Session A39: Superconducting Qubits: Entanglement and Feedback

Entanglement stabilization by synchronous and asynchronous feedback

Quantum feedback for error correction is now an important component of superconducting quantum information processing. We have implemented two feedback schemes, autonomous (AT) and measurement-based (MB), to stabilize entanglement between two transmon qubits coupled to a cavity. The two qubits are coupled with nearly equal dispersive shifts to the cavity, such that a cavity drive maps the qubits' parity onto the cavity state. Entanglement is autonomously stabilized by applying continuous photon-number-selective Rabi drives on qubit transitions with phases conditioned on the cavity state. Alternatively, entanglement can be stabilized in a measurement-based scheme by directing the cavity output via a high-fidelity measurement chain to an FPGA (Field Programmable Gate Array) which applies pi/2 pulses to the qubits with phases conditioned on the measured signal. A synchronous protocol stabilizes entanglement for a fixed duration using either scheme resulting in a target Bell state with an unconditioned fidelity in excess of 55 {\%} for MB and 77 {\%} for AT. Furthermore, we have enhanced the fidelity of the entanglement by implementing an asynchronous ``wait until success'' protocol conditioning the tomography on a parity measurement in real time.   

Steady-state entanglement of distant transmons, stabilised against high transmission loss

Being able to stabilise entanglement over long distances and long times provides numerous advantages over pulsed experiments (avoiding variability, synchronisation, and calibration issues) while providing an important resource on-demand, which can then be potentially distilled and used to construct a quantum network. We show how existing superconducting technologies can be entangled over distances of tens of meters providing resilient stabilisation even in the presence of high inefficiency of the transmission channel. This can be achieved both in the dispersive and near-resonant cavity regimes using simple protocols that employ correlated environmental interactions and symmetry breaking. These require only a single-frequency drive that interacts sequentially with each cavity-qubit system. The dispersive regime protocol uses feedback while the near-resonant regime protocol is autonomous.   

Stochastic master equation approach for analysis of remote entanglement with Josephson parametric converter amplifier

In the remote entanglement process, two distant stationary qubits are entangled with separate flying qubits and the which-path information is erased from the flying qubits by interference effects. As a result, an observer cannot tell from which of the two sources a signal came and the probabilistic measurement process generates perfect heralded entanglement between the two signal sources. Notably, the two stationary qubits are spatially separated and there is no direct interaction between them. We study two transmon qubits in superconducting cavities connected to a Josephson Parametric Converter (JPC). The qubit information is encoded in the traveling wave leaking out from each cavity. Remarkably, the quantum-limited phase-preserving amplification of two traveling waves provided by the JPC can work as a which-path information eraser. By using a stochastic master approach we demonstrate the probabilistic production of heralded entangled states and that unequal qubit-cavity pairs can be made indistinguishable by simple engineering of driving fields. Additionally, we will derive measurement rates, measurement optimization strategies and discuss the effects of finite amplification gain, cavity losses, and qubit relaxations and dephasing.   

Optimizing JPC-based remote entanglement of transmon qubits via stochastic master equation simulations

Remote entanglement of two superconducting qubits may be accomplished by first entangling them with flying coherent microwave pulses, and then erasing the which-path information of these pulses by using a non-degenerate parametric amplifier such as the Josephson Parametric Converter (JPC). Crucially, this process requires no direct interaction between the two qubits. The JPC, however, will fail to completely erase the which-path information if the flying microwave pulses encode any difference in dynamics of the two qubit-cavity systems. This which-path information can easily arise from mismatches in the cavity linewidths and the cavity dispersive shifts from their respective qubits. Through analysis of the Stochastic Master Equation for this system, we have found a strategy for shaping the measurement pulses to eliminate the effect of these mismatches on the entangling measurement. We have then confirmed the effectiveness of this strategy by numerical simulation.   

Entanglement of remote transmon qubits by concurrent measurement using Fock states

A requirement of any modular quantum computer is the ability to maintain individual qubits in isolated environments while also being able to entangle arbitrary distant qubits on demand. For superconducting qubits, such a protocol can be realized by first entangling the qubits with flying microwave coherent states which are then concurrently detected by a parametric amplifier. This protocol has a 50\% success probability but is vulnerable to losses between the qubits and the amplifier which reduce the entanglement fidelity. An alternative is to use itinerant Fock states, since losses now tend to reduce the success probability of creating an entangled state but not its fidelity. Such single-photon protocols have been implemented in trapped-ion and NV-center experiments. We present such a protocol tailored for entangling two transmon qubits in the circuit QED architecture. Each qubit is entangled with a Fock state of its cavity using sideband pulses. The Fock states leak out of the cavity, interfere on a beam-splitter which erases their which-path information, and are subsequently detected using a novel photo-detector realized by another qubit-cavity system. Simulations suggest that we can realize a high-fidelity entangled state with a success probability as large as 1\%.   

Generation of multi-qubit entanglement in a superconducting quantum circuit by parallelized parity measurements

We present the generation of multi-qubit entanglement using parallelized ancilla-based parity measurements in a five qubit superconducting processor. Two-qubit Bell states and three-qubit GHZ-type states are generated by single and double two-qubit parity measurements on superposition states, respectively, and characterized by both witnessing and state tomography. The protocol for generation of GHZ-type states can be used as the encoding step in the three-qubit bit-flip quantum error correction code, and made deterministic by digital feedback control. We assess its performance by state tomography of the six encoded cardinal states, and compare to the traditional method of encoding by gates.   

Geometry of two-qubit evolution and entanglement protection via local operations

The two-qubit pure state evolution under unitary (Hamiltonian) and measurement transformations is conveniently described via a Hopf fibration map, where an S4 sphere is the two-qubit analog of the single-qubit Bloch sphere and entanglement is encoded in an S4 subspace. We show that there exist two-qubit entanglement protection protocols based only on local operations that are as efficient as in the case of a single qubit state protection. We consider several examples related to current or near future experiments in superconducting circuits. We discuss the relevance of such ideas for quantum computing.   

Toward Resource-Efficient Deterministic Entanglement in 3D Superconducting Qubits

We present progress towards deterministic entanglement in a three-dimensional, superconducting qubit architecture, using only one continuous drive to engineer steady-state entanglement [1]. Our protocol uses two (nominally) identical copper waveguide cavities that each contain a transmon qubit. The cavities are directly coupled to one another and hybridize into symmetric and antisymmetric modes. The coupling of the cavities introduces a cavity-mediated qubit coupling that splits the degeneracy between the singlet and triplet states in the odd-parity subspace. By driving the cavities symmetrically and carefully tuning the single drive amplitude and frequency to take advantage of the hybridized cavity density of states, we aim to achieve bipartite entanglement that is stable against both dephasing and finite qubit lifetime. In this talk, we present experimental progress towards this cavity-assisted, bath engineered generation of entanglement. \\[4pt] [1] C. Aron, M. Kulkarni, \& H.E. Tureci. http://arxiv.org/abs/1403.6474   

Optimal feedback for remote entanglement

Recent experiments in superconducting qubits have demonstrated measurement as a resource for entanglement, even when qubits are spatially separated to a significant degree. We consider the problem of using measurement combined with feedback to deterministically entangle remote qubits. This constraint forces us to consider only local feedback, which leads us to an interesting control-theory problem. Within this constraint, we derive a series of protocols for this system which generate entanglement as quickly as possible. We find that even in the presence of expected experimental imperfections, it should be possible to achieve high-fidelity entanglement with currently accessible experimental parameters.   

Robust quantum state transfer using flying microwave qubits

We analyze the transfer of a quantum state between two superconducting microwave resonators connected by a transmission line. Nearly perfect state-transfer efficiency can be achieved by using adjustable couplers to cancel the back-reflection from the receiving coupler with destructive interference. We show that the transfer protocol is robust to parameter variations affecting the transmission amplitudes of the couplers. We also show that the effects of Gaussian filtering, pulse-shape noise, and multiple reflections on the transfer efficiency are not significant for experimentally realistic parameters. However, the transfer protocol is very sensitive to a frequency mismatch between the two resonators. Moreover, the type of coupler we consider produces a time-dependent detuning, which requires active compensation with sufficient accuracy to yield high-efficiency state transfer.   

Scaleable two and four qubit parity measurement with a photon counter

Multi-qubit parity readout is a central ingredient to quantum information processing, with applications ranging from quantum error correction to entanglement generation. As the physical implementation of QIP technologies grows in size, so too does the need for scalable readout protocols. Here we present a scalable, high-fidelity, quantum non-demolition readout protocol for the parity of two or four qubits using a single dispersively coupled cavity and a photon counter. By selectively populating the cavity dependent on the qubit parity, it is possible to non-destructively read out the qubit parity using a phase insensitive photon counter, without gaining any further qubit-state resolving information. We describe our protocol in the context of superconducting integrated circuits, where the cavity is a microwave resonator, and as an example photon counter we choose the Josephson photomultiplier (PRL 107, 217401 (2011)).   

Analytical approach to swift non-leaky entangling gates in superconducting qubits

We develop schemes for designing pulses that implement fast and precise entangling quantum gates in superconducting qubit systems despite the presence of nearby harmful transitions. Our approach is based on purposely involving the nearest harmful transition in the quantum evolution instead of trying to avoid it. Using analytical tools, we design simple microwave control fields that implement maximally entangling gates with fidelities exceeding 99 percent in times as low as 40 ns. We demonstrate our approach in a two-qubit circuit QED system by designing the two most important quantum entangling gates: a conditional-NOT gate and a conditional-Z gate. Our results constitute an important step toward overcoming the problem of spectral crowding, one of the primary challenges in controlling multi-qubit systems.   

3D Multimode Cavity QED

Scalable quantum computing architectures require many long-lived and highly coherent, yet easily addressable quantum states. Photonic qubits in 3D superconducting microwave cavities are a promising approach because they are highly insensitive to decoherence and single photon lifetimes exceeding 10 ms have been demonstrated$\footnote{M. Reagor, Appl. Phys. Lett. \textbf{102}, 192604 (2013) }$. However, the plurality of current 3D cavity devices are engineered to address single photon modes. In this talk, we introduce our implementation of a multimode 3D cavity that can store greater than 20 distinct, long-lived photon modes. To perform single- and two-qubit gates between photons, each of the modes are coupled to a single flux-tunable superconducting transmon qubit. We will discuss our preliminary results towards a controlled phase gate between any pair of photons modes. This multimode circuit QED architecture may also be used as a many-body bosonic system for quantum simulation, to study multimode quantum optics, and for quantum memories as part of a larger quantum network.   

Design and processing considerations for superconducting qubits coupled to multiple 3D cavities

Arrays of three-dimensional waveguide microwave cavities with superconducting transmon qubits bridging between adjacent cavities form a promising architecture for implementing quantum information processing. The performance of these transmon bridge qubits can be limited by multiple factors including losses on the substrate surface and in the metal traces, as well as surfaces losses in the cavities and couplings to spurious modes in the cavity structures. To address these issues, we are investigating a variety of cavity designs, materials, and surface coatings. We are also exploring different geometries for the qubit electrodes as well as different materials for the substrates and qubit capacitor metallization and various device-processing techniques. We will present low-temperature measurements of some of the resulting qubits.