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

arXiv:2602.15180

Efficient quantum circuits for high-dimensional representations of SU(n) and Ramanujan quantum expanders

Vishnu Iyer, Siddhartha Jain, Stephen Jordan, Rolando Somma

quant-phcs.CCSubmitted Feb 16, 2026• Announced Feb 18, 2026• 0 citations
arXivHTMLPDF

Abstract

We present efficient quantum circuits that implement high-dimensional unitary irreducible representations (irreps) of $SU(n)$, where $n \ge 2$ is constant. For dimension $N$ and error $ε$, the number of quantum gates in our circuits is polynomial in $\log(N)$ and $\log(1/ε)$. Our construction relies on the Jordan-Schwinger representation, which allows us to realize irreps of $SU(n)$ in the Hilbert space of $n$ quantum harmonic oscillators. Together with a recent efficient quantum Hermite transform, which allows us to map the computational basis states to the eigenstates of the quantum harmonic oscillator, this allows us to implement these irreps efficiently. Our quantum circuits can be used to construct explicit Ramanujan quantum expanders, a longstanding open problem. They can also be used to fast-forward the evolution of certain quantum systems.

Comments
39 pages, 2 figures

Related papers

Ramanujan quantum expanders from the Weil representation

Siddhartha Jain

For every odd prime power $q$ and $D=q+1$, we construct an infinite family of Ramanujan quantum expanders of degree $D$. The construction transfers Morgenstern's Ramanujan Cayley graphs on $\operatorname{PSL}_2(\mathbb{F}_{p})$ for $p$ which is an even power of $q$, through the odd irreducible subrepresentation of the Weil representation of $\operatorname{SL}_2(\mathbb{F}_{p})$. For a quantum expander of dimension $N$, our implementation uses $O(\log^2 N)$ elementary gates, and $O(\log N)$ ancilla qudits, using a fixed finite gate set depending on $q$. An advantage compared to the previous work of Iyer, Jain, Jordan, and Somma (FOCS 2026) is that assuming the quantum circuit is implemented exactly, we satisfy the $2\sqrt{D-1}/D$ singular value bound exactly without any additive error.

quant-pharXiv:2609.38075•Sep 30, 2026•0 citations

The Quantum Schur Transform: I. Efficient Qudit Circuits

Dave Bacon, Isaac L. Chuang, Aram W. Harrow

We present an efficient family of quantum circuits for a fundamental primitive in quantum information theory, the Schur transform. The Schur transform on n d-dimensional quantum systems is a transform between a standard computational basis to a labelling related to the representation theory of the symmetric and unitary groups. If we desire to implement the Schur transform to an accuracy of epsilon, then our circuit construction uses a number of gates which is polynomial in n, d and log(1/epsilon). The important insights we use to perform this construction are the selection of the appropriate subgroup adapted basis and the Wigner-Eckart theorem. Our efficient circuit construction renders numerous protocols in quantum information theory computationally tractable and is an important new efficient quantum circuit family which goes significantly beyond the standard paradigm of the quantum Fourier transform.

arXiv:quant-ph/0601001•Dec 30, 2005•23 citations

An Efficient Quantum Algorithm and Circuit to Generate Eigenstates of SU(2) and SU(3) Representations

U. Satya Sainadh

This thesis presents an efficient quantum algorithm and explicit circuits for generating eigenstates of arbitrary SU(2) and SU(3) representations. These include a wide variety of highly entangled states. The algorithm uses Schur transform that rotates the input computational basis states to the output Schur basis states with resources polynomial in number of qudits n. Using the fact that quantum logic is reversible, we accomplish the desired result using the inverse Schur transform. The algorithm can be easily generalized to any arbitrary higher groups.

arXiv:1309.2736•Sep 11, 2013•2 citations

The quantum Schur and Clebsch-Gordan transforms: I. efficient qudit circuits

Dave Bacon, Isaac L. Chuang, Aram W. Harrow

We present an efficient family of quantum circuits for a fundamental primitive in quantum information theory, the Schur transform. The Schur transform on n d-dimensional quantum systems is a transform between a standard computational basis to a labelling related to the representation theory of the symmetric and unitary groups. If we desire to implement the Schur transform to an accuracy of e-1, then our circuit construction uses a number of gates which is polynomial in n, d and log(e-1). The key tool in our construction is a poly(d, log n, log(e-1)) algorithm for the ud Clebsch-Gordan transform. Our efficient circuit construction renders numerous protocols in quantum information theory computationally tractable and yields a new possible approach to quantum algorithms which is distinct from the standard paradigm of the quantum Fourier transform.

Symposium on Discrete Algorithms•Jan 7, 2007•19 citations

Fast quantum algorithms for approximating some irreducible representations of groups

Stephen P. Jordan

We consider the quantum complexity of estimating matrix elements of unitary irreducible representations of groups. For several finite groups including the symmetric group, quantum Fourier transforms yield efficient solutions to this problem. Furthermore, quantum Schur transforms yield efficient solutions for certain irreducible representations of the unitary group. Beyond this, we obtain poly(n)-time quantum algorithms for approximating matrix elements from all the irreducible representations of the alternating group A_n, and all the irreducible representations of polynomial highest weight of U(n), SU(n), and SO(n). These quantum algorithms offer exponential speedup in worst case complexity over the fastest known classical algorithms. On the other hand, we show that average case instances are classically easy, and that the techniques analyzed here do not offer a speedup over classical computation for the estimation of group characters.

arXiv:0811.0562•Nov 4, 2008•8 citations

Quantum Circuits for Matrix-Product Unitaries

Georgios Styliaris, Rahul Trivedi, J. Ignacio Cirac

Matrix-product unitaries (MPUs) are many-body unitary operators that, as a consequence of their tensor-network structure, preserve the entanglement area law in 1D systems. However, it is unknown how to implement an MPU as a quantum circuit since the individual tensors describing the MPU are not unitary. In this Letter, we show that a large class of MPUs can be implemented with a polynomial-depth quantum circuit. For an N-site MPU built from a repeated bulk tensor with open boundary, we explicitly construct a quantum circuit of polynomial depth T=O(N^{α}) realizing the MPU, where the constant α depends only on the bulk and boundary tensor and not the system size N. We show that this class includes nontrivial unitaries that generate long-range entanglement and, in particular, contains a large class of unitaries constructed from representations of C^{*}-weak Hopf algebras. Furthermore, we also adapt our construction to nonuniform translationally varying MPUs and show that they can be implemented by a circuit of depth O(N^{β}polyD) where β≤1+log_{2}sqrt[D]/s_{min}, with D being the bond dimension and s_{min} the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU.

arXiv:2508.08160•Physical Review Letters•Dec 3, 2025•1 citation
We present efficient quantum circuits that implement high-dimensional unitary irreducible representations (irreps) of SU(n)SU(n)SU(n), where n≥2n \ge 2n≥2 is constant. For dimension NNN and error εεε, the number of quantum gates in our circuits is polynomial in log⁡(N)\log(N)log(N) and log⁡(1/ε)\log(1/ε)log(1/ε). Our construction relies on the Jordan-Schwinger representation, which allows us to realize irreps of SU(n)SU(n)SU(n) in the Hilbert space of nnn quantum harmonic oscillators. Together with a recent efficient quantum Hermite transform, which allows us to map the computational basis states to the eigenstates of the quantum harmonic oscillator, this allows us to implement these irreps efficiently. Our quantum circuits can be used to construct explicit Ramanujan quantum expanders, a longstanding open problem. They can also be used to fast-forward the evolution of certain quantum systems.
For every odd prime power qqq and D=q+1D=q+1D=q+1, we construct an infinite family of Ramanujan quantum expanders of degree DDD. The construction transfers Morgenstern's Ramanujan Cayley graphs on PSL⁡2(Fp)\operatorname{PSL}_2(\mathbb{F}_{p})PSL2​(Fp​) for ppp which is an even power of qqq, through the odd irreducible subrepresentation of the Weil representation of SL⁡2(Fp)\operatorname{SL}_2(\mathbb{F}_{p})SL2​(Fp​). For a quantum expander of dimension NNN, our implementation uses O(log⁡2N)O(\log^2 N)O(log2N) elementary gates, and O(log⁡N)O(\log N)O(logN) ancilla qudits, using a fixed finite gate set depending on qqq. An advantage compared to the previous work of Iyer, Jain, Jordan, and Somma (FOCS 2026) is that assuming the quantum circuit is implemented exactly, we satisfy the 2D−1/D2\sqrt{D-1}/D2D−1​/D singular value bound exactly without any additive error.
We present an efficient family of quantum circuits for a fundamental primitive in quantum information theory, the Schur transform. The Schur transform on n d-dimensional quantum systems is a transform between a standard computational basis to a labelling related to the representation theory of the symmetric and unitary groups. If we desire to implement the Schur transform to an accuracy of epsilon, then our circuit construction uses a number of gates which is polynomial in n, d and log(1/epsilon). The important insights we use to perform this construction are the selection of the appropriate subgroup adapted basis and the Wigner-Eckart theorem. Our efficient circuit construction renders numerous protocols in quantum information theory computationally tractable and is an important new efficient quantum circuit family which goes significantly beyond the standard paradigm of the quantum Fourier transform.
This thesis presents an efficient quantum algorithm and explicit circuits for generating eigenstates of arbitrary SU(2) and SU(3) representations. These include a wide variety of highly entangled states. The algorithm uses Schur transform that rotates the input computational basis states to the output Schur basis states with resources polynomial in number of qudits n. Using the fact that quantum logic is reversible, we accomplish the desired result using the inverse Schur transform. The algorithm can be easily generalized to any arbitrary higher groups.
We present an efficient family of quantum circuits for a fundamental primitive in quantum information theory, the Schur transform. The Schur transform on n d-dimensional quantum systems is a transform between a standard computational basis to a labelling related to the representation theory of the symmetric and unitary groups. If we desire to implement the Schur transform to an accuracy of e-1, then our circuit construction uses a number of gates which is polynomial in n, d and log(e-1). The key tool in our construction is a poly(d, log n, log(e-1)) algorithm for the ud Clebsch-Gordan transform. Our efficient circuit construction renders numerous protocols in quantum information theory computationally tractable and yields a new possible approach to quantum algorithms which is distinct from the standard paradigm of the quantum Fourier transform.
We consider the quantum complexity of estimating matrix elements of unitary irreducible representations of groups. For several finite groups including the symmetric group, quantum Fourier transforms yield efficient solutions to this problem. Furthermore, quantum Schur transforms yield efficient solutions for certain irreducible representations of the unitary group. Beyond this, we obtain poly(n)-time quantum algorithms for approximating matrix elements from all the irreducible representations of the alternating group A_n, and all the irreducible representations of polynomial highest weight of U(n), SU(n), and SO(n). These quantum algorithms offer exponential speedup in worst case complexity over the fastest known classical algorithms. On the other hand, we show that average case instances are classically easy, and that the techniques analyzed here do not offer a speedup over classical computation for the estimation of group characters.
Matrix-product unitaries (MPUs) are many-body unitary operators that, as a consequence of their tensor-network structure, preserve the entanglement area law in 1D systems. However, it is unknown how to implement an MPU as a quantum circuit since the individual tensors describing the MPU are not unitary. In this Letter, we show that a large class of MPUs can be implemented with a polynomial-depth quantum circuit. For an N-site MPU built from a repeated bulk tensor with open boundary, we explicitly construct a quantum circuit of polynomial depth T=O(N^{α}) realizing the MPU, where the constant α depends only on the bulk and boundary tensor and not the system size N. We show that this class includes nontrivial unitaries that generate long-range entanglement and, in particular, contains a large class of unitaries constructed from representations of C^{*}-weak Hopf algebras. Furthermore, we also adapt our construction to nonuniform translationally varying MPUs and show that they can be implemented by a circuit of depth O(N^{β}polyD) where β≤1+log_{2}sqrt[D]/s_{min}, with D being the bond dimension and s_{min} the smallest nonzero Schmidt value of the normalized Choi state corresponding to the MPU.