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arXiv:1608.04565 · Physical Review A

Superconducting quantum simulator for topological order and the toric code

Mahdi Sameti, Anton Potočnik, Dan E. Browne, Andreas J. Wallraff, Michael J. Hartmann

Published Apr 21, 2017• 50 citations
arXivHTMLPDF

Abstract

Topological order is now being established as a central criterion for characterizing and classifying ground states of condensed matter systems and complements categorizations based on symmetries. Fractional quantum Hall systems and quantum spin liquids are receiving substantial interest because of their intriguing quantum correlations, their exotic excitations, and prospects for protecting stored quantum information against errors. Here, we show that the Hamiltonian of the central model of this class of systems, the toric code, can be directly implemented as an analog quantum simulator in lattices of superconducting circuits. The four-body interactions, which lie at its heart, are in our concept realized via superconducting quantum interference devices (SQUIDs) that are driven by a suitably oscillating flux bias. All physical qubits and coupling SQUIDs can be individually controlled with high precision. Topologically ordered states can be prepared via an adiabatic ramp of the stabilizer interactions. Strings of qubit operators, including the stabilizers and correlations along noncontractible loops, can be read out via a capacitive coupling to read-out resonators. Moreover, the available single-qubit operations allow to create and propagate elementary excitations of the toric code and to verify their fractional statistics. The architecture we propose allows to implement a large variety of many-body interactions and thus provides a versatile analog quantum simulator for topological order and lattice gauge theories.

DOI
10.1103/physreva.95.042330

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Topological order is now being established as a central criterion for characterizing and classifying ground states of condensed matter systems and complements categorizations based on symmetries. Fractional quantum Hall systems and quantum spin liquids are receiving substantial interest because of their intriguing quantum correlations, their exotic excitations, and prospects for protecting stored quantum information against errors. Here, we show that the Hamiltonian of the central model of this class of systems, the toric code, can be directly implemented as an analog quantum simulator in lattices of superconducting circuits. The four-body interactions, which lie at its heart, are in our concept realized via superconducting quantum interference devices (SQUIDs) that are driven by a suitably oscillating flux bias. All physical qubits and coupling SQUIDs can be individually controlled with high precision. Topologically ordered states can be prepared via an adiabatic ramp of the stabilizer interactions. Strings of qubit operators, including the stabilizers and correlations along noncontractible loops, can be read out via a capacitive coupling to read-out resonators. Moreover, the available single-qubit operations allow to create and propagate elementary excitations of the toric code and to verify their fractional statistics. The architecture we propose allows to implement a large variety of many-body interactions and thus provides a versatile analog quantum simulator for topological order and lattice gauge theories.
The Landau Fermi liquid theory and the Ginzburg-Landau phase transition theory stand as two pivotal cornerstones in traditional condensed matter physics, achieving significant success in addressing crucial physical phenomena such as BCS superconductors and liquid helium superfluids. However, marked by the discoveries of the quantum Hall effect and high-temperature superconductivity in the 1980s, it gradually became evident that for a broad class of novel quantum states, such as fractional quantum Hall states and quantum spin liquids, their properties transcend the Landau Fermi liquid theory and Ginzburg-Landau phase transition theory. Topological order and its related concepts of long-range many-body quantum entanglement and fractionalized excitation have become the key concepts to understand these exotic quantum states. Designing and identifying topologically ordered states of matter in quantum materials and quantum simulation systems, and probing and manipulating their fractionalized excitations, are important research directions in modern condensed matter physics. In recent years, great progress has been made in the quantum simulation and manipulation of topological order on highly controllable quantum simulation platforms, such as Rydberg atomic systems, superconducting quantum processors, and two-dimensional moiré superlattices. This article provides a brief overview of recent research advances and challenges in the study of topological order in traditional condensed matter systems and quantum simulation experimental platforms. It also provides prospects for the future developments of this field.
Kitaev's toric code is an exactly solvable model with Z2{\mathbb{Z}}_{2}Z2​-topological order, which has potential applications in quantum computation and error correction. However, a direct experimental realization remains an open challenge. Here, we propose a building block for Z2{\mathbb{Z}}_{2}Z2​ lattice gauge theories coupled to dynamical matter and demonstrate how it allows for an implementation of the toric-code ground state and its topological excitations. This is achieved by introducing separate matter excitations on individual plaquettes, whose motion induce the required plaquette terms. The proposed building block is realized in the second-order coupling regime and is well suited for implementations with superconducting qubits. Furthermore, we propose a pathway to prepare topologically nontrivial initial states during which a large gap on the order of the underlying coupling strength is present. This is verified by both analytical arguments and numerical studies. Moreover, we outline experimental signatures of the ground-state wave function and introduce a minimal braiding protocol. Detecting a $\ensuremath{\pi}$-phase shift between Ramsey fringes in this protocol reveals the anyonic excitations of the toric-code Hamiltonian in a system with only three triangular plaquettes. Our work paves the way for realizing non-Abelian anyons in analog quantum simulators.
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We introduce a framework of constructing topological subsystem codes based on the class of anticommuting quantum spin liquids described in [Phys. Rev. B 113, 064402 (2026)]. A canonical model from this class can be considered as a spatial modification of the toric code that voids its stabilizer code property. Rather, these models contain an extensive set of anticommuting local conserved operators that lead to an extensive ground state degeneracy. This degeneracy forms the basis of the subsystem degrees of freedom in the associated quantum error correcting code. The code inherits the many-body topological order of the quantum spin liquid, making it a topological subsystem code. We present two concrete and detailed examples for constructing these codes on a square lattice and a kagome lattice geometry, requiring weight-4 and weight-3 local check operator measurements respectively. In contrast to other subsystem codes, a unique property of these codes is the presence of an extensive number of local gauge qubits that are left undisturbed by the check operators apart from the logical qubits. Our construction provides a template for generating this new category of topological subsystem codes on different lattice or graph geometries, suitable for implementation on various quantum hardware platforms.
Quantum phases with topological order, such as quantum spin liquids, have been the focus of explorations for several decades. Such phases feature a number of remarkable properties including long-range quantum entanglement. Moreover, they can be potentially exploited for the realization of robust quantum computation, as exemplified by the paradigmatic toric code model. While some indications that such phases may be present in frustrated condensed matter systems have been previously reported, so far quantum spin liquids have eluded direct experimental detection. In this talk, I will show how a programmable quantum simulator based on Rydberg atom arrays can be used to realize and probe quantum spin liquid states. In our approach, atoms are placed on the links of a kagome lattice and coherent evolution under Rydberg blockade enables the transition into frustrated quantum states with no local order. We detect the onset of a quantum spin liquid phase of the toric code type by measuring topological string operators in two complementary bases. The properties of this state are further revealed using a lattice with non-trivial topology, representing a step towards the realization of a topological qubit. Our observations open the door to the controlled experimental exploration of topological quantum matter, and could enable the investigation of new methods for topologically protected quantum information processing.