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arXiv:2007.07381 · New Journal of Physics

Ultrafast critical ground state preparation via bang–bang protocols

Luca Innocenti, Gabriele De Chiara, Mauro Paternostro, Ricardo Puebla

Published Aug 24, 2020• 12 citations
arXivHTMLPDF

Abstract

Abstract The fast and faithful preparation of the ground state of quantum systems is a challenging but crucial task for several applications in the realm of quantum-based technologies. Decoherence limits the maximum time-window allowed to an experiment to faithfully achieve such desired states. This is of particular significance in systems featuring a quantum phase transition, where the vanishing energy gap challenges an adiabatic ground state preparation. We show that a bang–bang protocol, consisting of a time evolution under two different values of an externally tunable parameter, allows for a high-fidelity ground state preparation in evolution times no longer than those required by the application of standard optimal control techniques, such as the chopped-random basis quantum optimization. In addition, owing to their reduced number of variables, such bang–bang protocols are very well suited to optimization tasks, reducing the high computational cost of other optimal control protocols. We benchmark the performance of such approach through two paradigmatic models, namely the Landau–Zener and the Lipkin–Meshkov–Glick model. Remarkably, we find that the critical ground state of the latter model, i.e. its ground state at the critical point, can be prepared with a high fidelity in a total evolution time that scales slower than the inverse of the vanishing energy gap.

DOI
10.1088/1367-2630/abb1df

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Abstract The fast and faithful preparation of the ground state of quantum systems is a challenging but crucial task for several applications in the realm of quantum-based technologies. Decoherence limits the maximum time-window allowed to an experiment to faithfully achieve such desired states. This is of particular significance in systems featuring a quantum phase transition, where the vanishing energy gap challenges an adiabatic ground state preparation. We show that a bang–bang protocol, consisting of a time evolution under two different values of an externally tunable parameter, allows for a high-fidelity ground state preparation in evolution times no longer than those required by the application of standard optimal control techniques, such as the chopped-random basis quantum optimization. In addition, owing to their reduced number of variables, such bang–bang protocols are very well suited to optimization tasks, reducing the high computational cost of other optimal control protocols. We benchmark the performance of such approach through two paradigmatic models, namely the Landau–Zener and the Lipkin–Meshkov–Glick model. Remarkably, we find that the critical ground state of the latter model, i.e. its ground state at the critical point, can be prepared with a high fidelity in a total evolution time that scales slower than the inverse of the vanishing energy gap.
A bang-bang (BB) algorithm prepares the ground state of a two-dimensional (2D) quantum many-body Hamiltonian H=H1+H2H={H}_{1}+{H}_{2}H=H1​+H2​ by evolving an initial product state alternating between H1{H}_{1}H1​ and H2{H}_{2}H2​. We use the neighborhood tensor update to simulate the BB evolution with an infinite pair-entangled projected state (iPEPS). The alternating sequence is optimized with the final energy as a cost function. The energy is calculated with a tangent space power method for the sake of its stability. The method is benchmarked in the 2D transverse field quantum Ising model near its quantum critical point against a ground state obtained with variational optimization of the iPEPS. The optimal BB sequence differs nonperturbatively from a sequence simulating quantum annealing or adiabatic preparation (AP) of the ground state. The optimal BB energy converges with the number of bangs much faster than the optimal AP energy.
The fast and efficient preparation of quantum critical states is a challenging yet crucial task for various quantum technologies. This difficulty is most particularly for systems near a quantum phase transition, where the closure of the energy gap fundamentally limits the timescale of adiabatic processes and thus precludes rapid state preparation. We propose a framework using deep reinforcement learning (DRL) to rapidly prepare quantum critical states, with broad extendibility to light-matter interaction systems. Specifically, a DRL agent optimizes a set of time-dependent control Hamiltonians to drive the system from an initial noncritical state to a target critical state within a finite time and over experimentally accessible parameter ranges. As a concrete application, we focus on the quantum Rabi model. The DRL-optimized time-dependent control Hamiltonian yield a final state with high-fidelity (>0.999>0.999>0.999) to the target critical state. The protocol can be readily extended to other quantum critical systems described by light-matter interaction models, such as quantum Dicke model. This investigation provides a powerful new framework for preparing and manipulating quantum critical states.
We develop protocols for high-fidelity single-qubit gates that exploit and extend theoretical ideas for accelerated adiabatic evolution. Our protocols are compatible with qubit architectures where direct transitions between logical states are either vanishingly small or nonexistent; in such systems traditional approaches cannot be implemented. Prime examples are superconducting fluxonium qubits, which have highly localized states, and AMO systems, where there are no dipole allowed transitions between the ground states encoding the logical states. By using an accelerated adiabatic protocol we can enforce the desired adiabatic evolution while having gate times that are comparable to the inverse adiabatic energy gap (a scale that is ultimately set by the amount of power used in the control pulses). By modeling the effects of decoherence, we explore the trade-off between speed and robustness that is inherent to shortcuts-to-adiabaticity approaches.
We present an alternative protocol allowing for the preparation of critical states that instead of suffering from the critical slowing down benefits from the critical speeding up. Paradoxically, we prepare these states by going away from the critical point which allows for the speed-up. We apply the protocol to the paradigmatic quantum Rabi model and its classical oscillator limit as well as the Lipkin-Meshkov-Glick model. Subsequently, we discuss the application of the adiabatic speed-up protocol in quantum metrology and compare its performance with critical quantum metrology. We show that critical quantum metrology with the Lipkin-Meshkov-Glick model cannot even overcome the standard quantum limit, and we argue that, even though critical metrology protocols can overcome it in some cases, critical metrology is a suboptimal metrological strategy. Finally, we conclude that systems exhibiting a phase transition are indeed interesting from the viewpoint of quantum technologies; however, it may not be the critical point that should attract the most attention.
Ground-state preparation, energy estimation, and property estimation are fundamental tasks in quantum simulation. Existing ground-state preparation algorithms achieve near-optimal complexity using block encodings or controlled Hamiltonian evolution, raising a natural question: is controlled Hamiltonian evolution necessary, or can the same scaling be attained in the weaker access model that provides access to only uncontrolled Hamiltonian evolution? We resolve this question by giving two algorithms that access the input Hamiltonian HHH only through queries to UHU_HUH​ and its inverse UH†U_H^\daggerUH†​, where ∥H∥≤αH\|H\|\leα_H∥H∥≤αH​ and τ=Θ(1/αH)τ=Θ(1/α_H)τ=Θ(1/αH​). Given an input state with ground-state weight at least ηηη and a spectral gap of HHH of at least γγγ, our first algorithm uses fresh copies of the input state and requires O~(αH/(γη))\widetilde O(α_H/(γη))O(αH​/(γη)) queries to UHU_HUH​ and UH†U_H^\daggerUH†​, while our second algorithm uses coherent access to the input-state preparation unitary and its inverse to improve this complexity to O~(αH/(γη))\widetilde O(α_H/(γ\sqrtη))O(αH​/(γη​)). The latter bound matches the known lower bound up to logarithmic factors, showing that algorithms with additional access to controlled Hamiltonian evolution cannot improve the query complexity for ground-state preparation beyond logarithmic factors. Both algorithms are based on a simple mechanism that we call the spectral descent principle, in which the algorithm starts from the input state ∣ψ⟩|ψ\rangle∣ψ⟩ and progressively descends through the energy spectrum, moving closer to the ground state until the desired fidelity is reached. We also extend this framework to ground-state property and energy estimation and give explicit complexity guarantees for both tasks.
We model the bang-bang optimization protocol as a shortcut to adiabaticity in the ground-state preparation of a trapped-ion quantum simulator. Compared to a locally adiabatic evolution, the bang-bang protocol typically produces a lower ground-state probability, but its implementation is so much simpler than the locally adiabatic approach, that it can become a competitive choice to use for maximizing ground-state preparation in systems that cannot be solved with conventional computers. We describe how one can optimize the shortcut and provide specific details for how it can be implemented with current trapped-ion quantum simulators. However, when frustration is strong enough, no method appears to work well for adiabatic state preparation within the experimental time frames, and one must confront the issue of dealing with diabatic excitations within the simulation.