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Nondestructive Detections of Qubits by Cavity Transmissions: Effects of Quantum Correlations

L. F. Wei, Jianhua Huang, Xun‐Li Feng, Changhun Oh

Published May 14, 2010• 0 citations
OpenAlex

Abstract

We propose an approach to nondestructively detect $N$ qubits by measuring the transmissions of a dispersively-coupled cavity. By taking into account all the qubit-cavity correlations (i.e., beyond the usual mean-field approximations), it is suggested that for an unknown normalized $N$-qubit state $|\psi_N>=\sum_{k=0}^{2^N-1}\beta_k|k>_N$, each detected peak of the cavity's transmission marks one of the logic states $|k>_N$ and the relative height of such a peak is related to the corresponding superposed-probability $|\beta_k|^2$. % Specifically, we show that these generic predications could be verified experimentally by the current circuit-quantum-electrodynamics systems with few superconducting qubits.

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We propose an approach to nondestructively detect NNN qubits by measuring the transmissions of a dispersively-coupled cavity. By taking into account all the qubit-cavity correlations (i.e., beyond the usual mean-field approximations), it is suggested that for an unknown normalized NNN-qubit state ∣ψN>=∑k=02N−1βk∣k>N|\psi_N>=\sum_{k=0}^{2^N-1}\beta_k|k>_N∣ψN​>=∑k=02N−1​βk​∣k>N​, each detected peak of the cavity's transmission marks one of the logic states ∣k>N|k>_N∣k>N​ and the relative height of such a peak is related to the corresponding superposed-probability ∣βk∣2|\beta_k|^2∣βk​∣2. % Specifically, we show that these generic predications could be verified experimentally by the current circuit-quantum-electrodynamics systems with few superconducting qubits.
We propose an approach to nondestructively detect NNN qubits by measuring the transmissions of a dispersively-coupled cavity. By taking into account all the cavity-qubits quantum correlations (i.e., beyond the usual coarse-grained/mean-field approximations), it is revealed that for an unknown normalized NNN-qubit state ∣ψN>=∑k=02N−1βk∣k>N|ψ_N>=\sum_{k=0}^{2^N-1}β_k|k>_N∣ψN​>=∑k=02N−1​βk​∣k>N​, each detected peak in the cavity transmitted spectra marks one of the basis states ∣k>N|k>_N∣k>N​ and the relative height of such a peak is related to the corresponding superposed-probability ∣βk∣2|β_k|^2∣βk​∣2. Our results are able to unambiguously account for the intriguing multi-peak structures of the spectra observed in a very recent circuit-quantum-electrodynamics experiment [Phys. Rev. A {\bf 81}, 062325 (2010)] with two superconducting qubits.
We present an efficient proposal to realize quantum nondestructive determination (QNDD) of unknown single-qubit states in two types of low-Q cavities, i.e., single-sided and double-sided cavity QED systems. In the dispersive regime, we demonstrate that the QNDD of single-qubit states can be realized by detecting qubit-state-dependent phase shifts (QSDPSs) of the reflected or transmitted photons from the cavity in the single-photon input-output process. Our proposal could be straightforwardly extended to the case of multiple-qubit states. Furthermore, the experimental feasibility of our proposal is also analyzed in experimentally-demonstrated circuit QED systems. The distinct feature of our proposal is that our proposal works in the dispersive regime of low-Q cavities and it is robust to both cavity decay and atomic spontaneous emission.
In the early 1980’s, it was suggested that a computer obeying the laws of quantum mechanics would be able to solve problems beyond the capabilities of a classical computer. The novel ways in which such a quantum computer works relies on the quantum properties of the quantum bits (qubits) used to store the information. Any candidate for a quantum computer must thus be able to sustain these properties and offer means to manipulate and read-out the information. One such candidate are quantum mechanical, superconducting circuits, where the logical bit is encoded in the energy eigenstates of the system. In circuit quantum electrodynamics, such a qubit is coupled to a microwave cavity allowing the qubit to be coherently controlled and read-out by probing the cavity. In this thesis, we theoretically investigate the destructive effects of noise which couple to the system as one tries to measure and control the qubits. We study the so called quantum capacitance read-out scheme, where the state of the qubit is mapped onto an equivalent capacitance of the circuit. It is shown that this is quantum limited, in the sense that the state of the qubit can be determined while simultaneously adding a minimum amount of noise to the system. Apart from the added noise, a measurement on a qubit will perturb it, causing the state to collapse to one of the measurement eigenstates. Such a state collapse can be utilized to generate entanglement between qubits by measuring on the cavity. We show that high-fidelity entangled states can be produced in this way and discuss the potential of the measurement to violate a bound given by local hidden variable theories. The possibility to prolong the life-time of the quantum state by active error correction is also investigated and we discuss limits on gate operation times to benefit from such a code, given realistic values for the error probabilities. This work was supported by the European Commission through IST-015708 EuroSQIP integrated pro ject, and by the Swedish Research Council.
In this thesis, experimentally relevant aspects and open questions of quantum information processing and measurement in circuit quantum electrodynamics have been investigated theoretically. Circuit quantum electrodynamics is a relatively young field combining superconducting transmission line resonators on-chip with superconducting quantum bits which serve as artificial atoms. Remarkable experiments have underlined the prospects of circuit QED as a possible architecture for quantum information processing as well as a framework within which quantum optics experiments can be performed. In contrast to their optical counterparts, these experiments reach the ultra-strong coupling limit and allow e.g. the generation of Fock states, the study of decoherence and the observation of quantum jumps. We present the physics of superconducting circuits based on the Josephson junction as a non-linear circuit element. We discuss the various types of quantum bits (qubits) before turning to an introduction on cavity QED and conclude the introduction with a brief presentation of the most relevant experiments and a short introduction to the principles and prospects of quantum information processing. Building on the pioneering theoretical and experimental work in which up to three qubits have been integrated on a chip and successfully coupled via a cavity mode, we propose a scheme to overcome the natural limits concerning scalability in these systems. Usually, scalability is severely limited by the resonance width of the qubit transitions and the attainable frequency range for both qubit transition and cavity resonance frequencies. This limits the present setups to at most ten qubits. As a possible solution, we propose and investigate a cross-bar grid layout of cavities with qubits at the intersections. It turns out that this setup, allowing for up to about 1000 qubits, offers significant advantages over e.g. nearest-neighbor coupling schemes. In addition, it can serve as a building block for a truly scalable scheme for fault-tolerant quantum information processing with superconducting circuits by combining 7x7 arrays in a staircase manner. Furthermore, we discuss and simulate different possibilities to improve upon the suggested scheme by e.g. using resonant gates and optimal control theory. Another focus of the present thesis is measurement physics. In order to investigate e.g. the possibilities of using a circuit QED system to dispersively detect single microwave photons or to generate multi-qubit entangled states, we introduce in a pedagogical manner the basics of stochastic master equations which generate quantum trajectories before turning to concrete applications. Quantum trajectories have proven to be a state-of-the-art tool to analyze measurement situations in a very realistic manner by giving access to both the measurement record as well as the internal quantum dynamics conditioned on this record. Using the method of quantum trajectories as generated by stochastic master equations, we discuss a scheme to efficiently generate multi-qubit entangled states in a very flexible way. The method is based on a collective, dispersive qubit readout. By this measurement, a suitable product state of N qubits can be reliably converted into e.g. an N-qubit Greenberger-Horne-Zeilinger or W state. We propose and discuss the scheme and investigate the effects of decoherence and parameter spread as they would be encountered in experiment. Last but not least, we propose and fully analyze a scheme to dispersively detect single itinerant microwave photons. While experiments to generate single microwave photons have already been successfully performed, their single-shot quantum non-demolition detection has not been possible so far. The scheme presented in this thesis closes this gap in the circuit QED toolbox and reaches detection efficiencies of 30% only limited by fundamental quantum mechanics in this case the Quantum Zeno effect. This detection scheme could be used in quantum communication or quantum cryptography applications.
Quantum computers have the capability to improve the efficiency and speed of many computational tasks. Among different candidates for physical implementation of quantum bits (qubits), superconducting qubits are currently one of the most promising candidates due to their accessible fabrication process and recent rapid developments. Measurement of these qubits is usually done in a circuit quantum electrodynamics (QED) setup, for which experimental and theoretical research is conducted to improve the accuracy and speed of the qubit readout. In this dissertation we study some aspects of the dispersive readout of superconducting qubits, and introduce tools and methods for studying these systems.In Chapter 2 we show that in presence of neighboring qubits, the system is typically measured in the basis of joint eigenstates of qubits, in contrast to what is expected from the textbook collapse postulate. In such setups, the excitation can switch between the eigenstates, leading to measurement error. In Chapter 3 we study the joint state of the qubit-resonator system during the measurement, and show that the qubit-induced nonlinearity of the resonator squeezes its state, and within the rotating wave approximation (RWA) the system mostly remains in the joint eigenladder that is associated with the qubit's initial state. In Chapter 4 we show that built-in energy resonances in the qubit-resonator Jaynes-Cummings ladder occur at specific resonator populations, and the couplings between these resonant levels are provided by the usually neglected non-RWA terms. Such resonances lead to measurement deterioration by exciting the qubit out of the computational subspace. In Chapter 5 we provide a hybrid phase-space-Fock-space approach for studying the evolution and squeezing of driven nonlinear resonators within Gaussian approximation, which is numerically efficient and sufficiently accurate. In Chapter 6 we study the propagating squeezed field that leaks out of the resonator, and write evolution equations for the correlators of the measured field quadrature. These equations are easy to simulate and can describe the squeezing of the propagating field during the transient, which can be used to optimize the fidelity and speed of the quadrature measurement in the dispersive readout of superconducting qubits.
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.