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arXiv:1903.06139 · Physical Review Applied

Demonstration of a Nonstoquastic Hamiltonian in Coupled Superconducting Flux Qubits

Isil Ozfidan, C. Deng, Anatoly Yu. Smirnov, Trevor M. Lanting, R. Harris, Loren J. Swenson, Jed D. Whittaker, Fabio Altomare, Michael C. Babcock, Catia Baron, A. J. Berkley, Kelly Boothby, Holly Christiani, P. Bunyk, Colin Enderud, Bram Evert, Markus Hager, A. Hajda, Jeremy Hilton, Shuiyuan Huang, Emile Hoskinson, Mark W. Johnson, Kais Jooya, E. Ladizinsky, N. Ladizinsky, R. Li, A. MacDonald, D. Marsden, G. Marsden, T. Medina, Reza Molavi, R. Neufeld, Mathias Stokkebye Nissen, Mana Norouzpour, T. Oh, I. Pavlov, I. Perminov, Gabriel Poulin-Lamarre, M. Reis, T. Prescott, C. Rich, Yuki Sato, George Sterling, Nicholas Tsai, Mark H. Volkmann, Warren Wilkinson, J. Yao, M. H. S. Amin

Published Mar 16, 2020• 63 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevapplied.13.034037

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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.
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.
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.
An effective Hamiltonian is derived for two coupled three-Josephson-junction (3JJ) qubits. This is not quite trivial, for the customary ``free'' 3JJ Hamiltonian is written in the limit of zero inductance LLL. Neglecting the self-flux is already dubious for one qubit when it comes to readout, and becomes untenable when discussing inductive coupling. First, inductance effects are analyzed for a single qubit. For small LLL, the self-flux is a ``fast variable,'' which can be eliminated adiabatically. However, the commonly used junction phases are not appropriate ``slow variables,'' and instead one introduces degrees of freedom that are decoupled from the loop current to leading order. In the quantum case, the zero-point fluctuations (LCLCLC oscillations) in the loop current diverge as $L\ensuremath{\rightarrow}0$. While their effect thus formally dominates over the classical self-flux, it merely renormalizes the Josephson couplings of the effective (two-phase) theory. In the coupled case, the strong zero-point fluctuations render the full (six-phase) wave function significantly entangled in leading order. However, in going to the four-phase theory, this uncontrollable entanglement is integrated out completely, leaving a computationally usable mutual-inductance term of the expected form as the effective interaction.
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.
Although stoquastic Hamiltonians are known to be simulable via sign-problem-free quantum Monte Carlo (QMC) techniques, the nonstoquasticity of a Hamiltonian does not necessarily imply the existence of a QMC sign problem.We give a sufficient and necessary condition for the QMC-simulability of Hamiltonians in a given basis: We prove that a QMC simulation will be sign-problem-free if and only if all the overall total phases along the chordless cycles of the weighted graph whose adjacency matrix is the Hamiltonian are zero (modulo 2π ).We use our findings to provide a construction for nonstoquastic, yet sign-problem-free and hence QMC-simulable, quantum many-body models.We also demonstrate why the simulation of truly sign-problematic models using the QMC weights of the stoquasticized Hamiltonian is generally suboptimal.We offer a superior alternative.