here, we study the irreversible entropy produced by the classical apparatus generating the control fields, thus providing a direct link between the cost of a control protocol and dissipation. We focus, in particular, on the case of time-dependent magnetic fields and shortcuts to adiabaticity. Our results are showcased with two experimentally realizable case studies: the Landau-Zener model of a spin-12\frac{1}{2}21​ particle in a magnetic field and an ion confined in a Penning trap.
Quantum systems are typically controlled by classical devices that necessarily dissipate energy to the surrounding environment. By the fluctuation-dissipation theorem, this implies noise in the control field produced. We investigate the relationship between the resulting errors and the entropy produced by the macroscopic controller, working in the context of a concrete minimal model. In the most experimentally relevant regime of fast control and high fidelity, we find a tradeoff between dissipation and control errors. These findings are exemplified by perturbative calculations of the gate fidelity for simple gate implementations and of the fidelity of state transfer via shortcuts to adiabaticity in the Landau-Zener model. Our results quantitatively demonstrate that suppressing errors in quantum control typically comes at the cost of macroscopic dissipation, which is needed to tame fluctuations in the classical controller.
Abstract We quantitatively assess the energetic cost of several well-known control protocols that achieve a finite time adiabatic dynamics, namely counterdiabatic and local counterdiabatic driving, optimal control, and inverse engineering. By employing a cost measure based on the norm of the total driving Hamiltonian, we show that a hierarchy of costs emerges that is dependent on the protocol duration. As case studies we explore the Landau–Zener model, the quantum harmonic oscillator, and the Jaynes–Cummings model and establish that qualitatively similar results hold in all cases. For the analytically tractable Landau–Zener case, we further relate the effectiveness of a control protocol with the spectral features of the new driving Hamiltonians and show that in the case of counterdiabatic driving, it is possible to further minimize the cost by optimizing the ramp.
Motivated by the need for precise, energy-efficient, and experimentally realistic quantum control protocols, we investigate the thermodynamic cost of performing quantum step-equilibration processes under the influence of classical stochastic control fields. Whereas purely deterministic protocols exhibit dissipation that scales inversely with the number of steps, we show that weak Gaussian noise in the control variables induces dissipative contributions that grow linearly with the number of steps. Consequently, we derive the finite optimal number of steps and minimal achievable average dissipated work and its variance using the quantum thermodynamic length. These results are demonstrated using two paradigmatic examples: a Landau-Zener sweep of a qubit strongly coupled to a thermal bath and the erasure of a transverse-field Ising model.
Adiabaticity is a cornerstone of many promising approaches to quantum control, computing, and simulation. In practice, however, there is always a trade-off. Although the deleterious effects of noise can be diminished by running a control schedule more quickly, this benefit comes at the expense of nonadiabaticity. To put these two unwanted effects on the same theoretical footing, we analyze the nonadiabatic error in qubit control as a form of entropy production, examining the mechanism by which fine-grained information is effectively lost despite the dynamics being fundamentally unitary. A crucial issue here is the question of how to define equilibrium under a time-dependent Hamiltonian. Using the Landau-Zener protocol as a test case, we show that entropy increases nearly monotonically when equilibrium is defined with respect to the effective Hamiltonian in the optimal superadiabatic frame. We then consider single-passage Landau-Zener-St\"uckelberg-Majorana interferometry, in which the initial state of the qubit is arbitrary. Violations of the second law of thermodynamics are possible but require exquisite control to achieve deliberately.
Irreversibility has a fundamental thermodynamic cost, erasing information inevitably generates heat. This connection is quantified by the Landauer bound, which gives the minimum dissipation needed to erase a single bit of information. While this bound applies in both classical and quantum settings, it is saturated only in idealised limits of infinite resources. Here, we provide a unified first principles description of finite-resource erasure in both classical and quantum systems. We begin by proving the communal folklore that in the idealised regime the erasure cost of a bit encoded in a quantum or classical system is the same. Despite this, we show that their practical implementation differs substantially: achieving comparable erasure quality in quantum systems requires more control, larger accessible energy gaps and longer operation times. Classical protocols can achieve the erasure of a comparable quantum protocol under far weaker constraints which we expose in trade-off relations. Our results explain why practical erasure schemes fall short of Landauer's bound and show that classical systems enjoy several fundamental thermodynamic advantages.
Achieving effectively adiabatic dynamics is a ubiquitous goal in almost all areas of quantum physics. Here, we study the speed with which a quantum system can be driven when employing transitionless quantum driving. As a main result, we establish a rigorous link between this speed, the quantum speed limit, and the (energetic) cost of implementing such a shortcut to adiabaticity. Interestingly, this link elucidates a trade-off between speed and cost, namely, that instantaneous manipulation is impossible as it requires an infinite cost. These findings are illustrated for two experimentally relevant systems-the parametric oscillator and the Landau-Zener model-which reveal that the spectral gap governs the quantum speed limit as well as the cost for realizing the shortcut.