ACM Transactions on Algorithms

Generic quantum Fourier transforms

Cristopher Moore, Daniel N. Rockmore, Alexander C. Russell

Published Oct 1, 2006• 66 citations

Abstract

The quantum Fourier transform (QFT) is a principal ingredient appearing in many efficient quantum algorithms. We present a generic framework for the construction of efficient quantum circuits for the QFT by “quantizing” the highly successful separation of variables technique for the construction of efficient classical Fourier transforms. Specifically, we apply Bratteli diagrams, Gel'fand-Tsetlin bases, and strong generating sets of small adapted diameter to provide efficient quantum circuits for the QFT over a wide variety of finite Abelian and non-Abelian groups, including all families of groups for which efficient QFTs are currently known and many new families as well. Moreover, our method provides the first subexponential-size quantum circuits for the QFT over the linear groups GL k ( q ), SL k ( q ), and the finite groups of Lie type, for any fixed prime power q .

DOI: 10.1145/1198513.1198525

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ACM Transactions on Algorithms

Generic quantum Fourier transforms

Cristopher Moore, Daniel N. Rockmore, Alexander C. Russell

Published Oct 1, 2006• 66 citations

Publisher

Abstract

DOI: 10.1145/1198513.1198525

Related papers

Generic Quantum Fourier Transforms

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The quantum Fourier transform (QFT) is the principal algorithmic tool underlying most efficient quantum algorithms. We present a generic framework for the construction of efficient quantum circuits for the QFT by ``quantizing'' the separation of variables technique that has been so successful in the study of classical Fourier transform computations. Specifically, this framework applies the existence of computable Bratteli diagrams, adapted factorizations, and Gel'fand-Tsetlin bases to offer efficient quantum circuits for the QFT over a wide variety a finite Abelian and non-Abelian groups, including all group families for which efficient QFTs are currently known and many new group families. Moreover, the method gives rise to the first subexponential-size quantum circuits for the QFT over the linear groups GL_k(q), SL_k(q), and the finite groups of Lie type, for any fixed prime power q.The quantum Fourier transform (QFT) is the principal algorithmic tool underlying most efficient quantum algorithms. We present a generic framework for the construction of efficient quantum circuits for the QFT by ``quantizing'' the separation of variables technique that has been so successful in the study of classical Fourier transform computations. Specifically, this framework applies the existence of computable Bratteli diagrams, adapted factorizations, and Gel'fand-Tsetlin bases to offer efficient quantum circuits for the QFT over a wide variety a finite Abelian and non-Abelian groups, including all group families for which efficient QFTs are currently known and many new group families. Moreover, the method gives rise to the first subexponential-size quantum circuits for the QFT over the linear groups GL_k(q), SL_k(q), and the finite groups of Lie type, for any fixed prime power q.

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Quantum Fourier transforms (QFTs) are essential primitives in quantum algorithms. While abelian groups admit efficient QFT circuits, with circuit size polynomial in the logarithm of the group order, efficient constructions are known for relatively few non-abelian families. We develop two new approaches to QFT circuit construction, based on Mackey theory and Clifford theory, respectively, and use them to show exponential improvement in circuit cost for specific group families. Using the Mackey-theoretic approach, we obtain explicit quantum circuits for the QFT over $\mathrm{GL}_2(F_q)$ that scale polynomially in $\log q$ , rather than polynomially in $q$ . Using the Clifford-theoretic approach, we obtain QFT circuits for wreath products $F\wr S_n$ , whose cost depends on the cost of a QFT over $F$ and the size of its representation registers. This removes the restriction $|F|=\operatorname{poly}(n)$ required by previous generic constructions and can yield exponential improvements when $F$ itself has an efficient QFT. Together, these methods provide new systematic tools to construct QFTs for broad classes of finite groups.Quantum Fourier transforms (QFTs) are essential primitives in quantum algorithms. While abelian groups admit efficient QFT circuits, with circuit size polynomial in the logarithm of the group order, efficient constructions are known for relatively few non-abelian families. We develop two new approaches to QFT circuit construction, based on Mackey theory and Clifford theory, respectively, and use them to show exponential improvement in circuit cost for specific group families. Using the Mackey-theoretic approach, we obtain explicit quantum circuits for the QFT over $\mathrm{GL}_2(F_q)$ that scale polynomially in $\log q$, rather than polynomially in $q$. Using the Clifford-theoretic approach, we obtain QFT circuits for wreath products $F\wr S_n$, whose cost depends on the cost of a QFT over $F$ and the size of its representation registers. This removes the restriction $|F|=\operatorname{poly}(n)$ required by previous generic constructions and can yield exponential improvements when $F$ itself has an efficient QFT. Together, these methods provide new systematic tools to construct QFTs for broad classes of finite groups.

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The quantum Fourier transform (QFT) is a fundamental primitive in quantum computation and quantum information. In this work, we extend the framework for the QFT from finite groups to finite-dimensional semisimple algebras, and give efficient QFTs for the partition algebra $P_n(d)$ , Brauer algebra $B_n(d)$ , and walled Brauer algebra $B_{r,s}(d)$ . These algebras play important roles in generalized Schur-Weyl duality, statistical physics and many-body systems, and have recently found several applications in quantum algorithms. Unlike the group case, the Fourier transform over a semisimple algebra can be non-unitary. Nevertheless, we show that when the parameter $d$ is sufficiently large, the Fourier transform is well approximated by a unitary operator. Furthermore, we show that for each of the algebras $A$ from above, such an approximate Fourier transform can be implemented efficiently: we give a quantum algorithm with gate complexity $\mathrm{poly}(n,\log d,\log(1/\varepsilon))$ for approximating the Fourier transform to error $(d^{-1/2} + \varepsilon) \cdot \mathrm{poly}(|A|)$ . Along the way, we establish several properties of the Fourier basis of semisimple algebras that may be of independent interest.The quantum Fourier transform (QFT) is a fundamental primitive in quantum computation and quantum information. In this work, we extend the framework for the QFT from finite groups to finite-dimensional semisimple algebras, and give efficient QFTs for the partition algebra $P_n(d)$, Brauer algebra $B_n(d)$, and walled Brauer algebra $B_{r,s}(d)$. These algebras play important roles in generalized Schur-Weyl duality, statistical physics and many-body systems, and have recently found several applications in quantum algorithms. Unlike the group case, the Fourier transform over a semisimple algebra can be non-unitary. Nevertheless, we show that when the parameter $d$ is sufficiently large, the Fourier transform is well approximated by a unitary operator. Furthermore, we show that for each of the algebras $A$ from above, such an approximate Fourier transform can be implemented efficiently: we give a quantum algorithm with gate complexity $\mathrm{poly}(n,\log d,\log(1/\varepsilon))$ for approximating the Fourier transform to error $(d^{-1/2} + \varepsilon) \cdot \mathrm{poly}(|A|)$. Along the way, we establish several properties of the Fourier basis of semisimple algebras that may be of independent interest.

quant-pharXiv:2605.05337May 8, 20260 citations

Highly-efficient quantum Fourier transformations for certain non-Abelian groups

Edison M. Murairi, M. Sohaib Alam, Henry Lamm +2 more

Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. While the standard Abelian QFrT is well studied, important variants corresponding to non-Abelian groups of interest have seen less development. In particular, fast non-Abelian Fourier transformations are important components for both quantum simulations of field theories as well as approaches to the non-Abelian hidden subgroup problem. In this work, we present fast quantum Fourier transformations for a number of non-Abelian groups of interest for high energy physics, $\mathbb{B}\mathbb{T}$ , $\mathbb{B}\mathbb{O}$ , $6\mathrm{\ensuremath{\Delta}}(27)$, $\mathrm{\ensuremath{\Delta}}(54)$, and $\mathrm{\ensuremath{\Sigma}}(36\ifmmode\times\else\texttimes\fi{}3)$. For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations. Our work shows that the development of a fast Fourier transformation can substantively reduce simulation costs by an up to three orders of magnitude for the finite groups that we have investigated.Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. While the standard Abelian QFrT is well studied, important variants corresponding to non-Abelian groups of interest have seen less development. In particular, fast non-Abelian Fourier transformations are important components for both quantum simulations of field theories as well as approaches to the non-Abelian hidden subgroup problem. In this work, we present fast quantum Fourier transformations for a number of non-Abelian groups of interest for high energy physics, $\mathbb{B}\mathbb{T}$, $\mathbb{B}\mathbb{O}$, $6\mathrm{\ensuremath{\Delta}}(27)$, $\mathrm{\ensuremath{\Delta}}(54)$, and $\mathrm{\ensuremath{\Sigma}}(36\ifmmode\times\else\texttimes\fi{}3)$. For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations. Our work shows that the development of a fast Fourier transformation can substantively reduce simulation costs by an up to three orders of magnitude for the finite groups that we have investigated.

Physical review. D/Physical review. D.Oct 1, 202414 citations

Highly-efficient quantum Fourier transformations for some nonabelian groups

Murairi, Edison M., M. Sohaib Alam, Henry Lamm +2 more

Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. While the standard abelian QFT is well-studied, important variants corresponding to \emph{nonabelian} groups of interest have seen less development. In particular, fast nonabelian Fourier transformations are important components for both quantum simulations of field theories as well as approaches to the nonabelian hidden subgroup problem. In this work, we present fast quantum Fourier transformations for a number of nonabelian groups of interest for high energy physics, $\mathbb{BT}$ , $\mathbb{BO}$ , $Δ(27)$ , $Δ(54)$ , and $Σ(36\times3)$ . For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations. Our work shows that the development of a fast Fourier transformation can substantively reduce simulation costs by up to three orders of magnitude for the finite groups that we have investigated.Quantum Fourier transformations are an essential component of many quantum algorithms, from prime factoring to quantum simulation. While the standard abelian QFT is well-studied, important variants corresponding to \emph{nonabelian} groups of interest have seen less development. In particular, fast nonabelian Fourier transformations are important components for both quantum simulations of field theories as well as approaches to the nonabelian hidden subgroup problem. In this work, we present fast quantum Fourier transformations for a number of nonabelian groups of interest for high energy physics, $\mathbb{BT}$, $\mathbb{BO}$, $Δ(27)$, $Δ(54)$, and $Σ(36\times3)$. For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations. Our work shows that the development of a fast Fourier transformation can substantively reduce simulation costs by up to three orders of magnitude for the finite groups that we have investigated.

arXiv:2408.00075Jul 31, 20240 citations

\mathrm{GL}_2(F_q)

that scale polynomially in

\log q

, rather than polynomially in

q

. Using the Clifford-theoretic approach, we obtain QFT circuits for wreath products

F\wr S_n

, whose cost depends on the cost of a QFT over

F

and the size of its representation registers. This removes the restriction

|F|=\operatorname{poly}(n)

required by previous generic constructions and can yield exponential improvements when

F

itself has an efficient QFT. Together, these methods provide new systematic tools to construct QFTs for broad classes of finite groups.

P_n(d)

, Brauer algebra

B_n(d)

, and walled Brauer algebra

B_{r,s}(d)

. These algebras play important roles in generalized Schur-Weyl duality, statistical physics and many-body systems, and have recently found several applications in quantum algorithms. Unlike the group case, the Fourier transform over a semisimple algebra can be non-unitary. Nevertheless, we show that when the parameter

d

is sufficiently large, the Fourier transform is well approximated by a unitary operator. Furthermore, we show that for each of the algebras

A

from above, such an approximate Fourier transform can be implemented efficiently: we give a quantum algorithm with gate complexity

\mathrm{poly}(n,\log d,\log(1/\varepsilon))

for approximating the Fourier transform to error

(d^{-1/2} + \varepsilon) \cdot \mathrm{poly}(|A|)

. Along the way, we establish several properties of the Fourier basis of semisimple algebras that may be of independent interest.

\mathbb{B}\mathbb{T}

\mathbb{B}\mathbb{O}

, $6\mathrm{\ensuremath{\Delta}}(27)$, $\mathrm{\ensuremath{\Delta}}(54)$, and $\mathrm{\ensuremath{\Sigma}}(36\ifmmode\times\else\texttimes\fi{}3)$. For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations. Our work shows that the development of a fast Fourier transformation can substantively reduce simulation costs by an up to three orders of magnitude for the finite groups that we have investigated.

\mathbb{BT}

\mathbb{BO}

Δ(27)

Δ(54)

, and

Σ(36\times3)

. For each group, we derive explicit quantum circuits and estimate resource scaling for fault-tolerant implementations. Our work shows that the development of a fast Fourier transformation can substantively reduce simulation costs by up to three orders of magnitude for the finite groups that we have investigated.