Limits...
A precise error bound for quantum phase estimation.

Chappell JM, Lohe MA, von Smekal L, Iqbal A, Abbott D - PLoS ONE (2011)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: School of Chemistry and Physics, University of Adelaide, Adelaide, South Australia, Australia. james.m.chappell@adelaide.edu.au

ABSTRACT
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.

Show MeSH
Defining the limits of summation for the phase estimation                            error.For the cases , we show                            the measurements which are accepted as lying within the required                            distance of , shown by                            the vertical arrow, which define the limits of summation used in Eq.                            (13).
© Copyright Policy
Related In: Results  -  Collection


getmorefigures.php?uid=PMC3091865&req=5

pone-0019663-g001: Defining the limits of summation for the phase estimation error.For the cases , we show the measurements which are accepted as lying within the required distance of , shown by the vertical arrow, which define the limits of summation used in Eq. (13).

Mentions: Thus we consider the angle to be successfully measured accurate to bits, if the estimated lies in the range . Considering our previous definition Eq. (10), due to the fact that is defined to be always less than , then compared to the previous definition of , we lose the outermost state at the lower end of the summation in Eq. (11) as shown in Fig. (1). For example for , the upper bracket in Fig. (1) (representing the error bound) can only cover two states instead of three, and so the sum in Eq. (11) will now sum from 0 to 1, instead of 1 to 1, for this case.


A precise error bound for quantum phase estimation.

Chappell JM, Lohe MA, von Smekal L, Iqbal A, Abbott D - PLoS ONE (2011)

Defining the limits of summation for the phase estimation                            error.For the cases , we show                            the measurements which are accepted as lying within the required                            distance of , shown by                            the vertical arrow, which define the limits of summation used in Eq.                            (13).
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3091865&req=5

pone-0019663-g001: Defining the limits of summation for the phase estimation error.For the cases , we show the measurements which are accepted as lying within the required distance of , shown by the vertical arrow, which define the limits of summation used in Eq. (13).
Mentions: Thus we consider the angle to be successfully measured accurate to bits, if the estimated lies in the range . Considering our previous definition Eq. (10), due to the fact that is defined to be always less than , then compared to the previous definition of , we lose the outermost state at the lower end of the summation in Eq. (11) as shown in Fig. (1). For example for , the upper bracket in Fig. (1) (representing the error bound) can only cover two states instead of three, and so the sum in Eq. (11) will now sum from 0 to 1, instead of 1 to 1, for this case.

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: School of Chemistry and Physics, University of Adelaide, Adelaide, South Australia, Australia. james.m.chappell@adelaide.edu.au

ABSTRACT
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.

Show MeSH