Authors: Jan-Michael Reiner, Frank Wilhelm-Mauch, Gerd Schön, Michael Marthaler
Journal reference: Quantum Sci. Technol. 4, 035005 (2019)

A key goal of digital quantum computing is the simulation of fermionic systems such as molecules or the Hubbard model. Unfortunately, for present and near-future quantum computers the use of quantum error correction schemes is still out of reach. Hence, the finite error rate limits the use of quantum computers to algorithms with a low number of gates. The variational Hamiltonian ansatz (VHA) has been shown to produce the ground state in good approximation in a manageable number of steps. Here we study explicitly the effect of gate errors on its performance. The VHA is inspired by the adiabatic quantum evolution under the influence of a time-dependent Hamiltonian, where the - ideally short - fixed Trotter time steps are replaced by variational parameters. The method profits substantially from quantum variational error suppression, e.g., unitary quasi-static errors are mitigated within the algorithm. We test the performance of the VHA when applied to the Hubbard model in the presence of unitary control errors on quantum computers with realistic gate fidelities.

Authors: Ping Yang, Jan David Brehm, Juha Leppäkangas, Lingzhen Guo, Michael Marthaler, Isabella Boventer, Alexander Stehli, Tim Wolz, Alexey V. Ustinov, Martin Weides
Journal reference: Phys. Rev. Applied 14, 024025 (2020)

We demonstrate the local control of up to eight two-level systems interacting strongly with a microwave cavity. Following calibration, the frequency of each individual two-level system (qubit) is tunable without influencing the others. Bringing the qubits one by one on resonance with the cavity, we observe the collective coupling strength of the qubit ensemble. The splitting scales up with the square root of the number of the qubits, which is the hallmark of the Tavis-Cummings model. The local control circuitry causes a bypass shunting the resonator, and a Fano interference in the microwave readout, whose contribution can be calibrated away to recover the pure cavity spectrum. The simulator's attainable size of dressed states with up to five qubits is limited by reduced signal visibility, and -- if uncalibrated -- by off-resonance shifts of sub-components. Our work demonstrates control and readout of quantum coherent mesoscopic multi-qubit system of intermediate scale under conditions of noise.

Authors: Juha Leppäkangas, Jan David Brehm, Ping Yang, Lingzhen Guo, Michael Marthaler, Alexey V. Ustinov, Martin Weides
Journal reference: Phys. Rev. A 99, 063804 (2019)

We demonstrate how the heating of an environment can invert the line shape of a driven cavity. We consider a superconducting coplanar cavity coupled to multiple artificial atoms. The measured cavity transmission is characterized by Fano-type resonances with a shape that is continuously tunable by bias current through nearby (magnetic flux) control lines. In particular, the same dispersive shift of the microwave cavity can be observed as a peak or a dip. We find that this Fano-peak inversion is possible due to a tunable interference between a microwave transmission through a background, with reactive and dissipative properties, and through the cavity, affected by bias-current induced heating. The background transmission occurs due to crosstalk between the control and transmission lines. We show how such background can be accounted for by Jaynes-Cummings type models via modified boundary conditions between the cavity and transmission lines. We find generally that whereas resonance positions determine system energy levels, resonance shapes give information on system fluctuations and dissipation.

Authors: Jan-Michael Reiner, Sebastian Zanker, Iris Schwenk, Juha Leppäkangas, Frank Wilhelm-Mauch, Gerd Schön, Michael Marthaler
Journal reference: Quantum Sci. Technol. 3, 045008 (2018)

Digital quantum simulations offer exciting perspectives for the study of fermionic systems such as molecules or lattice models. However, with quantum error correction still out of reach with present-day technology, a non-vanishing error rate is inevitable. We study the influence of gate errors on simulations of the Trotterized time evolution of the quantum system with a focus on the fermionic Hubbard model. Specifically, we consider the effect of stochastic over-rotations in the applied gates. Depending on the particular algorithm implemented such gate errors may lead to a time evolution that corresponds to a disordered fermionic system, or they may correspond to unphysical errors, e.g., violate particle number conservation. We substantiate our analysis by numerical simulations of model systems. In addition, we establish the relation between the gate fidelity and the strength of the over-rotations in a Trotterized quantum simulation. Based on this we provide estimates for the maximum number of Trotter steps which can be performed with sufficient accuracy for a given algorithm. This in turn implies, apart from obvious limitations on the maximum time of the simulation, also limits on the system size that can be handled.

Authors: Juha Leppäkangas, Michael Marthaler
Journal reference: Phys. Rev. B 98, 224511 (2018)

We study the scattering of propagating microwave fields by a DC-voltage-biased Josephson junction. At sub-gap voltages, a small Josephson junction works merely as a non-linear boundary that can absorb, amplify, and diversely convert propagating microwaves. In the leading-order perturbation theory of the Josephson coupling energy, the spectral density and quadrature fluctuations of scattered thermal and coherent radiation can be described in terms of the well-known P(E) function. Applying this, we study how thermal and coherent radiation is absorbed and amplified in an Ohmic transmission line and in a circuit with a resonance frequency. We show when a coherent input can create a two-mode squeezed output. In addition, we evaluate scattering amplitudes between arbitrary photon-number (Fock) states, characterizing individual photon multiplication and absorption processes occuring at the junction.

Authors: Juha Leppäkangas, Jochen Braumüller, Melanie Hauck, Jan-Michael Reiner, Iris Schwenk, Sebastian Zanker, Lukas Fritz, Alexey V. Ustinov, Martin Weides, Michael Marthaler
Journal reference: Phys. Rev. A 97, 052321 (2018)

We consider superconducting circuits for the purpose of simulating the spin-boson model. The spin-boson model consists of a single two-level system coupled to bosonic modes. In most cases, the model is considered in a limit where the bosonic modes are sufficiently dense to form a continuous spectral bath. A very well-known case is the ohmic bath, where the density of states grows linearly with the frequency. In the limit of weak coupling or large temperature, this problem can be solved numerically. If the coupling is strong, the bosonic modes can become sufficiently excited to make a classical simulation impossible. Here, we discuss how a quantum simulation of this problem can be performed by coupling a superconducting qubit to a set of microwave resonators. We demonstrate a possible implementation of a continuous spectral bath with individual bath resonators coupling strongly to the qubit. Applying a microwave drive scheme potentially allows us to access the strong coupling regime of the spin-boson model. We discuss how the resulting spin relaxation dynamics with different initialization conditions can be probed by standard qubit-readout techniques from circuit quantum electrodynamics.

Authors: Sebastian Zanker, Iris Schwenk, Jan-Michael Reiner, Juha Leppäkangas, Michael Marthaler
Journal reference: Phys. Rev. B 97, 214301 (2018)

Simulation of interacting electron systems is one of the great challenges of modern quantum chemistry and solid state physics. Controllable quantum systems offer the opportunity to create artificial structures which mimic the system of interest. An interesting quantity to extract from these quantum simulations is the spectral function. We map a noisy quantum simulator onto a fermionic system and investigate the influence of decoherence on the simulation of the spectral density using a diagrammatic approach on Keldysh contour. We show that features stronger than the single-qubit decoherence rate can be resolved, while weaker features wash out. For small systems, we compare our Keldysh approach to master-equation calculations.

Authors: Iris Schwenk, Jan-Michael Reiner, Sebastian Zanker, Lin Tian, Juha Leppäkangas, Michael Marthaler
Journal reference: Phys. Rev. A 97, 042310 (2018)

Well-controlled quantum systems can potentially be used as quantum simulators. However, a quantum simulator is inevitably perturbed by coupling to additional degrees of freedom. This constitutes a major roadblock to useful quantum simulations. So far there are only limited means to understand the effect of perturbation on the results of quantum simulation. Here, we present a method which, in certain circumstances, allows for the reconstruction of the ideal result from measurements on a perturbed quantum simulator. We consider extracting the value of the correlator ⟨O^i(t)O^j(0)⟩ from the simulated system, where O^i are the operators which couple the system to its environment. The ideal correlator can be straightforwardly reconstructed by using statistical knowledge of the environment, if any n-time correlator of operators O^i of the ideal system can be written as products of two-time correlators. We give an approach to verify the validity of this assumption experimentally by additional measurements on the perturbed quantum simulator. The proposed method can allow for reliable quantum simulations with systems subjected to environmental noise without adding an overhead to the quantum system.

Authors: Iris Schwenk, Sebastian Zanker, Jan-Michael Reiner, Juha Leppäkangas, Michael Marthaler
Journal reference: Phys. Rev. Lett. 119, 240502 (2017)

We study an analog quantum simulator coupled to a reservoir with a known spectral density. The reservoir perturbs the quantum simulation by causing decoherence. The simulator is used to measure an operator average, which cannot be calculated using any classical means. Since we cannot predict the result, it is difficult to estimate the effect of the environment. Especially, it is difficult to resolve whether the perturbation is small or if the actual result of the simulation is in fact very different from the ideal system we intend to study. Here, we show that in specific systems a measurement of additional correlators can be used to verify the reliability of the quantum simulation. The procedure only requires additional measurements on the quantum simulator itself. We demonstrate the method theoretically in the case of a single spin connected to a bosonic environment.

Authors: Alexander Bilmes, Sebastian Zanker, Andreas Heimes, Michael Marthaler, Gerd Schön, Georg Weiss, Alexey V. Ustinov, Jürgen Lisenfeld
Journal reference: Phys. Rev. B 96, 064504 (2017)

The sensitivity of superconducting qubits allows for spectroscopy and coherence measurements on individual two-level systems present in the disordered tunnel barrier of an Al/AlOx/Al Josephson junction. We report experimental evidence for the decoherence of two-level systems by Bogoliubov quasiparticles leaking into the insulating AlOx barrier. We control the density of quasiparticles in the junction electrodes either by the sample temperature or by injecting them using an on-chip dc-SQUID driven to its resistive state. The decoherence rates were measured by observing the two-level system's quantum state evolving under application of resonant microwave pulses and were found to increase linearly with quasiparticle density, in agreement with theory. This interaction with electronic states provides a noise and decoherence mechanism that is relevant for various microfabricated devices such as qubits, single-electron transistors, and field-effect transistors. The presented experiments also offer a possibility to determine the location of the probed two-level systems across the tunnel barrier, providing clues about the fabrication step in which they emerge.

Authors: Jan-Michael Reiner, Michael Marthaler, Jochen Braumüller, Martin Weides, Gerd Schön

The Jordan-Wigner transformation maps a one-dimensional spin-1/2 system onto a fermionic model without spin degree of freedom. A double chain of quantum bits with XX and ZZ couplings of neighboring qubits along and between the chains, respectively, can be mapped on a spin-full 1D Fermi-Hubbard model. The qubit system can thus be used to emulate the quantum properties of this model. We analyze physical implementations of such analog quantum simulators, including one based on transmon qubits, where the ZZ interaction arises due to an inductive coupling and the XX interaction due to a capacitive interaction. We propose protocols to gain confidence in the results of the simulation through measurements of local operators.

Authors: Sebastian Zanker, Michael Marthaler, Gerd Schön

It is frequently observed that even at very low temperatures the number of quasiparticles in superconducting materials is higher than predicted by standard BCS-theory. These quasiparticles can interact with two-level systems, such as superconducting qubits or two-level systems (TLS) in the amorphous oxide layer of a Josephson junction. This interaction leads to decay and decoherence of the TLS, with specific results, such as the time dependence, depending on the distribution of quasiparticles and the form of the interaction. We study the resulting decay laws for different experimentally relevant protocols.

Authors: Iris Schwenk, Michael Marthaler
Journal reference: Phys. Rev. B 93, 014305 (2016)

We study a system coupled to external degrees of freedom, called bath, where we assume that the total system, consisting of system and bath is in equilibrium. An expansion in the coupling between system and bath leads to a general form of the reduced density matrix of the system as a function of the bath selfenergy. The coupling to the bath results in a renormalization of the energies of the system and in a change of the eigenbasis. This theory is applicable to quantum emulators in thermal equilibrium. Undesired external degrees of freedom can affect their reliability. We study the influence of bosonic degrees of freedom on the state of a six qubit system.