Simulating the Factoring and Database Search Problems:

We have performed detailed simulation studies of circuits which factor the 15, 21, 35, and 57 as well as database search applied to the circuit SAT problem. The simulations show that random inaccuracies (noise) are more significant than fixed magnitude inaccuracies. Also errors in the duration of the laser pulse are more significant than errors in the phase of the laser. For problems of the size considered here, noise or constant magnitude inaccuracies of magnitude or greater cause a significant impact on the fidelity.
The factoring and database search benchmarks can tolerate a decoherence rate as high as 10^-5. For the ion trap computer this corresponds roughly to a decoherence lifetime of 50 milliseconds and a switching speed of 500 nanoseconds.
The fidelity of the circuits can be used to relate the physical quantities of inaccuracies and decoherence to the probability of error per gate. An error rate per gate on the order of 10^-6 corresponds to inaccuracies of less than per laser operation and a decoherence rate of 10^-6.

Error Correction Codes:

We have investigated the effectiveness of error correction codes by analyzing code that corrects a single error per block. This code encodes one qubit into seven. Using simulation we analyze correction and encoding circuits as well as a circuit that performs computation using encoded qubits. Quantum error correction codes increase the fidelity of the calculation by correcting bit flip errors and phase errors. Operations, that perform computation, are interspersed with error correction steps. The more error that occurs in a calculation the more often the correction steps need to be applied. However error correction itself also induces error, so performing too many correction steps may increase the total amount of error. For decoherence, our simulation results show that correction starts to help when the decoherence rate is 10^-4. Correction steps in this case are only applied when there is a spontaneous emission. For operational errors, correction improves the fidelity when applied after every 150 controlled-not gates, for an error rate per gate of about 10-5.

Optimized Gates and Circuits:

The most fundamental gate for quantum computation is the controlled-not gate. A gate that conditionally flips the state of the output qubit based on one or more input control qubits. In the ion trap quantum computer this controlled-not gate is implemented using a series of laser pulses. This however is not the only type of gate possible for quantum computing. More powerful gates can be constructed out of different sequences of laser pulses.

We have defined more powerful gates as well as assessing their impact on the implementation of the quantum factoring circuit. Using the enhanced gates the number of laser pulse is reduced by 40% as compared to the best circuit using only controlled-not gates. Because there are fewer gates, the enhanced circuit exhibits a lower amount of accumulated deviation operational error. The amount of mean operational error is greater however, because there is less cancellation of errors. For decoherence there is no improvement because the enhanced circuits do not reduce the number of pulses. Decoherence of the phonon mode only occurs during these pulses. Using the enhanced gates however allows more of the pulses to be combined, reducing the total number of laser operations. Therefore if other types of decoherence were modeled, or if the switching time between pulses becomes significant, enhanced gates may decrease the total decoherence.

Papers:

"Simulating the Effect of Decoherence and Inaccuracies on a Quantum Computer." Proceedings NASA Conference on Quantum Computation and Quantum Communication, Palm Springs, CA. Feb. 1998.

Using Simulation to Assess the Feasibility of Quantum Computing Kevin M. Obenland. Ph.D. Thesis University of Southern California May 1998.

Presentations:

Kevin Obenland, Palm Springs CA, February 1998.
"Simulating the Effect of Decoherence and Inaccuracies on a Quantum Computer"
NASA Conference on Quantum Computation and Quantum Communication.

Kevin Obenland, Boston MA, April 1998.
"A Parallel Quantum Computer Simulator"
High Performance Computing '98.

Kevin Obenland, Tucson Arizona, April 1998.
"Simulating the Effect of Decoherence and Inaccuracies on a Quantum Computer"
Ultra Scale Program Principal Investigators Meeting

Kevin Obenland, Houston TX, June 1998.
"Parallel Quantum Computer Simulation"
DoD High Performance Computing Users Group Conference