Realizing Quantum Convolutional Neural Networks on a Superconducting Quantum Processor to Recognize Quantum Phases

  1. Johannes Herrmann,
  2. Sergi Masot Llima,
  3. Ants Remm,
  4. Petr Zapletal,
  5. Nathan A. McMahon,
  6. Colin Scarato,
  7. Francois Swiadek,
  8. Christian Kraglund Andersen,
  9. Christoph Hellings,
  10. Sebastian Krinner,
  11. Nathan Lacroix,
  12. Stefania Lazar,
  13. Michael Kerschbaum,
  14. Dante Colao Zanuz,
  15. Graham J. Norris,
  16. Michael J. Hartmann,
  17. Andreas Wallraff,
  18. and Christopher Eichler
Quantum computing crucially relies on the ability to efficiently characterize the quantum states output by quantum hardware. Conventional methods which probe these states through direct
measurements and classically computed correlations become computationally expensive when increasing the system size. Quantum neural networks tailored to recognize specific features of quantum states by combining unitary operations, measurements and feedforward promise to require fewer measurements and to tolerate errors. Here, we realize a quantum convolutional neural network (QCNN) on a 7-qubit superconducting quantum processor to identify symmetry-protected topological (SPT) phases of a spin model characterized by a non-zero string order parameter. We benchmark the performance of the QCNN based on approximate ground states of a family of cluster-Ising Hamiltonians which we prepare using a hardware-efficient, low-depth state preparation circuit. We find that, despite being composed of finite-fidelity gates itself, the QCNN recognizes the topological phase with higher fidelity than direct measurements of the string order parameter for the prepared states.

In-situ tunable nonlinearity and competing signal paths in coupled superconducting resonators

  1. Michael Fischer,
  2. Qi-Ming Chen,
  3. Christian Besson,
  4. Peter Eder,
  5. Jan Goetz,
  6. Stefan Pogorzalek,
  7. Michael Renger,
  8. Edwar Xie,
  9. Michael J. Hartmann,
  10. Kirill G. Fedorov,
  11. Achim Marx,
  12. Frank Deppe,
  13. and Rudolf Gross
We have fabricated and studied a system of two tunable and coupled nonlinear superconducting resonators. The nonlinearity is introduced by galvanically coupled dc-SQUIDs. We simulate
the system response by means of a circuit model, which includes an additional signal path introduced by the electromagnetic environment. Furthermore, we present two methods allowing us to experimentally determine the nonlinearity. First, we fit the measured frequency and flux dependence of the transmission data to simulations based on the equivalent circuit model. Second, we fit the power dependence of the transmission data to a model that is predicted by the nonlinear equation of motion describing the system. Our results show that we are able to tune the nonlinearity of the resonators by almost two orders of magnitude via an external coil and two on-chip antennas. The studied system represents the basic building block for larger systems, allowing for quantum simulations of bosonic many-body systems with a larger number of lattice sites.

Many-body quantum circuits for quantum simulation and computing

  1. Samuel A. Wilkinson,
  2. and Michael J. Hartmann
Quantum simulators are attractive as a means to study many-body quantum systems that are not amenable to classical numerical treatment. A versatile framework for quantum simulation
is offered by superconducting circuits. In this perspective, we discuss how superconducting circuits allow the engineering of a wide variety of interactions, which in turn allows the simulation of a wide variety of model Hamiltonians. In particular we focus on strong photon-photon interactions mediated by nonlinear elements. This includes on-site, nearest-neighbour and four-body interactions in lattice models, allowing the implementation of extended Bose-Hubbard models and the toric code. We discuss not only the present state in analogue quantum simulation, but also future perspectives of superconducting quantum simulation that open up when concatenating quantum gates in emerging quantum computing platforms.

Floquet engineering in superconducting circuits: from arbitrary spin-spin interactions to the Kitaev honeycomb model

  1. Mahdi Sameti,
  2. and Michael J. Hartmann
We derive a theory for the generation of arbitrary spin-spin interactions in superconducting circuits via periodic time modulation of the individual qubits or the qubit-qubit interactions.
The modulation frequencies in our approach are in the microwave or radio frequency regime so that the required fields can be generated with standard generators. Among others, our approach is suitable for generating spin lattices that exhibit quantum spin liquid behavior such as Kitaev’s honeycomb model.

A Superconducting Quantum Simulator for Topological Order and the Toric Code

  1. Mahdi Sameti,
  2. Anton Potocnik,
  3. Dan E. Browne,
  4. Andreas Wallraff,
  5. and Michael J. Hartmann
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 non-contractible 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.

Quantum State Engineering with Circuit Electromechanical Three-Body Interactions

  1. Mehdi Abdi,
  2. Matthias Pernpeintner,
  3. Rudolf Gross,
  4. Hans Huebl,
  5. and Michael J. Hartmann
We propose a hybrid system with quantum mechanical three-body interactions between photons, phonons, and qubit excitations. These interactions take place in a circuit quantum electrodynamical
architecture with a superconducting microwave resonator coupled to a transmon qubit whose shunt capacitance is free to mechanically oscillate. We show that this system design features a three-mode polariton–mechanical mode and a nonlinear transmon–mechanical mode interaction in the strong coupling regime. Together with the strong resonator–transmon interaction, these properties provide intriguing opportunities for manipulations of this hybrid quantum system. We show, in particular, the feasibility of cooling the mechanical motion down to its ground state and preparing various nonclassical states including mechanical Fock and cat states and hybrid tripartite entangled states.

Steady-state phase diagram of a driven QED-cavity array with cross-Kerr nonlinearities

  1. Jiasen Jin,
  2. Davide Rossini,
  3. Martin Leib,
  4. Michael J. Hartmann,
  5. and Rosario Fazio
We study the properties of an array of QED-cavities coupled by nonlinear elements in the presence of photon leakage and driven by a coherent source. The main effect of the nonlinear
couplings is to provide an effective cross-Kerr interaction between nearest-neighbor cavities. Additionally correlated photon hopping between neighboring cavities arises. We provide a detailed mean-field analysis of the steady-state phase diagram as a function of the system parameters, the leakage and the external driving, and show the emergence of a number of different quantum phases. A photon crystal associated to a spatial modulation of the photon blockade appears. The steady state can also display oscillating behavior and bi-stability. In some regions the crystalline ordering may coexist with the oscillating behavior. Furthermore we study the effect of short-range quantum fluctuations by employing a cluster mean-field analysis. Focusing on the corrections to the photon crystal boundaries, we show that, apart for some quantitative differences, the cluster mean field supports the findings of the simple single-site analysis. In the last part of the paper we concentrate on the possibility to build up the class of arrays introduced here, by means of superconducting circuits of existing technology. We consider a realistic choice of the parameters for this specific implementation and discuss some properties of the steady-state phase diagram.

Synchronized Switching in a Josephson Junction Crystal

  1. Martin Leib,
  2. and Michael J. Hartmann
We consider a superconducting coplanar waveguide resonator where the central conductor is interrupted by a series of uniformly spaced Josephson junctions. The device forms an extended
medium that is optically nonlinear on the single photon level with normal modes that inherit the full nonlinearity of the junctions but are nonetheless accessible via the resonator ports. For specific plasma frequencies of the junctions a set of normal modes clusters in a narrow band and eventually become entirely degenerate. Upon increasing the intensity of a red detuned drive on these modes, we observe a sharp and synchronized switching from low occupation quantum states to high occupation classical fields, accompanied by a pronounced jump from low to high output intensity.

The Circuit Quantum Electrodynamical Josephson Interferometer

  1. Robert Jirschik,
  2. and Michael J. Hartmann
Arrays of circuit cavities offer fascinating perspectives for exploring quantum many-body systems in a driven dissipative regime where excitation losses are continuously compensated
by coherent input drives. Here we investigate a system consisting of three transmission line resonators, where the two outer ones are driven by coherent input sources and the central resonator interacts with a superconducting qubit. Whereas a low excitation number regime of such a device has been considered previously with a numerical integration, we here specifically address the high excitation density regime. This is of particular interest as intra cavity fields might undergo a transition from low excitation number quantum fields to high amplitude classical fields when increasing the input drives. We present analytical approximations to these regimes in the form of two methods. The first method is a Bogoliubov expansion in quantum fluctuations which can be understood as an approximation for weak nonlinearities. As the second method we introduce a combination of mean-field decoupling for the photon tunneling with an exact approach to a driven Kerr nonlinearity which can be understood as an approximation for low tunneling rates.

Photon solid phases in driven arrays of non-linearly coupled cavities

  1. Jiasen Jin,
  2. Davide Rossini,
  3. Rosario Fazio,
  4. Martin Leib,
  5. and Michael J. Hartmann
We introduce and study the properties of an array of QED cavities coupled by non-linear elements, in the presence of photon leakage and driven by a coherent source. The non-linear
couplings lead to photon hopping and to nearest-neighbor Kerr terms. By tuning the system parameters, the steady state of the array can exhibit a photon crystal associated to a periodic modulation of the photon blockade. In some cases the crystalline ordering may coexist with phase synchronisation. The class of cavity arrays we consider can be built with superconducting circuits of existing technology.