D. I. Kamenev

METHOD FOR IMPLEMENTATION OF UNIVERSAL QUANTUM LOGIC GATES IN A SCALABLE ISING SPIN QUANTUM COMPUTER

We present protocols for implementation of universal quantum gates on an arbitrary superposition of quantum states in a scalable solid-state Ising spin quantum computer. The spin chain is composed of identical spins 1/2 with the Ising interaction between the neighboring spins. The selective excitations of the spins are provided by the gradient of the external magnetic field. The protocols are built of rectangular radio-frequency pulses. Since the wavelength of the radio-frequency pulses is much larger than the distance between the spins, each pulse affects all spins in the chain and introduces the phase and probability errors, which occur even without the influence of the environment. These errors caused by the unwanted transitions are minimized and computed numerically.

Dynamical fidelity of a solid-state quantum computation

In this paper we analyze the dynamics in a spin model of quantum computer. Main attention is paid to the dynamical fidelity (associated with dynamical errors) of an algorithm that allows to create an entangled state for remote qubits. We show that in the regime of selective resonant excitations of qubits there is no danger of quantum chaos. Moreover, in this regime a modified perturbation theory gives an adequate description of the dynamics of the system. Our approach allows us to explicitly describe all peculiarities of the evolution of the system under time-dependent pulses corresponding to a quantum protocol. Specifically, we analyze, both analytically and numerically, how the fidelity decreases in dependence on the model parameters.

Perturbation theory for quantum computation with a large number of qubits

We develop a dynamical perturbation theory for solid-state quantum computation with many qubits. The errors in the implementation of simple quantum logic operations caused by nonresonant transitions are estimated. We verify our perturbation approach using exact numerical solution for relatively small

Modeling and Simulation of a Microstrip-SQUID Amplifier

(N=10)

MODELING FULL ADDER IN ISING SPIN QUANTUM COMPUTER WITH 1000 QUBITS USING QUANTUM MAPS

number of qubits. A preferred range of parameters is found in which the errors in processing quantum information are reasonably small. Our results can be useful for understanding the mechanisms of errors and for experimental testing of scalable solid-state quantum computers.

Influence of qubit displacements on quantum logic operations in a silicon-based quantum computer with constant interaction

Using a simple lumped-circuit model, we numerically study the dependence of the voltage gain and noise on the amplifier’s parameters. Linear, quasi-linear, and nonlinear regimes are studied. We have shown that the voltage gain of the amplifier cannot exceed a characteristic critical value, which decreases with the increase of the input power. We have also shown that the spectrum of the voltage gain depends significantly on the level of the Johnson noise generated by the SQUID resistors.

ANALYTIC SOLUTIONS FOR QUANTUM LOGIC GATES AND MODELING PULSE ERRORS IN A QUANTUM COMPUTER WITH A HEISENBERG INTERACTION

The quantum adder is an essential attribute of a quantum computer, just as classical adder is needed for operation of a digital computer. We model the quantum full adder as a realistic complex algorithm on a large number of qubits in an Ising spin quantum computer. Our results are an important step toward effective modeling of the quantum modular adder which is needed for Shors and other quantum algorithms. Our full adder has the following features. (i) The near-resonant transitions with small detunings are completely suppressed, which allows us to decrease errors by several orders of magnitude and to model a 1000-qubit full adder. (We add a 1000-bit number using 2001 spins.) (ii) We construct the full adder gates directly as sequences of radio-frequency pulses, rather than breaking them down into generalized logical gates, such as Control-Not and one qubit gates. This substantially reduces the number of pulses needed to implement the full adder. (The maximum number of pulses required to add one bit (F-gate) is 15.) (iii) Full adder is realized in a homogeneous spin chain. (iv) The phase error is minimized: the F-gates generate approximately the same phase for different states of the superposition. (v) Modeling of the full adder is performed using quantum maps instead of differential equations. This allows us to reduce the calculation time to a reasonable value.

MINIMIZATION OF NONRESONANT EFFECTS IN A SCALABLE ISING SPIN QUANTUM COMPUTER

The errors caused by qubit displacements from their prescribed locations in an ensemble of spin chains are estimated analytically and calculated numerically for a quantum computer based on phosphorus donors in silicon. We show that it is possible to polarize (initialize) the nuclear spins even with displaced qubits by using controlled-NOT gates between the electron and nuclear spins of the same phosphorus atom. However, a controlled-NOT gate between the displaced electron spins is implemented with large error because of the exponential dependence of exchange interaction constant on the distance between the qubits. If quantum computation is implemented on an ensemble of many spin chains, the errors can be small if the number of chains with displaced qubits is small.

IMPLEMENTATION OF QUANTUM LOGIC OPERATIONS AND CREATION OF ENTANGLEMENT BETWEEN TWO NUCLEAR SPIN QUBITS WITH CONSTANT INTERACTION

We analyze analytically and numerically quantum logic gates in a one-dimensional spin chain with Heisenberg interaction. Analytic solutions for basic one-qubit gates and swap gate are obtained for a quantum computer based on logical qubits. We calculated the errors caused by imperfect pulses which implement the quantum logic gates. It is numerically demonstrated that the probability error is proportional to e4, while the phase error is proportional to e, where e is the characteristic deviation from the perfect pulse duration.

Perturbation Approach for Quantum Computation

The errors caused by the transitions with large frequency offsets (nonresonant transitions) are calculated analytically for a scalable solid-state quantum computer based on a one-dimensional spin chain with Ising interactions between neighboring spins. Selective excitations of the spins are enabled by a uniform gradient of the external magnetic field. We calculate the probabilities of all unwanted nonresonant transitions associated with the flip of each spin with nonresonant frequency and with flips of two spins: one with resonant and one with nonresonant frequencies. It is shown that these errors oscillate with changing gradient of the external magnetic field. Choosing the optimal values of this gradient allows us to decrease these errors by 50%.