High contrast qubit interactions using multimode cavity QED
We introduce a new multimode cavity QED architecture for superconducting circuits which can be used to implement photonic memories, more efficient Purcell filters, and quantum simulations of photonic materials. We show that qubit interactions mediated by multimode cavities can have exponentially improved contrast for two qubit gates without sacrificing gate speed. Using two-qubits coupled via a three-mode cavity system we spectroscopically observe multimode strong couplings up to 102MHz and demonstrate suppressed interactions off-resonance of 10kHz when the qubits are 600MHz detuned from the cavity resonance. We study Landau-Zener transitions in our multimode systems and demonstrate quasi-adiabatic loading of single photons into the multimode cavity in 25ns. We introduce an adiabatic gate protocol to realize a controlled-Z gate between the qubits in 95ns and create a Bell state with 94.7% fidelity. This corresponds to an on/off ratio (gate contrast) of 1000.
Current address:]Yale Department of Applied Physics, New Haven, CT 06511
Circuit cavity quantum electrodynamics (cQED) using superconducting resonators and Josephson junction based qubits have demonstrated the essential building blocks of gate based quantum computing and quantum opticsDevoret and Schoelkopf (2013). Typically, cQED devices are engineered so that the qubits primarily couple to a single cavity mode, nonetheless, the true multimode nature of these devices is unavoidable. For example, a multimode treatment is required to correctly understand the Purcell effectHouck et al. (2008), and in 3D resonators, where the mode density is higher, it is necessary to take into account the full mode structure using the “black box quantization” approach to correctly model the device parametersNigg et al. (2012). Although these modes are usually treated as a nuisance, if properly utilized, they are a powerful asset. In this letter, we introduce an explicitly multimode QED architecture as a resource to study multimode quantum opticsEgger and Wilhelm (2013), as a many-body bosonic system for quantum simulationZhou et al. (2008); Schmidt and Koch (2013), as a photonic register for quantum memory, and to tailor coherent qubit-qubit interactions.
In the context of quantum computing, tailoring qubit interactions is of paramount importance for improving gate contrast. In the past several years much effort has been spent to improve gate fidelities; single qubit coherence times can approach 100s Rigetti et al. (2012), arbitrary rotations in the Bloch sphere are possible with gate fidelities higher than 99.8%Gustavsson et al. (2013), and elementary two-qubit gates have attained gate fidelities up to 99.4%Barends et al. (2014). There has been rapid progress towards constructing larger circuits to implement quantum algorithmsDiCarlo et al. (2010); Fedorov et al. (2012); Chow et al. (2013); Saira et al. (2014); Barends et al. (2014), photonic memoriesMariantoni et al. (2011), and nascent quantum simulationUnderwood et al. (2012). However, as strongly coupled circuits grow larger, issues inevitably arise due to residual cQED couplings.
Several methods have been developed to reduce unwanted interactions, however, they are not without their limitations. To counteract “always on” interactions in NMR quantum computing, decoupling pulse sequences have been developedCriger et al. (2012). These sequences can be applied to JJ qubits, but become onerous as the system size grows larger. Instead, another approach is to develop tunable interactions for high contrast gates, most commonly by coupling JJ qubits through a resonant interaction. In these experiments, interactions are controlled via the detuning from resonance, imposing a tradeoff between gate contrast and speed. Also, expanding beyond two qubits results in spectral crowding, which limits addressabilitySchutjens et al. (2013) and introduces spurious avoided crossings. While parametric gatesChow et al. (2011) sidestep some of these problems, in both cases the contrast is only linear in detuning. This limits the achievable off-rate since detuning is bounded. Alternatively, we can dynamically tune the coupling by destructive interference between two charge qubitsSrinivasan et al. (2011) or by flux tuning a JJ inductive couplerBialczak et al. (2011); Chen et al. (2014). However, dynamic coupling requires additional junctions, which introduces complexity and a new path for decoherence.
In this letter, we introduce a new multimode circuit QED architecture where qubits interact through a network of strongly coupled resonators, analogous to a multimode bandpass filter. The multimode architecture enables the off-resonant interactions to be suppressed exponentially in the number of modes (resonators) without any additional active elements. This multimode architecture could also be utilized to filter the qubit noise environment, i.e. a multimode Purcell filterReed et al. (2010); Jeffrey et al. (2014). To demonstrate the multimode architecture, we construct a circuit with two transmon-type qubits coupled via a three-mode (three-resonator) filter. We perform spectroscopy on our device and confirm the multimode circuit QED model. From spectroscopy, we observe multimode strong coupling when the qubit and filter are on-resonance and suppressed qubit-qubit interactions off-resonance. Next, we measure strong interaction dynamics by quickly tuning the qubit energy into resonance with the filter. We demonstrate fast loading of single photons into the lowest mode of the filter (ns) and measure a single photon Stark shift greater than 100MHz. Finally, we utilize the state-dependent Stark shift to realize a controlled-Z gate between the qubits in 95ns and create a Bell state with 94.7% fidelity.
A schematic of our circuit and the corresponding physical realization are illustrated in Fig. 1. Three identical lumped LC resonators of frequency are capacitively coupled to each other in a chain to form our multimode filter. Two flux-tunable transmonKoch et al. (2007) qubits (GHz) are capacitively coupled to the resonators at the end of the filters. For qubit frequencies , , the qubit-filter system (for -modes) is described by the Hamiltonian
where creates a photon in the resonator, is the raising (lowering) operator for the qubit, is the z-Pauli operator, is the filter-filter coupling and is the qubit-filter coupling.
Strong coupling between the bare filter resonators splits the three degenerate resonators into three “filter” modes with frequencies . Each of these filter modes are a superposition of photons in the bare resonators. Crucially, every filter mode has non-zero weight in the resonators at either end of the chain so that filter photons in mode strongly couple to qubit 1 (qubit 2) with coupling – this realizes our multimode strong coupling architecture. We fit the spectroscopy data in Fig. 1 (c) to extract bare qubit-filter parameters GHz, MHz, and ()MHz corresponding to multimode parameters GHz, GHz, GHz and MHz ().
When the qubits are detuned from all the filter modes and the filter is empty (analogous to the stop band of a classical filter) residual interactions are mediated by virtual photons through all modes and we can rewrite Eqn. 1 as
where is the exchange term and is the controlled-phase (c-phase) rate. If we consider identical qubit 1 and qubit 2 filter couplings and let be the averaged detuning of the qubit from the bare filter mode (i.e., ), then we can approximate and (for an -mode filter) as
Notably, these rates are suppressed exponentially in the number of filter modes , in terms of the small parameter . This is a result of destructive interference between multiple filter modes which sum coherently and enables the ability to turn off interactions with high constrast by detuning from the filter. To confirm the off-rate scaling predicted by Eqns. 6 and 7, we directly measure the exchange term from qubit spectroscopy, and numerically calculate the c-phase rate. The data plotted in Fig. 2 agrees well to the model with no free parameters, demonstrating the essential scaling of the multimode off-rate, and implying an off-rate less than 10kHz for a qubit-qubit detuning of 50MHz.
To enable strong interactions in the multimode architecture we tune the qubit frequency into resonance. In this limit, the expressions for the suppressed couplings given by Eqns. 6 and 7 are invalid because the virtual photon paths no longer destructively interfere. Since the qubit interacts primarily with the closest mode, the coupling strength is of order (the qubit-filter coupling). Interestingly, if there is a real photon in one of the filter modes the interference is also imbalanced, hence the Stark shift is not suppressed. In both cases, the qubits interact predominantly through a single mode and so the strong interaction physics is essentially that of two qubits interacting through a single cavity.
For our controlled-Z gate, we utilize these strong interactions by loading a real photon into the lowest filter mode and then employing a state-dependent one-photon Stark shift. Loading a single photon requires adiabatically traversing the qubit-filter avoided crossing shown in Fig 1, so we first study the dynamics of this crossing by performing the experiment illustrated in Fig. 3 (a). We excite qubit 1, raise the qubit energy quasi-linearly through the filter in time (the flux is ramped linearly), hold for time , ramp back in time , and then measure the excited state population. The total time is fixed to 110ns (see flux diagram inset). Because we traverse avoided crossings twice, we observe interference fringes. There are two types of fringes in Fig 3(a): fast fringes at short times and slower fringes dominant at longer times. The fast fringes correspond to ramp speeds larger than the total filter bandwidth (MHz) where a significant fraction of the excitation remains with the qubitKayanuma and Fukuchii (1985). The slower fringes correspond to the excitation being distributed over multiple filter modes and the fringe frequency is fixed by the filter mode splitting. The multimode nature of the crossing is advantageous; although the ramp is not adiabatic with the lowest filter mode unless it is slower than 25ns, the excitation remains in the filter for ramps 5ns. We exploit this multimode Landau-Zener physics to transfer population to the filter faster than the single mode adiabatic limit.
Next, we measure the Stark shift between a single photon and a qubit in the ground state by performing a Ramsey experiment on the photon while varying the length of the interaction at different detunings as illustrated by Fig. 3 (b). First, we prepare qubit 1 in the superposition state and then raise the qubit frequency through the filter to create a photon superposition state. Next, we raise the frequency of qubit 2 to for a variable time . After a fixed total time, we retrieve the photon from the filter, apply a pulse and measure the state of qubit 1. Because of the variable time interaction with qubit 2, we measure a Ramsey fringe versus . The frequency of the fringe is the Stark shift; sample data for one of the points is shown in the inset to Fig. 3(b). Approaching the filter from below, the Stark shift increases as , and then saturates at the maximum interaction (approximately the filter splitting MHz) as qubit 2 is brought through the filter. The data agree very well with a theory curve with no free parameters, thus validating that we are loading a single photon into the lowest filter mode and that we can generate strong interactions between a qubit and a single photon.
Finally, we combine the capabilities probed in the previous two experiments — loading a single photon into the filter and generating a strong Stark shift — to construct a quantum logic gate. The protocol for the gate is illustrated in Fig. 4(a). First, we convert the qubit 1 excitation into a photon, then we move qubit 2 close to the filter to acquire a state dependent Stark shift, and then we return the photon back to qubit 1. While the qubit energies cross during these ramps, we observe no evidence of an exchange process since our multimode filter strongly surpresses the off-resonance interaction (Eqn. 6). We realize a controlled-Z gate (CZ) because the conditional phase (calculated in Fig. 4 (b)) is . The qubits will also acquire trivial single qubit phases which we calibrate out by fine-tuning the flux pulse shape. Although the largest one-photon Stark shift occurs when we bring qubit 2 through the filter (Fig. 3 (b)), the largest state-dependent interaction occurs when we bring qubit 2 just below the filter because of the full state structure of the transmon (see supplementary information). The flux pulse sequence for our CZ gate is illustrated in Fig. 4. The total gate time, 95ns, is optimized to maximize gate fidelity. For 50MHz detuning between the qubits, this implies a gate contrast (on/off rate) greater than 1000 even for relatively small .
To demonstrate the gate we prepare a Bell state by applying single qubit pulses before and after the gate. Ideally this process creates the Bell state . To characterize the expected density matrix we perform state tomographyJames et al. (2001) on both qubits after the gate (see Fig. 4 (c)). The fidelity is 0.947 corresponding to concurrence of 0.926Wootters (1998). We also measure a full process fidelity of 0.89 (errors and tomography details are discussed in the supplementary information). Our fidelity is comparable to other contemporary results (two-qubit entangled states have been produced with state fidelities up to 99.5%Barends et al. (2014) and concurrence of 0.994 Chow et al. (2012)), and is limited by our lifetime, rather than the protocol. One advantage of our protocol is that our gate is relatively insensitive to inhomogeneous broadening due to flux noise; once the qubit excitation is a photon in the filter, the energy is not flux dependent. Several improvements are possible, for example, engineering a flux insensitive bias point below the filter for state preparation Strand et al. (2013), utilizing new materials Chang et al. (2013) and material processing for high Q resonators Megrant et al. (2012), as well as reducing the total gate time using techniques from optimal control for crossing the filter.
In conclusion, we have demonstrated a new multimode architecture for coupling superconducting qubits. We measured that the off-resonance coupling is suppressed exponentially in the number of modes, while still maintaining strong interactions when the qubits are tuned close to resonance. We used these capabilities to realize a high-contrast controlled-Z gate. Further, this work indicates a need to develop a microwave filter theory for coherent quantum systems. The multimode architecture is a promising platform for realizing lattice based quantum simulations and photonic registers for quantum information processing.
Acknowledgments: We acknowledge support from the University of Chicago MRSEC, Army Research Office under grant W911NF-12-1-0608, the Alfred P. Sloan Foundation, NSF under grant DMR-1151839, and DARPA grant N66001-11-1-4123. We acknowledge David Czaplewski at the Argonne Center for Nanomaterials for assistance with the optical lithography. We thank J. Chow and B. Johnson for discussions and J. Thywissen for manuscript comments.
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