  Quantum phase estimation is one of the key algorithms in the field of quantum
computing, but up until now, only approximate expressions have been derived for
the probability of error. We revisit these derivations, and find that by
ensuring symmetry in the error definitions, an exact formula can be found. This
new approach may also have value in solving other related problems in quantum
computing, where an expected error is calculated. Expressions for two special
cases of the formula are also developed, in the limit as the number of qubits
in the quantum computer approaches infinity and in the limit as the extra added
qubits to improve reliability goes to infinity. It is found that this formula
is useful in validating computer simulations of the phase estimation procedure
and in avoiding the overestimation of the number of qubits required in order to
achieve a given reliability. This formula thus brings improved precision in the
design of quantum computers.
