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arXiv:1005.2470

Joint quantum nondemolition measurements of qubits: beyond the mean-field theory

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

Published May 14, 2010• 0 citations
arXivHTMLPDF

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 cavity-qubits quantum correlations (i.e., beyond the usual coarse-grained/mean-field approximations), it is revealed that for an unknown normalized $N$-qubit state $|ψ_N>=\sum_{k=0}^{2^N-1}β_k|k>_N$, each detected peak in the cavity transmitted spectra marks one of the basis states $|k>_N$ and the relative height of such a peak is related to the corresponding superposed-probability $|β_k|^2$. 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.

DOI
10.48550/arxiv.1005.2470

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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.
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Dispersive readout, the current standard for measuring superconducting qubits, relies on relatively easy to engineer, static transverse coupling between a qubit and a detuned cavity mode. However, this method requires circuit design trade-offs between qubit-cavity coupling, cavity-feedline coupling, detuning, and the resultant decoherence introduced by the cavity mode. Additionally, the criteria for performing quantum nondemolition measurements, including reduced cavity photon numbers, further limits speed and fidelity. In this work, we introduce a circuit with pulse-tunable qubit-cavity and cavity-feedline couplings so that the entire qubit-measurement process can be fully optimized. Here, the core measurement process resembles a quantum gate operation between the qubit and the cavity, with the cavity acting like an ancilla, storing the qubit-state information. To fully explore time-dependent effects of the full qubit-cavity dynamics, we go beyond the rotating-wave, Jaynes-Cummings, or dispersive approximations, to develop a pulsed-measurement protocol. In addition, we propose a realistic circuit architecture that can provide a very fast, high-fidelity, QND measurement with near-unit efficiency.
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.
We propose a high-efficiency scheme to tomographically reconstruct an unknown quantum state by using a series of quantum nondemolition (QND) measurements. The proposed QND measurements of the qubits are implemented by probing the stationary transmissions through a driven dispersively coupled resonator. It is shown that only one kind of QND measurement is sufficient to determine all the diagonal elements of the density matrix of the detected quantum state. The remaining nondiagonal elements can be similarly determined by transferring them to the diagonal locations after a series of unitary operations. Compared with the tomographic reconstructions based on the usual destructive projective measurements (wherein one such measurement can determine only one diagonal element of the density matrix), the present reconstructive approach exhibits significantly high efficiency. Specifically, our generic proposal is demonstrated by the experimental circuit quantum electrodynamics systems with a few Josephson charge qubits.