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.
Citation: Chappell JM, Lohe MA, von Smekal L, Iqbal A, Abbott D (2011) A Precise Error Bound for Quantum Phase Estimation. PLoS ONE 6(5): e19663. https://doi.org/10.1371/journal.pone.0019663
Editor: Jürgen Kurths, Humboldt University, Germany
Received: January 10, 2011; Accepted: April 2, 2011; Published: May 10, 2011
Copyright: © 2011 Chappell et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The authors have no support or funding to report.
Competing interests: The authors have declared that no competing interests exist.
Phase estimation is an integral part of Shor's algorithm  as well as many other quantum algorithms , designed to run on a quantum computer, and so an exact expression for the maximum probability of error is valuable, in order to efficiently achieve a predetermined accuracy. Suppose we wish to determine a phase angle to an accuracy of bits, which hence could be in error, with regard to the true value of , by up to , then due to the probabilistic nature of quantum computers, to achieve this we will need to add extra qubits to the quantum register in order to succeed with a probability of . Quantum registers behave like classical registers upon measurement, returning a one or a zero from each qubit. Previously, Cleve et al.  determined the following upper bound:(1)
Thus the more confident we wish to be (a small ), for the output to achieve a given precision , the more qubits, , will need to be added to the quantum register. Formulas of essentially the same functional form as Eq. (1), are produced by two other authors, in  and , due to the use of similar approximations in their derivation. For example, we have , given in . As we show in the following, these approximate error formulas are unsatisfactory in that they overestimate the number of qubits required in order to achieve a given reliability.
The phase angle is defined as follows, given a unitary operator , we produce the eigenvalue equation , for some eigenvector , and we seek to determine the phase using the quantum phase estimation procedure . The first stage in phase estimation produces, in the measurement register with a qubit basis , the state (2)
In general however, when cannot be written in an exact bit binary expansion, the inverse Fourier transform in the final stage of the phase estimation procedure yields a state(4)from which we only obtain an estimate for . That is, the coefficients of the state in the qubit basis will yield probabilities which peak at the values of closest to .
Our goal now is to derive an upper bound which avoids the approximations used in the above formulas and hence obtain a precise result.
In order to derive an improved accuracy formula for phase estimation, we initially follow the procedure given in , where it is noted, that because of the limited resolution provided by the quantum register of qubits, the phase must be approximated by the fraction , where is an integer in the range to such that is the best bit approximation to , which is less than . We then definewhich is the difference between and and where clearly . The first stage of the phase estimation procedure produces the state given by Eq. (2). Applying the inverse quantum Fourier transform to this state produces(5)where(6)
Assuming the outcome of the final measurement is , we can bound the probability of obtaining a value of such that , where is a positive integer characterizing our desired tolerance to error, where and are integers such that and . The probability of observing such an is given by(7)
This is simply the sum of the probabilities of the states within of , where(8)which is the standard result obtained from Eq. (6), in particular see equation 5.26 in . Typically at this point approximations are now made to simplify , however we proceed without approximations. We have(9)
Suppose we wish to approximate to an accuracy of , that is, we choose , using , which can be compared with Eq. 5.35 in , and if we denote the probability of failure(10)then we have(11)
This formula assumes that for a measurement , we have a successful result if we measure a state either side of within a distance of , which is the conventional assumption.
This definition of error however is asymmetric because there will be unequal numbers of states summed about the phase angle to give the probability of a successful result, because an odd number of states is being summed. We now present a definition of the error which is symmetric about .
Modified definition of error
Given an actual angle that we are seeking to approximate in the phase estimation procedure, a measurement is called successful if it lies within a certain tolerance of the true value . That is, for a measurement of state out of a possible states, the probability of failure will be(12)
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.
An optimal bound
Next, we demonstrate that the right hand side of Eq. (14) takes its maximum value at . Since we know , and since we expect the maximum value of to lie about midway between the two nearest states to generate the largest error, that is at , we will substitute , where . To maximize we need to minimize(15)as a function of . Expanding to quadratic order with a Taylor series, we seek to minimize(16)where are the coefficients of the Taylor expansion of cosecant2 in . We find by the odd symmetry of the cotangent about that(17)and so we just need to minimize(18)
Differentiating, we see we have an extremum at , and therefore has a maximum at .
That is, given a desired accuracy of bits, then if we add more bits, we have a probability of success given by , of obtaining a measurement to at least bits of accuracy. Thus we have succeeded in deriving a best possible bound for the failure rate .
Numerical calculations show that quickly approaches its asymptotic value as , and this limit gives a fairly accurate upper bound for , for greater than about 10 qubits. Using which is valid for all , and is accurate for as ,(21)
An exact form for this can be found in terms of the trigamma function, being a special case of the polygamma function as shown in Abramowitz and Stegun , Eq. 6.4.5:(22)where is the trigamma function, is the digamma function, and is the standard gamma function.
Now considering the limit, which also includes the limit because , we can find an asymptotic form in the limit of large also from , Eq. 6.4.12, namely(23)which shows that the error rate drops off exponentially with extra qubits. The formula Eq. (23) can be re-arranged to give(24)which can be compared with the previous approximate formula shown in Eq. (1).
We have checked the new error formula through simulations, by running the phase estimation algorithm on a 2-dimensional rotation matrix, and undertaking a numerical search for the rotation angle that maximizes the error , which has confirmed Eq. (20) to six decimal places.
An exact formula is derived for the probability of error in the quantum phase estimation procedure, as shown in Eq. (20). That is, to calculate accurate to a required bits with a given probability of success we add extra qubits, where is given by Eq. (20). If we have a large number of qubits then we can use Eq. (22) valid at the limit. In the limit the asymptote is found as a simple exponential form Eq. (23).
The exact formula avoids overestimating the number of qubits actually required in order to achieve a given reliability for phase estimation and we have also found this formula to be useful in confirming the operation of classical simulators of the phase estimation procedure.
Discussions with Anthony G. Williams and Sanjeev Naguleswaran during the early stages of this work are gratefully acknowledged.
Analyzed the data: MAL LvS JMC AI DA. Contributed reagents/materials/analysis tools: DA. Wrote the paper: JMC AI. Proofreading: MAL AI DA.
- 1. Shor PW (1997) Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM, J Comp 26(5): 1484–1509.
- 2. Nielsen MA, Chuang IL (2002) Quantum Computation and Quantum Information. Cambridge UK: Addison-Wesley, first edition.
- 3. Cleve R, Ekert A (1998) Quantum algorithms revisited. Proc R Soc London A 454(1969): 339–354.
- 4. Imre S, Balazs F (2002) A tight bound for probability of error for quantum counting based multiuser detection. Proc ISIT 2002: 43.
- 5. Mosca M (1999) Quantum Computer Algorithms. Ph.D. thesis, University of Oxford.
- 6. Abramowitz M, Stegun IA (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover.