Half Flux Quantum Research Articles

Pure kinetic inductance coupling for cQED with flux qubits

We demonstrate a qubit-readout architecture where the dispersive coupling is entirely mediated by a kinetic inductance. This allows us to engineer the dispersive shift of the readout resonator independent of the qubit and resonator capacitances. We validate the pure kinetic coupling concept and demonstrate various generalized flux qubit regimes from plasmon to fluxon, with dispersive shifts ranging from 60 kHz to 2 MHz at the half-flux quantum sweet spot. We achieve readout performances comparable to conventional architectures with quantum state preparation fidelities of 99.7% and 92.7% for the ground and excited states, respectively, and below 0.1% leakage to non-computational states.

NbN-based tunnel-type π-junctions for low-power half-flux-quantum circuits

We have developed half-flux-quantum (HFQ) circuits using all-π-junctions formed from an NbN/AlN/PdNi/NbN (SIFS) structure. The circuits were fabricated using a novel process that incorporated a ground plane on top of the chip, enabling the epitaxial growth of NbN-based junctions from the substrate. The π-state of the junctions was demonstrated through a HFQ shift. Notably, these π-junctions exhibited self-overdamped current-voltage characteristics, enabling them to function as switching components without the need for shunt resistors. The elimination of shunt resistors and the high sheet inductance of NbN are expected to enhance the density of HFQ circuits. To evaluate the performance and power consumption of the all-π-junctions HFQ circuits, we designed and fabricated an HFQ toggle flip-flop (HFQ-TFF) circuit utilizing π-π-π SQUIDs as the fundamental components. Our findings reveal that the NbN-based HFQ-TFF circuit correctly operates as a frequency divider while consuming only around 30% of the power compared to single-flux-quantum TFF (SFQ-TFF) circuits. These results suggest that the HFQ circuit using SIFS-π-junctions has promising potential for integrated circuits requiring low-power consumption at cryogenic temperatures, such as qubit control.

NbN-based half-flux-quantum element for integration with superconducting qubits

Half-flux-quantum (HFQ) logic is a superconductor logic family comprising conventional Josephson junctions (0-JJs) and π Josephson junctions (π-JJs). The energy scale of HFQ logic can be flexibly reduced by phase shifts owing to π-JJs; thus, HFQ circuits are a promising building block for qubit interface circuits. In this study, we demonstrate an NbN-based HFQ circuit element compatible with NbN-based superconducting qubits for all-NbN monolithic integration of qubits and HFQ circuits. The use of NbN is beneficial for both qubits and HFQ circuits owing to its high-quality junctions and high kinetic inductance. First, we developed a prototype of an NbN-based 0-JJ/π-JJ hybrid fabrication process for designing HFQ circuits. We evaluated 0-JJs and π-JJs fabricated through the hybrid process, which demonstrated that the 0-JJs had a small deviation with regard to critical current density and the π-JJs had a sufficiently high critical current density to work as π phase shifters. Furthermore, we fabricated an HFQ superconducting quantum interference device, which is one of the most fundamental elements in HFQ circuits, and observed clear HFQ-period modulation in the magnetic flux dependence of the maximum current at 4.2 K.

Large second-order Josephson effect in planar superconductor-semiconductor junctions

We investigate the current-phase relations of Al/InAs-quantum well planar Josephson junctions fabricated using nanowire shadowing technique. Based on several experiments, we conclude that the junctions exhibit an unusually large second-order Josephson harmonic, the \sin(2\varphi)sin(2φ) term. First, superconducting quantum interference devices (dc-SQUIDs) show half-periodic oscillations, tunable by gate voltages as well as magnetic flux. Second, Josephson junction devices exhibit kinks near half-flux quantum in supercurrent diffraction patterns. Third, half-integer Shapiro steps are present in the junctions. Similar phenomena are observed in Sn/InAs quantum well devices. We perform data fitting to a numerical model with a two-component current phase relation. Analysis including a loop inductance suggests that the sign of the second harmonic term is negative. The microscopic origins of the observed effect remain to be understood. We consider alternative explanations which can account for some but not all of the evidence.

Energy efficient half-flux-quantum circuit aiming at milli-kelvin stage operation

Half-flux-quantum (HFQ) circuits are based on 0–π superconducting quantum interference devices (SQUIDs) and is one of the energy-efficient superconductor digital circuits. The bit energy is determined by the critical current I cn of 0–π SQUID, which can be easily tuned with the loop inductance and junction critical current. In this work, an alternative π–π–π SQUID is adopted to demonstrate HFQ circuits to simplify the fabrication process and enhance circuit energy efficiency. The properties of superconductor/ferromagnet/insulator/superconductor Josephson junctions (π-JJs) are measured with temperature dependence from 4.2 K down to 10 mK. HFQ toggle flip-flops (TFFs) are successfully demonstrated at frequencies of up to 6.7 GHz and 44.5 GHz at temperatures of 4.2 K and 10 mK, respectively. Comparing the HFQ TFF with its rapid single-flux quantum counterpart under the same fabrication process, it is anticipated that the HFQ TFF will exhibit approximately 70% reduction in both static and dynamic energy dissipation. This research establishes the foundation for developing cryogenic interface control and readout circuits for large-scale quantum computing in the future.

Optimization of Half-Flux-Quantum Circuits Composed of π-Shift and Conventional Josephson Junctions

We report optimization of half-flux-quantum (HFQ) circuits composed of SQUIDs containing conventional Josephson junctions and π-shift magnetic Josephson junctions. The SQUID can act as a Josephson junction with an extremely small critical current and be easily switched by a weak driving force with the assistance of the circulating current that is induced due to the π-shift and quantization conditions. It leads to a drastic reduction of both static and dynamic power consumption. However, the parameter margins of the HFQ circuits obtained so far is narrow compared with conventional single-flux-quantum circuits. Since HFQ circuits use many circuit elements and require the two junctions in the SQUID to switch alternately and symmetrically, circuit parameter optimization is much more difficult. In this study, we developed an optimization tool for HFQ circuits, including operation judgment and margin calculation. We improved the critical margin of the HFQ Josephson transmission line to 39%. We also optimized the HFQ splitter, confluence buffer, and D flip-flop. We discuss design guidelines for HFQ circuits based on our systematic exploration of the optimized circuit parameter sets. We suggest that the optimized loop inductances are around 0.5 and 1.5 of Φ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> /2 for data transmission and storage, respectively.

Superconductor digital circuits with π junctions alone

We adopt superconductor/ferromagnet/insulator/superconductor (SFIS) Josephson junctions (JJs) as both switching JJs and intrinsic π phase shifters in superconductor digital circuits. The critical current density (Jc) and characteristic voltage (Vc) of the SFIS junctions are about 22 A/cm2 and 22 μV, respectively. The intrinsic π phase shift is confirmed by measuring the suppressed nominal critical current Icn and half-period-shifted modulation pattern of a π–π–π superconducting quantum interference device (SQUID) that contains three π-JJs in a superconducting loop. A single-flux-quantum (SFQ) circuit composed of a DC/SFQ, Josephson transmission line (JTL), and SFQ/DC converter based on SFIS JJs alone is demonstrated at 4.2 K. The energy dissipation of the SFQ/DC converter decreases by 80% because some JJs are self-biased by the π phase shifter. The intrinsic circulating currents induced by the π phase shifters lead to a narrow bias margin (±5%) and even error function, which can be solved by parameters optimization or circuit initialization in the future. The half-modulation period (Φ0/2) of a half-flux-quantum (HFQ) SQUID (a partial HFQ JTL) exhibits propagation of HFQ between π–π–π SQUIDs, indicating that more complex HFQ circuits can be developed with π-JJs alone in the future.

Direct observations of π-leaps of superconducting phase differences in π-junction-based SQUIDs

We directly observed π-leaps of superconducting phase differences in π-junction-based superconducting quantum interference devices (SQUIDs). The SQUIDs studied here are formed by introducing a π-junction to a conventional-junction (0-junction)-based direct current (DC)-SQUID, which is referred to as the 0-0-π SQUID. Either clockwise or counter-clockwise-circulating currents flow spontaneously in the 0-0-π SQUID because of a π-phase shift of the π-junction. In other words, the 0-0-π SQUID has a bistable state corresponding to the directions of circulating currents. π-leaps are generated by transiting between the two states of the bistable state. π-leaps are an ultra-fast phenomenon and are difficult to observe as they are. We prepared a half-flux quantum (HFQ)-SQUID that comprised two 0-0-π SQUIDs. π-leaps are reflected in a static characteristic, that is, a modulation pattern of the critical current in the HFQ-SQUID. We formed π-junctions with the PdNi layer on 0-junction-based circuits supplied by the National Institute of Advanced Industrial Science and Technology. The modulation pattern of the HFQ-SQUID had a period corresponding to π-leaps as expected, although some microstructures were observed. We demonstrated that the microstructures originated from the asymmetry inside each 0-0-π SQUID by analyzing the relationship between the phase change of 0-0-π SQUIDs and the modulation patterns.

Enhancing quantum coherence of a fluxonium qubit by employing flux modulation with tunable-complex-amplitude

We propose to protect fluxonium qubits that are away from half flux quantum against environmental noises, especially flux noise, by adopting a modulated flux with tunable-complex-amplitude. Using open-system Floquet theory, we derive a Lindblad equation and extract decoherent rates for pure-dephasing, excitation and relaxation. After examining intrinsic attributes of the flux driven fluxonium qubit, we put forward an analytic manner to locate dynamical sweet spots for fast and weak driving. Continuous families of dynamical sweet spots are found in the parameter plane of relative amplitude factor and relative phase. Around a family of dynamical sweet spots or between two families of dynamical sweet spots, there exist continuous regions with long coherent times that exceed . Taking advantage of the two noise-insensitive channels: relative amplitude factor and relative phase, a flux driven fluxonium qubit can become immune to flux noises from both the dc and ac flux amplitudes. And the optimal driving amplitudes are no longer isolated at a certain driving frequency, but become continuous. This is in sharp contrast to the usual schemes based on flux modulation with real-amplitude. As a result, there are plenty of manipulating flexibility in our flux driving scheme with tunable-complex-amplitude, which may be useful in logical operations among flux driven fluxonium qubits or other flux qubits.

Josephson junctions of two-dimensional time-reversal invariant superconductors: Signatures of the topological phase

We determine the current-phase relation (CPR) of two-terminal configurations of Josephson junctions containing two-dimensional (2D) time-reversal invariant topological superconductors (TRITOPS), including TRITOPS-TRITOPS, as well as junctions between topological and non-topological superconductors (TRITOPS-S). We focus on long junctions for which several channels intervene in the tunneling coupling through the junction. We present a description of the topological edge modes for different TRITOPS models including $p$-wave pairing and the combination of $s$-wave pairing with spin-orbit coupling. We derive effective low-energy Hamiltonians to describe the Josephson junction, which can be solved analytically to explain the contribution of the edge states to the Josephson current as a function of the phase bias. We find that edge-modes yield singular corrections to the CPR for both junction types. The primary effects occur for the response of the Majorana zero-modes at half-flux quantum phase $\phi\approx \pi$ in TRITOPS-TRITOPS junctions and for integer flux quantum phase $\phi \approx 0$ for TRITOPS-S junctions, respectively. The former effect is particularly strong two-component nematic superconductors. The latter effect leads to a spontaneously broken time-reversal symmetry in the TRITOPS-S junction and to a breakdown of the bulk-boundary correspondence.

Fast Flux Entangling Gate for Fluxonium Circuits

We analyze a high-fidelity two-qubit gate using fast flux pulses on superconducting fluxonium qubits. The gate is realized by temporarily detuning magnetic flux through fluxonium loop away from the half flux quantum sweet spot. We simulate dynamics of two capacitively coupled fluxoniums during the flux pulses and optimize the pulse parameters to obtain a highly accurate $\sqrt{i\mathrm{swap}}$-like entangling gate. We also evaluate the effect of the flux noise and qubit relaxation on the gate fidelity. Our results demonstrate that the gate error remains below $10^{-4}$ for currently achievable magnitude of the flux noise and qubit relaxation time.

Protected Hybrid Superconducting Qubit in an Array of Gate-Tunable Josephson Interferometers

We propose a protected qubit based on a modular array of superconducting islands connected by semiconductor Josephson interferometers. The individual interferometers realize effective $\cos2\phi$ elements that exchange `pairs of Cooper pairs' between the superconducting islands when gate-tuned into balance and frustrated by a half flux quantum. If a large capacitor shunts the ends of the array, the circuit forms a protected qubit because its degenerate ground states are robust to offset charge and magnetic field fluctuations for a sizable window around zero offset charge and half flux quantum. This protection window broadens upon increasing the number of interferometers if the individual elements are balanced. We use an effective spin model to describe the system and show that a quantum phase transition point sets the critical flux value at which protection is destroyed.

Superconductive Logic Using 2ϕ—Josephson Junctions With Half Flux Quantum Pulses

Superconductive logic based on Josephson junctions (JJ) is a promising technology for energy efficient supercomputers and cloud computing. This technology can deliver significant improvements in performance and energy efficiency as compared to CMOS. Superconductive circuits, however, suffer from low density integration as compared to CMOS, primarily due to the limited scalability of the inductors. To improve the scalability of superconductive logic, a logic family based on a novel JJ technology, <inline-formula> <tex-math notation="LaTeX">$2\phi $ </tex-math></inline-formula>-JJ, has been proposed that eliminates the inductors. In this brief, three circuits are presented which exploit this scalable inductor-less technology. This novel <inline-formula> <tex-math notation="LaTeX">$2\phi $ </tex-math></inline-formula>-JJ technology represents the data as half flux quantum (HFQ) pulses, which improves the energy efficiency and speed as compared to standard superconductive logic such as rapid single flux quantum (RSFQ). Unlike RSFQ, the proposed circuits dynamically switch upon receiving an HFQ pulse, saving energy. These <inline-formula> <tex-math notation="LaTeX">$2\phi $ </tex-math></inline-formula>-JJ logic circuits operate 2.25X faster and require 2.6X less energy as compared to RSFQ.

Multiterminal ballistic Josephson junctions coupled to normal leads

Multiterminal Josephson junctions have aroused considerable theoretical interest recently and numerous works aim at putting the predictions of correlations among Coopers (i.e., the so-called quartets) and simulation of topological matter to the test of experiments. This paper is motivated by recent experimental investigation from the Harvard group reporting $h/4e$-periodic quartet signal in a four-terminal configuration containing a loop pierced by magnetic flux, together with inversion controlled by the bias voltage, i.e., the quartet signal can be larger at half-flux quantum than in zero magnetic field. Here, we theoretically focus on devices consisting of finite-size quantum dots connected to four superconducting leads and to a normal lead. In addition to presenting numerical calculations of the quartet signal within a simplified modeling, we reduce the device to a non-Hermitian Hamiltonian in the infinite-gap limit. Then, relaxation has the surprising effect of producing sharp peaks and log-normal variations in the voltage sensitivity of the quartet signal in spite of the expected moderate fluctuations in the two-terminal DC Josephson current. The phenomenon is reminiscent of a resonantly driven harmonic oscillator having amplitude inverse proportional to the damping rate, and of the thermal noise of a superconducting weak link, inverse proportional to the Dynes parameter. Perspectives involve quantum bath engineering multiterminal Josephson junctions in the circuits of cavity quantum electrodynamics.

Tunable coupling scheme for implementing two-qubit gates on fluxonium qubits

A superconducting fluxonium circuit is an RF-superconducting quantum interference device-type flux qubit that uses a large inductance built from an array of Josephson junctions or a high kinetic inductance material. This inductance suppresses charge sensitivity exponentially and flux sensitivity quadratically. In contrast to the transmon qubit, the anharmonicity of fluxonium can be large and positive, allowing for better separation between the low energy qubit manifold of the circuit and higher-lying excited states. Here, we propose a tunable coupling scheme for implementing two-qubit gates on fixed-frequency fluxonium qubits, biased at half flux quantum. In this system, both qubits and couplers are coupled capacitively and implemented as fluxonium circuits with an additional harmonic mode. We investigate the performance of the scheme by simulating a universal two-qubit fSim gate. In the proposed approach, we rely on a planar on-chip architecture for the whole device. Our design is compatible with existing hardware for transmon-based devices with the additional advantage of lower qubit frequency facilitating high-precision gating.

Demonstration of Interface Circuits Between Half- and Single- Flux- Quantum Circuits

We demonstrated interface circuits between a half-flux-quantum (HFQ) circuit and a rapid single-flux-quantum (RSFQ) circuit to obtain SFQ/HFQ and HFQ/SFQ converters. These converters were created by simply inserting a resistor between an RSFQ circuit and the adjacent HFQ circuit, where both circuits are based on pulse logic. These converters enable us to create combinatory circuits that make the best use of the features of the two circuits. Using a DC/SFQ, SFQ/HFQ, HFQ/SFQ, and SFQ/DC converter, we measure an HFQ Josephson transmission line (JTL) composed of eight 0-0-π superconducting quantum interference devices (SQUIDs), which are the essential elements of an HFQ circuit. The 0-0-π SQUIDs are composed of two conventional Josephson junctions and a π-shifted ferromagnetic junction. The HFQ JTL demonstrated in this study is driven by a bias current 11-fold lower than that of a similar RSFQ JTL. These experimental results indicate that the HFQ circuit requires power that is two or more orders of magnitude lower than that of an RSFQ circuit.

Low-power high-speed half-flux-quantum circuits driven by low bias voltages

Half-flux-quantum (HFQ) circuits store and propagate half-flux quanta. The basic circuit element is a 0-π SQUID, which is a superconducting quantum interference device with a conventional Josephson junction (0-junction) and a π-shifted ferromagnetic junction (π-junction). A 0-π SQUID achieves a small critical current in the absence of an external magnetic field, thus reducing power consumption. It is easy to set up 0-0-π SQUIDs with two 0-junctions and a π-junction which serves as a π phase-shifter. We simulated 0-0-π SQUID-based HFQ circuits driven by low bias voltages, referred to as LV-HFQ circuits. In these circuits, shunt resistors are not required for switching junctions because there is no hysteresis in the current–voltage characteristics of 0-0-π SQUIDs. We estimated the power consumption and maximum operating frequency of an HFQ Josephson transmission line based on 0-0-π SQUIDs. When operating at 43.5 GHz, the power dissipation of a single element composed of a 0-0-π SQUID and a bias resistor fell to about 0.165 nW when biased at 60 μV. The LV-HFQ circuit is potentially more power-efficient than all other currently available superconducting logic circuits.

Tuning the topology of p -wave superconductivity in an analytically solvable two-band model

We introduce and solve a two-band model of spinless fermions with $p_x$-wave pairing on a square lattice. The model reduces to the well-known extended Harper-Hofstadter model with half-flux quanta per plaquette and weakly coupled Kitaev chains in two respective limits. We show that its phase diagram contains a topologically nontrivial weak pairing phase as well as a trivial strong pairing phase as the ratio of the pairing amplitude and hopping is tuned. Introducing periodic driving to the model, we observe a cascade of Floquet phases with well defined quasienergy gaps and featuring chiral Majorana edge modes at the zero- or $\pi$-gap, or both. Dynamical topological invariants are obtained to characterize each phase and to explain the emergence of edge modes in the anomalous phase where all the quasienergy bands have zero Chern number. Analytical solution is achieved by exploiting a generalized mirror symmetry of the model, so that the effective Hamiltonian is decomposed into that of spin-$1/2$ in magnetic field, and the loop unitary operator becomes spin rotations. We further show the dynamical invariants manifest as the Hopf linking numbers.

Superconducting ‘twin’ qubit

We study a flux qubit consisting of a symmetrical pair of superconducting loops, with two Josephson junctions in each, joined by a common Josephson junction---a `twin' flux qubit. The qubit is capacitively coupled to a transmission line, which allows us to characterize the spectrum of the device by measuring the scattering of propagated electromagnetic waves. We perform a detailed analytical analysis of the double-loop system, revealing its properties, and compare experimental results with numerical simulations. At half-flux quantum bias of both loops, the qubit is protected against global and local magnetic field fluctuations with much less sensitivity to the global field in the second order. The system selection rules allow even-odd transitions and prohibit transitions between even-even or odd-odd levels due to the symmetry of the device.

Fermion zero-energy modes and fractional fermion numbers in a fractional vortex-fermion model

Topological materials have attracted much attention from both physicists and mathematicians recently. Topological properties are closely related to the fermion number (index) of Dirac fermions. The fermion number is given by the η invariant introduced by Atiyah, Padoti and Singer. We discuss a system of Dirac fermions interacting with a vortex and a kink. This system will be realized as a layered material of superconductors and topological insulators, where the Dirac fermion exists on the surface of the topological insulator. The fermion number is fractionalized and the fermion zero-energy excitation mode emerges when Dirac fermions interact with vortices and kinks. Our discussion includes the case where there is a half-flux quantum vortex associated with a kink in a magnetic field in a bilayer superconductor. A normalizable single-valued fermion zero-energy mode does not exist in the core of the half-flux quantum vortex.