D. Coppersmith
We define an approximate version of the Fourier transform on $2^L$ elements, which is computationally attractive in a certain setting, and which may find application to the problem of factoring integers with a quantum computer as is currently under investigation by Peter Shor. (1994 IBM Internal Report)
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The scheme of Clauser and Dowling (Phys. Rev. A 53, 4587 (1996)) for factoring $N$ by means of an N-slit interference experiment is translated into an experiment with a single Mach-Zehnder interferometer. With dispersive phase shifters the ratio of the coherence length to wavelength limits the numbers that can be factored. A conservative estimate permits $N \approx 10^7$. It is furthermore shown, that sine and cosine Fourier coefficients of a real periodic function can be obtained with such an interferometer.
The preprocessing stage of Shor's algorithm generates a class of quantum states referred to as periodic states, on which the quantum Fourier transform is applied. Such states also play an important role in other quantum algorithms that rely on the quantum Fourier transform. Since entanglement is believed to be a necessary resource for quantum computational speedup, we analyze the entanglement of periodic states and the way it is affected by the quantum Fourier transform. To this end, we derive a formula that evaluates the Groverian entanglement measure for periodic states. Using this formula, …
The intermediate quantum states of multiple qubits, generated during the operation of Shor's factoring algorithm are analyzed. Their entanglement is evaluated using the Groverian measure. It is found that the entanglement is generated during the pre-processing stage of the algorithm and remains nearly constant during the quantum Fourier transform stage. The entanglement is found to be correlated with the speedup achieved by the quantum algorithm compared to classical algorithms.
We consider a version of Shor's quantum factoring algorithm such that the quantum Fourier transform is replaced by an extremely simple one where decomposition coefficients take only the values of $1,i,-1,-i$. In numerous calculations which have been carried out so far, our algorithm has been surprisingly stable and never failed. There are numerical indications that the probability of period finding given by the algorithm is a slowly decreasing function of the number to be factorized and is typically less than in Shor's algorithm. On the other hand, quantum computer (QC), capable of implementin…
Lectures on quantum computing. Contents: Algorithms. Quantum circuits. Quantum Fourier transform. Elements of number theory. Modular exponentiation. Shor`s algorithm for finding the order. Computational complexity of Schor`s algorithm. Factoring integers. NP-complete problems.
A Sharp Noise Threshold for Shor's Quantum Factoring and Discrete Log Algorithms
本研究为Shor量子因式分解与离散对数算法确定了尖锐且趋于零的噪声阈值，低于阈值算法在期望多项式时间内成功，高于阈值则对正密度素数集合可证明失败。
Amongst the most remarkable successes of quantum computation are Shor's efficient quantum algorithms for the computational tasks of integer factorisation and the evaluation of discrete logarithms. In this article we review the essential ingredients of these algorithms and draw out the unifying generalization of the so-called abelian hidden subgroup problem. This involves an unexpectedly harmonious alignment of the formalism of quantum physics with the elegant mathematical theory of group representations and fourier transforms on finite groups. Finally we consider the non-abelian hidden subgrou…
We study the results of a compiled version of Shor's factoring algorithm on the ibmqx5 superconducting chip, for the particular case of $N=15$, $21$ and $35$. The semi-classical quantum Fourier transform is used to implement the algorithm with only a small number of physical qubits and the circuits are designed to reduce the number of gates to the minimum. We use the square of the statistical overlap to give a quantitative measure of the similarity between the experimentally obtained distribution of phases and the predicted theoretical distribution one for different values of the period. This …
The quantum Fourier transform (QFT) is sometimes said to be the source of various exponential quantum speed-ups. In this paper we introduce a class of quantum circuits which cannot outperform classical computers even though the QFT constitutes an essential component. More precisely, we consider normalizer circuits. A normalizer circuit over a finite Abelian group is any quantum circuit comprising the QFT over the group, gates which compute automorphisms and gates which realize quadratic functions on the group. We prove that all normalizer circuits have polynomial-time classical simulations. Th…
The quantum Fourier transform (QFT) has been implemented on a three bit nuclear magnetic resonance (NMR) quantum computer, providing a first step towards the realization of Shor's factoring and other quantum algorithms. Implementation of the QFT is presented with fidelity measures, and state tomography. Experimentally realizing the QFT is a clear demonstration of NMR's ability to control quantum systems.
The quantum algorithms of Deutsch, Simon and Shor are described in a way which highlights their dependence on the Fourier transform. The general construction of the Fourier transform on an Abelian group is outlined and this provides a unified way of understanding the efficacy of these algorithms. Finally we describe an efficient quantum factoring algorithm based on a general formalism of Kitaev and contrast its structure to the ingredients of Shor's algorithm.
This is an expository talk written for the Bourbaki Seminar. After a brief introduction, Section 1 discusses in the categorical language the structure of the classical deterministic computations. Basic notions of complexity icluding the P/NP problem are reviewed. Section 2 introduces the notion of quantum parallelism and explains the main issues of quantum computing. Section 3 is devoted to four quantum subroutines: initialization, quantum computing of classical Boolean functions, quantum Fourier transform, and Grover's search algorithm. The central Section 4 explains Shor's factoring algorith…
Factoring an integer with three oscillators and a qubit
本文提出了一种基于连续变量傅里叶变换的因数分解算法，利用单量子比特和三个振子系统实现多项式时间复杂度的整数因数分解，不同于Shor算法及基于量子比特-振子编码的连续变量实现。
We isolate and generalize a technique implicit in many quantum algorithms, including Shor's algorithms for factoring and discrete log. In particular, we show that the distribution sampled after a Fourier transform over ${\mathbb Z}_p$ can be efficiently approximated by transforming over ${\mathbb Z}_q$ for any q in a large range. Our result places no restrictions on the superposition to be transformed, generalizing the result implicit in Shor which applies only to periodic superpositions. In addition, our proof easily generalizes to multi-dimensional transforms for any constant number of dimen…
Ideal quantum algorithms usually assume that quantum computing is performed continuously by a sequence of unitary transformations. However, there always exist idle finite time intervals between consecutive operations in a realistic quantum computing process. During these delays, coherent "errors" will accumulate from the dynamical phases of the superposed wave functions. Here we explore the sensitivity of Shor's quantum factoring algorithm to such errors. Our results clearly show a severe sensitivity of Shor's factorization algorithm to the presence of delay times between successive unitary tr…
We report an experimental demonstration of a complied version of Shor's algorithm using four photonic qubits. We choose the simplest instance of this algorithm, that is, factorization of N=15 in the case that the period $r=2$ and exploit a simplified linear optical network to coherently implement the quantum circuits of the modular exponential execution and semi-classical quantum Fourier transformation. During this computation, genuine multiparticle entanglement is observed which well supports its quantum nature. This experiment represents a step toward full realization of Shor's algorithm and…
We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n + log log (1/epsilon)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2^n with error bounded by epsilon. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log (n/epsilon)). We also give an upper bound of O(n (log n)^2 log log n) on the circuit size of the exact QFT modulo 2^n, for which t…
From Period Finding to Lattice Sampling: Experimental Insights into Shor's and Regev's Factoring Algorithms
在N=15的量子硬件上实验比较Shor和Regev因式分解算法，分析噪声、电路深度和有限采样对一维与高维傅里叶采样的影响，揭示算法鲁棒性与失效模式。
Since Shor's proposition of the method for factoring products of prime numbers using quantum computing, there has been a quest to implement efficient quantum arithmetic algorithms. These algorithms are capable of applying arithmetic operations simultaneously on large sets of values using quantum parallelism. Draper proposed an addition algorithm based on the quantum Fourier transform whose operands are two quantum registers, which I refer to as register-by-register addition. However, for cases where there is the need to be added a constant value to a target register, Draper's algorithm is more…
The remarkable capability of quantum Fourier transformation (QFT) to extract the periodicity of a given periodic function has been exhibited by using nuclear magnetic resonance (NMR) techniques. Two separate sets of experiments were performed. In a full QFT, the periodicity were validated with state tomography and fidelity measurements. For a simplified QFT, the three-qubit pseudo-pure state was created by introducting an additional observer spin, and the spectra recorded on the observer spin showed intuitively the power of QFT\ to find the periodicity. Experimentally realizing the QFT provide…
The conventional Quantum Fourier Transform, with exponential speedup compared to the classical Fast Fourier Transform, has played an important role in quantum computation as a vital part of many quantum algorithms (most prominently, the Shor's factoring algorithm). However, situations arise where it is not sufficient to encode the Fourier coefficients within the quantum amplitudes, for example in the implementation of control operations that depend on Fourier coefficients. In this paper, we detail a new quantum algorithm to encode the Fourier coefficients in the computational basis, with succe…
Shor's algorithm (SA) is a quantum algorithm for factoring integers. Since SA has polynomial complexity while the best classical factoring algorithms are sub-exponential, SA is cited as evidence that quantum computers are more powerful than classical computers. SA is critically dependent on the Quantum Fourier Transform (QFT) and it is known that the QFT is sensitive to errors in the quantum state input to it. In this paper, we show that the polynomial scaling of SA is destroyed by input errors to the QFT part of the algorithm. We also show that Quantum Error Correcting Codes (QECC) are not ca…
A novel quantum circuit for the quantum Fourier transform
本研究提出量子傅里叶变换（QFT）的新型因式分解方法，据此设计出效率优于传统方案的新型QFT量子电路，并基于该电路开发出性能更优的HHL算法与Shor算法改进版本。
The Fourier transform proves indispensable in the processing of classical information as well as in the quantum domain, where it finds many applications ranging from state reconstruction to prime factoring. An implementation scheme of the $d$-dimensional Fourier transform acting on single photons is known that uses the path encoding and requires $O(d \log d)$ optical elements. In this paper we present an alternative design that uses the orbital angular momentum as a carrier of information and needs only $O(\sqrt{d}\log d)$ elements, rendering the path-encoded design inefficient. The advantageo…
The quantum Fourier transform (QFT) has emerged as the primary tool in quantum algorithms which achieve exponential advantage over classical computation and lies at the heart of the solution to the abelian hidden subgroup problem, of which Shor's celebrated factoring and discrete log algorithms are a special case. We begin by addressing various computational issues surrounding the QFT and give improved parallel circuits for both the QFT over a power of 2 and the QFT over an arbitrary cyclic group. These circuits are based on new insight into the relationship between the discrete Fourier transf…
Many quantum algorithms, including Shor's celebrated factoring and discrete log algorithms, proceed by reduction to a hidden subgroup problem, in which a subgroup H of a group G must be determined from a quantum state y uniformly supported on a left coset of H. These hidden subgroup problems are then solved by Fourier sampling: the quantum Fourier transform of y is computed and measured. When the underlying group is non-Abelian, two important variants of the Fourier sampling paradigm have been identified: the weak standard method, where only representation names are measured, and the strong st…
The ability to implement the Quantum Fourier Transform (QFT) efficiently on a quantum computer facilitates the advantages offered by a variety of fundamental quantum algorithms, such as those for integer factoring, computing discrete logarithm over Abelian groups, solving systems of linear equations, and phase estimation, to name a few. The standard fault-tolerant implementation of an $n$-qubit unitary QFT approximates the desired transformation by removing small-angle controlled rotations and synthesizing the remaining ones into Clifford+T gates, incurring the T-count complexity of $O(n \log^…
Highly-efficient quantum Fourier transformations for some nonabelian groups
本文针对高能物理中若干非阿贝尔群提出了快速量子傅里叶变换，推导出显式量子电路并估计容错资源扩展，证明可将模拟成本降低多达三个数量级。
Fourier transform is an essential ingredient in Shor's factoring algorithm. In the standard quantum circuit model with the gate set $\{\U(2), \textrm{CNOT}\}$, the discrete Fourier transforms $F_N=(ω^{ij})_{N\times N},i,j=0,1,..., N-1, ω=e^{\frac{2πi}{N}}$, can be realized exactly by quantum circuits of size $O(n^2), n=\textrm{log}N$, and so can the discrete sine/cosine transforms. In topological quantum computing, the simplest universal topological quantum computer is based on the Fibonacci (2+1)-topological quantum field theory (TQFT), where the standard quantum circuits are replaced by unit…
We find a new integration transformation which can convert a chirplet function to fractional Fourier transformation kernel, this new transformation is invertible and obeys Parseval theorem. Under this transformation a new relationship between a phase space function and its Weyl-Wigner quantum correspondence operator is revealed.
The problem of bound states in a double delta potential is revisited by means of Fourier sine and cosine transforms
Quantum Fourier transformation is important in many quantum algorithms. In this paper, we generalize quantum Fourier transformation over the Abelian group $\mathbb{Z}_N$ from two different points to get more efficient unitary transformations. The obtained unitary transformations are given in concise and explicit formula which can be used directly. A relationship between the generalized quantum Fourier transformation and the dihedral hidden subgroup problem are discussed in this paper. This may lead a way to solve the dihedral hidden subgroup problem. The second goal of this paper is to give an…
This paper has been withdrawn.
NNR siganal will be enhanced by phase pre-whitening of presession of spin followed by quantum Fourier transform. FFT cannot the business as the phase is random.
Quantum Fourier transform is of primary importance in many quantum algorithms. In order to eliminate the destructive effects of decoherence induced by couplings between the quantum system and its environment, we propose a robust scheme for quantum Fourier transform over the intrinsic decoherence-free subspaces. The scheme is then applied to the circuit design of quantum Fourier transform over quantum networks under collective decoherence. The encoding efficiency and possible improvements are also discussed.
In this paper, we introduce an optical analogy to quantum Fourier tanformation based on a pseudorandom phase ensemble. The optical analogy also brings about exponential speedup over classical Fourier tanformation. Using the analogy, we demonstrate three classcial fields to realize Fourier transform similar to three quantum particles.
Fourier transforms are ubiquitous mathematical tools in basic and applied sciences. We here report classical and quantum optical realizations of the discrete fractional Fourier transform, a generalization of the Fourier transform. In the integrated configuration used in our experiments, the order of the transform is mapped onto the longitudinal coordinate, thus opening up the prospect of simultaneously observing all Transformation orders. In the context of classical optics, we implement discrete fractional Fourier transforms, both integer and fractional, of exemplary wave functions and experim…
We show, in a formal way, how a class of complex quasiprobability distribution functions may be introduced by using the fractional Fourier transform. This leads to the Fresnel transform of a characteristic function instead of the usual Fourier transform. We end the manuscript by showing a way in which the distribution we are introducing may be reconstructed by using atom-field interactions.
The Code Equivalence problem is that of determining whether two given linear codes are equivalent to each other up to a permutation of the coordinates. This problem has a direct reduction to a nonabelian hidden subgroup problem (HSP), suggesting a possible quantum algorithm analogous to Shor's algorithms for factoring or discrete log. However, we recently showed that in many cases of interest---including Goppa codes---solving this case of the HSP requires rich, entangled measurements. Thus, solving these cases of Code Equivalence via Fourier sampling appears to be out of reach of current famil…
Previously, Bennet and Feynman asked if Heisenberg's uncertainty principle puts a limitation on a quantum computer (Quantum Mechanical Computers, Richard P. Feynman, Foundations of Physics, Vol. 16, No. 6, p597-531, 1986). Feynman's answer was negative. In this paper, we will revisit the same question for the discrete time Fourier transform uncertainty principle. We will show that the discrete time Fourier transform uncertainty principle plays a fundamental role in showing that Shor's type of quantum algorithms has efficient running time and conclude that the discrete time uncertainty principl…
The present chapter [submitted for publication in "Fourier Transforms, Theory and Applications", G. Nikolic (Ed.), InTech (Open Access Publisher), Vienna, 2011] is concerned with the introduction and study of a quadratic discrete Fourier transform. This Fourier transform can be considered as a two-parameter extension, with a quadratic term, of the usual discrete Fourier transform. In the case where the two parameters are taken to be equal to zero, the quadratic discrete Fourier transform is nothing but the usual discrete Fourier transform. The quantum quadratic discrete Fourier transform plays…
Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical representation transformation and the method of integration within normal ordered product (IWOP) of operators, we find the key point for composing FrFT, and reveal the structure of FrFT. Following this procedure, a full family of generalized fractional transformations are discovered with the usual FrFT as one special case. The eigen-functions of arbitrary GFrT are de…
We introduce two methods for speeding up adiabatic quantum computations by increasing the energy between the ground and first excited states. Our methods are even more general. They can be used to shift a Hamiltonian's density of states away from the ground state, so that fewer states occupy the low-lying energies near the minimum, hence allowing for faster adiabatic passages to find the ground state with less risk of getting caught in an undesired low-lying excited state during the passage. Even more generally, our methods can be used to transform a discrete optimization problem into a new on…
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