Simon E. Nigg
ContactDepartment of PhysicsUniversity of Basel Klingelbergstrasse 82 CH4056 Basel, Switzerland

Research Interests
 Quantum information processing with superconducting circuits
 Mesoscopic electron transport
 Molecular magnetism
 Quantum chaos
Curriculum Vitae (CV)
 2013present: SNF Ambizione fellow in the Condensed Matter Theory department at University of Basel, (CH)
 20102013: Postdoctoral fellow in the physics department at Yale University, CT (USA)
 20052010: Graduate studies in the physics department at University of Geneva, (CH)
 20042005: Diploma work at the Max Planck Institute for Quantum Optics in Garching, (GE)
 20002004: Studies of physics at Technische Universitaet Muenchen, (GE)
Publications
Show all abstracts.1.  Superconducting gridbus surface code architecture for holespin qubits 
Simon E. Nigg, Andreas Fuhrer, and Daniel Loss. arXiv:1612.07292
We present a scalable hybrid architecture for the 2D surface code combining superconducting resonators and holespin qubits in nanowires with tunable direct Rashba spinorbit coupling. The backbone of this architecture is a square lattice of capacitively coupled coplanar waveguide resonators each of which hosts a nanowire holespin qubit. Both the frequency of the qubits and their coupling to the microwave field are tunable by a static electric field applied via the resonator center pin. In the dispersive regime, an entangling twoqubit gate can be realized via a third order process, whereby a virtual photon in one resonator is created by a first qubit, coherently transferred to a neighboring resonator, and absorbed by a second qubit in that resonator. Numerical simulations with stateoftheart coherence times yield gate fidelities approaching the 99% fault tolerance threshold.
 
2.  Robust quantum optimizer with full connectivity 
Simon E. Nigg, Niels Loerch, and Rakesh P. Tiwari. arXiv:1609.06282; Submitted (2016).
Quantum phenomena have the potential to speed up the solution of hard optimization problems. For example quantum annealing, based on the quantum tunneling effect, has recently been shown to scale exponentially better with system size as compared with classical simulated annealing. However, current realizations of quantum annealers with superconducting qubits face two major challenges. First, the connectivity between the qubits is limited, excluding many optimization problems from a direct implementation. Second, decoherence degrades the success probability of the optimization. We address both of these shortcomings and propose an architecture in which the qubits are robustly encoded in continuous variable degrees of freedom. Remarkably, by leveraging the phenomenon of flux quantization, alltoall connectivity is obtained without overhead. Furthermore, we demonstrate the robustness of this architecture by simulating the optimal solution of a small instance of the NPhard and fully connected number partitioning problem in the presence of dissipation.
 
3.  Decoherence of highenergy electrons in weakly disordered quantum Hall edge states 
Simon E. Nigg and Anders Mathias Lunde. arxiv:1606.08574; Phys. Rev. B 94, 041407(R) (2016).
We investigate theoretically the phase coherence of electron transport in edge states of the integer quantum
Hall effect at filling factor ν = 2, in the presence of disorder and interedge state Coulomb interaction. Within
a FokkerPlanck approach, we calculate analytically the visibility of the AharonovBohm oscillations of the
current through an electronic MachZehnder interferometer. In agreement with recent experiments, we find that
the visibility is independent of the energy of the currentcarrying electrons injected high above the Fermi sea.
Instead, it is the amount of disorder at the edge that sets the phase space available for interedge state energy
exchange and thereby controls the visibility suppression.
 
4.  Implementing and characterizing precise multiqubit measurements 
J. Z. Blumoff, K. Chou, C. Shen, M. Reagor, C. Axline, R. T. Bierley, M. P. Silveri, C. Wang, B. Vlastakis, S. E. Nigg, L. Frunzio, M. H. Devoret, L. Jiang, S. M. Girvin, and R. J. Schoelkopf. arXiv:1606.00817; (2016); Phys. Rev. X 6, 031041 (2016) .
There are two general requirements to harness the computational power of quantum mechanics: the ability to manipulate the evolution of an isolated system and the ability to faithfully extract information from it. Quantum error correction and simulation often make a more exacting demand: the ability to perform nondestructive measurements of specific correlations within that system. We realize such measurements by employing a protocol adapted from [S. Nigg and S. M. Girvin, Phys. Rev. Lett. 110, 243604 (2013)], enabling realtime selection of arbitrary registerwide Pauli operators. Our implementation consists of a simple circuit quantum electrodynamics (cQED) module of four highlycoherent 3D transmon qubits, collectively coupled to a highQ superconducting microwave cavity. As a demonstration, we enact all seven nontrivial subsetparity measurements on our threequbit register. For each we fully characterize the realized measurement by analyzing the detector (observable operators) via quantum detector tomography and by analyzing the quantum backaction via conditioned process tomography. No single quantity completely encapsulates the performance of a measurement, and standard figures of merit have not yet emerged. Accordingly, we consider several new fidelity measures for both the detector and the complete measurement process. We measure all of these quantities and report high fidelities, indicating that we are measuring the desired quantities precisely and that the measurements are highly nondemolition. We further show that both results are improved significantly by an additional errorheralding measurement. The analyses presented here form a useful basis for the future characterization and validation of quantum measurements, anticipating the demands of emerging quantum technologies.
 
5.  Statistical theory of relaxation of high energy electrons in quantum Hall edge states 
Ander Mathias Lunde and Simon E. Nigg. arXiv:1602.05039; Phys. Rev. B 94, 045409 (2016).
We investigate theoretically the energy exchange between electrons of two copropagating, outofequilibrium edge states with opposite spin polarization in the integer quantum Hall regime. A quantum dot tunnelcoupled to one of the edge states locally injects electrons at high energy. Thereby a narrow peak in the energy distribution is created at high energy above the Fermi level. A second downstream quantum dot performs an energy resolved measurement of the electronic distribution function. By varying the distance between the two dots, we are able to follow every step of the energy exchange and relaxation between the edge states  even analytically under certain conditions. In the absence of translational invariance along the edge, e.g. due to the presence of disorder, energy can be exchanged by nonmomentum conserving twoparticle collisions. For weakly broken translational invariance, we show that the relaxation is described by coupled FokkerPlanck equations. From these we find that relaxation of the injected electrons can be understood statistically as a generalized driftdiffusion process in energy space for which we determine the driftvelocity and the dynamical diffusion parameter. Finally, we provide a physically appealing picture in terms of individual edge state heating as a result of the relaxation of the injected electrons.
 
6.  Nonlocal quantum state engineering with the Cooper pair splitter beyond the Coulomb blockade regime 
Ehud Amitai, Rakesh Tiwari, Stefan Walter, Thomas Schmidt, and Simon E. Nigg. arXiv:1512.02952; Phys. Rev. B 93, 075421 (2016).
A Cooper pair splitter consists of two quantum dots sidecoupled to a conventional superconductor. Usually, the quantum dots are assumed to have a large charging energy compared to the superconducting gap, in order to suppress processes other than the coherent splitting of Cooper pairs. In this work, in contrast, we investigate the limit in which the charging energy is smaller than the superconducting gap. This allows us, in particular, to study the effect of a Zeeman field comparable to the charging energy. We find analytically that in this parameter regime the superconductor mediates an interdot tunneling term with a spin symmetry determined by the Zeeman field. Together with electrostatically tunable quantum dots, we show that this makes it possible to engineer a spin triplet state shared between the quantum dots. Compared to previous works, we thus extend the capabilities of the Cooper pair splitter to create entangled nonlocal electron pairs.
 
7.  Correlated voltage probe model of relaxation in two Coulombcoupled edge channels 
Simon E. Nigg Physica E: Lowdimensional Systems and Nanostructures, online (2015)
A phenomenological correlated voltage probe model is introduced to mimic the effects of inelastic scattering between particles in different conduction channels of a phase coherent conductor. As an illustration, the nonequilibrium distribution functions of two noisy copropagating chiral edge channels of the integer quantum Hall effect are calculated and compared with recent experiments. The method is further applied to calculate the linear response current noise through an interacting MachZehnder interferometer.
 
8.  Detecting nonlocal Cooper pair entanglement by optical Bell inequality violation 
Simon E. Nigg, Rakesh P. Tiwari, Stefan Walter, and Thomas L. Schmidt. arXiv:1411.3945; Phys. Rev. B 91, 094516 (2015).
Based on the BardeenCooperSchrieffer theory of superconductivity, the coherent splitting of Cooper pairs from a superconductor to two spatially separated quantum dots has been predicted to generate nonlocal pairs of entangled electrons. In order to test this hypothesis, we propose a scheme to transfer the spin state of a split Cooper pair onto the polarization state of a pair of optical photons. We show that the photon pairs produced can be used to violate a Bell inequality, unambiguously demonstrating the entanglement of the split Cooper pairs.
 
9.  Deterministic Hadamard gate for microwave catstate qubits in circuit QED 
Simon E. Nigg arXiv:1401.6884; Phys. Rev. A 89, 022340 (2014).
We propose the implementation of a deterministic Hadamard gate for logical photonic qubits encoded in superpositions of coherent states of a harmonic oscillator. The proposed scheme builds on a recently introduced set of conditional operations in the strong dispersive regime of circuit QED [Z. Leghtas et al., Phys. Rev. A 87, 042315 (2013)]. We further propose an architecture for coupling two such logical qubits and provide a universal set of deterministic quantum gates. Based on parameter values taken from the current state of the art, we give estimates for the achievable gate fidelities accounting for fundamental gate imperfections and finite coherence time due to photon loss.
 
10.  Deterministically Encoding Quantum Information Using 100Photon Schroedinger Cat States 
Brian Vlastakis, Gerhard Kirchmair, Zaki Leghtas, Simon E. Nigg, Luigi Frunzio, S. M. Girvin, Mazyar Mirrahimi, M. H. Devoret, and R. J. Schoelkopf. Science 342 no. 6158 pp. 607610 (2013)
In contrast to a single quantum bit, an oscillator can store multiple excitations and coherences provided one has the ability to generate and manipulate complex multiphoton states. We demonstrate multiphoton control by using a superconducting transmon qubit coupled to a waveguide cavity resonator with a highly ideal offresonant coupling. This dispersive interaction is much greater than decoherence rates and higherorder nonlinearities to allow simultaneous manipulation of hundreds of photons. With a tool set of conditional qubitphoton logic, we mapped an arbitrary qubit state to a superposition of coherent states, known as a “cat state.” We created cat states as large as 111 photons and extended this protocol to create superpositions of up to four coherent states. This control creates a powerful interface between discrete and continuous variable quantum computation and could enable applications in metrology and quantum information processing.
 
11.  Observation of quantum state collapse and revival due to the singlephoton Kerr effect 
Gerhard Kirchmair, Brian Vlastakis, Zaki Leghtas, Simon E. Nigg, Hanhee Paik, Eran Ginossar, Mazyar Mirrahimi, Luigi Frunzio, S. M. Girvin, and R. J. Schoelkopf. Nature 495, 205 (2013)
Photons are ideal carriers for quantum information as they can have a long
coherence time and can be transmitted over long distances. These properties are
a consequence of their weak interactions within a nearly linear medium. To
create and manipulate nonclassical states of light, however, one requires a
strong, nonlinear interaction at the single photon level. One approach to
generate suitable interactions is to couple photons to atoms, as in the strong
coupling regime of cavity QED systems. In these systems, however, one only
indirectly controls the quantum state of the light by manipulating the atoms. A
direct photonphoton interaction occurs in socalled Kerr media, which
typically induce only weak nonlinearity at the cost of significant loss. So
far, it has not been possible to reach the singlephoton Kerr regime, where the
interaction strength between individual photons exceeds the loss rate. Here,
using a 3D circuit QED architecture, we engineer an artificial Kerr medium
which enters this regime and allows the observation of new quantum effects. We
realize a Gedankenexperiment proposed by Yurke and Stoler, in which the
collapse and revival of a coherent state can be observed. This time evolution
is a consequence of the quantization of the light field in the cavity and the
nonlinear interaction between individual photons. During this evolution
nonclassical superpositions of coherent states, i.e. multicomponent
Schroedinger cat states, are formed. We visualize this evolution by measuring
the Husimi Qfunction and confirm the nonclassical properties of these
transient states by Wigner tomography. The singlephoton Kerr effect could be
employed in QND measurement of photons, single photon generation, autonomous
quantum feedback schemes and quantum logic operations.
 
12.  Frequencydependent admittance of a short superconducting weak link 
F. Kos, S. E. Nigg, and L. I. Glazman. arXiv:1303.2918; Phys. Rev. B 87, 174521 (2013).
We consider the linear and nonlinear electromagnetic responses of a nanowire
connecting two bulk superconductors. Andreev states appearing at a finite phase
bias substantially affect the finitefrequency admittance of such a wire
junction. Electron transitions involving Andreev levels are easily saturated,
leading to the nonlinear effects in photon absorption for the subgap photon
energies. We evaluate the complex admittance analytically at arbitrary
frequency and arbitrary, possibly nonequilibrium, occupation of Andreev
levels. Special care is given to the limits of a singlechannel contact and a
disordered metallic weak link. We also evaluate the quasistatic fluctuations
of admittance induced by fluctuations of the occupation factors of Andreev
levels. In view of possible qubit applications, we compare properties of a weak
link with those of a tunnel Josephson junction. Compared to the latter, a weak
link has smaller lowfrequency dissipation. However, because of the deeper
Andreev levels, the lowtemperature quasistatic fluctuations of the inductance
of a weak link are exponentially larger than of a tunnel junction. These
fluctuations limit the applicability of nanowire junctions in superconducting
qubits.
 
13.  Stabilizer quantum error correction toolbox for superconducting qubits 
Simon E. Nigg and Steven M. Girvin. arXiv:1212.4000; Phys. Rev. Lett. 110, 243604 (2013).
We present a general protocol for stabilizer measurements and pumping in a
system of N superconducting qubits. We assume alwayson, equal dispersive
couplings to a single mode of a highQ microwave resonator in the ultrastrong
dispersive limit where the dispersive shifts largely exceed the spectral
linewidth. In this limit, we show how to map the two eigenvalues of an
arbitrary weight M < N Pauli operator, onto two quasiorthogonal coherent
states of the cavity. Together with a fast cavity readout, this enables the
efficient measurement of stabilizer operators.
 
14.  Blackbox superconducting circuit quantization 
Simon E. Nigg, Hanhee Paik, Brian Vlastakis, Gerhard Kirchmair, Shyam Shankar, Luigi Frunzio, Michel Devoret, Robert Schoelkopf, and Steven Girvin. arXiv:1204.0587; Phys. Rev. Lett. 108, 240502 (2012).
We present a semiclassical method for determining the effective lowenergy
quantum Hamiltonian of weakly anharmonic superconducting circuits containing
mesoscopic Josephson junctions coupled to electromagnetic environments made of
an arbitrary combination of distributed and lumped elements. A convenient
basis, capturing the multimode physics, is given by the quantized eigenmodes
of the linearized circuit and is fully determined by a classical linear
response function. The method is used to calculate numerically the lowenergy
spectrum of a 3Dtransmon system, and quantitative agreement with measurements
is found.
 
15.  Decoherence of superconducting qubits caused by quasiparticle tunneling 
G. Catelani, Simon E. Nigg, S. M. Girvin, R. J. Schoelkopf, and L. I. Glazman. arXiv:1207.7084; Phys. Rev. B 86, 184514 (2012).
In superconducting qubits, the interaction of the qubit degree of freedom
with quasiparticles defines a fundamental limitation for the qubit coherence.
We develop a theory of the pure dephasing rate \Gamma_{\phi} caused by
quasiparticles tunneling through a Josephson junction and of the inhomogeneous
broadening due to changes in the occupations of Andreev states in the junction.
To estimate \Gamma_{\phi}, we derive a master equation for the qubit dynamics.
The tunneling rate of free quasiparticles is enhanced by their large density of
states at energies close to the superconducting gap. Nevertheless, we find that
\Gamma_{\phi} is small compared to the rates determined by extrinsic factors in
most of the current qubit designs (phase and flux qubits, transmon, fluxonium).
The split transmon, in which a single junction is replaced by a SQUID loop,
represents an exception that could make possible the measurement of
\Gamma_{\phi}. Fluctuations of the qubit frequency leading to inhomogeneous
broadening may be caused by the fluctuations in the occupation numbers of the
Andreev states associated with a phasebiased Josephson junction. This
mechanism may be revealed in qubits with smallarea junctions, since the
smallest relative change in frequency it causes is of the order of the inverse
number of transmission channels in the junction.
 
16.  Realization of ThreeQubit Quantum Error Correction with Superconducting Circuits 
M. D. Reed, L. DiCarlo, S. E. Nigg, L. Sun, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf. Nature 482, 382385 (2012)
Quantum computers promise to solve certain problems exponentially faster than
possible classically but are challenging to build because of their increased
susceptibility to errors. Remarkably, however, it is possible to detect and
correct errors without destroying coherence by using quantum error correcting
codes [1]. The simplest of these are the threequbit codes, which map a
onequbit state to an entangled threequbit state and can correct any single
phaseflip or bitflip error of one of the three qubits, depending on the code
used [2]. Here we demonstrate both codes in a superconducting circuit by
encoding a quantum state as previously shown [3,4], inducing errors on all
three qubits with some probability, and decoding the error syndrome by
reversing the encoding process. This syndrome is then used as the input to a
threequbit gate which corrects the primary qubit if it was flipped. As the
code can recover from a single error on any qubit, the fidelity of this process
should decrease only quadratically with error probability. We implement the
correcting threequbit gate, known as a conditionalconditional NOT (CCNot) or
Toffoli gate, using an interaction with the third excited state of a single
qubit, in 63 ns. We find 85\pm1% fidelity to the expected classical action of
this gate and 78\pm1% fidelity to the ideal quantum process matrix. Using it,
we perform a single pass of both quantum bit and phaseflip error correction
with 76\pm0.5% process fidelity and demonstrate the predicted firstorder
insensitivity to errors. Concatenating these two codes and performing them on a
ninequbit device would correct arbitrary singlequbit errors. When combined
with recent advances in superconducting qubit coherence times [5,6], this may
lead to scalable quantum technology.
 
17.  Interaction induced edge channel equilibration 
Anders Mathias Lunde, Simon E. Nigg, and Markus Buttiker. arXiv:0910.2476; Phys. Rev. B 81, 041311(R) (2010).
The electronic distribution functions of two Coulomb coupled chiral edge
states forming a quasi1D system with broken translation invariance are found
using the equation of motion approach. We find that relaxation and thereby
energy exchange between the two edge states is determined by the shot noise of
the edge states generated at a quantum point contact (QPC). In close vicinity
to the QPC, we derive analytic expressions for the distribution functions. We
further give an iterative procedure with which we can compute numerically the
distribution functions arbitrarily far away from the QPC. Our results are
compared with recent experiments of Le Sueur et al..
 
18.  Universal detector efficiency of a mesoscopic capacitor 
Simon E. Nigg and Markus Buttiker. arXiv:0902.0686; Phys. Rev. Lett. 102, 236801 (2009).
We investigate theoretically a novel type of high frequency quantum detector
based on the mesoscopic capacitor recently realized by Gabelli et al., [Science
{\bf 313}, 499 (2006)], which consists of a quantum dot connected via a single
channel quantum point contact to a single lead. We show that the state of a
double quantum dot charge qubit capacitively coupled to this detector can be
read out in the GHz frequency regime with near quantum limited efficiency. To
leading order, the quantum efficiency is found to be universal owing to the
universality of the charge relaxation resistance of the mesoscopic capacitor.
 
19.  Mesoscopic Capacitance Oscillations 
Markus Buttiker and Simon E. Nigg. arXiv:condmat/0608417; Nanotechnology 18, 044029 (2007).
We examine oscillations as a function of Fermi energy in the capacitance of a
mesoscopic cavity connected via a single quantum channel to a metallic contact
and capacitively coupled to a back gate. The oscillations depend on the
distribution of single levels in the cavity, the interaction strength and the
transmission probability through the quantum channel. We use a HartreeFock
approach to exclude selfinteraction. The sample specific capacitance
oscillations are in marked contrast to the charge relaxation resistance, which
together with the capacitance defines the RCtime, and which for spin polarized
electrons is quantized at half a resistance quantum. Both the capacitance
oscillations and the quantized charge relaxation resistance are seen in a
strikingly clear manner in a recent experiment.
 
20.  Role of coherence in resistance quantization 
Markus Buttiker and Simon E. Nigg. arXiv:0806.1821; Eur. Phys. J. Special Topics 172, 247  255 (2009).
The quantization of resistances in the quantum Hall effect and ballistic
transport through quantum point contacts is compared with the quantization of
the charge relaxation resistance of a coherent mesoscopic capacitor. While the
former two require the existence of a perfectly transmitting channel, the
charge relaxation resistance remains quantized for arbitrary backscattering.
The quantum Hall effect and the quantum point contact require only local phase
coherence. In contrast quantization of the charge relaxation resistance
requires global phase coherence.
 
21.  Quantum to Classical Transition of the Charge Relaxation Resistance of a Mesoscopic Capacitor 
Simon E. Nigg and Markus Buttiker. arXiv:0709.3956; Phys. Rev. B 77, 085312 (2008).
We present an analysis of the effect of dephasing on the single channel
charge relaxation resistance of a mesoscopic capacitor in the linear low
frequency regime. The capacitor consists of a cavity which is via a quantum
point contact connected to an electron reservoir and Coulomb coupled to a gate.
The capacitor is in a perpendicular high magnetic field such that only one
(spin polarized) edge state is (partially) transmitted through the contact. In
the coherent limit the charge relaxation resistance for a single channel
contact is independent of the transmission probability of the contact and given
by half a resistance quantum. The loss of coherence in the conductor is modeled
by attaching to it a fictitious probe, which draws no net current. In the
incoherent limit one could expect a charge relaxation resistance that is
inversely proportional to the transmission probability of the quantum point
contact. However, such a two terminal result requires that scattering is
between two electron reservoirs which provide full inelastic relaxation. We
find that dephasing of a single edge state in the cavity is not sufficient to
generate an interface resistance. As a consequence the charge relaxation
resistance is given by the sum of one constant interface resistance and the
(original) Landauer resistance. The same result is obtained in the high
temperature regime due to energy averaging over many occupied states in the
cavity. Only for a large number of open dephasing channels, describing
spatially homogenous dephasing in the cavity, do we recover the two terminal
resistance, which is inversely proportional to the transmission probability of
the QPC. We compare different dephasing models and discuss the relation of our
results to a recent experiment.
 
22.  Mesoscopic Charge Relaxation 
Simon E. Nigg, Rosa Lopez, and Markus Buttiker. arXiv:condmat/0606603; Phys. Rev. Lett. 97, 206804 (2006).
We consider charge relaxation in the mesoscopic equivalent of an RC circuit.
For a singlechannel, spinpolarized contact, selfconsistent scattering theory
predicts a universal charge relaxation resistance equal to half a resistance
quantum independent of the transmission properties of the contact. This
prediction is in good agreement with recent experimental results. We use a
tunneling Hamiltonian formalism and show in HartreeFock approximation, that at
zero temperature the charge relaxation resistance is universal even in the
presence of Coulomb blockade effects. We explore departures from universality
as a function of temperature and magnetic field.
