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arXiv:2011.01109 · Physical Review A

Stoquasticity in circuit QED

Alessandro Ciani, Barbara M. Terhal

Published Apr 1, 2021• 8 citations
arXivHTMLPDF

Abstract

We analyze whether circuit QED Hamiltonians are stoquastic, focusing on systems of coupled flux qubits. We show that scalable sign-problem-free path integral Monte Carlo simulations can typically be performed for such systems. Despite this, we corroborate the recent finding [I. Ozfidan et al., Phys. Rev. Appl. 13, 034037 (2020)] that an effective, nonstoquastic qubit Hamiltonian can emerge in a system of capacitively coupled flux qubits. We find that if the capacitive coupling is sufficiently small, this nonstoquasticity of the effective qubit Hamiltonian can be avoided if we perform a canonical transformation prior to projecting onto an effective qubit Hamiltonian. Our results shed light on the power of circuit QED Hamiltonians for the use of quantum adiabatic computation and the subtlety of finding a representation which cures the sign problem in these systems.

DOI
10.1103/physreva.103.042401

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We analyze whether circuit QED Hamiltonians are stoquastic, focusing on systems of coupled flux qubits. We show that scalable sign-problem-free path integral Monte Carlo simulations can typically be performed for such systems. Despite this, we corroborate the recent finding [I. Ozfidan et al., Phys. Rev. Appl. 13, 034037 (2020)] that an effective, nonstoquastic qubit Hamiltonian can emerge in a system of capacitively coupled flux qubits. We find that if the capacitive coupling is sufficiently small, this nonstoquasticity of the effective qubit Hamiltonian can be avoided if we perform a canonical transformation prior to projecting onto an effective qubit Hamiltonian. Our results shed light on the power of circuit QED Hamiltonians for the use of quantum adiabatic computation and the subtlety of finding a representation which cures the sign problem in these systems.
Demonstration of a $n\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}o\phantom{\rule{0}{0ex}}q\phantom{\rule{0}{0ex}}u\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}s\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}c$ Hamiltonian---one for which there exists no local basis in which all off-diagonal elements are nonpositive---is an important step toward the development of quantum annealers that are more computationally powerful, or even universal. (This condition is tied to the ``sign problem'' in quantum Monte Carlo techniques.) This work presents the implementation of such a Hamiltonian by coupling two flux qubits both inductively and capacitively. The signature of nonstoquastic behavior is observed through destructive interference in quantum coherent oscillations. This result would seem to bear strong implications for scalable quantum computing.
There is a tremendous interest in fabricating superconducting flux circuits that are nonstoquastic -- i.e., have positive off-diagonal matrix elements -- in their qubit representation, as these circuits are thought to be unsimulable by classical approaches due to the presence of a sign problem and thus could play a key role in the demonstration of speedups in quantum annealing protocols. We show, however, that the elimination of the sign problem in these systems is possible by the direct simulation of the flux circuits. Our approach not only obviates the reduction of flux circuits to their qubit representation but also produces results that are more in the spirit of the experimental setup. We discuss the implications of our work, arguing that our findings cast doubt on the conception that superconducting flux circuits represent the correct avenue for universal adiabatic quantum computers.
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 (Yurke B and Denker J S 1984 Phys. Rev. A 29 1419, Reiter F and Sørensen A S 2012 Phys. Rev. A 85 032111 and Amin M H et al 2012 Phys. Rev. A 86 052314), 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 report specific examples of superconducting qubits, including the capacitively-shunted flux qubit, where the standard reduction approaches fail. 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.
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 (Yurke B and Denker J S 1984 Phys. Rev. A 29 1419, Reiter F and Sørensen A S 2012 Phys. Rev. A 85 032111 and Amin M H et al 2012 Phys. Rev. A 86 052314), 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 report specific examples of superconducting qubits, including the capacitively-shunted flux qubit, where the standard reduction approaches fail. 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.
A flux qubit can interact strongly when it is capacitively coupled to other circuit elements. This interaction can be separated into two parts, one acting on the qubit subspaces and one in which excited states mediate the interaction. The first term dominates the interaction between the flux qubit and an LCLCLC resonator, leading to ultrastrong couplings of the form ${\ensuremath{\sigma}}^{y}(a+{a}^{\ifmmode\dagger\else\textdagger\fi{}})$, which complement the inductive ${\ensuremath{\sigma}}^{x}i({a}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{-}a)$ coupling. However, when coupling two flux qubits capacitively, all terms need to be taken into account, leading to complex nonstoquastic ultrastrong interaction of types ${\ensuremath{\sigma}}_{1}^{y}{\ensuremath{\sigma}}_{2}^{y}$, ${\ensuremath{\sigma}}_{1}^{z}{\ensuremath{\sigma}}_{2}^{z}$, and ${\ensuremath{\sigma}}_{1}^{x}{\ensuremath{\sigma}}_{2}^{x}$. Our theory explains all these interactions, describing them in terms of general circuit properties---coupling capacitances, qubit gaps, inductive, Josephson and capacitive energies---that apply to a wide variety of circuits and flux qubit designs.
Quantum many-body systems whose Hamiltonians are non-stoquastic, i.e., have positive off-diagonal matrix elements in a given basis, are known to pose severe limitations on the efficiency of Quantum Monte Carlo algorithms designed to simulate them, due to the infamous sign problem. We study the computational complexity associated with 'curing' non-stoquastic Hamiltonians, i.e., transforming them into sign-problem-free ones. We prove that if such transformations are limited to single-qubit Clifford group elements or general single-qubit orthogonal matrices, finding the curing transformation is NP-complete. We discuss the implications of this result.