Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Regional conference series in mathematics

Topological Quantum Computation

Zhenghan Wang

Published Apr 21, 2010• 574 citations
Publisher

Abstract

A polynomial depth quantum circuit effects, by definition a polylocal unitary transformation of tensor product state space. It is a physically reasonable belief [Fy][L][FKW] that these are precisely the transformations which will be available from physics to help us solve computational problems. The poly-locality of discrete Fourier transform on cyclic groups is at the heart of Shor’s factoring algorithm. We describe a class of poly-local transformations, including all the discrete orthogonal wavelet transforms in the hope that these may be helpful in constructing new quantum algorithms. We also observe that even a rather mild violation of poly-locality leads to a model without oneway functions, giving further evidence that poly-locality is an essential concept. 1

DOI
10.1090/cbms/112

Related papers

Large quantum Fourier transforms are never exactly realized by braiding conformal blocks

Michael Hartley Freedman, Zhenghan Wang

Fourier transform is an essential ingredient in Shor's factoring algorithm. In the standard quantum circuit model with the gate set {$\mathbb{U}(2)$, controlled-NOT}, the discrete Fourier transforms ${F}_{N}={({\ensuremath{\omega}}^{ij})}_{N\ifmmode\times\else\texttimes\fi{}N}$, $i,j=0,1,\dots{},N\ensuremath{-}1$, $\ensuremath{\omega}={e}^{2\ensuremath{\pi}i∕N}$, can be realized exactly by quantum circuits of size $O({n}^{2})$, $n=\mathrm{ln}\phantom{\rule{0.2em}{0ex}}N$, and so can the discrete sine or 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 unitary transformations realized by braiding conformal blocks. We report here that the large Fourier transforms ${F}_{N}$ and the discrete sine or cosine transforms can never be realized exactly by braiding conformal blocks for a fixed TQFT. It follows that an approximation is unavoidable in the implementation of Fourier transforms by braiding conformal blocks.

arXiv:cond-mat/0609411•Physical Review A•Mar 15, 2007•58 citations

The Quantum Fourier Transform and Extensions of the Abelian Hidden Subgroup Problem

Lisa Hales

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 transform over different cyclic groups. We then exploit this insight to extend the class of hidden subgroup problems with efficient quantum solutions. First we relax the condition that the underlying hidden subgroup function be distinct on distinct cosets of the subgroup in question and show that this relaxation can be solved whenever G is a finitely-generated abelian group. We then extend this reasoning to the hidden cyclic subgroup problem over the reals, showing how to efficiently generate the bits of the period of any sufficiently piecewise-continuous function on R. Finally, we show that this problem of period-finding over R, viewed as an oracle promise problem, is strictly harder than its integral counterpart. In particular, period-finding over R lies outside the complexity class MA, a class which contains period-finding over the integers.

arXiv:quant-ph/0212002•Nov 30, 2002•25 citations

Low-depth quantum architectures for factoring

Aram W. Harrow, Paul Pham

Quantum computing is a new field which combines computer science and quantum physics. Its most famous result, Shor's factoring algorithm, would enable us to one day compromise the widely-used RSA cryptosystem if we are able to design efficient quantum architectures. Studying the depth of these architectures would allow us to solve this human problem within a human lifetime. Toward that end, we contribute two hybrid factoring architectures in poly-logarithmic and sub-logarithmic depths, which are both exponential improvements over previous known works. We also present an improved procedure for generating quantum Fourier states useful for quantum compiling. Finally, we invent a new circuit resource called coherence which upper bounds the error-correcting effort needed in a future quantum computer. We use this to characterize a better time-space tradeoff for factoring as well as to provide configurable-depth factoring architectures.

•Jan 1, 2013•0 citations

Efficient Quantum Transforms

Peter Høyer

Quantum mechanics requires the operation of quantum computers to be unitary, and thus makes it important to have general techniques for developing fast quantum algorithms for computing unitary transforms. A quantum routine for computing a generalized Kronecker product is given. Applications include re-development of the networks for computing the Walsh-Hadamard and the quantum Fourier transform. New networks for two wavelet transforms are given. Quantum computation of Fourier transforms for non-Abelian groups is defined. A slightly relaxed definition is shown to simplify the analysis and the networks that computes the transforms. Efficient networks for computing such transforms for a class of metacyclic groups are introduced. A novel network for computing a Fourier transform for a group used in quantum error-correction is also given.

arXiv:quant-ph/9702028•Feb 12, 1997•77 citations

Quantum Fourier transform

Alexander M. Dalzell, Sam McArdle, Mario Berta +10 more

This chapter covers the quantum Fourier transform, which is an essential quantum algorithmic primitive that efficiently applies a discrete Fourier transform to the amplitudes of a quantum state. It features prominently in quantum phase estimation and Shor’s algorithm for factoring and computing discrete logarithms.

Cambridge University Press eBooks•Apr 24, 2025•0 citations

Algebraic and Mathematical Foundations of Quantum Computation: Hilbert Spaces, Group Structures, and Quantum Algorithms

Bhavesh B. Prajapati

Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.

Communications on Applied Nonlinear Analysis•Jan 24, 2025•0 citations
A polynomial depth quantum circuit effects, by definition a polylocal unitary transformation of tensor product state space. It is a physically reasonable belief [Fy][L][FKW] that these are precisely the transformations which will be available from physics to help us solve computational problems. The poly-locality of discrete Fourier transform on cyclic groups is at the heart of Shor’s factoring algorithm. We describe a class of poly-local transformations, including all the discrete orthogonal wavelet transforms in the hope that these may be helpful in constructing new quantum algorithms. We also observe that even a rather mild violation of poly-locality leads to a model without oneway functions, giving further evidence that poly-locality is an essential concept. 1
Fourier transform is an essential ingredient in Shor's factoring algorithm. In the standard quantum circuit model with the gate set {U(2)\mathbb{U}(2)U(2), controlled-NOT}, the discrete Fourier transforms ${F}_{N}={({\ensuremath{\omega}}^{ij})}_{N\ifmmode\times\else\texttimes\fi{}N}$, $i,j=0,1,\dots{},N\ensuremath{-}1$, $\ensuremath{\omega}={e}^{2\ensuremath{\pi}i∕N}$, can be realized exactly by quantum circuits of size O(n2)O({n}^{2})O(n2), n=lnNn=\mathrm{ln}\phantom{\rule{0.2em}{0ex}}Nn=lnN, and so can the discrete sine or cosine transforms. In topological quantum computing, the simplest universal topological quantum computer is based on the Fibonacci (2+1)(2+1)(2+1)-topological quantum field theory (TQFT), where the standard quantum circuits are replaced by unitary transformations realized by braiding conformal blocks. We report here that the large Fourier transforms FN{F}_{N}FN​ and the discrete sine or cosine transforms can never be realized exactly by braiding conformal blocks for a fixed TQFT. It follows that an approximation is unavoidable in the implementation of Fourier transforms by braiding conformal blocks.
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 transform over different cyclic groups. We then exploit this insight to extend the class of hidden subgroup problems with efficient quantum solutions. First we relax the condition that the underlying hidden subgroup function be distinct on distinct cosets of the subgroup in question and show that this relaxation can be solved whenever G is a finitely-generated abelian group. We then extend this reasoning to the hidden cyclic subgroup problem over the reals, showing how to efficiently generate the bits of the period of any sufficiently piecewise-continuous function on R. Finally, we show that this problem of period-finding over R, viewed as an oracle promise problem, is strictly harder than its integral counterpart. In particular, period-finding over R lies outside the complexity class MA, a class which contains period-finding over the integers.
Quantum computing is a new field which combines computer science and quantum physics. Its most famous result, Shor's factoring algorithm, would enable us to one day compromise the widely-used RSA cryptosystem if we are able to design efficient quantum architectures. Studying the depth of these architectures would allow us to solve this human problem within a human lifetime. Toward that end, we contribute two hybrid factoring architectures in poly-logarithmic and sub-logarithmic depths, which are both exponential improvements over previous known works. We also present an improved procedure for generating quantum Fourier states useful for quantum compiling. Finally, we invent a new circuit resource called coherence which upper bounds the error-correcting effort needed in a future quantum computer. We use this to characterize a better time-space tradeoff for factoring as well as to provide configurable-depth factoring architectures.
Quantum mechanics requires the operation of quantum computers to be unitary, and thus makes it important to have general techniques for developing fast quantum algorithms for computing unitary transforms. A quantum routine for computing a generalized Kronecker product is given. Applications include re-development of the networks for computing the Walsh-Hadamard and the quantum Fourier transform. New networks for two wavelet transforms are given. Quantum computation of Fourier transforms for non-Abelian groups is defined. A slightly relaxed definition is shown to simplify the analysis and the networks that computes the transforms. Efficient networks for computing such transforms for a class of metacyclic groups are introduced. A novel network for computing a Fourier transform for a group used in quantum error-correction is also given.
Quantum computation is, at bottom, a branch of applied algebra and functional analysis: the state of a quantum register is a unit vector in a finite-dimensional complex Hilbert space, computation is the action of the unitary group on that space, and the structure of the most important quantum algorithms is dictated by the representation theory of finite and continuous groups. This paper develops the mathematical scaffolding of the field from first principles and uses it to explain why quantum computers can outperform classical ones. We review complex inner-product spaces, self-adjoint and unitary operators, the spectral theorem, and the tensor-product construction that gives rise to entanglement and the exponential dimension of the state space. We then describe the qubit and its geometry, the group-theoretic classification of quantum gates (the special unitary groups, the Pauli and Clifford groups, and their Lie-algebraic generators), and the universality results that justify finite gate sets together with the Solovay–Kitaev theorem. With this apparatus we analyse the quantum Fourier transform, phase estimation, and the hidden subgroup problem, showing how Shor’s factoring algorithm, Grover’s search, and the quantum linear-systems algorithm arise as instances of a small number of algebraic templates. Finally we treat the stabilizer formalism of quantum error correction—where finite abelian groups over GF(2) play the central role—and the variational, near-term (NISQ) algorithms that dominate current practice. Throughout, the emphasis is on the algebraic invariants that make quantum speed-ups possible.