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arXiv:2608.28573

Quantum Fourier transform toolbox

Carli Bruinsma, Pietro M. Posta, Joppe Stokvis, Dmitry Grinko, Maris Ozols

quant-phmath.RTSubmitted Aug 28, 2026• Announced Aug 31, 2026• 0 citations
v1 · Aug 28, 2026v2 · Sep 30, 2026
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Abstract

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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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 GL2(Fq)\mathrm{GL}_2(F_q)GL2​(Fq​) that scale polynomially in log⁡q\log qlogq, rather than polynomially in qqq. Using the Clifford-theoretic approach, we obtain QFT circuits for wreath products F≀SnF\wr S_nF≀Sn​, whose cost depends on the cost of a QFT over FFF and the size of its representation registers. This removes the restriction ∣F∣=poly⁡(n)|F|=\operatorname{poly}(n)∣F∣=poly(n) required by previous generic constructions and can yield exponential improvements when FFF itself has an efficient QFT. Together, these methods provide new systematic tools to construct QFTs for broad classes of finite groups.
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, BT\mathbb{BT}BT, BO\mathbb{BO}BO, Δ(27)Δ(27)Δ(27), Δ(54)Δ(54)Δ(54), and Σ(36×3)Σ(36\times3)Σ(36×3). 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 up to three orders of magnitude for the finite groups that we have investigated.
The quantum Fourier transform (QFT) is an important ingredient in various quantum algorithms which achieve superpolynomial speed-ups over classical computers. In this paper we study under which conditions the QFT can be simulated efficiently classically. We introduce a class of quantum circuits, called \emph{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. In our main result we prove that all normalizer circuits have polynomial-time classical simulations. The proof uses algorithms for linear diophantine equation solving and the monomial matrix formalism introduced in our earlier work. Our result generalizes the Gottesman-Knill theorem: in particular, Clifford circuits for ddd-level qudits arise as normalizer circuits over the group Zdm{\mathbf Z}_d^mZdm​. We also highlight connections between normalizer circuits and Shor's factoring algorithm, and the Abelian hidden subgroup problem in general. Finally we prove that quantum factoring cannot be realized as a normalizer circuit owing to its modular exponentiation subroutine.
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, BT\mathbb{B}\mathbb{T}BT, BO\mathbb{B}\mathbb{O}BO, $6\mathrm{\ensuremath{\Delta}}(27)$, $\mathrm{\ensuremath{\Delta}}(54)$, 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.
Abstract 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 log2 (n))O(n\,{\mathrm{log}}^{2}\,(n))O(nlog2(n)) O ( n log 2 ( n ) ) . In this paper, we show how to obtain approximate QFT with the T-count of O(n log (n))O(n\,{\mathrm{log}}\,(n))O(nlog(n)) O ( n log ( n ) ) . For brevity, the above figures omit the dependence on the approximation error ε, assuming the error is fixed. Our approach relies on quantum circuits with measurements and feedforward, and on reusing a special quantum state that induces the phase gradient transformation. We report asymptotic analysis as well as concrete circuits, demonstrating significant advantages in both theory and practice.
Quantum Fourier transforms (QFT) for general groups were recognized to be fundamental already early in the field. A canonical example of non-abelian QFT for the symmetric group was outlined by Beals (1997). Later, a more detailed analysis of this algorithm was carried out by Kawano and Sekigawa (2016). In this paper, we revisit that construction. After a careful analysis, we revise their gate complexity to O~(n3.5)\widetilde{\mathcal{O}}(n^{3.5})O(n3.5) and circuit depth to O~(n3)\widetilde{\mathcal{O}}(n^3)O(n3). Moreover, we observe that their construction is not optimal in the choice of transversal elements, so we propose simpler realization of the symmetric group QFT.
A Clifford template is a finite ordered family of repeatable Clifford operations, and an instantiation specifies how many times each operation is applied. The Clifford template compilation problem asks how to choose these repetition numbers so that the template realizes a target transformation of Pauli operators. This problem arises, for example, when searching for logical operations in quantum error correction using only Clifford operations permitted by physical or fault-tolerance constraints. Although forward Clifford dynamics is efficiently classically simulable, this inverse problem has sharp complexity transitions. For commuting templates with unrestricted integer exponents, feasibility lies in NP∩BQP\mathrm{NP}\cap\mathrm{BQP}NP∩BQP and a constructive quantum algorithm returns a particular solution together with the full exponent-relation lattice; already at k=1k=1k=1, recovering the repetition number contains finite-field discrete logarithm over F2r×\mathbb{F}_{2^r}^{\times}F2r×​. In general, restricting every exponent to {0,1}\{0,1\}{0,1} removes the Abelian-group closure and makes feasibility NP-complete for variable kkk, even for exactly commuting CNOT-only operations and X-type Paulis. For commuting self-inverse Clifford actions, both binary feasibility and recovery of one solution are classically polynomial-time solvable, but imposing a bound on the total repetition count is NP-complete, even for CNOT-only operations. These results reveal a rich complexity landscape within Clifford template compilation, spanning classically tractable cases, problems admitting quantum polynomial-time algorithms, and NP-complete variants.