Improved Bounds for Approximate Quantum Fourier Transforms

1. Improved Bounds for Approximate Quantum Fourier Transforms Donny Cheung Institute for Quantum Computing University of Waterloo Background: Quantum Fourier Transform and Phase Estimation Previous Approximate QFT Results The quantum Fourier transform (QFT) is a quantum circuit which performs a (normalized) discrete Fourier transform on the complex-valued vector of 2n probability amplitudes associated with an n-qubit quantum system. The following diagram gives a quantum circuit which performs a QFT on an n-qubit basis state |xñ = |x1x2…xnñ. The gates labelled H are Hadamard gates, and the gate Rk is a 2pi/2k phase rotation gate. Hadamard gates and controlled-U2k gates to prepare the states |0ñ + e2pi(2kf)|1ñ. When f is an integer multiple of 1/2n, these states are also the output of the QFT given f as input. So, to retrieve the value of f, we simply perform the inverse QFT on these states. This gives us an exact quantum algorithm for finding f when it is an integer multiple of 1/2n. Griffiths and Niu (1996) observed that it is not necessary for the entire inverse QFT to be performed at once. Instead, since each output qubit |xkñ is controlled only by higher-index qubits, they suggested a semi-classical version of the inverse QFT circuit in which each output qubit is obtained one at a time, staring with |xnñ, using the results from measuring previous qubits to classically determine whether a particular phase rotation gate will be applied to the current qubit. This does not affect the final result of the algorithm, but has the advantage of replacing the controlled phase rotation gates with regular phase rotation gates. Also, we can analyze the phase estimation algorithm simply by looking at each individual trial. When f is not an integer multiple of 1/2n, the quantum phase estimation algorithm will return the nearest integer multiple of 1/2n to f with probability 4/p2. With probability 8/p2, it will return one of the two nearest integer multiples of 1/2n. Barenco, Ekert, Suominen, and Törmä (1996) Barenco, et al., showed that with a threshold of m ≥ log2n+4, the AQFT-based phase estimation algorithm returned the nearest estimate to f with probability This implies that O(n3/m3) repetitions of the AQFT circuit are required in order to achieve the same success probability as the original QFT. From this analysis, they concluded that the AQFT was only useful in the presence of decoherence. Cleve and Watrous (2000) Cleve and Watrous give a lower bound of W(log n) on the depth of any circuit which could approximate a QFT operation. They demonstrate circuits with size O(n log(n/e)) and depth O(log n+log log(1/e)) which approximate the QFT with error O(e). The logarithmic-depth AQFT, where m=O(log n), also achieves the depth bound given by Cleve and Watrous. The new AQFT analysis also gives an improved value of O(e). A New Analysis of the Approximate QFT One of the drawbacks of analyzing the AQFT in terms of fidelity is that it always presumes that errors accumulate in the worst possible manner. However, in a circuit such as the AQFT, different errors are correlated to each other, and sometimes cancel each other out. The AQFT-based phase estimation algorithm can also be separated into individual semi-classical trials in order to aid analysis. In general, the probability of obtaining the correct bit from a trial, given a possible error of e2pid in the phase rotation amount is cos2(pd). In the AQFT, we distinguish two types of trials. For inputs |xkñ where k ≥ n-m+1, no rotation gates were removed between the QFT and the AQFT, so |d| can be at most the difference between f and the nearest multiple of 1/2n, magnified by a factor of 2k-1 by the controlled-U2k-1 gate. Thus, we have |d|≤ 2(k-1)/2(n+1). For the other inputs, all rotations smaller than Rm are removed, and we have |d|≤ 1/2m. We can express the probability of AQFT-based phase estimation obtaining the nearest estimate for f as: If m grows too slowly in comparison to n, this expression will approach 0 asymptotically as n approaches ¥. We are also constrained by the lower bound m=W(log n) given by Cleve and Watrous. So assume that m ≥ log2n+2, so that 2m ≥ 4n. This yields Since the restriction m ≥ log2n+2 does not differentiate the AQFT from the regular QFT until n ≥ 4, we can also give an explicit bound of Note also the bound approaches 4/2 as n approaches ¥. This indicates that the difference between the AQFT and the QFT are negligible for sufficiently large n, and are essentially equivalent in performance. This is a significant improvement over the bound given by Barenco, et al., which does not have this asymptotic property. For any real-world application of quantum phase estimation, such as integer factorization, we would expect n to be quite large. This AQFT analysis suggests some more general results. For an individual trial, the phase rotation can actually be any rotation which falls within the given bound d. This may allow for more flexibility in the implementation, as instead of implementing a specific combination of rotation gates, we may instead implement one rotation gate to within an error of d. Also, the flexibility of the individual trials allows us to repeat only those trials which have greater error probabilities, which increases the probability of correctness more efficiently. We can construct a more general model for AQFT circuits based on these ideas. The QFT is useful in solving the Phase Estimation problem: Given a unitary operator U, and eigenstate, |uñ, find the corresponding eigenvalue, l=e2i. In order to solve the Phase Estimation problem, we use The Approximate Quantum Fourier Transform The idea of an approximate quantum Fourier transform was first introduced by Coppersmith, who proposed that if the QFT could be sped up at expense of the accuracy of the algorithm, the trade-off may yield an improvement in the phase estimation algorithm. In Coppersmith’s circuits, instead of performing all phase rotations Rk that are used in the QFT circuit, we ignore any phase rotation gates which do not alter the phase of a qubit by a significant amount. Specifically, we set a positive integer, m, as a threshold parameter and form the approximate QFT circuit AQFTm by removing the phase rotation gates Rk, for which k > m. In order to evaluate this, we use the following lemma (which can be proven using induction): For any integer n ≥ 1, From this, we can derive: References Barenco, A., Ekert, A., Suominen, K.-A., Törmä, P., Approximate Quantum Fourier Transform and Decoherence, Physics Review A, 54(1), pp. 139-146 (1996). (http://www.arxiv.org/quant-ph/9601018) Cheung, D., Using Generalized Quantum Fourier Transforms in Quantum Phase Estimation Algorithms, M.Math Thesis, University of Waterloo (2003). Cheung, D., Improved Bounds for the Approximate QFT, Winter International Symposium on Information and Communications Technologies, pp. 192-197 (2004).(http://www.arxiv.org/quant-ph/0403071) Cleve, R., Watrous, J., Fast Parallel Circuits for the Quantum Fourier Transform, IEEE Symposium on Foundations of Computer Science, pp. 526-526 (2000). (http://www.arxiv.org/quant-ph/0006004) Coppersmith, D., An Approximate Fourier Transform Useful in Quantum Factoring, IBM Research Report RC19642 (1994). (http://www.arxiv.org/quant-ph/0201067) Griffiths, R. B., Niu, C.-S., Semiclassical Fourier Transform for Quantum Computation, Physical Review Letters 76, pp. 3228-3231 (1996). (http://www.arxiv.org/quant-ph/9511007)