Non-ideal monitoring of a qubit state using a quantum tunnelling device

We propose a model for non-ideal monitoring of the state of a coupled quantum dot qubit by a quantum tunnelling device. The non-ideality is modelled using an equivalent measurement circuit. This allows realistically available measurement results to be related to the state of the quantum system (qubit). We present a quantum trajectory that describes the stochastic evolution of the qubit state conditioned by tunnelling events (i.e. current) through the device. We calculate and compare the noise power spectra of the current in an ideal and a non-ideal measurement. The results show that when the two qubit dots are strongly coupled the non-ideal measurement cannot detect the qubit state precisely. The limitation of the ideal model for describing a realistic system maybe estimated from the noise spectra.

Practicality of time-optimal two-qubit Hamiltonian simulation

What is the time-optimal way of using a set of control Hamiltonians to obtain a desired interaction? Vidal, Hammerer, and Cirac [Phys. Rev. Lett. 88, 237902 (2002)] have obtained a set of powerful results characterizing the time-optimal simulation of a two-qubit quantum gate using a fixed interaction Hamiltonian and fast local control over the individual qubits. How practically useful are these results? We prove that there are two-qubit Hamiltonians such that time-optimal simulation requires infinitely many steps of evolution, each infinitesimally small, and thus is physically impractical. A procedure is given to determine which two-qubit Hamiltonians have this property, and we show that almost all Hamiltonians do. Finally, we determine some bounds on the penalty that must be paid in the simulation time if the number of steps is fixed at a finite number, and show that the cost in simulation time is not too great.

Spin squeezing as a measure of entanglement in a two-qubit system

We show that the two definitions of spin squeezing extensively used in the literature [M. Kitagawa and M. Ueda, Phys. Rev. A 47, 5138 (1993) and D.J. Wineland , Phys. Rev. A 50, 67 (1994)] give different predictions of entanglement in the two-atom Dicke system. We analyze differences between the definitions and show that the spin squeezing parameter of Kitagawa and Ueda is a better measure of entanglement than the commonly used spectroscopic spin squeezing parameter. We illustrate this relation by examining different examples of a driven two-atom Dicke system in which spin squeezing and entanglement arise dynamically. We give an explanation of the source of the difference using the negativity criterion for entanglement.

Fast nonadiabatic two-qubit gates for the Kane quantum computer

In this paper, we apply the canonical decomposition of two-qubit unitaries to find pulse schemes to control the proposed Kane quantum computer. We explicitly find pulse sequences for the controlled-NOT, swap, square root of swap, and controlled Z rotations. We analyze the speed and fidelity of these gates, both of which compare favorably to existing schemes. The pulse sequences presented in this paper are theoretically faster, with higher fidelity, and simpler. Any two-qubit gate may be easily found and implemented using similar pulse sequences. Numerical simulation is used to verify the accuracy of each pulse scheme.

Measuring a photonic qubit without destroying it

Measuring the polarization of a single photon typically results in its destruction. We propose, demonstrate, and completely characterize a quantum nondemolition (QND) scheme for realizing such a measurement nondestructively. This scheme uses only linear optics and photodetection of ancillary modes to induce a strong nonlinearity at the single-photon level, nondeterministically. We vary this QND measurement continuously into the weak regime and use it to perform a nondestructive test of complementarity in quantum mechanics. Our scheme realizes the most advanced general measurement of a qubit to date: it is nondestructive, can be made in any basis, and with arbitrary strength.

Quantum-error correction on linear-nearest-neighbor qubit arrays

A quantum circuit implementing 5-qubit quantum-error correction on a linear-nearest-neighbor architecture is described. The canonical decomposition is used to construct fast and simple gates that incorporate the necessary swap operations allowing the circuit to achieve the same depth as the current least depth circuit. Simulations of the circuit's performance when subjected to discrete and continuous errors are presented. The relationship between the error rate of a physical qubit and that of a logical qubit is investigated with emphasis on determining the concatenated error correction threshold.

Fungible dynamics: There are only two types of entangling multiple-qubit interactions

What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? It has been shown that all two-body Hamiltonian evolutions can be simulated using any fixed two-body entangling n-qubit Hamiltonian and fast local unitaries. By entangling we mean that every qubit is coupled to every other qubit, if not directly, then indirectly via intermediate qubits. We extend this study to the case where interactions may involve more than two qubits at a time. We find necessary and sufficient conditions for an arbitrary n-qubit Hamiltonian to be dynamically universal, that is, able to simulate any other Hamiltonian acting on n qubits, possibly in an inefficient manner. We prove that an entangling Hamiltonian is dynamically universal if and only if it contains at least one coupling term involving an even number of interacting qubits. For odd entangling Hamiltonians, i.e., Hamiltonians with couplings that involve only an odd number of qubits, we prove that dynamic universality is possible on an encoded set of n-1 logical qubits. We further prove that an odd entangling Hamiltonian can simulate any other odd Hamiltonian and classify the algebras that such Hamiltonians generate. Thus, our results show that up to local unitary operations, there are only two fundamentally different types of entangling Hamiltonian on n qubits. We also demonstrate that, provided the number of qubits directly coupled by the Hamiltonian is bounded above by a constant, our techniques can be made efficient.

Quantum trajectories for the realistic measurement of a solid-state charge qubit

We present a new model for the continuous measurement of a coupled quantum dot charge qubit. We model the effects of a realistic measurement, namely adding noise to, and filtering, the current through the detector. This is achieved by embedding the detector in an equivalent circuit for measurement. Our aim is to describe the evolution of the qubit state conditioned on the macroscopic output of the external circuit. We achieve this by generalizing a recently developed quantum trajectory theory for realistic photodetectors [P. Warszawski, H. M. Wiseman, and H. Mabuchi, Phys. Rev. A 65, 023802 (2002)] to treat solid-state detectors. This yields stochastic equations whose (numerical) solutions are the realistic quantum trajectories of the conditioned qubit state. We derive our general theory in the context of a low transparency quantum point contact. Areas of application for our theory and its relation to previous work are discussed.

Qubit channels that achieve capacity with two states

This paper considers a class of qubit channels for which three states are always sufficient to achieve the Holevo capacity. For these channels, it is known that there are cases where two orthogonal states are sufficient, two nonorthogonal states are required, or three states are necessary. Here a systematic theory is given which provides criteria to distinguish cases where two states are sufficient, and determine whether these two states should be orthogonal or nonorthogonal. In addition, we prove a theorem on the form of the optimal ensemble when three states are required, and present efficient methods of calculating the Holevo capacity.

Numerical analysis of capacities for two-qubit unitary operations

We present numerical results on the capacities of two-qubit unitary operations for performing communication and creating entanglement. The capacities for communication considered are based upon the increase in Holevo information of an ensemble. Our results indicate that the capacity may be accurately estimated using ensemble sizes and ancilla dimensions of 4. In addition, the calculated values of these capacities were close to, and in some cases equal to, the similarly defined entangling capacities; this result indicates connections between these capacities.

Numerical method for determination of the NMR frequency of the single-qubit operation in a silicon-based solid-state quantum computer

A numerical method is introduced to determine the nuclear magnetic resonance frequency of a donor (P-31) doped inside a silicon substrate under the influence of an applied electric field. This phosphorus donor has been suggested for operation as a qubit for the realization of a solid-state scalable quantum computer. The operation of the qubit is achieved by a combination of the rotation of the phosphorus nuclear spin through a globally applied magnetic field and the selection of the phosphorus nucleus through a locally applied electric field. To realize the selection function, it is required to know the relationship between the applied electric field and the change of the nuclear magnetic resonance frequency of phosphorus. In this study, based on the wave functions obtained by the effective-mass theory, we introduce an empirical correction factor to the wave functions at the donor nucleus. Using the corrected wave functions, we formulate a first-order perturbation theory for the perturbed system under the influence of an electric field. In order to calculate the potential distributions inside the silicon and the silicon dioxide layers due to the applied electric field, we use the multilayered Green's functions and solve an integral equation by the moment method. This enables us to consider more realistic, arbitrary shape, and three-dimensional qubit structures. With the calculation of the potential distributions, we have investigated the effects of the thicknesses of silicon and silicon dioxide layers, the relative position of the donor, and the applied electric field on the nuclear magnetic resonance frequency of the donor.

Dark periods and revivals of entanglement in a two-qubit system

In a recent paper Yu and Eberly [Phys. Rev. Lett. 93, 140404 (2004)] have shown that two initially entangled and afterward not interacting qubits can become completely disentangled in a finite time. We study transient entanglement between two qubits coupled collectively to a multimode vacuum field, assuming that the two-qubit system is initially prepared in an entangled state produced by the two-photon coherences, and find the unusual feature that the irreversible spontaneous decay can lead to a revival of the entanglement that has already been destroyed. The results show that this feature is independent of the coherent dipole-dipole interaction between the atoms but it depends critically on whether or not collective damping is present.

Teleportation via Multi-Qubit Channels

We investigate the problem of teleporting an unknown qubit state to a recipient via a channel of 2L qubits. In this procedure a protocol is employed whereby L Bell state measurements are made and information based on these measurements is sent via a classical channel to the recipient. Upon receiving this information the recipient determines a local gate which is used to recover the original state. We find that the 2(2L)-dimensional Hilbert space of states available for the channel admits a decomposition into four subspaces. Every state within a given subspace is a perfect channel, and each sequence of Bell measurements projects 2L qubits of the system into one of the four subspaces. As a result, only two bits of classical information need be sent to the recipient for them to determine the gate. We note some connections between these four subspaces and ground states of many-body Hamiltonian systems, and discuss the implications of these results towards understanding entanglement in multi-qubit systems.

Noise in detection of qubit states using a quantum point contact

We present a model for detection of the states of a coupled quantum dots (qubit) by a quantum point contact. Most proposals for measurements of states of quantum systems are idealized. However in a real laboratory the measurements cannot be perfect due to practical devices and circuits. The models using ideal devices are not sufficient for describing the detection information of the states of the quantum systems. Our model therefore includes the extension to a non-ideal measurement device case using an equivalent circuit. We derive a quantum trajectory that describes the stochastic evolution of the state of the system of the qubit and the measuring device. We calculate the noise power spectrum of tunnelling events in an ideal and a non-ideal quantum point contact measurement respectively. We found that, for the strong coupling case it is difficult to obtain information of the quantum processes in the qubit by measurements using a non-ideal quantum point contact. The noise spectra can also be used to estimate the limits of applicability of the ideal model.

Scaling of decoherence effects in quantum computers

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.