Quantum-mechanical effects at the macroscopic level were first explored in Josephson-junction-based superconducting circuits in the 1980s. In recent decades, the emergence of quantum information science has intensified research toward using these circuits as qubits in quantum information processors. The realization that superconducting qubits can be made to strongly and controllably interact with microwave photons, the quantized electromagnetic fields stored in superconducting circuits, led to the creation of the field of circuit quantum electrodynamics (QED), the topic of this review. While atomic cavity QED inspired many of the early developments of circuit QED, the latter has now become an independent and thriving field of research in its own right. Circuit QED allows the study and control of light-matter interaction at the quantum level in unprecedented detail. It also plays an essential role in all current approaches to gate-based digital quantum information processing with superconducting circuits. In addition, circuit QED provides a framework for the study of hybrid quantum systems, such as quantum dots, magnons, Rydberg atoms, surface acoustic waves, and mechanical systems interacting with microwave photons. Here the coherent coupling of superconducting qubits to microwave photons in high-quality oscillators focusing on the physics of the Jaynes-Cummings model, its dispersive limit, and the different regimes of light-matter interaction in this system are reviewed. Also discussed is coupling of superconducting circuits to their environment, which is necessary for coherent control and measurements in circuit QED, but which also invariably leads to decoherence. Dispersive qubit readout, a central ingredient in almost all circuit QED experiments, is also described. Following an introduction to these fundamental concepts that are at the heart of circuit QED, important use cases of these ideas in quantum information processing and in quantum optics are discussed. Circuit QED realizes a broad set of concepts that open up new possibilities for the study of quantum physics at the macro scale with superconducting circuits and applications to quantum information science in the widest sense.

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Circuit quantum electrodynamics

A time-resolved, continuous parity measurement of two superconducting qubits in a three-dimensional circuit quantum electrodynamics architecture is reported; by further implementing feedback control, entanglement is generated ‘on demand’. In quantum computing, parity measurement projects a register of qubits to a state with an even or odd total number of excitations. Despite numerous proposals, realizing a parity meter that creates entanglement for both even and odd measurement results has remained an outstanding challenge. Here Leonardo DiCarlo and colleagues report the realization of a time-resolved, continuous parity measurement of two superconducting qubits in a three-dimensional circuit quantum electrodynamics architecture. By further implementing feedback control, entanglement generation can be achieved deterministically, or 'on demand'. The results provide key ingredients for active quantum error correction in the solid state. The stochastic evolution of quantum systems during measurement is arguably the most enigmatic feature of quantum mechanics. Measuring a quantum system typically steers it towards a classical state, destroying the coherence of an initial quantum superposition and the entanglement with other quantum systems. Remarkably, the measurement of a shared property between non-interacting quantum systems can generate entanglement, starting from an uncorrelated state. Of special interest in quantum computing is the parity measurement1, which projects the state of multiple qubits (quantum bits) to a state with an even or odd number of excited qubits. A parity meter must discern the two qubit-excitation parities with high fidelity while preserving coherence between same-parity states. Despite numerous proposals for atomic2, semiconducting1,3,4,5,6,7 and superconducting qubits8,9, realizing a parity meter that creates entanglement for both even and odd measurement results has remained an outstanding challenge. Here we perform a time-resolved, continuous parity measurement of two superconducting qubits using the cavity in a three-dimensional circuit quantum electrodynamics10,11 architecture and phase-sensitive parametric amplification12. Using postselection, we produce entanglement by parity measurement reaching 88 per cent fidelity to the closest Bell state. Incorporating the parity meter in a feedback-control loop, we transform the entanglement generation from probabilistic to fully deterministic, achieving 66 per cent fidelity to a target Bell state on demand. These realizations of a parity meter and a feedback-enabled deterministic measurement protocol provide key ingredients for active quantum error correction in the solid state13,14,15.

Deterministic entanglement of superconducting qubits by parity measurement and feedback

The quantum state of a superconducting qubit nonresonantly coupled to a transmission line resonator can be determined by measuring the quadrature amplitudes of an electromagnetic field transmitted through the resonator. We present experiments in which we analyze in detail the dynamics of the transmitted field as a function of the measurement frequency for both weak continuous and pulsed measurements. We find excellent agreement between our data and calculations based on a set of Bloch-type differential equations for the cavity field derived from the dispersive Jaynes-Cummings Hamiltonian including dissipation. We show that the measured system response can be used to construct a measurement operator from which the qubit population can be inferred accurately. Such a measurement operator can be used in tomographic methods to reconstruct single and multiqubit states in ensemble-averaged measurements.

Dynamics of dispersive single-qubit readout in circuit quantum electrodynamics

We propose methods for the preparation and entanglement detection of multi-qubit Greenberger–Horne–Zeilinger (GHZ) states in circuit quantum electrodynamics. Using quantum trajectory simulations appropriate for the situation of a weak continuous measurement, we show that the joint dispersive readout of several qubits can be utilized for the probabilistic production of high-fidelity GHZ states. When employing a nonlinear filter on the recorded homodyne signal, the selected states are found to exhibit values of the Bell–Mermin operator exceeding 2 under realistic conditions. We discuss the potential of the dispersive readout to demonstrate a violation of the Mermin bound, and present a measurement scheme avoiding the necessity for full detector tomography.

https://iopscience.iop.org/article/10.1088/1367-2630/11/7/073040/pdf

Proposal for generating and detecting multi-qubit GHZ states in circuit QED

Joint measurements of multiple qubits open new possibilities for quantum information processing. Here we present an approach based on homodyne detection to realize such measurements in the dispersive regime of cavity or circuit QED. By changing details of the measurement, the readout can be tuned from extracting only single-qubit to extracting only multiqubit properties. We obtain a reduced stochastic master equation describing this measurement and its effect on the qubits. As an example, we present results showing parity measurements of two qubits. In this situation, measurement of an initially unentangled state can yield, with near unit probability, a state of significant concurrence.

Tunable joint measurements in the dispersive regime of cavity QED

In this paper we derive an effective master equation and quantum trajectory equation for multiple qubits in a single resonator and in the large resonator decay limit. We show that homodyne measurem...

https://epiq.physique.usherbrooke.ca/data/files/publications/Hutchison2009225.pdf

Quantum trajectory equation for multiple qubits in circuit QED: Generating entanglement by measurementThis paper was presented at the Theory CANADA 4 conference, held at Centre de recherches mathématiques, Montréal, Québec, Canada on 4–7 June 2008.

F. K. WilhelmF.K. Wilhelm

Papers

Quantum trajectory equation for multiple qubits in circuit QED: Generating entanglement by measurementThis paper was presented at the Theory CANADA 4 conference, held at Centre de recherches mathématiques, Montréal, Québec, Canada on 4–7 June 2008.