Session P48: Quantum Control in Superconducting Circuits

Quantum Control of Cavity Resonators, Part I: Control Algorithms

Harmonic oscillators are linear systems with equally spaced energy levels, which makes them hard to control. We have previously explored a constructive control approach mediated by a far off-resonantly coupled two-level ancilla. Here we present an extension to that method which relies on optimal control algorithms to allow much more efficient quantum control of a combined resonator – ancilla system. We show that full control of the resonator is possible on a time-scale of order 1/chi, the dispersive shift. In practice this means that a unitary operation on the Hilbert space of our superconducting resonator truncated to 8 levels can be performed using a pulse of around a microsecond.   

Quantum Control of Cavity Resonators, Part II:Experiment

Harmonic oscillators offer a large Hilbert space that can potentially be used to encode multiple bits of quantum information. The long lifetime of superconducting cavity resonators make them a suitable candidate to explore this direction. Due to the linearity of harmonic oscillators it is not directly obvious how to manipulate them. Here we show that pulses designed using optimal control methods allow us to manipulate the combined cavity -- transmon system on a time-scale of order 1/chi, the dispersive shift; in practice pulses of about a microsecond long. Several example unitary operations addressing the first 8 levels of the resonator are described and characterized.   

Optimal control of single flux quantum (SFQ) pulse sequences

Single flux quantum (SFQ) pulses are a natural candidate for on-chip control of superconducting qubits [1]. High accuracy quantum gates are accessible with quantum optimal control methods. We apply trains of SFQ pulses to operate single qubit gates, under the constraint of fixed amplitude and duration of each pulse. Timing of the control pulses is optimized using genetic algorithms and simulated annealing, decreasing the average fidelity errorby several orders of magnitude. Furthermore we are able to reduce the gate time to the quantum speed limit. Leakage out of the qubit subspace as well as timing errors of the pulses are considered, exploring the robustness of our optimized sequence.This takes usone step further to a scalable quantum processor. [1] R. McDermott and M.G. Vavilov, Phys. Rev. Appl. 2, 014007 (2014)   

Robust Control of a Two-Qubit Operation in 3D Circuit Quantum Electrodynamics

Superconducting qubits have shown great improvement in coherence times with the introduction of 3D cavities. In order to control the qubits in 3D a microwave drive is usually coupled to the common mode of the cavity, which makes individual addressability a challenge and causes additional unwanted single and two-qubit dynamics when performing two qubit operations. Quantum information processing requires precise control of the system dynamics in the presence of potential uncertainties in the estimated system parameters. We use optimal control theory to develop pulse shapes that are able to implement an all-microwave two-qubit gate, while mitigating extra unwanted interaction terms, with $\mathcal{F}=0.9964$. In addition we develop pulses which are robust to errors in the two qubit transition frequencies. This is demonstrated with experimentally relevant parameters and includes realistic constraints in the possible pulse shapes, presenting pulses that can be implemented in experiment.   

Gradient Optimization for Analytic conTrols - GOAT

Quantum optimal control becomes a necessary step in a number of studies in the quantum realm. Recent experimental advances showed that superconducting qubits can be controlled with an impressive accuracy. However, most of the standard optimal control algorithms are not designed to manage such high accuracy. To tackle this issue, a novel quantum optimal control algorithm have been introduced: the Gradient Optimization for Analytic conTrols (GOAT). It avoids the piecewise constant approximation of the control pulse used by standard algorithms. This allows an efficient implementation of very high accuracy optimization. It also includes a novel method to compute the gradient that provides many advantages, e.g. the absence of backpropagation or the natural route to optimize the robustness of the control pulses. This talk will present the GOAT algorithm and a few applications to transmons systems.   

Fast resonator reset in circuit QED using open quantum system optimal control

Practical implementations of quantum information processing requires repetitive qubit readout. In circuit QED, where readout is performed using a resonator dispersively coupled to the qubits, the measurement repetition rate is limited by the resonator reset time. This reset is usually performed passively by waiting several resonator decay times. Alternatively, it was recently shown that a simple pulse sequence allows to decrease the reset time to twice the resonator decay time [1]. In this work, we show how to further optimize the ring-down pulse sequence by using optimal control theory for open quantum systems. Using a new implementation of the open GRAPE algorithm that is well suited to large Hilbert spaces, we find active resonator reset procedures that are faster than a single resonator decay time. Simple quantum speed limits for this kind of active reset processes will be discussed. [1] McClure et al., arXiv 1503.01456   

Analysis of non-adiabatic effects in circuit QED measurement of a transmon

In a circuit QED setup with a transmon qubit dispersively coupled to a driven resonator, we investigate whether rapid resonator ringup will cause nonadiabatic effects that disturb the qubit state. We show that only unrealistically fast high-power pulses will produce significant deviations from adiabatic behavior, while typically the qubit-resonator dynamics is well described by coherent evolution in the joint eigenbasis. Nevertheless, even in typical parameter regimes we show that the qubit nonlinearity still produces a dynamical shearing effect that squeezes the state of the resonator field.   

A method of extracting operating parameters of a quantum circuit

Rigorous simulation-driven design methods are an essential component of traditional integrated circuit design. We adapt these techniques to the design and development of superconducting quantum integrated circuits by combining classical finite element analysis in the microwave domain with Brune circuit synthesis by Solgun [PhD thesis 2014] and BKD Hamiltonian analysis by Burkard et al. [Phys. Rev. B \textbf{69}, 064503 (2004)]. Using the Hamiltonian of the quantum circuit, constructed using the synthesized equivalent linear circuit and the nonlinear Josephson junctions' contributions, we extract operating parameters of the quantum circuit such as resonance coupling strength, dispersive shift, qubit anharmonicitiy, and decoherence rates for single-and multi-port quantum circuits. This approach has been experimentally validated and allows the closed-loop iterative simulation-driven development of quantum information processing devices.   

Critical fluctuations near excitation threshold of a quantum parametric oscillator

A weakly damped parametrically driven oscillator has several vibrational states already for weak driving. These are stable and unstable states with twice the modulation period and also the steady state. At the critical point all states merge. We show that this leads to anomalously strong quantum fluctuations. These fluctuations are similar whether the friction, in the classical picture, is linear or nonlinear. The critical region is $\propto [\hbar (2\bar n +1)]^{1/3}$ along the field frequency axis and $\propto [\hbar (2\bar n +1)]^{2/3}$ along the field amplitude axis, where $\bar n$ is the Planck number. The correlation time scales as $[\hbar (2\bar n +1)]^{-2/3}$. The number of photons for $\bar n=0$ scales as $\hbar^{-2/3}$. It is determined by the oscillator nonlinearity and decay rate. Above the threshold, quantum fluctuations induce transitions between the period-two states over the quasienergy barrier. We find the effective quantum activation energies for such transitions and their scaling with the difference of the driving amplitude from its critical value. We also present the results of relevant experimental observations obtained with a circuit QED system.   

Fluctuations of a parametric oscillator: from the semiclassical to a strongly quantum regime

A semiclassical parametric oscillator has two dynamically stable vibrational states with equal amplitudes and with phases differing by $\pi$. The rate of switching between these states is exponentially small, and the oscillator displays fluctuations with the reciprocal correlation time given by this rate. It also displays critical slowing down near the excitation threshold. The parameter of the ``quantumness" is the ratio of the nonequidistance $\hbar V$ of the oscillator energy levels due to the nonlinearity and the level width $\hbar\Gamma$ due to decay. In the strongly-quantum regime where $V/\Gamma\gg 1$ and driving is not too strong, the picture of coexisting vibrational states with opposite phases does not apply. An insight into the transition from the semiclassical to strongly-quantum regime can be gained by studying the quasienergy spectrum and the decay of quantum fluctuations. An analogue of the critical slowing down in the strongly-quantum regime is a sharp increase of the fluctuation correlation time that occurs at a hypersurface in the oscillator parameter space. We find that the quasienergy spectrum and the ratio of the level spacing to their width also sensitively depend on the parameters, in particular on $V/\Gamma$.   

Robust tomography of microwave resonator arrays for quantum simulation with light

We are interested in using a bottom-up approach to create topologically non-trivial states of light via Hamiltonian engineering in coupled microwave cavities. Characterization and reduction of disorder is paramount to realizing and studying idealized many-body Hamiltonians. Our tight-binding lattices are made of arrays of evanescently coupled three-dimensional microwave resonators. From the spectroscopic response measured at specific lattice sites, we develop methods to fully map out the underlying tight-binding Hamiltonian, including onsite energies, nearest-neighbor couplings and the local dissipation on all sites. We show that for a 1D system, one reflection measurement off of the site at the end of the chain is sufficient, while for 2D only measurements along one edge of the system is sufficient for complete tomography of the lattice Hamiltonian. The transmission between neighboring sites also reveals the phase of the tunnel coupling, thereby allow direct measurement of the flux in lattices with time-reversal breaking synthetic gauge fields. These methods can be readily applied to many other physical systems for the characterization of quantum processes or the validation of quantum simulators.   

Analysis of qubit dynamics under strong resonant pulses using Floquet theory

Resonant driving is the most common way of implementing single-qubit gates in various quantum systems. Most of the experiments and optimization of such gates are performed in the weak-driving regime, where the qubit dynamics is relatively slow and well described using the rotating wave approximation. However, the implementation of qubit gates with strong driving, which in principle promises a higher speed, has not been studied extensively. In this work, we consider the dynamics of a qubit driven by strong resonant pulses in the framework of Floquet theory. We analyze the role of pulse shaping in the dynamics, as determined by nonadiabatic transitions between the Floquet states. By suppressing the nonadiabatic transitions, we show that high-fidelity single-qubit operations can be achieved in very short times. This work provides the theoretical basis for optimizing strong pulses for single-qubit gates. These results are particularly relevant for the implementation of single-qubit gates in superconducting qubits, where strong driving with shaped pulses has been demonstrated experimentally.   

Flexible, low-latency architecture for qubit control and measurement in circuit QED.

Increasing qubit numbers in circuit QED requires an extensible architecture for digital waveform generation of qubit control and measurement signals. For quantum error correction, the ability to select from a number of predetermined waveforms based on measurement results will become paramount. We present a room-temperature architecture with very low latency from measurement to waveform output. This modular FPGA-based system can generate both baseband and RF modulated signals using DACs clocked at 1 GHz. A backplane that interconnects several modules allows exchange of (measurement) information between modules and maintains deterministic timing across those modules. We replace the typical line based sequencer used in arbitrary waveform generators by a user programmable processor that treats waveforms and measurements as instructions added to a conventional CPU architecture. This allows for flexible coding of triggering, repetitions, delays and interactions between measurement and signal generation. We acknowledge funding from the Dutch Research Organization (NWO), an ERC Synergy Grant, and European project SCALEQIT.   

Fock State Generator

Using a single junction Al/AlO$_\text{x}$/Al transmon qubit coupled to a superconducting Al cavity (at a temperature 15 mK), we have used a Raman technique to produce a single Fock state in the cavity. The technique requires 3 microwave tones to drive the system from the ground state of the cavity/qubit system. We achieve an experimental fidelity of the final Fock state of around 90\%, limited by thermal photons in the cavity and by decay during the operation time. Using this technique, we have also generated an arbitrary superposition of Fock states and a superposition of qubit and cavity states. Results, simulations and applications of this technique will be discussed.