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arXiv:1909.06333 · Physical Review A

Validating a two-qubit nonstoquastic Hamiltonian in quantum annealing

Tameem Albash

Published Jan 9, 2020• 3 citations
arXivHTMLPDF

Abstract

We propose a two-qubit experiment for validating tunable antiferromagnetic XX interactions in quantum annealing. Such interactions allow the time-dependent Hamiltonian to be nonstoquastic, and the instantaneous ground state can have negative amplitudes in the computational basis. Our construction relies on how the degeneracy of the Ising Hamiltonian's ground states is broken away from the end point of the anneal: above a certain value of the antiferromagnetic XX interaction strength, the perturbative ground state at the end of the anneal changes from a symmetric to an antisymmetric state. This change is associated with a suppression of one of the Ising ground states, which can then be detected using solely computational basis measurements. We show that a semiclassical approximation of the annealing protocol fails to reproduce this feature, making it a candidate ``quantum signature'' of the evolution.

DOI
10.1103/physreva.101.012310

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We propose a two-qubit experiment for validating tunable antiferromagnetic XX interactions in quantum annealing. Such interactions allow the time-dependent Hamiltonian to be nonstoquastic, and the instantaneous ground state can have negative amplitudes in the computational basis. Our construction relies on how the degeneracy of the Ising Hamiltonian's ground states is broken away from the end point of the anneal: above a certain value of the antiferromagnetic XX interaction strength, the perturbative ground state at the end of the anneal changes from a symmetric to an antisymmetric state. This change is associated with a suppression of one of the Ising ground states, which can then be detected using solely computational basis measurements. We show that a semiclassical approximation of the annealing protocol fails to reproduce this feature, making it a candidate ``quantum signature'' of the evolution.
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Quantum annealing (QA) is a generic method for solving optimization problems using fictitious quantum fluctuation. The current device performing QA involves controlling the transverse field; it is classically simulatable by using the standard technique for mapping the quantum spin systems to the classical ones. In this sense, the current system for QA is not powerful despite utilizing quantum fluctuation. Hence, we developed a system with a time-dependent Hamiltonian consisting of a combination of the formulated Ising model and the "driver" Hamiltonian with only quantum fluctuation. In the previous study, for a fully connected spin model, quantum fluctuation can be addressed in a relatively simple way. We proved that the fully connected antiferromagnetic interaction can be transformed into a fluctuating transverse field and is thus classically simulatable at sufficiently low temperatures. Using the fluctuating transverse field, we established several ways to simulate part of the nonstoquastic Hamiltonian on classical computers. We formulated a message-passing algorithm in the present study. This algorithm is capable of assessing the performance of QA with part of the nonstoquastic Hamiltonian having a large number of spins. In other words, we developed a different approach for simulating the nonstoquastic Hamiltonian without using the quantum Monte Carlo technique. Our results were validated by comparison to the results obtained by the replica method.
Quantum dynamics of two XXXXXX interacting PT-symmetric non-Hermitian qubits: enhancement of quantum annealing
Quantum information platforms enable analog quantum simulations, such as quantum annealing, offering a promising route to solving complex combinatorial optimization problems. Here, we propose a quantum information architecture based on networks of interacting parity-time (PT)-symmetric non-Hermitian qubits. While the dynamics of individual PT-symmetric qubits have been experimentally demonstrated across multiple platforms including NV centers, superconducting circuits, and trapped-ion systems yet coherent dynamics in interacting systems remain largely unexplored. To address this issue we theoretically investigate stationary and time-dependent Hamiltonians relevant to quantum annealing using a minimal model of two interacting XX-coupled PT-symmetric non-Hermitian qubits. We analyze both symmetry-preserving and symmetry-broken regimes and demonstrate that adding even tiny PT-symmetric non-Hermitian terms in the qubits Hamiltonian allows to greatly enhance the probability of reaching the ground state after annealing.
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Quantum annealing (QA) is a promising method for solving combinatorial optimization problems whose solutions are embedded into a ground state of the Ising Hamiltonian. This method employs two types of Hamiltonians: a driver Hamiltonian and a problem Hamiltonian. After a sufficiently slow change from the driver Hamiltonian to the problem Hamiltonian, we can obtain the target ground state that corresponds to the solution. The inclusion of non-stoquastic terms in the driver Hamiltonian is believed to enhance the efficiency of the QA. Meanwhile, decoherence is regarded as of the main obstacles for QA. Here, we present examples showing that non-stoaquastic Hamiltonians can lead to catastrophic failure of QA, whereas a certain decoherence process can be used to avoid such failure. More specifically, when we include anti-ferromagnetic interactions (i.e., typical non-stoquastic terms) in the Hamiltonian, we are unable to prepare the target ground state even with an infinitely long annealing time for some specific cases. In our example, owing to a symmetry, the Hamiltonian is block-diagonalized, and a crossing occurs during the QA, which leads to a complete failure of the ground-state search. Moreover, we show that, when we add a certain type of decoherence, we can obtain the ground state after QA for these cases. This is because, even when symmetry exists in isolated quantum systems, the environment breaks the symmetry. Our counter intuitive results provide a deep insight into the fundamental mechanism of QA.