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arXiv:2011.03831

Stoquastic simulations of non-stoquastic superconducting flux circuits

Kol-Namer, Harel, Tom Halverson, Gupta, Lalit, Moshe Goldstein, Itay Hen

Published Nov 7, 2020• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2011.03831

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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.
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.
Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like XX and XXXXX with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian non-stoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.
The applicability of quantum annealing to various problems can be improved by expressing the Hamiltonian using a circuit satisfiability problem. We investigate the detailed characteristics of the NOR/NAND functions of a superconducting quantum circuit, which are the basic building blocks to implementing various types of problem Hamiltonians. The circuit is composed of superconducting flux qubits with all-to-all connectivity, where direct magnetic couplers are utilized instead of the variable couplers in the conventional superconducting quantum circuit. This configuration provides efficient scalability because the problem Hamiltonian is implemented using fewer qubits. We present an experiment with a complete logic operation of NOR/NAND, in which the circuit produces results with a high probability of success for arbitrary combinations of inputs. The features of the quantum circuit agree qualitatively with the theory, especially the mechanism for an operation under external flux modulation. Moreover, by calibrating the bias conditions to compensate for the offset flux from the surrounding circuit, the quantum circuit quantitatively agrees with the theory. To achieve true quantum annealing, we discuss the effects of the reduction in electric noise in quantum annealing.
Abstract We argue that a complete description of quantum annealing implemented with continuous variables must take into account the non-adiabatic Aharonov-Anandan geometric phase that arises when the system Hamiltonian changes during the anneal. We show that this geometric effect leads to the appearance of non-stoquasticity in the effective quantum Ising Hamiltonians that are typically used to describe quantum annealing with flux qubits. We explicitly demonstrate the effect of this geometric non-stoquasticity when quantum annealing is performed with a system of one and two coupled flux qubits. The realization of non-stoquastic Hamiltonians has important implications from a computational complexity perspective, since it is believed that in many cases quantum annealing with stoquastic Hamiltonians can be efficiently simulated via classical algorithms such as Quantum Monte Carlo. It is well known that the direct implementation of non-stoquastic Hamiltonians with flux qubits is particularly challenging. Our results suggest an alternative path for the implementation of non-stoquasticity via geometric phases that can be exploited for computational purposes.
The field of quantum information has been a fast growing field of research in the last few decades. The reason for this development is the potentially immense power of quantum computers, capable of solving hard problems in a fraction of the time it would take a classical computer. Quantum annealers are a type of quantum computer that seek to solve computational problems by finding the ground state of a quantum system. This type of quantum computer has been implemented on a large scale using superconducting flux quantum bits, but the computational power is limited. One potential improvement to help in this context is the implementation of multi-body interactions between quantum bits. Many-body interactions would be useful in allowing quantum annealers to solve more complex problems that are relevant in various areas of quantum information and science and implementing quantum error correction. \n \nWe present a superconducting device that implements a strong and tunable in sign and magnitude three-body interaction between superconducting flux quantum bits. The circuit design proposed has vanishing two-body interactions and robustness against noise and parameter variations. This circuit behaves as an ideal computational basis ZZZ coupler in a simulated three-qubit quantum annealing experiment. These properties are confirmed by calculations based on the Born-Oppenheimer approximation, a two-level spin model for the coupling circuit, and full numerical diagonalization. \n \nThe proposed circuit is based on already available technology and therefore could be readily implemented in the next generation of quantum annealing hardware. This work will be relevant for advanced quantum annealing protocols and future developments of high-order many-body interactions in quantum computers and simulators.