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arXiv:cond-mat/0104513 · Physical Review A

Decoherence and dissipation during a quantum XOR gate operation

Michael Thorwart, Peter Hänggi A

Published Dec 11, 2001• 117 citations
arXivHTMLPDF

Abstract

The dynamics of a generic quantum XOR gate operation involving two interacting qubits being coupled to a bath of quantum harmonic oscillators is explored. By use of the formally exact quasiadiabatic-propagator path-integral methodology we study the time-resolved evolution of this interacting and decohering two-qubit system in presence of time-dependent external fields. The quality of the XOR gate operation is monitored by evaluating the four characteristic gate quantifiers: fidelity, purity, the quantum degree, and the entanglement capability of the gate. Two different types of errors for the XOR operation have been modeled, i.e., (i) bit-flip errors and (ii) phase errors. The various quantifiers are systematically investigated vs the strength of the interqubit coupling and vs both, the environmental temperature and the (Ohmic-like) bath-interaction strength. Our main findings are that these four gate quantifiers depend only very weakly on temperature, but are extremely sensitive to the bath-interaction strength. Interestingly enough, however, we find that the XOR gate operation deteriorates only weakly upon decreasing the interqubit coupling strength. This generic case study yields lower bounds on the quality of realistic XOR gate operations.

DOI
10.1103/physreva.65.012309

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The dynamics of a generic quantum XOR gate operation involving two interacting qubits being coupled to a bath of quantum harmonic oscillators is explored. By use of the formally exact quasiadiabatic-propagator path-integral methodology we study the time-resolved evolution of this interacting and decohering two-qubit system in presence of time-dependent external fields. The quality of the XOR gate operation is monitored by evaluating the four characteristic gate quantifiers: fidelity, purity, the quantum degree, and the entanglement capability of the gate. Two different types of errors for the XOR operation have been modeled, i.e., (i) bit-flip errors and (ii) phase errors. The various quantifiers are systematically investigated vs the strength of the interqubit coupling and vs both, the environmental temperature and the (Ohmic-like) bath-interaction strength. Our main findings are that these four gate quantifiers depend only very weakly on temperature, but are extremely sensitive to the bath-interaction strength. Interestingly enough, however, we find that the XOR gate operation deteriorates only weakly upon decreasing the interqubit coupling strength. This generic case study yields lower bounds on the quality of realistic XOR gate operations.
In this PhD thesis we have been concerned with dissipative and decoherence processes which take place specifically for quantum systems in a dissipative bistable potential. This potential describes a Hamiltonian system whose spectrum is composed by a ladder of levels, grouped as doublets under the potential barrier and being almost equally spaced above it. The system interacts with a thermal environment, fully characterized by the so-called bath spectral density. The dynamical quantity we looked at is the so-called reduced density matrix (RDM), obtained by tracing out the full density matrix the environmental degrees of freedom. The matrix elements for the RDM are exactly evaluated by means of the Feynman-Vernon real-time path-integral technique. Depending on which regime of parameters we are interested in, i.e. temperature and coupling strength with the reservoir, a few or several energy levels contribute to the dynamics. In particular, a two-level system (TLS) is first analyzed, corresponding to the regime where only the lowest doublet is energetically accessible. We discussed in the beginning the case of a TLS interacting with Ohmic and super-Ohmic baths and then the situation in which a symmetric TLS is assumed to be coupled with an external detector, like a dc-SQUID, the latter being coupled to an Ohmic environment. This model provides a better description of a real qubit, as frequently discussed in various experiments. In the former case, we developed a theory called Weakly-Interacting Blip Approximation (WIBA), which is nonperturbative in the coupling strength as well as in the tunneling frequency and gives reliable results over a wide range of temperatures and coupling strength to the thermal environment. In the latter case, the dynamics was evaluated upon starting from the Non-Interacting Blip Approximation (NIBA) and approximating the NIBA kernels up to first order in the detector-bath coupling strength. Analytical results, valid also at finite detuning, were obtained and we showed that our theory correctly reproduces the onset of multiple oscillation frequencies in the dynamics due to the entanglement between TLS and detector. Finally, we discussed the more general case of an M-level system coupled with an Ohmic environment. This situation describes the case of a bistable potential where the temperature is high enough that higher-lying energy levels are being populated and the system cannot be reduced to a simple TLS. Many are the experimental situations which can be described by our theory. We developed a novel approximation called Weakly-Interacting VR-Blip Approximation (VR-WIBA) for the dynamics in a driven dissipative bistable potential. In the regimes of low to moderate temperatures and coupling strengths, we exploited a separation of time scales between tunneling and vibrational relaxation events and we could derive analytical kernels for the intra- and inter-well dynamics.
We use numerically exact iterative path integral methods to investigate the decoherence and entanglement dynamics of a tunneling pair of two coupled qubits (spins) system interacting with a dissipative bath. We find that decoherence is generally accompanied by the destruction of entanglement, although the specifics of this destruction depend sensitively on the parameters of the Hamiltonian (qubit-qubit coupling and/or energy bias), the strength of dissipation, the temperature, and the choice of initial condition. We also observe that dissipation can in some cases generate a substantial amount of entanglement. Finally, if an entangled eigenstate exists which does not couple to the environment, the long-time entanglement can significantly exceed the value corresponding to the Boltzmann equilibrium state.
We examine how dissipation and gate timing errors affect the fidelity of a sequence of swap gates on a chain of interacting qubits in comparison to noise in the interqubit interaction. Although interqubit interaction noise and gate timing errors are always present in any qubit platform, dissipation is a special case that can arise in multivalley semiconductor spin qubit systems, such as Si-based qubits, where dissipation may be used as a general model for valley leakage. In our Hamiltonian, each qubit is coupled via Heisenberg exchange to every other qubit in the chain, with the strength of the exchange interaction decreasing exponentially with distance between the qubits. Dissipation is modeled through the term $\ensuremath{-}i\ensuremath{\gamma}\mathbf{1}$ in the Hamiltonian, and $\ensuremath{\gamma}$ is chosen so as to be consistent with the experimentally observed intervalley tunneling in Si. We show that randomness in the dissipation parameter should have little to no effect on the swap gate fidelity in the currently fabricated Si circuits. We introduce quasistatic noise in the interqubit interaction and random gate timing error and average the fidelities over 10 000 realizations for each set of parameters. The fidelities are then plotted against JSWAP{J}_{\mathrm{SWAP}}JSWAP​, the strength of the exchange coupling corresponding to the swap gate. We find that dissipation decreases the fidelity of the swap operation---though the effect is small compared to that of the known noise in the interqubit interaction---and that gate timing error creates an effective optimal value of JSWAP{J}_{\mathrm{SWAP}}JSWAP​, beyond which infidelity begins to increase.
We present an approach that allows quantifying decoherence processes in an open quantum system subject to external time-dependent control. Interactions with the environment are modeled by a standard bosonic heat bath. We develop two unitarity-preserving approximation schemes to calculate the reduced density matrix. One of the approximations relies on a short-time factorization of the evolution operator, while the other utilizes expansion in terms of the system-bath coupling strength. Applications are reported for two illustrative systems: an exactly solvable adiabatic model, and a model of a rotating-wave quantum-computing gate function. The approximations are found to produce consistent results at short and intermediate times.
We perform a theoretical study of composed superconducting qubit systems for the case of a coupled qubit configuration based on a hybrid qubit circuit made of both charge and phase qubits, which are coupled via a σ x ⊗ σ z interaction. We compute the system's eigen-energies in terms of the qubit transition frequencies and the strength of the inter-qubit coupling, and describe the sensitivity of the energy crossing/anti-crossing features to such coupling. We compute the hybrid system's dissipative dynamics for the cases of i) collective and ii) independent decoherence, whereby the system interacts with one common and two different baths of harmonic oscillators, respectively. The calculations have been performed within the Bloch-Redfield formalism and we report the solutions for the populations and the coherences of the system's reduced density matrix. The dephasing and relaxation rates are explicitly calculated as a function of the heat bath temperature.
Abstract Using a dissipative quantum Rabi model, we study the dynamics of a slow qubit coupled to a fast quantum harmonic oscillator interacting with a bosonic bath from weak to strong and ultra-strong coupling regimes. Solving the quantum Heisenberg equations of motion, perturbative in the internal coupling between qubit and oscillator, we derive functional relationships directly linking the qubit coordinates in the Bloch sphere to oscillator observables. We then perform accurate time-dependent Matrix Product State simulations and compare our results both with the analytical solutions of the Heisenberg equations of motion, and with numerical solutions of a Lindblad master equation, perturbative in the external coupling between oscillator and environment. Indeed, we show that, up to the strong coupling regime, the qubit state accurately fulfils the derived functional relationships. We analyse in detail the case of a qubit starting with generic coordinates on the Bloch sphere of which we evaluate the three components of the Bloch vector through the averages of oscillator observables. Interestingly, a weak to intermediate oscillator coupling to the bath is able to simplify the Bloch vector evaluation since qubit-oscillator relationships are more immediate. Moreover, by monitoring the qubit fidelity with respect to free limit, we find the parameter regime where the combined effect of internal and external couplings is able to hinder the reliable evaluation of the qubit Bloch vector. Finally, in the ultra-strong coupling regime, non-Markovian effects become robust and the dynamics of qubit and oscillator are inextricably entangled making the qubit Bloch vector evaluation difficult. Graphical Abstract