Coppersmith, D., 1994, "An Approximate Fourier Transform Useful in Quantum Factoring", IBM Research Report No. RC19642.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....In order for Shor s algorithm to be a polynomial algorithm, the quantum Fourier transform must be performed in polynomial time. This requires [Sho94] 1) N can be represented with a polynomial number of bits, and (2) that N must be smooth, i.e. must have small prime factors. Coppersmith [Cop94] and Deutsch (unpublished, see [EJ96] independently found an e#cient construction for the QFT based on the fast Fourier transform (FFT) algorithm [Knu81] The QFT is a variant of the FFT which is based on powers of two, and only gives approximate results for periods which are not a power of ....

Coppersmith. An approximate Fourier transform useful in quantum factoring. Research Report RC 19642, IBM, 1994.

....Uf j x ij y i = j x ij y f(x) i where is the bitwise XOR, the two registers having respectively size n and m. These operators, which implement a classical function in a reversible fashion, are always self adjoint 16 , generally 15 the best known example is the quantum Fourier transform, see [15], where the least signi cant qubits of the input are transformed into the most signi cant qubits of the output and vice versa. 16 Since U 2 f j x ij y i = U f j x ij y f(x) i = j x ij y f(x) f(x) i = j x ij y i 8 create entanglement between the input and output registers and are ....

....adder This example illustrates how operator compositions, permutations and adjoining can be used in the classical preprocessing stage in order to build a complex parametric quantum operator by reusing smaller circuits. The following circuit implements the core of the quantum Fourier transform [15] for a four qubit register, where j ( i stands for 1 p 2 j 0 i e 2 i j 1 i . The circuit is di erent from that usually reported on quantum computing textbooks since the nal rearrangement of qubit lines is not performed. For this reason the corresponding unitary operator is named e ....

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D.Coppersmith, \An Approximate Fourier Transform Useful in Quantum Factoring", unpublished, Technical report IBM, Research report

....the quantum Fourier transform over Z p and runs in time O(n log n log log n log 2 1 ) The previous algorithm [36] for computing the Fourier transform for an arbitrary p takes time O(n 2 ) so Algorithm 5.1 is strictly faster for polynomial approximations. Proof: Coppersmith [16] gives an algorithm to approximate the Fourier transform over Z 2 n in time O(n log( n ) so choose s and q to be powers of two. Step 3 requires multiplying two n bit numbers which takes time n log n log log n. 5.4.2 A Better Fourier Sampling Theorem Theorem 5.3 can also be applied to ....

D. Coppersmith. An approximate fourier transform useful in quantum factoring. Technical Report RC

....diagonal or o diagonal. p 2 p 4 p 4 p 2 p 8 p 2 p 2 p 2 p 4 R R R R R R R R p 2 p 4 = p 8 Fig. 19. The standard circuit for the exact quantum Fourier transform on n qubits can be carried out in 2n 1 layers. Can it be parallelized to less than linear depth O(n 2 ) gates [6, 21]. Careful inspection shows that the QFT can in fact be parallelized to O(n) depth as shown in Figure 19 (an upside down version of which appears in [12] but it seems dicult to do any better. Cleve and Watrous [5] have shown that fast parallel circuits exist for an approximate QFT, with error ....

D. Coppersmith, \An approximate Fourier transform useful in quantum factoring." IBM Research Report RC 19642.

....we exhibit a circuit with depth O(n) for performing the QFT. Griffiths and Niu have already done this, in fact in a more natural way [3] However, perhaps the reader will enjoy a new construction using slightly different ideas. The standard quantum algorithm for the QFT takes n(n Gamma 1) 2 gates [2, 6]. One way to construct it is to reshuffle the rows of the matrix by putting the digits of the input in reverse order. Then for n = 3, for instance, we have 0 B B B B B B B B B B 1 1 1 1 1 1 1 1 1 Gamma1 1 Gamma1 1 Gamma1 1 Gamma1 1 i Gamma1 Gammai 1 i Gamma1 Gammai 1 Gammai ....

D. Coppersmith, "An approximate Fourier transform useful in quantum factoring." IBM Research Report RC 19642.

....error models. The first circuit is the Los Alamos factor 4[6] network which consists of 16 operations including a four bit Quantum FFT. Figure 1 shows a schematic of the circuit constructed from gates defined in table 1. Figure 2 shows the four bit Quantum FFT operation as defined by Coppersmith[2]. We have also developed optimized circuits for the factor 15 and factor 21 problems. The complexity of these circuits is exponential in the number of input bits (N ) but for factoring 15 and 21 N is small. Therefore the resulting size of these circuits is much smaller than can be designed using ....

Coppersmith, Don, "An Approximate Fourier Transform Useful in Quantum Factoring", Research Report RC 19642, IBM Research Division (7/12/94).

.... unitary transformation that maps the standard basis to the Fourier basis: QFT: jai j a i: It is known that if q is smooth (meaning that all factors of q are O(log q) for instance q is a power of 2) then the QFT can be implemented on a quantum computer using O( log q) 2 ) elementary gates [Cop94, Cle94, CEMM98] 5.3 Easy Case: r Divides q Assume we have picked a random x as in Section 5.1, and we want to find the corresponding period r. We can always efficiently pick some smooth q such that N 2 q 2N 2 (for instance take q a power of 2) The QFT for Z q can be implemented ....

D. Coppersmith. An approximate Fourier transform useful in quantum factoring. IBM

....i . This transformation is called the discrete quantum Fourier transform. The fact that one can efficiently implement such a quantum gate is not immediately clear, if only for the fact that the amplitudes seem to require increasing precision as m grows large. However, Deutsch and Coppersmith [Cop94] independently found an efficient solution based on the Fast Fourier Transform algorithm [Knu81] which only requires O(m 2 ) elementary quantum gates. The gate array for Shor s algorithm to find the order r of an element x (mod n) is: 0 m ( 0 m ( Sm E x n Am Upsilon Sigma Xi Pi ....

D. Coppersmith. An approximate fourier transform useful in quantum computing. Technical report, IBM Research Division, 1994.

....6 QNC F NaN 6= QP The staircase circuit A simple, perhaps minimal, example of a quantum circuit that seems hard to parallelize is the staircase circuit shown in figure 12. This kind of structure appears in the standard circuit for the quantum Fourier transform, which has O(n 2 ) gates [2, 8]. Careful inspection shows that the QFT can in fact be parallelized to O(n) depth as shown in figure 13 [5] but it seems difficult to do any better. Clearly, any fast parallel circuit for the QFT would be relevant to prime factoring and other problems the QFT is used for. If we define QP as the ....

D. Coppersmith, "An approximate Fourier transform useful in quantum factoring." IBM Research Report RC 19642.

....Fourier transform modulo 2 m can be done using O(m 2 ) quantum gates. An ffl approximation of the transform (in the 2 norm, see [5] for a general treatment of approximations) requires at most O(m log(m=ffl) quantum gates. The approximate quantum Fourier transform is due to Coppersmith [3]. Approximation of rationals by continued fractions. A variant of the Euclidean algorithm can be used to obtain the convergents of a rational by continued fractions. A careful analysis shows that even using long hand division, this algorithm takes O(n 2 ) time. See [2] Section 1.3. The ....

D. Coppersmith. An approximate Fourier transform useful in quantum factoring. Unpublished manuscript, 1994.

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Coppersmith, D., 1994, "An Approximate Fourier Transform Useful in Quantum Factoring", IBM Research Report No. RC19642.

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Coppersmith, D., 1994, "An Approximate Fourier Transform Useful in Quantum Factoring", IBM Research Report No. RC19642.

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D. Coppersmith. An approximate Fourier transform useful in quantum factoring. Technical Report RC1964.

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D. Coppersmith. An approximate Fourier transform useful in quantum factoring. Technical Report RC1964.

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D. Coppersmith, "An approximate Fourier transform useful in quantum factoring", IBM T. J. Watson Research Report RC 19642 (1994).

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D. Coppersmith, "An approximate Fourier transform useful in quantum factoring", IBM T. J. Watson Research Report RC 19642 (1994).

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D. Coppersmith, An approximate Fourier transform useful in quantum computing, IBM Technical Report RC 19642 (1994), quant-ph/0201067.

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D. Coppersmith, "An Approximate Fourier Transform Useful in Quantum Factoring", unpublished, Technical report IBM, Research report 19642, IBM, 07/12/1994

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Coppersmith D, An approximate Fourier transform useful in quantum factoring, IBM Research Report RC 1964.

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Coppersmith. An approximate Fourier transform useful in quantum factoring. Research Report RC 19642, IBM, 1994.

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Coppersmith, Don, An approximate Fourier transform useful in quantum factoring, Workshop on Quantum Computing and Communication, Gaitherburh, MD, August 18-19, 1994, (preprint, 9 pages).

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