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arxiv: 2605.05337 · v1 · submitted 2026-05-06 · 🪐 quant-ph

Efficient Quantum Fourier Transforms For Semisimple Algebras

Ben Foxman , Barak Nehoran , Yongshan Ding This is my paper

Pith reviewed 2026-05-08 16:45 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum Fourier transformsemisimple algebraspartition algebraBrauer algebrawalled Brauer algebraquantum algorithmsapproximate unitariesSchur-Weyl duality
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The pith

Quantum algorithms can efficiently approximate the Fourier transform over semisimple algebras like the partition and Brauer algebras by a unitary when d is large.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes the quantum Fourier transform from finite groups to finite-dimensional semisimple algebras and constructs explicit efficient quantum algorithms for the partition algebra, Brauer algebra, and walled Brauer algebra. It proves that for large enough d the transform, which may otherwise be non-unitary, lies close to a unitary operator. The resulting circuits use a number of gates polynomial in n, log d, and log(1/ε) while keeping the approximation error at most (d^{-1/2} + ε) times a polynomial in the algebra dimension. A reader would care because these algebras arise in models of many-body physics and generalized Schur-Weyl duality, so the new primitive opens the door to quantum algorithms that manipulate their representation theory directly.

Core claim

We generalize the quantum Fourier transform to finite-dimensional semisimple algebras and give quantum algorithms that approximate the Fourier transform over the partition algebra P_n(d), the Brauer algebra B_n(d), and the walled Brauer algebra B_{r,s}(d). When d is sufficiently large the transform is close to unitary, and each of these algebras admits an efficient circuit implementation with gate complexity poly(n, log d, log(1/ε)) that achieves error (d^{-1/2} + ε) poly(|A|). Several structural properties of the Fourier basis are established along the way.

What carries the argument

The approximate unitary Fourier transform over a semisimple algebra, realized by explicit quantum circuits whose complexity is polynomial in the rank n and log d.

If this is right

  • Quantum algorithms that rely on representation theory or Schur-Weyl duality can now invoke an efficient Fourier transform step over these algebras.
  • Simulations of statistical-mechanical models whose Hamiltonians are expressed in Brauer or partition algebra bases become feasible on quantum hardware.
  • The same circuit techniques may be reused whenever a new semisimple algebra satisfies the large-d unitary approximation condition.
  • Properties of the Fourier basis derived in the paper can be used to design further quantum primitives such as convolution or sampling algorithms over these algebras.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approximation strategy may extend to other families of semisimple algebras that appear in quantum information but are not treated here.
  • When d is too small for the unitary approximation to hold, one could explore quantum dilation or embedding techniques to implement the exact non-unitary map.
  • Classical precomputation of the circuit descriptions for fixed small n could be combined with the quantum part to yield hybrid algorithms for algebraic problems.

Load-bearing premise

When the parameter d is sufficiently large, the Fourier transform over the semisimple algebra is well approximated by a unitary operator.

What would settle it

For a concrete small-n instance of the partition algebra with large d, compute the exact Fourier matrix, measure its distance to the nearest unitary matrix, and check whether that distance exceeds the claimed error bound (d^{-1/2} + ε) poly(|A|).

Figures

Figures reproduced from arXiv: 2605.05337 by Barak Nehoran, Ben Foxman, Yongshan Ding.

Figure 1
Figure 1. Figure 1: Diagram multiplication in the symmetric group algebra. view at source ↗
Figure 2
Figure 2. Figure 2: Example elements of the partition algebra view at source ↗
Figure 3
Figure 3. Figure 3: The Fourier transform block-diagonalizes the natural view at source ↗
Figure 4
Figure 4. Figure 4: Actions of the partition algebra generators on a graph view at source ↗
Figure 5
Figure 5. Figure 5: An example of multiplying two partition diagrams. view at source ↗
Figure 6
Figure 6. Figure 6: An example of a walled Brauer diagram in view at source ↗
Figure 7
Figure 7. Figure 7: Examples of the Schur representation for different partition diagrams. view at source ↗
Figure 8
Figure 8. Figure 8: The Fourier transform maps diagrams with propagating number view at source ↗
Figure 9
Figure 9. Figure 9: The Bratteli diagram of 𝐵3,2 (𝑑). The dimension of an irrep 𝜆 ∈ 𝐵œ3,2 (𝑑) is equivalent to the number of paths from the root (∅, ∅) ∈ 𝐵œ0,0 (𝑑) to the label 𝜆. Note that the sum of the squares of the dimensions is 120 = |𝐵3,2 (𝑑)|. We will sometimes refer to the subalgebra adapted basis vectors as Bratteli paths, and use the letters 𝑃, 𝑄, 𝑅, . . . to denote such paths. For a Bratteli path 𝑃, we write 𝑃(𝑖) … view at source ↗
Figure 10
Figure 10. Figure 10: A circuit for the quantum Fourier transform, following the separation of variables approach. On view at source ↗
Figure 11
Figure 11. Figure 11: The circuit implementing Algorithm 1, equivalent to the postprocessing gate in view at source ↗
read the original abstract

The quantum Fourier transform (QFT) is a fundamental primitive in quantum computation and quantum information. In this work, we generalize the QFT for finite groups to a QFT for finite-dimensional semisimple algebras, and give efficient quantum Fourier transforms 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper generalizes the quantum Fourier transform from finite groups to finite-dimensional semisimple algebras. It establishes properties of the Fourier basis and gives quantum algorithms for approximating the (possibly non-unitary) Fourier transform over the partition algebra P_n(d), Brauer algebra B_n(d), and walled Brauer algebra B_{r,s}(d) to error (d^{-1/2} + ε) · poly(|A|) with gate complexity poly(n, log d, log(1/ε)), provided d is large enough that the transform is well approximated by a unitary.

Significance. If the approximation result holds with a useful (sub-constant) error bound, this would be a significant extension of the QFT primitive to algebraic structures arising in generalized Schur-Weyl duality, statistical physics, and many-body systems. The efficient implementations for these concrete algebras and the auxiliary properties of the Fourier basis could support new quantum algorithms in those domains.

major comments (1)
  1. [Abstract] Abstract (main complexity and approximation claim): the stated error is (d^{-1/2} + ε) · poly(|A|). For P_n(d) the algebra dimension |A| equals the Bell number B(2n) ∼ exp(Θ(n log n)); the same exponential growth holds for the Brauer and walled Brauer algebras. Consequently poly(|A|) is exponential in n, so the product exceeds 1 (and grows unbounded) for any fixed d and ε once n is large. This renders the approximation guarantee vacuous and supplies no nontrivial guarantee that the output is close to the Fourier transform.
minor comments (1)
  1. [Abstract] The abstract asserts that 'several properties of the Fourier basis of semisimple algebras' are established but does not enumerate them; a short list or forward reference to the relevant theorem would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting this issue with the error bound. We address the comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (main complexity and approximation claim): the stated error is (d^{-1/2} + ε) · poly(|A|). For P_n(d) the algebra dimension |A| equals the Bell number B(2n) ∼ exp(Θ(n log n)); the same exponential growth holds for the Brauer and walled Brauer algebras. Consequently poly(|A|) is exponential in n, so the product exceeds 1 (and grows unbounded) for any fixed d and ε once n is large. This renders the approximation guarantee vacuous and supplies no nontrivial guarantee that the output is close to the Fourier transform.

    Authors: We agree that the referee's observation is correct: as written, the factor of poly(|A|) renders the stated error bound greater than 1 for sufficiently large n and therefore vacuous as a guarantee of closeness in operator norm. The poly(|A|) term arises in our current proof from a loose bound on the maximum matrix-element size combined with a union bound over the (exponentially many) basis elements when controlling the deviation between the non-unitary Fourier transform and its unitary approximation. We will revise the abstract, the statement of the main theorem, and the error analysis in Section 4 to replace this factor with a polynomial in n and log d (specifically, we expect a revised bound of the form (d^{-1/2} + ε) poly(n, log d)). The gate complexity poly(n, log d, log(1/ε)) is unaffected. These changes will make the approximation guarantee nontrivial and restore the claimed utility for the target algebras. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper generalizes the known group QFT to semisimple algebras via explicit algebraic constructions and quantum circuit implementations for the partition, Brauer, and walled Brauer algebras. The approximation result (unitary approximation to the possibly non-unitary Fourier transform for large d) and the gate-complexity bound poly(n, log d, log(1/ε)) are derived from properties of the Fourier basis and standard quantum algorithmic techniques rather than from any fitted parameter, self-definition, or load-bearing self-citation that reduces the central claim to its inputs by construction. The error expression (d^{-1/2} + ε) · poly(|A|) is an explicit (if loose) bound obtained from norm estimates; it does not constitute a tautological renaming or a prediction forced by prior fitting. The derivation therefore remains self-contained against external algebraic and quantum-computing benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existence of a Fourier basis for semisimple algebras and on the statement that the transform becomes approximately unitary for large d; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Finite-dimensional semisimple algebras admit a Fourier transform that can be approximated by a unitary operator when the parameter d is sufficiently large
    Directly stated in the abstract as the condition under which the approximation holds.

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