A Precise Error Bound for Quantum Phase Estimation

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.


Introduction
Phase estimation is an integral part of Shor's algorithm [1] as well as many other quantum algorithms [2], 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 w to an accuracy of s bits, which hence could be in error, with regard to the true value of w, by up to 2 {s , then due to the probabilistic nature of quantum computers, to achieve this we will need to add p extra qubits to the quantum register in order to succeed with a probability of 1{e. Quantum registers behave like classical registers upon measurement, returning a one or a zero from each qubit. Previously, Cleve et al. [3] determined the following upper bound: Thus the more confident we wish to be (a small e), for the output to achieve a given precision s, the more qubits, p, 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 [2] and [4], due to the use of similar approximations in their derivation. For example, we have p~qlog 2 1 2e z2 z log 2 pr, given in [4]. 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 U, we produce the eigenvalue equation UDuT~e 2piw DuT, for some eigenvector DuT, and we seek to determine the phase w[½0,1) using the quantum phase estimation procedure [5]. The first stage in phase estimation produces, in the measurement register with a t qubit basis fDkTg, the state [2] Dw wT Stage1~1 If w~b=2 t for some integer b~0, 1 , . .
is the discrete Fourier transform of the basis state DbT, that is, the state with amplitudes x k~dkb . We then read off the exact phase w~b=2 t from the inverse Fourier transform as DbT~F { Dw wT.
In general however, when w cannot be written in an exact t bit binary expansion, the inverse Fourier transform in the final stage of the phase estimation procedure yields a state from which we only obtain an estimate for w. That is, the coefficients x k of the state DwT in the t qubit basis fDkTg will yield probabilities which peak at the values of k closest to w.
Our goal now is to derive an upper bound which avoids the approximations used in the above formulas and hence obtain a precise result.

Results
In order to derive an improved accuracy formula for phase estimation, we initially follow the procedure given in [3], where it is noted, that because of the limited resolution provided by the quantum register of t qubits, the phase w must be approximated by the fraction b 2 t , where b is an integer in the range 0 to 2 t {1 such that b=2 t~0 :b 1 . . . b t is the best t bit approximation to w, which is less than w. We then define d~w{b=2 t , which is the difference between w and b=2 t and where clearly 0ƒdv2 {t . 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 where Assuming the outcome of the final measurement is m, we can bound the probability of obtaining a value of m such that Dm{bDƒe, where e is a positive integer characterizing our desired tolerance to error, where m and b are integers such that 0ƒmv2 t and 0ƒbv2 t . The probability of observing such an m is given by This is simply the sum of the probabilities of the states within e of b, where which is the standard result obtained from Eq. (6), in particular see equation 5.26 in [2]. Typically at this point approximations are now made to simplify x ' , however we proceed without approximations. We have Suppose we wish to approximate w to an accuracy of 2 {s , that is, we choose e~2 t{s{1~2p{1 , using t~szp, which can be compared with Eq. 5.35 in [2], and if we denote the probability of failure then we have This formula assumes that for a measurement m, we have a successful result if we measure a state either side of b within a distance of e, 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 w 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 w.

Modified definition of error
Given an actual angle w that we are seeking to approximate in the phase estimation procedure, a measurement is called successful if it lies within a certain tolerance e of the true value w. That is, for a measurement of state m out of a possible 2 t states, the probability of failure will be E~p D2p Thus we consider the angle to be successfully measured accurate to s bits, if the estimated w lies in the range w+ 1 2 2p 2 s . Considering our previous definition Eq. (10), due to the fact that b is defined to be always less than w, then compared to the previous definition of E, we lose the outermost state at the lower end of the summation in Eq. (11) as shown in Fig. (1). For example for p~1, 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
Based on this new definition then for all cases we need to add 1 to the lower end of the summation giving and if we define a~2 t d and rearrange the cosine term in the summation we find Next, we demonstrate that the right hand side of Eq. (14) takes its maximum value at a~1 2 . Since we know 0ƒav1, and since we expect the maximum value of E~E(a,t,p) to lie about midway between the two nearest states to generate the largest error, that is at a~1=2, we will substitute a~1 2 zD, where D% 1 2 .
To maximize E we need to minimize as a function of D. Expanding to quadratic order with a Taylor series, we seek to minimize where c i are the coefficients of the Taylor expansion of cosecant 2 in D. We find by the odd symmetry of the cotangent about '~1 2 that and so we just need to minimize Differentiating, we see we have an extremum at D~0, and therefore E(a,t,p) has a maximum at a~1=2.
Substituting a~1 2 we obtain We note that the summation is symmetrical about '~1=2, and substituting t~pzs, we obtain for our final result E(s,p)~1{ 1 2 2(pzs){2 That is, given a desired accuracy of s bits, then if we add p more bits, we have a probability of success given by 1{E, of obtaining a measurement to at least s bits of accuracy. Thus we have succeeded in deriving a best possible bound for the failure rate E~E(s,p).