Superconducting Qubits: Quantum computing with Josephson Junctions

The scalability of a qubit is a key condition for its potential suitability as a building block for a quantum computer. The first step is to couple two qubits. This should be done in such a way that the coupling can be chosen by design, allowing weak or strong interactions to be implemented as required from a systems design point of view. Flux qubits should be coupled employing the flux parameter. To augment the mutal inductance by the kinetic contribution we introduced an additional Josephson junction as the actual coupling element. The coupling energy can be chosen by the critical current of the junction. From the results may draw a few very important conclusions: we have been able to trace the full spectrum; the experimental diagram closely matches the predicted design; the qubit-qubit coupling strength is very close to the design value; the visibility of transitions between excited states indicates initially non-zero occupation of excited states. The results were analysed by comparing with the standard simple model Hamiltonian. Remarkably good agreement was obtained between spectroscopic data and the model, indicating that indeed the flux qubits behave as simple two level systems. In our first measurements on two-coupled flux qubits the qubits could not be tuned to optimal conditions independently. The applied flux could be set to a good value for inducing Rabi oscillations on one transition involving one of the qubits. At that same value of the external magnetic field the flux setting for the other qubit was not optimal and no Rabi oscillation could be induced there. The lesson from these first experiments is that future systems must necessarily have independent tuning. Until now the only good coherence is obtained when a qubit is biased at its symmetry point. In a system with multiple qubits therefore, each qubit must be tuned to its symmetry point with independent bias lines. Tunable coupling is possible with a dynamic procedure. Rabi oscillations could be induced for some of the transitions at some of the flux values. Not at all positions was coherence high enough. Conditional spectroscopy was performed by first sending a pi-pulse to occupy one of the excited states and subsequently scanning a certain frequency range with a longer RF pulse. In doing this, it was found that the transitions that could be observed in the scan were consistent with a full transition to the excited state in question. No complete two-qubit operations could yet be performed due to limited coherence.

It is now established that several types of super-conducting circuits based on Josephson junctions are sufficiently quantum that simple manipulations of their quantum state can be performed. These quantum bit circuits are the basic building blocks of a quantum processor. The coherence time of the quantum state is an essential figure of merit, being related to the number of qubit operations that can be performed without error. Despite significant advances in coherence times during recent years, with coherence times of order of a fraction of microsecond reached, decoherence due to the coupling between the quantum circuit and the degrees of freedom of the environment still severely hinders using these circuits for the development of a quantum processor, even with a small number of qubits. Thus the quantitative characterization and understanding of decoherence processes is presently a central issue for the development of qubit circuits. In this result, we present experiments on the quantronium qubit, and we develop a general framework for their theoretical analysis. We show that a simple model for the spectral densities of the noise sources coupled to the qubit allows accounting for the experimental findings. The framework developed can be applied to all super-conducting qubits.

Sample and hold readout of qubits. We implemented a pulse and hold strategy for reading out qubits. Here the non-linear dynamics of the circuit is used to latch the output, holding the result long enough for a measurement to be made. We studied the case of switching current detector with over-damped dynamics and high frequencies and discovered that it is possible to suppress phase diffusion and make a circuit with good latching characteristics. The readout of qubits was not successful, but the technique by is adapted to other problems in sensitive measurement where over-damped high frequency dynamics is desired, such as fast SQUID readout.

The sluice is a device where charges can be moved fast and synchronously. As a building block it opens a way to create a metrological source of electrical current and to investigate geometric phases in quantum mechanics. The key benefits are the controllability of both the charge and phase degrees of freedom, which renders the speed of the device without sacrificing its accuracy.

We analysed the decoherence in Josephson Qubits in the charge regime induced by background charges in the substrate, which are also responsible for the 1/f noise. We proposed a microscopic model and study its dynamics pointing out the crucial role played by slow moving charges. Far away from the degeneracy we showed that this model for the dephasing can be solved exactly. We also studied dynamical decoupling of a qubit from non-Gaussian quantum noise due to discrete sources, as bistable fluctuators and 1/f noise. We obtain analytic and numerical results for generic operating points. For very large pulse frequency, where dynamic decoupling compensates decoherence, we found universal behaviour. At intermediate frequencies noise can be compensated or enhanced, depending on the nature of the fluctuators and on the operating point.

Delsing and co-workers have developed an rf-SET (radio frequency single-electron transistor) readout for the charge qubit, i.e. the single Cooper-pair box (SCB) two-level system, and successfully detected free oscillations and studied the detailed behaviour of relaxation and de-phasing. A major task and activity at Chalmers has been to investigate the coherence and mixing times for the charge qubit. This is described in detail in an extensive paper accounting for the Chalmers work on the single Cooper-pair box as a charge qubit using dc-pulse operation and RF-SET detection.

Flux qubit quantum state readout is based on measuring the value of the magnetic moment resulting from the clockwise or anti-clockwise circulating current in the SQUID loop containing the three Josephson junctions. Equivalently one may readout the phase set up by the circulating current over a certain part of the loop. In both cases a DC-SQUID detector is employed, comprising two Josephson junctions in a loop. Commonly a "click"-type detection scheme is employed, where the SQUID is either not switched (OFF) or switched into the phase-evolving running state (ON). This discrimination yields a YES/NO output of the system, depending on its state. This scheme is attractive because of its simplicity and potential large effective gain. It however suffers from a few serious drawbacks. Most pronounced, the vigorous perturbation resulting from the SQUID once it is in the ON state. The perturbation onto the qubit being measured is uncontrolled in strength and frequency spectrum, and leads to a back action onto the quantum state of that qubit way beyond anything close to the minimum quantum limit enforced by the fundamental quantum uncertainty. However, more serious, it also may affect nearby other qubits, the quantum evolution of which should not be influenced at all. As a (serious) side effect the switching of the SQUID leads to the generation of heat and of quasi particles. These two should be allowed to decay away, and bringing the system into the ground state again, before being able to redo a quantum operation. This limits the throughput rate. The inductive dispersive readout allows a fully controlled approach, where the degree of system-detector interaction can be chosen at will, even in-situ during the experiment. The SQUID is assumed to be coupled to the qubit to be readout. Effectively the harmonic oscillator, formed by the SQUID (acting as a flux-dependent inductor) and the capacitor, transforms the qubit state dependent magnetic moment into a shift of the resonator frequency, and so in an amplitude and/or phase shift available in the ac voltage or the seusceptibility. The ac current driving strength of the harmonic oscillator is the knob allowing control on the detection strength of the SQUID, and so also on the degree of back action from detector to measured object (=qubit). The system can also be operated in the non-linear driving mode. Following the developments at Yale (Devoret), this allows a strong enhancement of the sensitivity, augmented by an intrinsic latching capability due to a bifurcation phenomenon. We have used this second approach to reach our most optimal fidelity readout results. To check the fidelity of the readout a DC flux signal of the value generated by the qubit is applied, and the shift in switching is measured to be 98% at its optimum. If now operating the qubit, an optimal difference of 87% is found, implying a loss of only 11%. This is one of the best values reached so far in the community. A careful analyses of the various contributions resulting from the different steps in the actual state measurement process yields the following numbers: 2% during (but not due to) the relaxation time T1; 7% from the adiabatic shift to bring the qubit from its optimal point to the readout point; and 3% from the dynamical readout itself. Within the errors this sums correctly. These results demonstrate the great capabilities of this readout system. Based we will introduce a similar detection system in our second measurement setup as well. The major change will be to enhance the frequency of the resonator. Based on the work of Patrice Bertet in our group and this current work it is a strong advantage to implement the next system at ~ 2GHz, in this way reducing de-phasing effects resulting from the thermal occupation variations of the harmonic oscillator.

We have developed original technology of fabricating super-conducting circuits with small Josephson junctions from the Nb/Al/AlOx/Nb sandwich. The minimum linear size of the junctions can be made as small as 70 nm. The pre-gap leakage current measured in these junctions at low temperature is very low, while the value of the superconductor gap is close to the bulk-Nb value. The circuits fabricated with the help of this technology can be used for single-electron, single-Cooper-pair and Josephson quantum computing devices. The advantage of the developed technology consists in extending the working frequency of the devices up to one order (due to using the super-conducting material with high value of the energy gap) and in possibility of designing the circuits without stray metallic-layer shadows, which are unavoidable in the traditional shadow evaporation technique.

We have fabricated and measured a novel type of super-conducting single-electron transistor in which the island is not in physical contact with the substrate: the island in fact is suspended above the SiN2-on-Si chip. As a result, the noise coming from fluctuators in the substrate is eliminated. We have demonstrated that the devices fabricated with our method are robust and they display all the functionality associated with standard (non-suspended) single-electron transistors. This circuit element could have applications in quantum computing and nanoelectromechanics.

TU Delft has fabricated and investigated a system of a flux qubit coupled to a harmonic oscillator, observing Rabi oscillations of the composite system. The qubit is the standard TU Delft three Josephson junction flux qubit, and the harmonic oscillator is a DC SQUID with attached capacitor. The potential for this scheme is very large, providing a strongly local system with controlled locality. In this respect it strongly differs from the approach taken by others (e.g. Yale), where the harmonic oscillator has to be taken of a size comparable to the wavelength associated with the qubit level splitting frequency. The actual implementation employs the SQUID in two distinctly different manners: as the detecor (as usual), and as the harmonic oscillator (which makes the system to be a composite one). The SQUID with the capacitor attached to it, may act quantum mechanically. This requires the level separation (or plasma frequency) to be sufficiently large, and the damping to be sufficiently small, such that the decay rate is smaller than the energy of the qubit-SQUID coupling. The results imply successful entanglement of a 2-level qubit and an n-level harmonic oscillator, demonstrating for the first time the quantum dynamics of a solid state two-component system through controlled and conditional Rabi flopping employing the lowest four levels of the composite system.

Non-Abelian geometric phases can be generated and detected in certain super-conducting nanocircuits. Here we consider an example where the holonomies are related to the adiabatic charge dynamics of the Josephson network. We demonstrate that such a device can be applied both for adiabatic charge pumping and as an implementation of a quantum computer.

Prior to 2005, the qubits fabricated and studied at Chalmers incorporated a radio-frequency single-electron transistor (RF-SET), which functioned as an electrometer to measure the charge of the single Cooper-pair box (SCB) two-level system. For this type of projective measurement, one must necessarily move away from the optimal point (charge degeneracy) to perform a charge measurement. Another drawback of this design concerns scalability. In a multiple qubit system, each of the qubits of this type requires its own RF-SET. In addition to the DC line for the SCB voltage gate, one needs two DC lines (voltage gate and bias), and a radio-frequency coax for operation of the RF-SET. One must also synchronize the SCB gate and RF-SET gate to cancel cross capacitances. Our newly designed readout scheme utilizes the effective capacitance of the SCB by placing the SCB in a resonant LC circuit. In this way, one can combine the two-level system and measurement apparatus into a single device. Furthermore, unlike measuring charge, the effective capacitance readout is most sensitive at the charge degeneracy---the optimal point for qubit operation. The effective capacitance of a single Cooper-pair box can be defined as the derivative of the induced charge with respect to gate voltage and has two parts, the geometric capacitance and the quantum capacitance. The latter is due to the level anti-crossing caused by the Josephson coupling and is dual to the Josephson inductance. It depends parametrically on the gate voltage and its magnitude may be substantially larger than. Calculations show that such methods are capable of single-shot readout of the quantum state.

We have investigated theoretically a number of quantum manipulations in charge-flux (quantronium) devices, namely how to create interaction-free measurements and how to couple two qubits. For the first topic, a sequence of pulses can be designed so that a quantronium device can assess the presence of one of them without absorbing any energy. For the second topic, we have shown that it is possible to produce entangled states and CNOT gates by irradiating the qubits with pulses of microwave radiation.

The Chalmers theory group has investigated the design and functionality of a network of loop-shaped charge qubits with switchable nearest-neighbour coupling. The qubit coupling is achieved by placing large Josephson junctions (JJs) at the intersections of the qubit loops and selectively applying bias currents. The network is scalable and makes it possible to perform a universal set of quantum gates. The coupling scheme allows gate operation at the charge degeneracy point of each qubit, and also applies to charge-phase qubits. Additional JJs included in the qubit loops for qubit readout can also be employed for qubit coupling.

Several breakthrough experiments with Josephson-junction circuits performed by, e.g., SQUBIT-2 partners from Saclay, Delft, and Chalmers showed that decoherence can be dramatically suppressed at optimal operation points where the linear coupling to low-frequency fluctuations vanishes. We analysed the influence of various noise sources on the dynamics of two-level systems at optimal operation points due to non-linear (quadratic) coupling. We focused on the experimentally relevant 1/f and Ohmic noise power spectra. For 1/f noise strong higher-order effects influence the evolution. As a result the amplitude of coherent oscillations decays according to a power law rather than exponentially. This prediction has been verified experimentally. Characterization of decoherence at optimal points is very important for understanding the fundamental limitation on operation of Josephson circuits as qubits.

Chalmers theory group has investigated the problem of decoherence of Andreev levels in super-conducting junctions due to interaction with the quantum fluctuations of the super-conducting phase across the junction. We have derived a general kinetic equation for the Andreev level de-phasing and relaxation including quasiparticle exchange with the bands. Our main finding is that the fermionic nature of the Andreev levels considerably affect the decoherence terms in the kinetic equation leading to a substantial suppression of the decoherence rate. This many-body effect even leads to a non-exponential relaxation of Andreev levels at small temperature. Recently we have specifically considered the case of a large Josephson junction inserted in the qubit loop forming an oscillator for dispersive readout of the qubit. The junction is permanently connected via small capacitor to a line providing measurement rf signal. We have applied the derived kinetic equation to the present case of interaction of the Andreev levels with EM fluctuations in the line and found that the decoherence rate is very small. Namely, under a generic for qubit operation condition that the plasma frequency of the measurement junction exceeds the qubit frequency, the decoherence time may be in a milliseconds range for properly chosen circuit parameters.

Experiments of the NEC (Japan) group indicated a connection between the low- and high-frequency noises affecting super-conducting quantum systems. In addition experiments of the group of J. Martinis (NIST and UCSB) showed that coherent two-level fluctuators are present in the Josephson junction of the phase qubits. We proposed that both noises can be produced by one ensemble of microscopic modes, made up by sufficiently coherent two-level systems (TLS's). This implies a relation between the noise powers in different frequency domains, which depends on the distribution of the parameters of the TLS's. We showed that a distribution, natural for tunneling TLS's, with a log-uniform distribution in the tunnel splitting and linear distribution in the bias, accounts for experimental observations. The work is important for understanding decoherence in Josephson qubits and for improving coherence by, e.g., better choice of materials.

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