Effective Hamiltonians for interacting superconducting qubits: local basis reduction and the Schrieffer–Wolff transformation

Abstract

An open question in designing superconducting quantum circuits is how best to reduce the full circuit Hamiltonian which describes their dynamics to an effective two-level qubit Hamiltonian which is appropriate for manipulation of quantum information. Despite advances in numerical methods to simulate the spectral properties of multi-element superconducting circuits[1, 2, 3], the literature lacks a consistent and effective method of determining the effective qubit Hamiltonian. Here we address this problem by introducing a novel local basis reduction method. This method does not require any ad hoc assumption on the structure of the Hamiltonian such as its linear response to applied fields. We numerically benchmark the local basis reduction method against other Hamiltonian reduction methods in the literature and show that it is applicable over a wider parameter range, particularly for superconducting qubits with reduced anharmonicity, including the capacitively-shunted flux qubit. By combining the local basis reduction method with the Schrieffer-Wolff transformation we further extend its applicability to systems of interacting qubits and use it to extract both non-stoquastic two-qubit Hamiltonians and three-local interaction terms in three-qubit Hamiltonians. arXiv:1912.00464v1 [quant-ph] 1 Dec 2019

Stoquasticity in circuit QED

“…This finding was further confirmed in Ref. [47], where the authors put forward a more refined analysis based on the perturbative Schrieffer-Wolff (SW) transformation [48].…”

“…We remark that, for simplicity of exposition, we present the discussion assuming the validity of the projection, in order to highlight the mechanism that leads to a nonstoquastic behavior. By refining the perturbation theory, i.e., using higher order SW transformation for instance [47], the Hamiltonian can still be nonstoquastic even in the absence of an inductive coupling. In particular, it is shown in Ref.…”

“…Note that this was also used in Ref. [47] to obtain the effective Hamiltonian of two capacitively and inductively coupled flux qubits, and to study its stoquasticity.…”

Locally suppressed transverse-field protocol for diabatic quantum annealing

“…The method could also be used to elucidate the structure of a black-box problem, an application which we also wish to investigate in further work. Owing to recent work on the accurate mapping of superconducting circuit Hamiltonians to Ising Hamiltonians [32], and efficient methods for finding physical annealing schedules that accurately reproduce the desired scheduling of the Ising terms [33], we anticipate that the schedules proposed in Eqs. ( 9)-( 12), or similar variants, can be readily implemented provided the qubit design is suitable (e.g., it allows a near-zero transverse field).…”

“…To this end, we first introduce a simple model of magnetic frustration to show that quantum tunneling effects can be leveraged to create a small avoided level crossing. We then introduce a means of creating an additional avoided crossing via customized annealing schedules [32][33][34] that allows us to sketch a heuristic algorithm for finding lower-energy eigenstates of the problem than in AQA under certain circumstances. We also provide a statistical analysis of randomly generated examples to demonstrate the performance of LSTF-DQA and we analyze in detail a single instance to discuss the consequences of its application.…”