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arXiv:1807.02053 · New Journal of Physics

Rapid counter-diabatic sweeps in lattice gauge adiabatic quantum computing

Andreas Hartmann, Wolfgang Lechner

Published Mar 29, 2019• 52 citations
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Abstract

We present a coherent counter-diabatic quantum protocol to prepare ground states in the lattice gauge mapping of all-to-all Ising models (LHZ) with considerably enhanced final ground state fidelity compared to a quantum annealing protocol.We make use of a variational method to find approximate counter-diabatic Hamiltonians that has recently been introduced by Sels and Polkovnikov (2017 Proc.Natl.Acad.Sci.114 3909).The resulting additional terms in our protocol are time-dependent local onsite y-magnetic fields.These additional Hamiltonian terms do not increase the minimal energy gap, but instead compensate for the Berry curvature.A single free parameter is introduced which is optimized via classical updates.The protocol consists only of local and nearest-neighbor terms which makes it attractive for implementation in near term experiments.

DOI
10.1088/1367-2630/ab14a0

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We present a coherent counter-diabatic quantum protocol to prepare ground states in the lattice gauge mapping of all-to-all Ising models (LHZ) with considerably enhanced final ground state fidelity compared to a quantum annealing protocol.We make use of a variational method to find approximate counter-diabatic Hamiltonians that has recently been introduced by Sels and Polkovnikov (2017 Proc.Natl.Acad.Sci.114 3909).The resulting additional terms in our protocol are time-dependent local onsite y-magnetic fields.These additional Hamiltonian terms do not increase the minimal energy gap, but instead compensate for the Berry curvature.A single free parameter is introduced which is optimized via classical updates.The protocol consists only of local and nearest-neighbor terms which makes it attractive for implementation in near term experiments.
Abstract Counterdiabatic driving can suppress diabatic losses during adiabatic ground state preparation. As its implementation necessitates the generation of an adiabatic gauge potential, which requires knowledge of the spectral gap of instantaneous Hamiltonians and involves highly non-local controls in many-body systems, local counterdiabatic driving with approximate adiabatic gauge potential has been widely adopted. In this work, using transverse-field Ising model as an example, we present an in-depth study of the performance of local counterdiabatic protocols and provide a closed-form analysis of ground-state-fidelity optimization. We then propose a two-stage protocol based on local counterdiabatic and simple local single-body control to further improve the performance. The practical implementation of these protocols does not require diagonalization of instantaneous Hamiltonians or spectral gaps, and only additional local single-body driving is involved. To benchmark their performance, we experimentally implement digitized adiabatic quantum evolution in a trapped-ion system. We characterize the quality of the prepared states and explore the scaling behavior with system size up to 14 qubits. Our demonstration of quantum shortcut to adiabaticity opens a path towards preparing ground states of complex systems with accessible local controls.
We investigate the role of local counter-diabatic (CD) terms in enhancing the performance of discrete-time digital protocols for a frustrated Ising ring, a system with an exponentially small spectral gap that acts as a bottleneck for conventional quantum annealing. The techniques investigated range from a digitised version of a fixed-schedule protocol, including lowest-order analytical CD terms, all the way to a full-fledged Quantum Approximate Optimisation Algorithm (QAOA), including variational CD terms (DC-QAOA). By analyzing the resulting residual energy, we show that DC-QAOA combined with Chopped RAndom Basis (CRAB) quantum control techniques for optimizing the circuit parameters, can outperform all the other strategies. By monitoring the ground-state population during the dynamics we learn that DC-QAOA finds effective shortcuts towards the target state through intermediate excited states, a process that is unexpectedly enhanced by the inclusion of local CD unitaries. Our results highlight the importance of flexible variational control of CD dynamics and demonstrate that digital optimization can explore operator dynamics that remain inaccessible to standard analytical CD constructions.
The strategy of shortcuts to adiabaticity enables us to realize adiabatic dynamics in finite time. In the counter-diabatic driving approach, an auxiliary Hamiltonian which is called the counter-diabatic Hamiltonian is appended to an original Hamiltonian to cancel out diabatic transitions. The counter-diabatic Hamiltonian is constructed by using the eigenstates of the original Hamiltonian. Therefore, it is in general difficult to construct the counter-diabatic Hamiltonian for quantum many-body systems. Even if the counter-diabatic Hamiltonian for quantum many-body systems is obtained, it is generally non-local and even diverges at critical points. We construct an approximated counter-diabatic Hamiltonian for the infinite-range Ising model by making use of the mean-field approximation. An advantage of this method is that the mean-field counter-diabatic Hamiltonian is constructed by only local operators. We numerically demonstrate the effectiveness of this method through quantum annealing processes going the vicinity of the critical point. It is also confirmed that the mean-field counter-diabatic Hamiltonian is still well-defined in the limit to the critical point. The present method can take higher order contributions into account and is consistent with the variational approach for local counter-diabatic driving.
We introduce a two-parameter approximate counter-diabatic term into the Hamiltonian of the transverse-field Ising model for quantum annealing to accelerate convergence to the solution, generalizing an existing single-parameter approach. The protocol is equivalent to unconventional diabatic control of the longitudinal and transverse fields in the transverse-field Ising model and thus makes it more feasible for experimental realization than an introduction of new terms such as non-stoquastic catalysts toward the same goal of performance enhancement. We test the idea for the ppp-spin model with p=3p=3p=3, which has a first-order quantum phase transition, and show that our two-parameter approach leads to significantly larger ground-state fidelity and lower residual energy than those by traditional quantum annealing as well as by the single-parameter method. We also find a scaling advantage in terms of the time to solution as a function of the system size in a certain range of parameters as compared to the traditional methods.
Exploring the ground state properties of many-body quantum systems conventionally involves adiabatic processes, alongside exact diagonalization, in the context of quantum annealing or adiabatic quantum computation. Shortcuts to adiabaticity by counter-diabatic driving serve to accelerate these processes by suppressing energy excitations. Motivated by this, we develop variational quantum algorithms incorporating the auxiliary counter-diabatic interactions, comparing them with digitized adiabatic algorithms. These algorithms are then implemented on gate-based quantum circuits to explore the ground states of the Fermi-Hubbard model on honeycomb lattices, utilizing systems with up to 26 qubits. The comparison reveals that the counter-diabatic inspired ansatz is superior to traditional Hamiltonian variational ansatz. Furthermore, the number and duration of Trotter steps are analyzed to understand and mitigate errors. Given the model’s relevance to materials in condensed matter, our study paves the way for using variational quantum algorithms with counterdiabaticity to explore quantum materials in the noisy intermediate-scale quantum era.
Using counterdiabatic (CD) driving-aiming at suppression of diabatic transition-in digitized adiabatic evolution has garnered immense interest in quantum protocols and algorithms. However, improving the approximate CD terms with a nested commutator ansatz is a challenging task. In this work, we propose a technique of finding optimal coefficients of the CD terms using a variational quantum circuit. By classical optimization routines, the parameters of this circuit are optimized to provide the coefficients corresponding to the CD terms. Then their improved performance is exemplified in Greenberger-Horne-Zeilinger state preparation on the nearest-neighbour Ising model. Finally, we also show the advantage over the usual quantum approximation optimization algorithm, in terms of fidelity with bounded time. This article is part of the theme issue 'Shortcuts to adiabaticity: theoretical, experimental and interdisciplinary perspectives'.