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AIP conference proceedings

Decoherence Rates in Large Scale Quantum Computers

Bryan J. Dalton

Published Jan 1, 2004• 0 citations
Publisher

Abstract

Markovian regime decoherence effects in quantum computers are studied in terms of the fidelity for the situation where the number of qubits N becomes large. A standard model involving three‐state lambda system ionic qubits is considered, with qubits localised around well‐separated positions via trapping potentials. The centre of mass vibrations of the qubits act as a reservoir. Coherent one and two qubit gating processes are controlled by time dependent localised classical EM fields that address specific qubits, the two qubit gating processes being facilitated by a cavity mode ancilla, which permits state interchange between qubits. With suitably chosen parameters, the decoherence time can be made essentially independent of N.

DOI
10.1063/1.1834463

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Markovian regime decoherence effects in quantum computers are studied in terms of the fidelity for the situation where the number of qubits N becomes large. A standard model involving three‐state lambda system ionic qubits is considered, with qubits localised around well‐separated positions via trapping potentials. The centre of mass vibrations of the qubits act as a reservoir. Coherent one and two qubit gating processes are controlled by time dependent localised classical EM fields that address specific qubits, the two qubit gating processes being facilitated by a cavity mode ancilla, which permits state interchange between qubits. With suitably chosen parameters, the decoherence time can be made essentially independent of N.
Markovian regime decoherence effects in quantum computers are studied in terms of the fidelity for the situation where the number of qubits N becomes large. A general expression giving the decoherence time scale in terms of Markovian relaxation elements and expectation values of products of system fluctuation operators is obtained, which could also be applied to study decoherence in other macroscopic systems such as Bose condensates and superconductors. A standard circuit model quantum computer involving three-state lambda system ionic qubits is considered, with qubits localized around well-separated positions via trapping potentials. The centre of mass vibrations of the qubits act as a reservoir. Coherent one and two qubit gating processes are controlled by time-dependent localized classical electromagnetic fields that address specific qubits, the two qubit gating processes being facilitated by a cavity mode ancilla, which permits state interchange between qubits. With a suitable choice of parameters, it is found that the decoherence time can be made essentially independent of N.
The scaling of decoherence rates with qubit number N is studied for a simple model of a quantum computer in the situation where N is large. The two state qubits are localized around well-separated positions via trapping potentials and vibrational centre of mass motion of the qubits occurs. Coherent one and two qubit gating processes are controlled by external classical fields and facilitated by a cavity mode ancilla. Decoherence due to qubit coupling to a bath of spontaneous modes, cavity decay modes and to the vibrational modes is treated. A non-Markovian treatment of the short time behaviour of the fidelity is presented, and expressions for the characteristic decoherence time scales obtained for the case where the qubit/cavity mode ancilla is in a pure state and the baths are in thermal states. Specific results are given for the case where the cavity mode is in the vacuum state and gating processes are absent and the qubits are in (a) the Hadamard state (b) the GHZ state.
This thesis is concerned with various aspects of ion traps and their use as a quantum simulation and computation device. In its first part we investigate various sources of noise and decoherence in ion traps. As quantum information is very fragile, a detailed knowledge of noise and decoherence sources in a quantum computation device is essential. In the special case of an ion trap quantum computer we investigate the effects of intensity and phase noise in the laser, which is used to perform the gate operations. We then look at other sources of noise which are present without a laser being switched on. These are fluctuations in the trapping frequency caused by noise in the electric potentials applied to the trap and fluctuating electrical fields which will cause heating of the centre-of-mass vibrational state of the ions in the trap. For the case of fluctuating electrical fields we estimate the effect on a quantum gate operation. We then propose a scheme for performing quantum gates without having the ions cooled down to their motional ground state. The second part deals with various aspects of the use of ion traps as a device for quantum computation. We start with the use of ionic qubits as a measurement device for the centre-of-mass vibrational mode and investigate in detail the effect these measurements will have on the vibrational mode. If one wants to use quantum computation devices as systems to simulate quantum mechanics, it is of interest to know how to simulate say a k-level system with N qubits. We investigate the easiest case of this wider problem and look at how to simulate a three-level system (a so called trit) with two qubits in an ion trap quantum computer. We show how to get and measure a SU (3) geometric phase with this toy model. Finally we investigate how to simulate collective angular momentum models with a string of qubits in an ion trap. We assume that the ionic qubits are coupled to a thermal reservoir and derive a master equation for this case. We investigate the semiclassical limit of this master equation and, in the case for two qubits in the trap, determine the entanglement of the steady state. We also outline a way to find the steady state for the master equation using coherence vectors.
Effective quantum-information processing requires coherent control of large numbers of qubits on a time scale that is short compared to the decoherence time of the system. It is therefore important to extrapolate and measure decoherence times for large quantum registers and to determine the effect of different couplings between system and environment on the decoherence rate. For this purpose, we have experimentally realized a system that allows one to generate model quantum registers with more than 100 qubits and measure the decay of the information in these states while adjusting the strength of the interaction between the quantum register and the environment. Our results indicate a power-law dependence of the decoherence rate on the number of qubits in the system, with an exponent of the order of 0.5. This behavior remains qualitatively unchanged when the coupling strength to the environment is reduced by about an order of magnitude.
We investigate the scaling of decoherence rates and their dynamical suppression with the number NNN of qubits in various entangled states. Remarkably, for sufficiently large NNN, coherence time is always in the Zeno regime. This changes the scaling to the square root of the Markov-regime scaling. We find that a simple and effective control strategy is to locally modulate the individual qubits and thereby not only suppress the decoherence rate of each qubit, but also reduce the decoherence scaling of entangled states that are particularly fragile, from N2{N}^{2}N2 to NNN, resulting in a dramatic reduction of the decoherence (by orders of magnitude). Surprisingly, the conditions for the effectiveness of such decoherence control are independent of NNN.
An essential action in quantum information processing is the manipulation (control) of a single qubit, ideally a closed two-level system. However, in realistic applications, quantum processes are often under the influence of the external environment, e.g., presenting some degree of dissipation and decoherence. In this paper we address the emerging difficulties in the (tracking) quantum control of a two-level system under the influence of both Markovian and non-Markovian noise. We employ a same framework to treat both situations, a Lindblad-type equation, but considering that for the former (latter) case, the decay rate $\mathrm{\ensuremath{\Gamma}}$ is time independent (dependent). We discuss the conditions leading to a breakdown of the quantum control and eventual ways to overcome the problem, like employing a fast control scheme or controlling the off-diagonal terms of the system density matrix. Surprisingly, for Markovian noise such breakdown time decreases with $\mathrm{\ensuremath{\Gamma}}$ not as an exponential but as a power law. This indicates that the quantum control should be possible for a coupling between the system and the environment stronger than previously expected. Moreover, we find that for non-Markovian noise, the breakdown time is longer when there is backflow, i.e., $\mathrm{\ensuremath{\Gamma}}(t)$ can be negative. The present theoretical results point to certain favorable scenarios to operate qubits even in a noisy medium.