low-weight operations with an ancilla to extract information about errors without causing backaction on the encoded system. Essentially, ancilla errors must not propagate to the encodedsystem and induce errors beyond those which can be corrected. The current schemes for achieving this fault-tolerance to ancilla errors come at the cost of increased overhead requirements. An efficient way to extract error syndromes in a fault-tolerant manner is by using a single ancilla with strongly biased noise channel. Typically, however, required elementary operations can become challenging when the noise is extremely biased. We propose to overcome this shortcoming by using a bosonic-cat ancilla in a parametrically driven nonlinear cavity. Such a cat-qubit experiences only bit-flip noise and is stabilized against phase-flips. To highlight the flexibility of this approach, we illustrate the syndrome extraction process in a variety of codes such as qubit-based toric codes, bosonic cat- and Gottesman-Kitaev-Preskill (GKP) codes. Our results open a path for realizing hardware-efficient, fault-tolerant error syndrome extraction.

The realization of robust universal quantum computation with any platform ultimately requires both the coherent storage of quantum information and (at least) one entangling operationbetween 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 [UE(θc)] can coherently transfer the states between two bosonic modes, regardless of the chosen encoding, realizing a deterministic entangling operation for certain θc. Here, we develop an efficient circuit to implement UE(θc) 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.

Radio Frequency driven Josephson circuits provide a rich platform to engineer a variety of nonlinear Hamiltonians for superconducting quantum circuits. While Josephson junctions mediatestrong interactions between microwave photons, some particular types of interaction Hamiltonians can only be obtained through the application of microwave drives (pumps) at well-chosen frequencies. For various applications, it is important to increase the pump strength without introducing undesired couplings and interferences that limit the fidelity of the operations. In this Letter, we analyze these limitations through the theoretical study of the steady state behavior of the driven-dissipative systems. Our general analysis, based on the Floquet-Markov theory, indicates that the ubiquitous circuit consisting of a transmon coupled to a harmonic oscillator suffers from strong limitations in this regard. In accordance with a parallel experimental study, we find that above a fairly low critical pump power the transmon state escapes the Josephson potential confinement and is sent to a statistical mixture of free-particle like states. Next, we illustrate that by diluting the non-linearity of the Josephson junction through a parallel inductive shunt, the picture changes significantly and one achieves very large dynamic ranges in the pump power. This theoretical study provides the ground for drastic modifications in Josephson circuit designs to be used in parametric Hamiltonian engineering experiments.

Strong microwave drives, referred to as pumps, are widely applied to superconducting circuits incorporating Josephson junctions in order to induce couplings between electromagneticmodes. This offers a variety of applications, from quantum-limited amplification, to quantum state and manifold stabilization. These couplings scale with the pump power, therefore, seeking stronger couplings requires a detailed understanding of the behavior of such circuits in the presence of stronger pumps. In this work, we probe the dynamics of a transmon qubit in a 3D cavity, for various pump powers and frequencies. For all pump frequencies, we find a critical pump power above which the transmon is driven into highly excited states, beyond the first seven states which we individually resolve through cavity spectroscopy. This observation is compatible with our theory describing the escape of the transmon state out of its Josephson potential well, into states resembling those of a free particle which does not induce any non-linear couplings.

We introduce a driven-dissipative two-mode bosonic system whose reservoir causes simultaneous loss of two photons in each mode and whose steady states are superpositions of pair-coherent/Barut-Girardellocoherent states. We show how quantum information encoded in a steady-state subspace of this system is exponentially immune to phase drifts (cavity dephasing) in both modes. Additionally, it is possible to protect information from arbitrary photon loss in either (but not simultaneously both) of the modes by continuously monitoring the difference between the expected photon numbers of the logical states. Despite employing more resources, the two-mode scheme enjoys two advantages over its one-mode counterpart with regards to implementation using current circuit QED technology. First, monitoring the photon number difference can be done without turning off the currently implementable dissipative stabilizing process. Second, a lower average photon number per mode is required to enjoy a level of protection at least as good as that of the cat-codes. We discuss circuit QED proposals to stabilize the code states, perform gates, and protect against photon loss via either active syndrome measurement or an autonomous procedure. We introduce quasiprobability distributions allowing us to represent two-mode states of fixed photon number difference in a two-dimensional complex plane, instead of the full four-dimensional two-mode phase space. The two-mode codes are generalized to multiple modes in an extension of the stabilizer formalism to non-diagonalizable stabilizers. The M-mode codes can protect against either arbitrary photon losses in up to M−1 modes or arbitrary losses or gains in any one mode.

Encoding quantum states in complex multiphoton fields can overcome loss during signal transmission in a quantum network. Transmitting quantum information encoded in this way requiresthat locally stored states can be converted to propagating fields. Here we experimentally show the controlled conversion of multiphoton quantum states, like „Schr\“odinger cat“ states, from a microwave cavity quantum memory into propagating modes. By parametric conversion using the nonlinearity of a single Josephson junction, we can release the cavity state in ~500 ns, about 3 orders of magnitude faster than its intrinsic lifetime. This `catapult‘ faithfully converts arbitrary cavity fields to traveling signals with an estimated efficiency of > 90%, enabling on-demand generation of complex itinerant quantum states. Importantly, the release process can be controlled precisely on fast time scales, allowing us to generate entanglement between the cavity and the traveling mode by partial conversion. Our system can serve as the backbone of a microwave quantum network, paving the way towards error-correctable distribution of quantum information and the transfer of highly non-classical states to hybrid quantum systems.

Quantum channels can describe all transformations allowed by quantum mechanics. We provide an explicit universal protocol to construct all possible quantum channels, using a singlequbit 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.

A central requirement for any quantum error correction scheme is the ability to perform quantum non-demolition measurements of an error syndrome, corresponding to a special symmetryproperty of the encoding scheme. It is in particular important that such a measurement does not introduce extra error mechanisms, not included in the error model of the correction scheme. In this letter, we ensure such a robustness by designing an interaction with a measurement device that preserves the degeneracy of the measured observable. More precisely, we propose a scheme to perform continuous and quantum non-demolition measurement of photon-number parity in a microwave cavity. This corresponds to the error syndrome in a class of error correcting codes called the cat-codes, which have recently proven to be efficient and versatile for quantum information processing. In our design, we exploit the strongly nonlinear Hamiltonian of a high-impedance Josephson circuit, coupling a high-Q cavity storage cavity mode to a low-Q readout one. By driving the readout resonator at its resonance, the phase of the reflected/transmitted signal carries directly exploitable information on parity-type observables for encoded cat-qubits of the high-Q mode.

The article is a short opinionated review of the quantum treatment of electromagnetic circuits, with no pretension to exhaustiveness. This review, which is an updated and modernizedversion of a previous set of Les Houches School lecture notes, has 3 main parts. The first part describes how to construct a Hamiltonian for a general circuit, which can include dissipative elements. The second part describes the quantization of the circuit, with an emphasis on the quantum treatment of dissipation. The final part focuses on the Josephson non-linear element and the main linear building blocks from which superconducting circuits are assembled. It also includes a brief review of the main types of superconducting artificial atoms, elementary multi-level quantum systems made from basic circuit elements.

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 ofthe 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 (99%, 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.