Evolution Of Qubit Research Articles

Admissible memory kernels for random unitary qubit evolution

We analyze the random unitary evolution of a qubit within the memory kernel approach. We provide sufficient conditions which guarantee that the corresponding memory kernel generates physically legitimate quantum evolution. Interestingly, we are able to recover several well-known examples and to generate new classes of nontrivial qubit evolution. Surprisingly, it turns out that a class of quantum evolutions with a memory kernel generated by our approach gives rise to the vanishing of a non-Markovianity measure based on the distinguishability of quantum states.

Nonadiabatic dynamics of a slowly driven dissipative two-level system

We study the dynamics of a two-level system described by a slowly varying Hamiltonian and weakly coupled to the Ohmic environment. We follow the Bloch--Redfield perturbative approach to include the effect of the environment on qubit evolution and take into account modification of the spectrum and matrix elements of qubit transitions due to time-dependence of the Hamiltonian. This formalism is applied to two problems. (1) We consider a qubit, or a spin-1/2, in a rotating magnetic field. We show that once the rotation starts, the spin has a component perpendicular to the rotation plane of the field that initially wiggles and eventually settles to the value proportional to the product of angular rotation velocity of the field and the Berry curvature. (2) We re-examine the Landau--Zener transition for a system coupled to environment at arbitrary temperature. We show that as temperature increases, the thermal excitation and relaxation become leading processes responsible for transition between states of the system. We also apply the Lindblad master equations to these two problems and compare results with those obtained from the Bloch--Redfield equations.

Qubit dephasing affected by preparation-induced initial correlation with stochastic environment

The reduced time evolution of a single qubit under the influence of a classical environment is studied by means of the stochastic Liouville equation. The qubit initially correlates with the environment, which is created by a preparation process of an initial state. The phase fluctuation caused by the environment is characterized by means of the Gauss-Markov process and the two-state-jump Markov process. It is shown that depending on values of the parameters, the initial correlation enhances or reduces the decoherence of the qubit. In the case of the slow modulation, the effect of the initial preparation of the qubit becomes significant and it cannot be ignored in the reduced time evolution of the qubit.

Improving quantum gate fidelities by using a qubit to measure microwave pulse distortions.

We present a new method for determining pulse imperfections and improving the single-gate fidelity in a superconducting qubit. By applying consecutive positive and negative π pulses, we amplify the qubit evolution due to microwave pulse distortions, which causes the qubit state to rotate around an axis perpendicular to the intended rotation axis. Measuring these rotations as a function of pulse period allows us to reconstruct the shape of the microwave pulse arriving at the sample. Using the extracted response to predistort the input signal, we are able to reduce the average error per gate by 37%, which enables us to reach an average single-qubit gate fidelity higher than 0.998.

Qubit purification speed-up for three complementary continuous measurements

We consider qubit purification under simultaneous continuous measurement of the three non-commuting qubit operators σ(x), σ(y), σ(z). The purification dynamics is quantified by (i) the average purification rate and (ii) the mean time of reaching a given level of purity, 1-ε. Under ideal measurements (detector efficiency η=1), we show in the first case an asymptotic mean purification speed-up of 4 when compared with a standard (classical) single-detector measurement. However, by the second measure-the mean time of first passage T(ε) of the purity-the corresponding speed-up is only 2. We explain these speed-ups using the isotropy of the qubit evolution that provides an equivalence between the original measurement directions and three simultaneous measurements, one with an axis aligned along the Bloch vector and the other with axes in the two complementary directions. For inefficient detectors, η=1 - δ < 1, the mean time of first passage T(δ,ε)increases because qubit purification competes with an isotropic qubit dephasing. In the asymptotic high-purity limit (ε,δ≪1), we show that the increase possesses a scaling behaviour: ΔT(δ,ε) is a function only of the ratio δ/ε. The increase ΔT(δ,ε) is linear for small arguments, but becomes exponential ~exp(δ,2ε) for δ/ε large.

Complete control of a matter qubit using a single picosecond laser pulse

We demonstrate that a matter physical two-level system, a qubit, can be fully controlled using one ultrafast step. We show that the spin state of an optically excited electron, an exciton, confined in a quantum dot, can be rotated by any desired angle, about any desired axis, during such a step. For this we use a single, resonantly tuned, picosecond long, polarized optical pulse. The polarization of the pulse defines the rotation axis, while the pulse detuning from a nondegenerate absorption resonance defines the magnitude of the rotation angle. We thereby achieve a high fidelity, universal gate operation, applicable to other spin systems, using only one short optical pulse. The operation duration equals the pulse temporal width, orders of magnitude shorter than the qubit evolution life and coherence times.

Relaxation equations for the qubit in the tomographic representation

Within the framework of the tomographic-probability representation of quantum mechanics, we revisit the problem of the qubit evolution and show that the dynamics can be efficiently separated into two processes, namely, the unitary-like rotation and the relaxation. We study both types of evolution and derive analogs of the master equation for tomographic probabilities. We derive the tomographic relaxation equation for the qubit spin tomogram and investigate the properties of its solution for both anisotropic and isotropic cases of the relaxation. We analyze the relation between the spin tomographic description and the symmetric informationally complete description of the qubit time evolution.

Informational Description of the Flux Qubit Evolution

Lindblad-type equation is used to describe flux qubit evolution. In the frame of informational approach two types of flux qubit - three-junction qubit based on low-inductive π -interferometer and two-junction qubit based on low-inductive φ-interferometer - are considered and compared. Such approach to flux qubit analysis can be used to study the coherence state decay in the measurement procedure.

Anomalous decoherence and absence of thermalization in a photonic many-body system

The intention of this work is twofold, first to present a most simple system capable of simulating the intrinsic bosonic Josephson effect with photons, and second to study various outcomes deriving from inherent or external decoherence. A qubit induces an effective coupling between two externally pumped cavity modes. Without cavity losses and in the dispersive regime, intrinsic Josephson oscillations of photons between the two modes occurs. In this case, contrary to regular Markovian decoherence, the qubit purity shows a Gaussian decay and recurrence of its coherence. Due to intrinsic non-linearities, both the Josephson oscillations as well as the qubit properties display a rich collapse-revival structure, where, however, the complexity of the qubit evolution is in some sense stronger. The qubit as a meter of the photon dynamics is considered, and it is shown that qubit dephasing, originating for example from non-demolition measurements, results in an exponential destruction of the oscillations which manifests the collectiveness of the Josephson effect. Non-selective qubit measurements, on the other hand, render a Zeno effect seen in a slowing down of the Josephson oscillations. Contrary to dephasing, cavity dissipation results in a Gaussian decay of the scaled Josephson oscillations. Finally, following Ponomarev et al. [Phys. Rev. Lett. 106, 010405 (2011)] we analyze aspects of thermalization. In particular, despite similarities with the generic model studied by Ponomarev {\it et al.}, our system does not seem to thermalize.

Influence of external fields and environment on the dynamics of a phase-qubit–resonator system

We analyze the dynamics of a qubit-resonator system coupled with a thermal bath and external electromagnetic fields. Using the evolution equations for the set of Heisenberg operators, that describe the whole system, we derive an expression for the resonator field, accounting for the resonator-drive,-bath, and -qubit interaction. The renormalization of the resonator frequency, caused by the qubit-resonator interaction, is accounted for. Using solutions for the resonator field, we derive the equation describing qubit dynamics. The influence of the qubit evolution during the measurement time on the fidelity of a single-shot measurement is studied. The relation between the fidelity and measurement time is shown explicitly. Also, an expression describing relaxation of the superposition qubit state towards its stationary value is derived. The possibility of controlling this state, by varying the amplitude and frequency of drive, is shown.

Memory-keeping effects and forgetfulness in the dynamics of a qubit coupled to a spin chain

Using recently proposed measures for non-Markovianity [H.-P. Breuer, E. M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009)], we study the dynamics of a qubit coupled to a spin environment via an energy-exchange mechanism. We show the existence of a point, in the parameter space of the system, where the qubit dynamics is effectively Markovian and that such a point separates two regions with completely different dynamical behaviors. Indeed, our study demonstrates that the qubit evolution can in principle be tuned from a perfectly forgetful one to a deep non-Markovian regime where the qubit is strongly affected by the dynamical backaction of the environmental spins. By means of a theoretical quantum process tomography analysis, we provide a complete and intuitive characterization of the qubit channel.

Entanglement evolution in a five qubit error correction code

In this paper I explore the entanglement evolution of qubits that are part of a five qubit quantum error correction code subject to various decohering environments. Specifically, I look for possible parallels between the entanglement degradation and the fidelity of the logical qubit of quantum information stored in the physical qubits. In addition, I note the possible exhibition of entanglement sudden death (ESD) due to decoherence and question whether ESD is actually a roadblock to successful quantum computation.

Towards optimized suppression of dephasing in systems subject to pulse timing constraints

We investigate the effectiveness of different dynamical decoupling protocols for storage of a single qubit in the presence of a purely dephasing bosonic bath, with emphasis on comparing quantum coherence preservation under uniform vs. non-uniform delay times between pulses. In the limit of instantaneous bit-flip pulses, this is accomplished by establishing a new representation of the controlled qubit evolution, where the resulting decoherence behaviour is directly expressed in terms of the free evolution. Simple analytical expressions are given to approximate the long- and short- term coherence behaviour for both ohmic and supra-ohmic environments. We focus on systems with physical constraints on achievable time delays, with emphasis on pure dephasing of excitonic qubits in quantum dots. Our analysis shows that little advantage of high-level decoupling schemes based on concatenated or optimal design is to be expected if operational constraints prevent pulses to be applied sufficiently fast. In such constrained scenarios, we demonstrate how simple modifications of repeated periodic echo protocols can offer significantly improved coherence preservation in realistic parameter regimes.

Spin in CdTe/ZnTe Quantum Dot: Its Potential for Information Storage

Inventions of transistor [1] and integrated circuit [2] established a close link between semiconductor physics and information processing. Rapid development of the field, aptly described by Moore’s law [3], based on optimization of growth and processing techniques as well as on progress in designing complex CPU circuits. Quantum phenomena, inevitable in nanoscale systems, have been considered a limitation to further development of classical electronics according to Moore’s law. Simultaneously, quantum mechanics was recognized as an opportunity to extend the bounds of classical computational paradigm through an idea of quantum computer, described by Deutsch in 1985 [4]. Physical realization of proper building blocks for quantum computer — qubits conform to certain requirements [5] — exhibits major experimental difficulty. Many different approaches were proposed, including trapped ions [6], cavity QED systems [7], liquid state NMR [8], gated quantum dots [9], superconductors [10], crystal lattice impurities [11], and others. Among all these possible solutions, solid-state systems are particularly promising due to relative ease of qubit manipulation, e.g., by electric, optical or magnetic fields. Prospects of future integration of qubits with existing classical electronics also constitute a significant advantage. Drawbacks of the solid state-based systems are related mainly to short coherence times limited by the interaction with the environment. The interaction with the surrounding crystal matrix may be suppressed by e.g. constraining otherwise mobile carriers inside potential well of a quantum dot. This paper presents a selection of our studies of spins in self-assembled CdTe/ZnTe quantum dots from the point of view of quantum information processing. We will discuss three basic processes related to a single quantum dot used as a qubit, that is read-out of encoded information, writing (initialization) the qubit state and evolution of the qubit state which provides means to manipulate it. 2. Samples

Unusual decoherence in qubit measurements with a Bose-Einstein condensate

We consider an electrostatic qubit located near a Bose-Einstein condensate (BEC) of noninteracting bosons in a double-well potential, which is used for qubit measurements. Tracing out the BEC variables we obtain a simple analytical expression for the qubit's density matrix. The qubit's evolution exhibits a slow $(\ensuremath{\propto}1/\sqrt{t})$ damping of the qubit's coherence term, which however turns to be a Gaussian one in the case of static qubit. This is in contrast to the exponential damping produced by most classical detectors. The decoherence is, in general, incomplete and strongly depends on the initial state of the qubit.

Quantum decoherence of a charge qubit in a spin-fermion model

We consider quantum decoherence in solid-state systems by studying the transverse dynamics of a single qubit interacting with a fermionic bath and driven by external pulses. Our interest is in investigating the extent to which the lost coherence can be restored by the application of external pulses to the qubit. We show that the qubit evolution under various pulse sequences can be mapped onto Keldysh path integrals. This approach allows a simple diagrammatic treatment of different bath excitation processes contributing to qubit decoherence. We apply this theory to the evolution of the qubit coupled to the Andreev-fluctuator bath in the context of widely studied superconducting qubits. We show that charge fluctuations within the Andreev-fluctuator model lead to a $1/f$ noise spectrum with a characteristic temperature depedence. We discuss the strategy for suppression of decoherence by the application of higher-order (beyond spin echo) pulse sequences.

Dynamical creation of entanglement by homodyne-mediated feedback

For two two-level atoms coupled to a single Bosonic mode that is driven and heavily damped, the steady state can be entangled by resonantly driving the system [S. Schneider and G. J. Milburn, Phys. Rev. A 65, 042107 (2002)]. We present a scheme to significantly increase the steady-state entanglement by using homodyne-mediated feedback, in which the Bosonic mode is that of an electromagnetic cavity, the output of which is measured and the resulting homodyne photocurrent is used to modulate the field driving the qubits. Such feedback can increase the nonlinear response to both the decoherence process of the two-qubit system and the coherent evolution of individual qubits. We present the properties of the entangled states using the SO(3) $Q$ function.

Moving atom-field interaction: Quantum motional decoherence and relaxation

The reduced dynamics of an atomic qubit coupled both to its own quantized center of mass motion through the spatial mode functions of the electromagnetic field, as well as the vacuum modes, is calculated in the influence functional formalism. The formalism chosen can describe the entangled non-Markovian evolution of the system with a full account of the coherent back-action of the environment on the qubit. We find a slight increase in the decoherence due to the quantized center of mass motion and give a condition on the mass and qubit resonant frequency for which the effect is important. In optically resonant alkali-metal atom systems, we find the effect to be negligibly small. The framework presented here can nevertheless be used for general considerations of the coherent evolution of qubits in moving atoms in an electromagnetic field.

Thermal noise in a solid state quantum computer

The Kane proposal for the construction of a solid state quantum computer calls for the placement of a metallic A-gate above each nuclear qubit. Voltage biases applied to these A-gates are to be used to control the evolution of the qubits. In this paper, we consider the decohering effect of thermal fluctuations, the so-called Johnson noise, in the A-gate biases, on the average evolution of qubits undergoing single qubit rotations, as well as those that are not undergoing any operation at all. We find that although the errors introduced are of a different nature for each case, phase errors for the static qubits and both phase and bit-flip errors for qubits undergoing operations, the net error probability is similar for both, and low enough when compared to the timescale of single qubit operations for them to be corrected efficiently.

Energy barrier to decoherence

We formulate a novel ground state quantum computation approach that requires no unitary evolution of qubits in time: the qubits are fixed in stationary states of the Hamiltonian. This formulation supplies a completely time-independent approach to realizing quantum computers. We give a concrete suggestion for a ground state quantum computer involving linked quantum dots.