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arXiv:cond-mat/0611445 · Physical Review B

Simulations of quantum gates with decoherence

Valentina Corato, P. Silvestrini, A. Görlich, Piotr Korcyl, J. Wosiek, Leo Stodolsky

Published May 7, 2007• 5 citations
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Abstract

Methods and results for numerical simulations of one and two interacting rf-SQUID systems suitable for adiabatic quantum gates are presented. These are based on high accuracy numerical solutions to the static and time dependent Schr\"odinger equation for the full SQUID Hamiltonian in one and two variables. Among the points examined in the static analysis is the range of validity of the effective two-state or ``spin-$\frac{1}{2}$'' picture. A range of parameters is determined where the picture holds to good accuracy as the energy levels undergo gate manipulations. Some general points are presented concerning the relations between device parameters and ``good'' quantum mechanical state spaces. The time dependent simulations allow the examination of suitable conditions for adiabatic behavior, and permits the introduction of a random noise to simulate the effects of decoherence. A formula is derived and tested relating the random noise to the decoherence rate. Sensitivity to device and operating parameters for the logical gates NOT and CNOT are examined, with particular attention to values of the tunnel parameter $\ensuremath{\beta}$ slightly above one. It appears that with values of $\ensuremath{\beta}$ close to 1, a quantum CNOT gate is possible even with rather short decoherence times. Many of the methods and results will apply to coupled double-potential well systems in general.

DOI
10.1103/physrevb.75.184507

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Methods and results for numerical simulations of one and two interacting rf-SQUID systems suitable for adiabatic quantum gates are presented. These are based on high accuracy numerical solutions to the static and time dependent Schr"odinger equation for the full SQUID Hamiltonian in one and two variables. Among the points examined in the static analysis is the range of validity of the effective two-state or spin-$\frac{1}{2}$'' picture. A range of parameters is determined where the picture holds to good accuracy as the energy levels undergo gate manipulations. Some general points are presented concerning the relations between device parameters and good'' quantum mechanical state spaces. The time dependent simulations allow the examination of suitable conditions for adiabatic behavior, and permits the introduction of a random noise to simulate the effects of decoherence. A formula is derived and tested relating the random noise to the decoherence rate. Sensitivity to device and operating parameters for the logical gates NOT and CNOT are examined, with particular attention to values of the tunnel parameter \ensuremathβ\ensuremath{\beta}\ensuremathβ slightly above one. It appears that with values of \ensuremathβ\ensuremath{\beta}\ensuremathβ close to 1, a quantum CNOT gate is possible even with rather short decoherence times. Many of the methods and results will apply to coupled double-potential well systems in general.
We present a general method for studying coupled qubits driven by adiabatically changing external parameters. Extended calculations are provided for a two-bit Hamiltonian whose eigenstates can be used as logical states for a quantum CNOT gate. From a numerical analysis of the stationary Schr\"odinger equation we find a set of parameters suitable for representing CNOT, while from a time-dependent study the conditions for adiabatic evolution are determined. Specializing to a concrete physical system involving superconducting quantum interference devices (SQUID's), we determine reasonable parameters for experimental purposes. The dissipation for SQUID's is discussed by fitting experimental data. The low dissipation obtained supports the idea that adiabatic operations could be performed on a time scale shorter than the decoherence time.
We examine the realization of a quantum CNOT gate by adiabatic operations.The principles of such systems and their analysis are briefly discussed and a model consisting of two weakly coupled double- potential well qubits is studied numerically. Regions of the parameter space with suitable well-defined sets of wavefunctions are found, in which then an adiabatic sweep of an external bias produces the switching behavior of CNOT. Results are presented on the adiabatic condition and the identification with the parameters of a flux-coupled two-SQUID system is given.For typical parameters adiabatic times in the nanosecond regime are obtained.
We study the effects of noise and decoherence for a double-potential well system, suitable for the fabrication of qubits and quantum logic elements. A random noise term is added to the hamiltonian, the resulting wavefunction found numerically and the density matrix obtained by averaging over noise signals. Analytic solutions using the two-state model are obtained and found to be generally in agreement. In particular, a simple formula for the decoherence rate in terms of the noise parameters in the two-state model is reviewed and verified for the full simulation with the multi-level system. The formalism is extended to describe multiple sources of noise or different "dephasing" axes at the same time. Furthermore, the old formula for the "Turing-Watched Pot" effect is generalized to the case where the environmental interactions do not conserve the "quality" in question. Various forms for the noise signal are investigated. An interesting result is the importance of the noise power at low frequency. If it vanishes there is, in leading order, no decoherence. This is verified in a numerical simulation where two apparently similar noise signals, but differing in the power at zero frequency, give strikingly different decoherence effects. A short discussion of situations dominated by low frequency noise is given.
We present a scheme to realize the basic two-qubit logic gates such as the quantum phase gate and SWAP gate using a detuned microwave cavity interacting with three-level superconducting-quantum-interference-device (SQUID) qubit(s), by placing SQUID(s) in a two-mode microwave cavity and using adiabatic passage methods. In this scheme, the two logical states of the qubit are represented by the two lowest levels of the SQUID and the cavity fields are treated as quantized. Compared with the previous method, the complex procedures of adjusting the level spacing of the SQUID and applying the resonant microwave pulse to the SQUID to create transformation are not required. Based on superconducting device with relatively long decoherence time and simplified operation procedure, the gates operate at a high speed, which is important in view of decoherence.
Systems of individual electrons electrostatically trapped on condensed noble gas surfaces have recently attracted considerable interest as potential platforms for quantum computing. The electrons serve as charge qubits in the system, and the purity of the noble gas surface protects the relevant quantum properties of each electron. Previous work has indicated that manipulation of a confining double-well potential for electrons on superfluid helium can generate entanglement suitable for two-qubit gate operations. In this work, we incorporate a time-dependent tuning of the potential shape to further explore operation of two-qubit gates with the superfluid helium system. Through numerical time evolution of the closed system (without decoherence), we show that control-induced errors can be minimized to allow for fast, high-fidelity two-qubit gates. In particular, we simulate operation of the iSWAP\sqrt{i\mathrm{SWAP}}iSWAP​ and CZ gates and obtain estimated fidelities of 0.999 and 0.996 with execution times of 2.9 ns and 9.4 ns, respectively. Furthermore, we examine the stability of these gate fidelities under non-ideal execution conditions, which reveals new properties to consider in the device design. Finally, we reflect on the impact of screening and decoherence on our results. The methodology presented here enables future efforts to isolate control-induced effects from environmental noise, which is an important step towards the realization of high-fidelity two-qubit gates with electrons on helium.