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Bosonic modes have wide applications in various quantum technologies, such as optical photons for quantum communication, magnons in spin ensembles for quantum information storage and mechanical modes for reversible microwave-to-optical quantum transduction. There is emerging interest in utilizing bosonic modes for quantum information processing, with circuit quantum electrodynamics (circuit QED) as one of the leading architectures. Quantum information can be encoded into subspaces of a bosonic superconducting cavity mode with long coherence time. However, standard Gaussian operations (e.g., beam splitting and two-mode squeezing) are insufficient for universal quantum computing. The major challenge is to introduce additional nonlinear control beyond Gaussian operations without adding significant bosonic loss or decoherence. Here we review recent advances in universal control of a single bosonic code with superconducting circuits, including unitary control, quantum feedback control, driven-dissipative control and holonomic dissipative control. Various approaches to entangling different bosonic modes are also discussed., 23 pages, 9 figures, 1 table (+2 pages supplement with 1 figure)

The realization of robust universal quantum computation with any platform ultimately requires both the coherent storage of quantum information and (at least) one entangling operation between individual elements. The use of continuous-variable bosonic modes as the quantum element is a promising route to preserve the coherence of quantum information against naturally-occurring errors. However, operations between bosonic modes can be challenging. In analogy to the exchange interaction between discrete-variable spin systems, the exponential-SWAP unitary [$\mathbf{U}_{\mathrm{E}}\left(\theta_c\right)$] can coherently transfer the states between two bosonic modes, regardless of the chosen encoding, realizing a deterministic entangling operation for certain $\theta_c$. Here, we develop an efficient circuit to implement $\mathbf{U}_{\mathrm{E}}\left(\theta_c\right)$ and realize the operation in a three-dimensional circuit QED architecture. We demonstrate high-quality deterministic entanglement between two cavity modes with several different encodings. Our results provide a crucial primitive necessary for universal quantum computation using bosonic modes.

Single-photon detectors are ubiquitous and integral components of photonic quantum cryptography, communication, and computation. Many applications, however, require not only detecting the presence of any photons, but distinguishing the number present with a single shot. Here, we implement a single-shot, high-fidelity photon number-resolving detector of up to 15 microwave photons in a cavity-qubit circuit QED platform. This detector functions by measuring a series of generalized parity operators which make up the bits in the binary decomposition of the photon number. Our protocol consists of successive, independent measurements of each bit by entangling the ancilla with the cavity, then reading out and resetting the ancilla. Photon loss and ancilla readout errors can flip one or more bits, causing nontrivial errors in the outcome, but these errors have a traceable form which can be captured in a simple hidden Markov model. Relying on the independence of each bit measurement, we mitigate biases in ensembles of measurements, showing good agreement with the predictions of the model. The mitigation improves the average total variation distance error of Fock states from $13.5\%$ to $1.1\%$. We also show that the mitigation is efficiently scalable to an $M$-mode system provided that the errors are independent and sufficiently small. Our work motivates the development of new algorithms that utilize single-shot, high-fidelity PNR detectors., Comment: 15 pages, 7 figures

Quantum bits are more robust to noise when they are encoded non-locally. In such an encoding, errors affecting the underlying physical system can then be detected and corrected before they corrupt the encoded information. In 2001, Gottesman, Kitaev and Preskill (GKP) proposed a hardware-efficient instance of such a qubit, which is delocalised in the phase-space of a single oscillator. However, implementing measurements that reveal error syndromes of the oscillator while preserving the encoded information has proved experimentally challenging: the only realisation so far relied on post-selection, which is incompatible with quantum error correction (QEC). The novelty of our experiment is precisely that it implements these non-destructive error-syndrome measurements for a superconducting microwave cavity. We design and implement an original feedback protocol that incorporates such measurements to prepare square and hexagonal GKP code states. We then demonstrate QEC of an encoded qubit with unprecedented suppression of all logical errors, in quantitative agreement with a theoretical estimate based on the measured imperfections of the experiment. Our protocol is applicable to other continuous variable systems and, in contrast with previous implementations of QEC, can mitigate all logical errors generated by a wide variety of noise processes, and open a way towards fault-tolerant quantum computation., Text and figures edited for clarity. The claims of the paper remain the same. Author list fixed

Ancilla systems are often indispensable to universal control of a nearly isolated quantum system. However, ancilla systems are typically more vulnerable to environmental noise, which limits the performance of such ancilla-assisted quantum control. To address this challenge of ancilla-induced decoherence, we propose a general framework that integrates quantum control and quantum error correction, so that we can achieve robust quantum gates resilient to ancilla noise. We introduce the path independence criterion for fault-tolerant quantum gates against ancilla errors. As an example, a path-independent gate is provided for superconducting circuits with a hardware-efficient design., Comment: 6 pages, 2 figures (+15 pages supplement)

Microwave photons are used to wire up modular quantum processors, but mitigating the effects of loss between modules remains a crucial challenge. We use a low-loss bus resonator to couple bosonic qubits across a superconducting network with protocols made robust to photon loss in the bus. We transfer a multiphoton qubit and track loss events, improving the fidelity to the break-even point with respect to the best uncorrectable encoding. We also demonstrate a entanglement protocol using Hong-Ou-Mandel interference and error detection to prepare a two-photon Bell state with fidelity 94% and success probability 0.79, halving the error obtained with a single photon. This network link also presents new opportunities for resource-efficient direct gates between modules., PRX Quantum, 2 (3), ISSN:2691-3399

A quantum computer has the potential to effciently solve problems that are intractable for classical computers. Constructing a large-scale quantum processor, however, is challenging due to errors and noise inherent in real-world quantum systems. One approach to this challenge is to utilize modularity--a pervasive strategy found throughout nature and engineering--to build complex systems robustly. Such an approach manages complexity and uncertainty by assembling small, specialized components into a larger architecture. These considerations motivate the development of a quantum modular architecture, where separate quantum systems are combined via communication channels into a quantum network. In this architecture, an essential tool for universal quantum computation is the teleportation of an entangling quantum gate, a technique originally proposed in 1999 which, until now, has not been realized deterministically. Here, we experimentally demonstrate a teleported controlled-NOT (CNOT) operation made deterministic by utilizing real-time adaptive control. Additionally, we take a crucial step towards implementing robust, error-correctable modules by enacting the gate between logical qubits, encoding quantum information redundantly in the states of superconducting cavities. Such teleported operations have significant implications for fault-tolerant quantum computation, and when realized within a network can have broad applications in quantum communication, metrology, and simulations. Our results illustrate a compelling approach for implementing multi-qubit operations on logical qubits within an error-protected quantum modular architecture., Comment: Main text: 7 pages, 4 figures; Supplementary Information: 26 pages, 13 figures

The ability to engineer and manipulate different types of quantum mechanical objects allows us to take advantage of their unique properties and create useful hybrid technologies. Thus far, complex quantum states and exquisite quantum control have been demonstrated in systems ranging from trapped ions to superconducting resonators. Recently, there have been many efforts to extend these demonstrations to the motion of complex, macroscopic objects. These mechanical objects have important applications as quantum memories or transducers for measuring and connecting different types of quantum systems. In particular, there have been a few experiments that couple motion to nonlinear quantum objects such as superconducting qubits. This opens up the possibility of creating, storing, and manipulating non-Gaussian quantum states in mechanical degrees of freedom. However, before sophisticated quantum control of mechanical motion can be achieved, we must realize systems with long coherence times while maintaining a sufficient interaction strength. These systems should be implemented in a simple and robust manner that allows for increasing complexity and scalability in the future. Here we experimentally demonstrate a high frequency bulk acoustic wave resonator that is strongly coupled to a superconducting qubit using piezoelectric transduction. In contrast to previous experiments with qubit-mechanical systems, our device requires only simple fabrication methods, extends coherence times to many microseconds, and provides controllable access to a multitude of phonon modes. We use this system to demonstrate basic quantum operations on the coupled qubit-phonon system. Straightforward improvements to the current device will allow for advanced protocols analogous to what has been shown in optical and microwave resonators, resulting in a novel resource for implementing hybrid quantum technologies., 21 pages, 12 figures, including supplementary information

The ability to transfer quantum information from a memory to a flying qubit is important for building quantum networks. The very fast release of a multiphoton state in a microwave cavity memory into propagating modes is demonstrated.

To solve classically hard problems, quantum computers need to be resilient to the influence of noise and decoherence. In such a fault-tolerant quantum computer, noise-induced errors must be detected and corrected in real-time to prevent them from propagating between components. This requirement is especially pertinent while applying quantum gates, when the interaction between components can cause errors to quickly spread throughout the system. However, the large overhead involved in most fault-tolerant architectures makes implementing these systems a daunting task, which motivates the search for hardware-efficient alternatives. Here, we present a gate enacted by a multilevel ancilla transmon on a cavity-encoded logical qubit that is fault-tolerant with respect to decoherence in both the ancilla and the encoded qubit. We maintain the purity of the encoded qubit in the presence of ancilla errors by detecting those errors in real-time, and applying the appropriate corrections. We show a reduction of the logical gate error by a factor of two in the presence of naturally occurring decoherence, and demonstrate resilience against ancilla bit-flips and phase-flips by observing a sixfold suppression of the gate error with increased energy relaxation, and a fourfold suppression with increased dephasing noise. The results demonstrate that bosonic logical qubits can be controlled by error-prone ancilla qubits without inheriting the ancilla's inferior performance. As such, error-corrected ancilla-enabled gates are an important step towards fully fault-tolerant processing of bosonic qubits., 15 pages, 8 figures

Hybrid quantum systems in which acoustic resonators couple to superconducting qubits are promising quantum information platforms. High quality factors and small mode volumes make acoustic modes ideal quantum memories, while the qubit-phonon coupling enables the initialization and manipulation of quantum states. We present a scheme for quantum computing with multimode quantum acoustic systems, and based on this scheme, propose a hardware-efficient implementation of a quantum random access memory (QRAM). Quantum information is stored in high-Q phonon modes, and couplings between modes are engineered by applying off-resonant drives to a transmon qubit. In comparison to existing proposals that involve directly exciting the qubit, this scheme can offer a substantial improvement in gate fidelity for long-lived acoustic modes. We show how these engineered phonon-phonon couplings can be used to access data in superposition according to the state of designated address modes--implementing a QRAM on a single chip., 24 pages, 10 figures; updated to include expanded supplementary material

Quantum physics was invented to account for two fundamental features of measurement results -- their discreetness and randomness. Emblematic of these features is Bohr's idea of quantum jumps between two discrete energy levels of an atom. Experimentally, quantum jumps were first observed in an atomic ion driven by a weak deterministic force while under strong continuous energy measurement. The times at which the discontinuous jump transitions occur are reputed to be fundamentally unpredictable. Can there be, despite the indeterminism of quantum physics, a possibility to know if a quantum jump is about to occur or not? Here, we answer this question affirmatively by experimentally demonstrating that the jump from the ground to an excited state of a superconducting artificial three-level atom can be tracked as it follows a predictable "flight," by monitoring the population of an auxiliary energy level coupled to the ground state. The experimental results demonstrate that the jump evolution when completed is continuous, coherent, and deterministic. Furthermore, exploiting these features and using real-time monitoring and feedback, we catch and reverse a quantum jump mid-flight, thus deterministically preventing its completion. Our results, which agree with theoretical predictions essentially without adjustable parameters, support the modern quantum trajectory theory and provide new ground for the exploration of real-time intervention techniques in the control of quantum systems, such as early detection of error syndromes., Clarified and expanded the introduction and conclusion. Reorganized Methods and Supplementary Information. Revised and expanded citations. Corrected a few remaining minor typos

A striking characteristic of superconducting circuits is that their eigenspectra and intermode coupling strengths are well predicted by simple Hamiltonians representing combinations of quantum circuit elements. Of particular interest is the Cooper-pair-box Hamiltonian used to describe the eigenspectra of transmon qubits, which can depend strongly on the offset-charge difference across the Josephson element. Notably, this offset-charge dependence can also be observed in the dispersive coupling between an ancillary readout mode and a transmon fabricated in the offset-charge-sensitive (OCS) regime. We utilize this effect to achieve direct, high-fidelity dispersive readout of the joint plasmon and charge-parity state of an OCS transmon, which enables efficient detection of charge fluctuations and nonequilibrium-quasiparticle dynamics. Specifically, we show that additional high-frequency filtering can extend the charge-parity lifetime of our device by two orders of magnitude, resulting in a significantly improved energy relaxation time $T_1\sim200~\mu\mathrm{s}$., Comment: 11 pages, 9 figures

Photonic states of superconducting microwave cavities controlled by transmon ancillas provide a platform for encoding and manipulating quantum information. A key challenge in scaling up the platform is the requirement to communicate on demand the information between the cavities. It has been recently demonstrated that a tunable bilinear interaction between two cavities can be realized by coupling them to a bichromatically-driven transmon ancilla, which allows swapping and interfering the multi-photon states of the cavities [Gao et al., Phys. Rev. X 8, 021073(2018)]. Here, we explore both theoretically and experimentally the regime of relatively strong drives on the ancilla needed to achieve fast SWAP gates but which can also lead to undesired non-perturbative effects that lower the SWAP fidelity. We develop a theoretical formalism based on linear response theory that allows one to calculate the rate of ancilla-induced interaction, decay and frequency shift of the cavities in terms of a susceptibility matrix. We treat the drives non-perturbatively using Floquet theory, and find that the interference of the two drives can strongly alter the system dynamics even in the regime where the rotating wave approximation applies. We identify two major sources of infidelity due to ancilla decoherence. i) Ancilla dissipation and dephasing leads to incoherent hopping among Floquet states which occurs even when the ancilla is at zero temperature, resulting in a sudden change of the SWAP rate. ii) The cavities inherit finite decay from the relatively lossy ancilla through the inverse Purcell effect; the effect can be enhanced when the drive-induced AC Stark shift pushes certain ancilla transition frequencies to the vicinity of the cavity frequencies. The theoretical predictions agree quantitatively with the experimental results, paving the way for using the theory to design future experiments., 43 pages, 18 figures. Extended discussions on transmon-induced cavity dephasing in Sec. IVD

The efficient simulation of quantum systems is a primary motivating factor for developing controllable quantum machines. For addressing systems with underlying bosonic structure, it is advantageous to utilize a naturally bosonic platform. Optical photons passing through linear networks may be configured to perform quantum simulation tasks, but the efficient preparation and detection of multiphoton quantum states of light in linear optical systems are challenging. Here, we experimentally implement a boson sampling protocol for simulating molecular vibronic spectra [J. Huh et al., Nat. Photonics 9, 615 (2015)NPAHBY1749-488510.1038/nphoton.2015.153] in a two-mode superconducting device. In addition to enacting the requisite set of Gaussian operations across both modes, we fulfill the scalability requirement by demonstrating, for the first time in any platform, a high-fidelity single-shot photon number resolving detection scheme capable of resolving up to 15 photons per mode. Furthermore, we exercise the capability of synthesizing non-Gaussian input states to simulate spectra of molecular ensembles in vibrational excited states. We show the reprogrammability of our implementation by extracting the spectra of photoelectron processes in H_{2}O, O_{3}, NO_{2}, and SO_{2}. The capabilities highlighted in this work establish the superconducting architecture as a promising platform for bosonic simulations, and by combining them with tools such as Kerr interactions and engineered dissipation, enable the simulation of a wider class of bosonic systems.

Nanofabrication techniques for superconducting qubits rely on resist-based masks patterned by electron-beam or optical lithography. We have developed an alternative nanofabrication technique based on free-standing silicon shadow masks fabricated from silicon-on-insulator wafers. These silicon shadow masks not only eliminate organic residues associated with resist-based lithography, but also provide a pathway to better understand and control surface-dielectric losses in superconducting qubits by decoupling mask fabrication from substrate preparation. We have successfully fabricated aluminum 3D transmon superconducting qubits with these shadow masks and found coherence quality factors comparable to those fabricated with standard techniques., Comment: 6 pages, 6 figures, additional experimental data - new figure, typos corrected, supplementary material (2 pages, 1 figure, 1 table)

Low-loss cavities are important in building high-coherence superconducting quantum computers. Generating high quality joints between parts is crucial to the realization of a scalable quantum computer using the circuit quantum electrodynamics (cQED) framework. In this paper, we adapt the technique of indium bump bonding to the cQED architecture to realize high quality superconducting microwave joints between chips. We use this technique to fabricate compact superconducting cavities in the multilayer microwave integrated quantum circuits (MMIQC) architecture and achieve single photon quality factor over 300 million or single-photon lifetimes approaching 5 ms. To quantify the performance of the resulting seam, we fabricate microwave stripline resonators in multiple sections connected by different numbers of bonds, resulting in a wide range of seam admittances. The measured quality factors combined with the designed seam admittances allow us to bound the conductance of the seam at $g_\text{seam} \ge 2\times 10^{10} /(\Omega \text{m})$. Such a conductance should enable construction of micromachined superconducting cavities with quality factor of at least a billion. These results demonstrate the capability to construct very high quality microwave structures within the MMIQC architecture.

High-fidelity qubit measurements play a crucial role in quantum computation, communication, and metrology. In recent experiments, it has been shown that readout fidelity may be improved by performing repeated quantum non-demolition (QND) readouts of a qubit's state through an ancilla. For a qubit encoded in a two-level system, the fidelity of such schemes is limited by the fact that a single error can destroy the information in the qubit. On the other hand, if a bosonic system is used, this fundamental limit could be overcome by utilizing higher levels such that a single error still leaves states distinguishable. In this work, we present a robust readout scheme, applicable to bosonic systems dispersively coupled to an ancilla, which leverages both repeated QND readouts and higher-level encodings to asymptotically suppress the effects of qubit/cavity relaxation and individual measurement infidelity. We calculate the measurement fidelity in terms of general experimental parameters, provide an information-theoretic description of the scheme, and describe its application to several encodings, including cat and binomial codes., 15 pages, 8 figures

Long-lived, high-frequency phonons are valuable for applications ranging from optomechanics to emerging quantum systems. For scientific as well as technological impact, we seek high-performance oscillators that offer a path toward chip-scale integration. Confocal bulk acoustic wave resonators have demonstrated an immense potential to support long-lived phonon modes in crystalline media at cryogenic temperatures. So far, these devices have been macroscopic with cm-scale dimensions. However, as we push these oscillators to high frequencies, we have an opportunity to radically reduce the footprint as a basis for classical and emerging quantum technologies. In this paper, we present novel design principles and simple microfabrication techniques to create high performance chip-scale confocal bulk acoustic wave resonators in a wide array of crystalline materials. We tailor the acoustic modes of such resonators to efficiently couple to light, permitting us to perform a non-invasive laser-based phonon spectroscopy. Using this technique, we demonstrate an acoustic Q-factor of 2.8 × 107 (6.5 × 106) for chip-scale resonators operating at 12.7 GHz (37.8 GHz) in crystalline z-cut quartz (x-cut silicon) at cryogenic temperatures., APL Photonics, 3 (6), ISSN:2378-0967

Quantum states of mechanical motion can be important resources for quantum information, metrology, and studies of fundamental physics. Recent demonstrations of superconducting qubits coupled to acoustic resonators have opened up the possibility of performing quantum operations on macroscopic motional modes, which can act as long-lived quantum memories or transducers. In addition, they can potentially be used to test for novel decoherence mechanisms in macroscopic objects and other modifications to standard quantum theory. Many of these applications call for the ability to create and characterize complex quantum states, putting demanding requirements on the speed of quantum operations and the coherence of the mechanical mode. In this work, we demonstrate the controlled generation of multi-phonon Fock states in a macroscopic bulk-acoustic wave resonator. We also perform Wigner tomography and state reconstruction to highlight the quantum nature of the prepared states. These demonstrations are made possible by the long coherence times of our acoustic resonator and our ability to selectively couple to individual phonon modes. Our work shows that circuit quantum acousto-dynamics (circuit QAD) enables sophisticated quantum control of macroscopic mechanical objects and opens the door to using acoustic modes as novel quantum resources., Includes main text and supplementary information

Entangling gates between qubits are a crucial component for performing algorithms in quantum computers. However, any quantum algorithm must ultimately operate on error-protected logical qubits encoded in high-dimensional systems. Typically, logical qubits are encoded in multiple two-level systems, but entangling gates operating on such qubits are highly complex and have not yet been demonstrated. Here, we realize a controlled NOT (CNOT) gate between two multiphoton qubits in two microwave cavities. In this approach, we encode a qubit in the high-dimensional space of a single cavity mode, rather than in multiple two-level systems. We couple two such encoded qubits together through a transmon, which is driven by an RF pump to apply the gate within 190 ns. This is two orders of magnitude shorter than the decoherence time of the transmon, enabling a high-fidelity gate operation. These results are an important step towards universal algorithms on error-corrected logical qubits., Comment: 10 pages, 11 figures (incl. Supplementary Information)

Interference experiments provide a simple yet powerful tool to unravel fundamental features of quantum physics. Here we engineer an RF-driven, time-dependent bilinear coupling that can be tuned to implement a robust 50:50 beamsplitter between stationary states stored in two superconducting cavities in a three-dimensional architecture. With this, we realize high contrast Hong-Ou- Mandel (HOM) interference between two spectrally-detuned stationary modes. We demonstrate that this coupling provides an efficient method for measuring the quantum state overlap between arbitrary states of the two cavities. Finally, we showcase concatenated beamsplitters and differential phase shifters to implement cascaded Mach-Zehnder interferometers, which can control the signature of the two-photon interference on-demand. Our results pave the way toward implementation of scalable boson sampling, the application of linear optical quantum computing (LOQC) protocols in the microwave domain, and quantum algorithms between long-lived bosonic memories., Comment: 6 pages, 5 figures

Through engineerable optomechanical coupling to bulk acoustic modes of plano-convex resonators fabricated on-chip, we demonstrate ultra-high acoustic Q-factors of 28 million (6.6 million) for phonons at high frequencies of 12.7 GHz (37.8 GHz) in quartz (silicon) at cryogenic temperatures.

International audience; Quantum error correction can allow quantum computers to operate despite the presence of noise and imperfections. A critical component of any error correcting scheme is the mapping of error syndromes onto an ancillary measurement system. However, errors occurring in the ancilla can propagate onto the logical qubit, and irreversibly corrupt the encoded information. Here, we demonstrate a fault-tolerant syndrome measurement scheme that dramatically suppresses forward propagation of ancilla errors. We achieve a fivefold reduction of the logical error probability per measurement, while maintaining the syndrome assignment fidelity. We use the same method to prevent the propagation of thermal ancilla excitations, increasing the logical qubit dephasing time by more than an order of magnitude. Our approach is hardware-efficient, as it uses a single multilevel transmon ancilla and a cavity-encoded logical qubit, whose interaction is engineered in situ using an off-resonant sideband drive. These results demonstrate that hardware-efficient approaches which exploit system-specific error models can yield practical advances towards fault-tolerant quantum computation.

Modular quantum computing architectures require fast and efficient distribution of quantum information through propagating signals. Here we report rapid, on-demand quantum state transfer between two remote superconducting cavity quantum memories through traveling microwave photons. We demonstrate a quantum communication channel by deterministic transfer of quantum bits with 76% fidelity. Heralding on errors induced by experimental imperfection can improve this to 87% with a success probability of 0.87. By partial transfer of a microwave photon, we generate remote entanglement at a rate that exceeds photon loss in either memory by more than a factor of three. We further show the transfer of quantum error correction code words that will allow deterministic mitigation of photon loss. These results pave the way for scaling superconducting quantum devices through modular quantum networks., main text 7 pages, 4 figures; supplement 16 pages, 16 figures

A logical qubit is a two-dimensional subspace of a higher dimensional system, chosen such that it is possible to detect and correct the occurrence of certain errors. Manipulation of the encoded information generally requires arbitrary and precise control over the entire system. Whether based on multiple physical qubits or larger dimensional modes such as oscillators, the individual elements in realistic devices will always have residual interactions, which must be accounted for when designing logical operations. Here we demonstrate a holistic control strategy which exploits accurate knowledge of the Hamiltonian to manipulate a coupled oscillator-transmon system. We use this approach to realize high-fidelity (98.5%, inferred), decoherence-limited operations on a logical qubit encoded in a superconducting cavity resonator using four-component cat states. Our results show the power of applying numerical techniques to control linear oscillators and pave the way for utilizing their large Hilbert space as a resource in quantum information processing., A logical qubit is a two-dimensional subspace of a higher dimensional system, whose manipulation requires precise control over the whole system. Here the authors demonstrate a control strategy which exploits precise knowledge of the Hamiltonian to manipulate a coupled oscillator-transmon system.

Controlling quasiparticle dynamics can improve the performance of superconducting devices. For example, it has been demonstrated effective in increasing lifetime and stability of superconducting qubits. Here we study how to optimize the placement of normal-metal traps in transmon-type qubits. When the trap size increases beyond a certain characteristic length, the details of the geometry and trap position, and even the number of traps, become important. We discuss for some experimentally relevant examples how to shorten the decay time of the excess quasiparticle density. Moreover, we show that a trap in the vicinity of a Josephson junction can reduce the steady-state quasiparticle density near that junction, thus suppressing the quasiparticle-induced relaxation rate of the qubit. Such a trap also reduces the impact of fluctuations in the generation rate of quasiparticles, rendering the qubit more stable., 16 pages, 7 figures; to appear in Phys. Rev. Applied

Large-scale quantum information processing networks will most probably require the entanglement of distant systems that do not interact directly. This can be done by performing entangling gates between standing information carriers, used as memories or local computational resources, and flying ones, acting as quantum buses. We report the deterministic entanglement of two remote transmon qubits by Raman stimulated emission and absorption of a traveling photon wavepacket. We achieve a Bell state fidelity of 73 %, well explained by losses in the transmission line and decoherence of each qubit.

We convert propagating qubits encoded as superpositions of zero and one photons to the motion of a micrometer-sized mechanical resonator. Using quantum state tomography, we determine the density matrix of both the propagating photons and the mechanical resonator. By comparing a sufficient set of states before and after conversion, we determine the average process fidelity to be $F_{\textrm{avg}} = 0.83\substack{+0.03-0.06}$ which exceeds the classical bound for the conversion of an arbitrary qubit state. This conversion ability is necessary for using mechanical resonators in emerging quantum communication and modular quantum computation architectures., Comment: 22 pages, 14 figures (including Supplementary Information)

The long-predicted suppression of quasiparticle dissipation in a Josephson junction when the phase difference across the junction is π is inferred from a sharp maximum in the energy relaxation time of a superconducting artificial atom. Josephson junctions, which consist of two superconductors connected by a weak link, have a central role in quantum electronic applications, such as in sensitive magnetic field detectors, high-speed processing and quantum information networks. However, a fundamental prediction concerning the Josephson effect has not yet been confirmed. It is known that the current flowing through a Josephson junction is made up from superconducting Cooper pairs as well as excitations called quasiparticles, which contribute in a few different ways. One contribution causes dissipation but can in theory be suppressed by tuning the phase difference between the superconductors. This has been achieved experimentally. Ioan Pop et al. have made a qubit comprising a Josephson junction. The energy relaxation time of this qubit increases by almost two orders of magnitude owing to the suppression of quasiparticle dissipation. This finding confirms the existence of a fundamental quantum phenomenon predicted over 50 years ago. Owing to the low-loss propagation of electromagnetic signals in superconductors, Josephson junctions constitute ideal building blocks for quantum memories, amplifiers, detectors and high-speed processing units, operating over a wide band of microwave frequencies. Nevertheless, although transport in superconducting wires is perfectly lossless for direct current, transport of radio-frequency signals can be dissipative in the presence of quasiparticle excitations above the superconducting gap1. Moreover, the exact mechanism of this dissipation in Josephson junctions has never been fully resolved experimentally. In particular, Josephson’s key theoretical prediction that quasiparticle dissipation should vanish in transport through a junction when the phase difference across the junction is π (ref. 2) has never been observed3. This subtle effect can be understood as resulting from the destructive interference of two separate dissipative channels involving electron-like and hole-like quasiparticles. Here we report the experimental observation of this quantum coherent suppression of quasiparticle dissipation across a Josephson junction. As the average phase bias across the junction is swept through π, we measure an increase of more than one order of magnitude in the energy relaxation time of a superconducting artificial atom. This striking suppression of dissipation, despite the presence of lossy quasiparticle excitations above the superconducting gap, provides a powerful tool for minimizing decoherence in quantum electronic systems and could be directly exploited in quantum information experiments with superconducting quantum bits.

We propose a systematic procedure to optimize quantum state tomography protocols for continuous variable systems based on excitation counting preceded by a displacement operation. Compared with conventional tomography based on Husimi or Wigner function measurement, the excitation counting approach can significantly reduce the number of measurement settings. We investigate both informational completeness and robustness, and provide a bound of reconstruction error involving the condition number of the sensing map. We also identify the measurement settings that optimize this error bound, and demonstrate that the improved reconstruction robustness can lead to an order of magnitude reduction of estimation error with given resources. This optimization procedure is general and can incorporate prior information of the unknown state to further simplify the protocol., 8 pages of main text, 8 figures, 20 pages including supplementary material

Entangling two remote quantum systems which never interact directly is an essential primitive in quantum information science and forms the basis for the modular architecture of quantum computing. When protocols to generate these remote entangled pairs rely on using traveling single photon states as carriers of quantum information, they can be made robust to photon losses, unlike schemes that rely on continuous variable states. However, efficiently detecting single photons is challenging in the domain of superconducting quantum circuits because of the low energy of microwave quanta. Here, we report the realization of a robust form of concurrent remote entanglement based on a novel microwave photon detector implemented in the superconducting circuit quantum electrodynamics (cQED) platform of quantum information. Remote entangled pairs with a fidelity of $0.57\pm0.01$ are generated at $200$ Hz. Our experiment opens the way for the implementation of the modular architecture of quantum computation with superconducting qubits., Main paper: 7 pages, 4 figures; Appendices: 14 pages, 9 figures

There are two general requirements to harness the computational power of quantum mechanics: the ability to manipulate the evolution of an isolated system and the ability to faithfully extract information from it. Quantum error correction and simulation often make a more exacting demand: the ability to perform non-destructive measurements of specific correlations within that system. We realize such measurements by employing a protocol adapted from [S. Nigg and S. M. Girvin, Phys. Rev. Lett. 110, 243604 (2013)], enabling real-time selection of arbitrary register-wide Pauli operators. Our implementation consists of a simple circuit quantum electrodynamics (cQED) module of four highly-coherent 3D transmon qubits, collectively coupled to a high-Q superconducting microwave cavity. As a demonstration, we enact all seven nontrivial subset-parity measurements on our three-qubit register. For each we fully characterize the realized measurement by analyzing the detector (observable operators) via quantum detector tomography and by analyzing the quantum back-action via conditioned process tomography. No single quantity completely encapsulates the performance of a measurement, and standard figures of merit have not yet emerged. Accordingly, we consider several new fidelity measures for both the detector and the complete measurement process. We measure all of these quantities and report high fidelities, indicating that we are measuring the desired quantities precisely and that the measurements are highly non-demolition. We further show that both results are improved significantly by an additional error-heralding measurement. The analyses presented here form a useful basis for the future characterization and validation of quantum measurements, anticipating the demands of emerging quantum technologies., 10 pages, 5 figures, plus supplement

Quantum computation is a completely new and different paradigm for how to store and process information. It offers the possibility of exponential computational advantage for certain types of hard problems, but the hardware for implementing quantum algorithms is still at an early stage. The basic idea of quantum computing is to replace the ordinary binary bits of conventional computing with their quantum equivalent, or qubit. A qubit can be any quantum system whose states can be prepared and controlled, provided that we can then fabricate them in large quantities and then couple them to one another in prescribed ways. While there are several candidate technologies for building a quantum computer, one of the most promising is superconducting quantum circuits, operated at cryogenic temperatures approaching 0.01 K.

We demonstrate and contrast two approaches to the stabilization of qubit entanglement by feedback. Our demonstration is built on a feedback platform consisting of two superconducting qubits coupled to a cavity which are measured by a nearly-quantum-limited measurement chain and controlled by high-speed classical logic circuits. This platform is used to stabilize entanglement by two nominally distinct schemes: a "passive" reservoir engineering method and an "active" correction based on conditional parity measurements. In view of the instrumental roles that these two feedback paradigms play in quantum error-correction and quantum control, we directly compare them on the same experimental setup. Further, we show that a second layer of feedback can be added to each of these schemes, which heralds the presence of a high-fidelity entangled state in realtime. This "nested" feedback brings about a marked entanglement fidelity improvement without sacrificing success probability., Comment: 40 pages, 12 figures

As experimental quantum information processing (QIP) rapidly advances, an emerging challenge is to design a scalable architecture that combines various quantum elements into a complex device without compromising their performance. In particular, superconducting quantum circuits have successfully demonstrated many of the requirements for quantum computing, including coherence levels that approach the thresholds for scaling. However, it remains challenging to couple a large number of circuit components through controllable channels while suppressing any other interactions. We propose a hardware platform intended to address these challenges, which combines the advantages of integrated circuit fabrication and long coherence times achievable in three-dimensional circuit quantum electrodynamics (3D cQED). This multilayer microwave integrated quantum circuit (MMIQC) platform provides a path toward the realization of increasingly complex superconducting devices in pursuit of a scalable quantum computer., Comment: 5 pages, 2 figures

Quantum error correction (QEC) can overcome the errors experienced by qubits1 and is therefore an essential component of a future quantum computer. To implement QEC, a qubit is redundantly encoded in a higher-dimensional space using quantum states with carefully tailored symmetry properties. Projective measurements of these parity-type observables provide error syndrome information, with which errors can be corrected via simple operations2. The ‘break-even’ point of QEC—at which the lifetime of a qubit exceeds the lifetime of the constituents of the system—has so far remained out of reach3. Although previous works have demonstrated elements of QEC4–16, they primarily illustrate the signatures or scaling properties of QEC codes rather than test the capacity of the system to preserve a qubit over time. Here we demonstrate a QEC system that reaches the break-even point by suppressing the natural errors due to energy loss for a qubit logically encoded in superpositions of Schrodinger-cat states17 of a superconducting resonator18–21. We implement a full QEC protocol by using real-time feedback to encode, monitor naturally occurring errors, decode and correct. As measured by full process tomography, without any post-selection, the corrected qubit lifetime is 320 microseconds, which is longer than the lifetime of any of the parts of the system: 20 times longer than the lifetime of the transmon, about 2.2 times longer than the lifetime of an uncorrected logical encoding and about 1.1 longer than the lifetime of the best physical qubit (the |0〉f and |1〉f Fock states of the resonator). Our results illustrate the benefit of using hardware-efficient qubit encodings rather than traditional QEC schemes. Furthermore, they advance the field of experimental error correction from confirming basic concepts to exploring the metrics that drive system performance and the challenges in realizing a fault-tolerant system.

Universal computation of a quantum system consisting of superpositions of well-separated coherent states of multiple harmonic oscillators can be achieved by three families of adiabatic holonomic gates. The first gate consists of moving a coherent state around a closed path in phase space, resulting in a relative Berry phase between that state and the other states. The second gate consists of "colliding" two coherent states of the same oscillator, resulting in coherent population transfer between them. The third gate is an effective controlled-phase gate on coherent states of two different oscillators. Such gates should be realizable via reservoir engineering of systems which support tunable nonlinearities, such as trapped ions and circuit QED., 6 pages, 3 figures + 4 page supplement; (v3 scheme now universal; v4 title changed to match published version); gate animations available at http://dancingblobs.krastanov.org/

We present a method for calculating the energy levels of superconducting circuits that contain highly anharmonic, inductively-shunted modes with arbitrarily strong coupling. Our method starts by calculating the normal modes of the linearized circuit and proceeds with numerical diagonalization in this basis. As an example, we analyze the Hamiltonian of a fluxonium qubit inductively coupled to a readout resonator. While elementary, this simple example is nontrivial because it cannot be efficiently treated by the method known as "black-box quantization," numerical diagonalization in the bare harmonic oscillator basis, or perturbation theory. Calculated spectra are compared to measured spectroscopy data, demonstrating excellent quantitative agreement between theory and experiment., Comment: 10 pages, 7 figures

Quantum jumps of a qubit are usually observed between its energy eigenstates, also known as its longitudinal pseudo-spin component. Is it possible, instead, to observe quantum jumps between the transverse superpositions of these eigenstates? We answer positively by presenting the first continuous quantum nondemolition measurement of the transverse component of an individual qubit. In a circuit QED system irradiated by two pump tones, we engineer an effective Hamiltonian whose eigenstates are the transverse qubit states, and a dispersive measurement of the corresponding operator. Such transverse component measurements are a useful tool in the driven-dissipative operation engineering toolbox, which is central to quantum simulation and quantum error correction., Comment: 6 pages, 4 figures and an additional supplementary material

Quantum channels can describe all transformations allowed by quantum mechanics. We provide an explicit universal protocol to construct all possible quantum channels, using a single qubit ancilla with quantum non-demolition readout and adaptive control. Our construction is efficient in both physical resources and circuit depth, and can be demonstrated using superconducting circuits and various other physical platforms. There are many applications of quantum channel construction, including system stabilization and quantum error correction, Markovian and exotic channel simulation, implementation of generalized quantum measurements and more general quantum instruments. Efficient construction of arbitrary quantum channels opens up exciting new possibilities for quantum control, quantum sensing and information processing tasks., Comment: 16 pages, 8 figures

We present a device demonstrating a lithographically patterned transmon integrated with a micromachined cavity resonator. Our two-cavity, one-qubit device is a multilayer microwave integrated quantum circuit (MMIQC), comprising a basic unit capable of performing circuit-QED (cQED) operations. We describe the qubit-cavity coupling mechanism of a specialized geometry using an electric field picture and a circuit model, and finally obtain specific system parameters using simulations. Fabrication of the MMIQC includes lithography, etching, and metallic bonding of silicon wafers. Superconducting wafer bonding is a critical capability that is demonstrated by a micromachined storage cavity lifetime $34.3~\mathrm{\mu s}$, corresponding to a quality factor of 2 million at single-photon energies. The transmon coherence times are $T_1=6.4~\mathrm{\mu s}$, and $T_2^{Echo}= 11.7~\mathrm{\mu s}$. We measure qubit-cavity dispersive coupling with rate $\chi_{q\mu}/2\pi=-1.17~$MHz, constituting a Jaynes-Cummings system with an interaction strength $g/2\pi=49~$MHz. With these parameters we are able to demonstrate cQED operations in the strong dispersive regime with ease. Finally, we highlight several improvements and anticipated extensions of the technology to complex MMIQCs.

Thorough control of quantum measurement is key to the development of quantum information technologies. Many measurements are destructive, removing more information from the system than they obtain. Quantum non-demolition (QND) measurements allow repeated measurements that give the same eigenvalue. They could be used for several quantum information processing tasks such as error correction, preparation by measurement, and one-way quantum computing. Achieving QND measurements of photons is especially challenging because the detector must be completely transparent to the photons while still acquiring information about them. Recent progress in manipulating microwave photons in superconducting circuits has increased demand for a QND detector which operates in the gigahertz frequency range. Here we demonstrate a QND detection scheme which measures the number of photons inside a high quality-factor microwave cavity on a chip. This scheme maps a photon number onto a qubit state in a single-shot via qubit-photon logic gates. We verify the operation of the device by analyzing the average correlations of repeated measurements, and show that it is 90% QND. It differs from previously reported detectors because its sensitivity is strongly selective to chosen photon number states. This scheme could be used to monitor the state of a photon-based memory in a quantum computer., 5 pages, 4 figures, includes supplementary material

The processing of the single-quantum-level signals produced by current nanoscale solid-state devices such as qubits and nanomechanical resonators would require the development of very sensitive active circuits, such as amplifiers or frequency up- and down-converters that could attain the ultimate performances limited by the laws of quantum mechanics, while remaining of practical use. Bergeal et al. now demonstrate a phase-preserving, superconducting parametric amplifier with ultra-low noise properties, following theoretical principles recently presented in Nature Physics ( http://go.nature.com/F7lwR2 ). Based on a Josephson ring modulator, the new device can operate within a factor of three of the quantum limit. Possible applications include quantum analog signal processing such as the production of entangled microwave signal pairs. Recent progress in solid-state quantum information processing has stimulated the search for amplifiers and frequency converters with quantum-limited performance in the microwave range. Here, a phase-preserving, superconducting parametric amplifier with ultra-low-noise properties has been experimentally realized. Recent progress in solid-state quantum information processing1 has stimulated the search for amplifiers and frequency converters with quantum-limited performance in the microwave range. Depending on the gain applied to the quadratures of a single spatial and temporal mode of the electromagnetic field, linear amplifiers can be classified into two categories (phase sensitive and phase preserving) with fundamentally different noise properties2. Phase-sensitive amplifiers use squeezing to reduce the quantum noise, but are useful only in cases in which a reference phase is attached to the signal, such as in homodyne detection. A phase-preserving amplifier would be preferable in many applications, but such devices have not been available until now. Here we experimentally realize a proposal3 for an intrinsically phase-preserving, superconducting parametric amplifier of non-degenerate type. It is based on a Josephson ring modulator, which consists of four Josephson junctions in a Wheatstone bridge configuration. The device symmetry greatly enhances the purity of the amplification process and simplifies both its operation and its analysis. The measured characteristics of the amplifier in terms of gain and bandwidth are in good agreement with analytical predictions. Using a newly developed noise source, we show that the upper bound on the total system noise of our device under real operating conditions is three times the quantum limit. We foresee applications in the area of quantum analog signal processing, such as quantum non-demolition single-shot readout of qubits4, quantum feedback5 and the production of entangled microwave signal pairs6.

Amplifiers are crucial in every experiment carrying out a very sensitive measurement. However, they always degrade the information by adding noise. Quantum mechanics puts a limit on how small this degradation can be. Theoretically, the minimum noise energy added by a phase-preserving amplifier to the signal it processes amounts at least to half a photon at the signal frequency. Here we propose a practical microwave device that can fulfil the minimal requirements to reach the quantum limit. The availability of such a device is of importance for the readout of solid-state qubits, and more generally for the measurement of very weak signals in various areas of science. We discuss how this device can be the basic building block for a variety of practical applications, such as amplification, noiseless frequency conversion, dynamic cooling and production of entangled signal pairs. The minimum noise energy that a phase-preserving amplifier adds to the signal is fundamentally limited to half a photon. A proposed parametric amplifier based on Josephson junctions should be able to reach this limit at microwave frequencies.

The emerging field of circuit quantum electrodynamics could pave the way for the design of practical quantum computers.

In quantum information science, the phase of a wavefunction plays an important role in encoding information. While most experiments in this field rely on dynamic effects to manipulate this information, an alternative approach is to use geometric phase, which has been argued to have potential fault tolerance. We demonstrate the controlled accumulation of a geometric phase, Berry's phase, in a superconducting qubit, manipulating the qubit geometrically using microwave radiation, and observing the accumulated phase in an interference experiment. We find excellent agreement with Berry's predictions, and also observe a geometry dependent contribution to dephasing., 5 pages, 4 figures, version with high resolution figures available at http://qudev.ethz.ch/content/science/PubsPapers.html