Linear parametric amplification is a key operation in information processing. Our interest here is quantum-limited parametric amplification, i.e., amplification of quantum signals whileadding the minimum amount of noise allowed by quantum mechanics, which is essential for any viable implementation of quantum information processing. We describe parametric amplifiers based on the dispersive nonlinearity of Josephson junctions driven with appropriate tones playing the role of pumps. We discuss two defining characteristics in the architecture of these amplifiers: the number of modes occupied by the signal, idler and pump waves and the number of independent ports through which these waves enter into the circuit. We discuss scattering properties of these amplifiers. This is followed by computations of the dynamic range and phase-space distributions of the fluctuations of the modes of the amplifiers.

Entangling gates between qubits are a crucial component for performing algorithms in quantum computers. However, any quantum algorithm will ultimately have to operate on error-protectedlogical qubits, which are effective qubits encoded in a high-dimensional Hilbert space. A common approach is to encode logical qubits in collective states of multiple two-level systems, but algorithms operating on multiple logical qubits are highly complex and have not yet been demonstrated. Here, we experimentally realize a controlled NOT (CNOT) gate between two multiphoton qubits in two microwave cavities. In this approach, we encode a qubit in the large Hilbert 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 with an RF pump to apply the CNOT gate within 190 ns. This is two orders of magnitude shorter than the decoherence time of any part of the system, enabling high-fidelity operations comparable to state-of-the-art gates between two-level systems. These results are an important step towards universal algorithms on error-corrected logical qubits.

This short and opinionated review starts with a concept of quantum signals at microwave frequencies and focuses on the principle of linear parametric amplification. The amplificationprocess arises from the dispersive nonlinearity of Josephson junctions driven with appropriate tones. We discuss two defining characteristics of these amplifiers: the number of modes receiving the signal, idler and pump waves and the number of independent ports through which these waves enter into the circuit.

Concurrent remote entanglement of distant, non-interacting quantum entities is a crucial function for quantum information processing. In contrast with the existing protocols which employaddition of signals to generate entanglement between two remote qubits, the protocol we present is based on multiplication of signals. This protocol can be straightforwardly implemented by a novel Josephson junction mixing circuit. Our scheme would be able to generate provable entanglement even in presence of practical imperfections: finite quantum efficiency of detectors and undesired photon loss in current state-of-the-art devices.

While dissipation is widely considered as being harmful for quantum coherence, it can, when properly engineered, lead to the stabilization of non-trivial pure quantum states. We proposea scheme for continuous generation and stabilization of Schr\“{o}dinger cat states in a cavity using dissipation engineering. We first generate non-classical photon states with definite parity by means of a two-photon drive and dissipation, and then stabilize these transient states against single-photon decay. The single-photon stabilization is autonomous, and is implemented through a second engineered bath, which exploits the photon number dependent frequency-splitting due to Kerr interactions in the strongly dispersive regime of circuit QED. Starting with the Hamiltonian of the baths plus cavity, we derive an effective model of only the cavity photon states along with analytic expressions for relevant physical quantities, such as the stabilization rate. The deterministic generation of such cat states is one of the key ingredients in performing universal quantum computation.

We have realized a microwave quantum-limited amplifier that is directional and can therefore function without the front circulator needed in many quantum measurements. The amplificationtakes place in only one direction between the input and output ports. Directionality is achieved by multi-pump parametric amplification combined with wave interference. We have verified the device noise performances by using it to readout a superconducting qubit and observed quantum jumps. With an improved version of this device, qubit and preamplifer could be integrated on the same chip.

Non-reciprocal devices, which have different transmission coefficients for
propagating waves in opposite directions, are crucial components in many low
noise quantum measurements. Inmost schemes, magneto-optical effects provide
the necessary non-reciprocity. In contrast, the proof-of-principle device
presented here, consists of two on-chip coupled Josephson parametric converters
(JPCs), which achieves directionality by exploiting the non-reciprocal phase
response of the JPC in the trans-gain mode. The non-reciprocity of the device
is controlled in-situ by varying the amplitude and phase difference of two
independent microwave pump tones feeding the system. At the desired working
point and for a signal frequency of 8.453 GHz, the device achieves a forward
power gain of 15 dB within a dynamical bandwidth of 9 MHz, a reverse gain of -6
dB and suppression of the reflected signal by 8 dB. We also find that the
amplifier adds a noise equivalent to less than one and a half photons at the
signal frequency (referred to the input). It can process up to 3 photons at the
signal frequency per inverse dynamical bandwidth. With a directional amplifier
operating along the principles of this device, qubit and readout preamplifier
could be integrated on the same chip.

We present a new theoretical framework to analyze microwave amplifiers based
on the dc SQUID. Our analysis applies input-output theory generalized for
Josephson junction devices biasedin the running state. Using this approach we
express the high frequency dynamics of the SQUID as a scattering between the
participating modes. This enables us to elucidate the inherently nonreciprocal
nature of gain as a function of bias current and input frequency. This method
can, in principle, accommodate an arbitrary number of Josephson harmonics
generated in the running state of the junction. We report detailed calculations
taking into account the first few harmonics that provide simple
semi-quantitative results showing a degradation of gain, directionality and
noise of the device as a function of increasing signal frequency. We also
discuss the fundamental limits on device performance and applications of this
formalism to real devices.

We present a semi-classical method for determining the effective low-energy
quantum Hamiltonian of weakly anharmonic superconducting circuits containing
mesoscopic Josephson junctionscoupled to electromagnetic environments made of
an arbitrary combination of distributed and lumped elements. A convenient
basis, capturing the multi-mode physics, is given by the quantized eigenmodes
of the linearized circuit and is fully determined by a classical linear
response function. The method is used to calculate numerically the low-energy
spectrum of a 3D-transmon system, and quantitative agreement with measurements
is found.