A Quantum Computer is a new type of computer which can solve problems such as factoring and database search very efficiently. The usefulness of a quantum computer is limited by the effect of two different types of errors, decoherence and inaccuracies. In this paper we show the results of simulations of a quantum computer which consider both decoherence and inaccuracies. We simulate circuits which factor the numbers 15, 21, 35, and 57 as well as circuits which use database search to solve the circuit satisfaction problem. Our simulations show that the error rate per gate is on the order of 10^-6 for a trapped ion quantum computer whose noise is kept below π/4096 per gate and with a decoherence rate of 10^-6. This is an important bound because previous studies have shown that a quantum computer can factor more efficiently than a classical computer if the error rate is of order 10^-6.