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

Physical Review A

Synthesis of arbitrary SU(3) transformations of atomic qutrits

Nikolay V. Vitanov

Published Mar 27, 2012• 55 citations
Publisher

Abstract

Several scenarios are proposed and analyzed for engineering of arbitrary preselected SU(3) transformations of laser-driven atomic qutrits. Two of the most natural implementations of qutrits are considered, in which the three qutrit states are coupled to each other via two-photon transitions through either (i) a common state in a tripod linkage pattern or (ii) two upper states in an M-shaped linkage chain. The SU(3) transformation for the tripod qutrit can be realized by 3 Givens SU(2) rotations, which require 9 consecutive interaction steps. Alternatively, because under certain conditions the propagator of the tripod system reduces to the Householder reflection operator, any SU(3) transformation can be constructed physically by 3 Householder reflections, each of which is implemented in a single interaction step. As an example, the discrete Fourier transform can be synthesized by 7 consecutive interaction steps with Givens rotations or, alternatively, by only a single Householder reflection and a phase gate. For the M-qutrit, the propagator is given by coupled Householder reflections and it cannot be reduced to Givens rotations or independent Householder reflections. By using these coupled Householder reflections it is shown that an arbitrary SU(3) transformation of the M qutrit can be realized with just two fields in at most 3 interaction steps; the discrete Fourier transform, in particular, requires only 2 interaction steps.

DOI
10.1103/physreva.85.032331

Related papers

Engineering of arbitrary U(N) transformations by quantum Householder reflections

Peter A. Ivanov, Elica S. Kyoseva, Nikolay V. Vitanov

We propose a simple physical implementation of the quantum Householder reflection (QHR) $\mathbf{M}(v)=\mathbf{I}\ensuremath{-}2\ensuremath{\mid}v⟩⟨v\ensuremath{\mid}$ in a quantum system of $N$ degenerate states (forming a qunit) coupled simultaneously to an ancillary (excited) state by $N$ resonant or nearly resonant pulsed external fields. We also introduce the generalized QHR $\mathbf{M}(v;\ensuremath{\varphi})=\mathbf{I}+({e}^{i\ensuremath{\varphi}}\ensuremath{-}1)\ensuremath{\mid}v⟩⟨v\ensuremath{\mid}$, which can be produced in the same $N$-pod system when the fields are appropriately detuned from resonance with the excited state. We use these two operators as building blocks in constructing arbitrary preselected unitary transformations. We show that the most general $\mathrm{U}(N)$ transformation can be factorized (and thereby produced) by either $N\ensuremath{-}1$ standard QHRs and an $N$-dimensional phase gate, or $N\ensuremath{-}1$ generalized QHRs and a one-dimensional phase gate. Viewed mathematically, these QHR factorizations provide parametrizations of the $\mathrm{U}(N)$ group. As an example, we propose a recipe for constructing the quantum Fourier transform (QFT) by at most $N$ interaction steps. For example, the QFT requires a single QHR for $N=2$, and only two QHRs for $N=3$ and 4.

arXiv:0708.2811•Physical Review A•Aug 25, 2006•82 citations

Synthesis of Arbitrary Unitary Transformations in Quantum Systems by Householder Reflections

Peter A. Ivanov, Nikolay V. Vitanov, Theodore E. Simos +1 more

We demonstrate that Householder reflections emerge naturally in the propagator of a coherently driven degenerate two‐level system. Such reflections are a very powerful tool for constructing arbitrary unitary transformations of an N‐state quantum system. We present examples for construction of discrete Fourier transforms, superposition‐to‐superposition navigation between pure states, engineering of arbitrary mixed states, and entanglement of trapped ions.

AIP conference proceedings•Jan 1, 2007•0 citations

Arbitrary qudit gates by adiabatic passage

Benjamin Rousseaux, S. Guérin, Nikolay V. Vitanov

We derive an adiabatic technique that implements the most general SU($d$) transformation in a quantum system of $d$ degenerate states, featuring a qudit. This technique is based on the factorization of the SU($d$) transformation into $d$ generalized quantum Householder reflections, each of which is implemented by a two-shot stimulated Raman adiabatic passage with appropriate static phases. The energy of the lasers needed to synthesize a single Householder reflection is shown to be remarkably constant as a function of $d$. This technique is directly applicable to a linear trapped ion system with $d+1$ ions. We implement the quantum Fourier transform numerically in a qudit with $d=4$ (defined as a quartit) as an example.

Physical Review A•Mar 25, 2013•67 citations

Complete Unitary Qutrit Control in Ultracold Atoms

Joseph Lindon, Arina Tashchilina, Logan Cooke +1 more

Physical quantum systems are commonly composed of more than two levels and offer the capacity to encode information in higher-dimensional spaces beyond the qubit, starting with the three-level qutrit. Here, we encode neutral-atom qutrits in an ensemble of ultracold ${}^{87}\mathrm{Rb}$ and demonstrate arbitrary single-qutrit SU(3) gates. We generate a full set of gates using only two resonant microwave tones, including synthesizing a gate that effects a direct coupling between the two disconnected levels in the three-level $\mathrm{\ensuremath{\Lambda}}$-scheme. Using two different gate sets, we implement and characterize the Walsh-Hadamard Fourier transform, and find similar final-state fidelity and purity from both approaches. This work establishes the ultracold neutral-atom qutrit as a promising platform for qutrit-based quantum information processing, extensions to $d$-dimensional qudits, and explorations in multilevel quantum state manipulations with nontrivial geometric phases.

arXiv:2208.00045•Physical Review Applied•Mar 28, 2023•31 citations

Synthesis of arbitrary unitary transformations of collective states of trapped ions by quantum Householder reflections

Peter A. Ivanov, Nikolay V. Vitanov

We propose an efficient technique for creation and manipulation of entangled states and for construction of arbitrary many-particle unitary transformations in a linear ion trap. We show that, under suitable conditions on the laser pulse areas and detunings, the propagator within the single-excitation manifold has the form of a Householder reflection (HR). We demonstrate that a single HR is sufficient to create an entangled $W$-type state or to transfer one entangled state into another. An arbitrary $N$-dimensional unitary transformation can be constructed by at most $N\ensuremath{-}1$ HRs, i.e., significantly faster than $O({N}^{2})$ steps in previous proposals, which makes this technique less sensitive to heating and other types of decoherence. We present examples of synthesis of quantum Fourier transforms for four, five, and six ions, which are constructed using only two, three, and four HRs, respectively.

Physical Review A•Jan 30, 2008•35 citations

Universal qutrit control in asymmetric-top molecules

Qian-Qian Hong, Zhi-Jian Zheng, Zhe-Jun Zhang +2 more

We present a theoretical framework for universal single-qutrit control in asymmetric-top molecules, advancing molecular quantum information processing. In this approach, the qutrit is encoded in three rotational eigenstates, with an auxiliary state providing independent phase control within the computational manifold. We explore an analytic protocol for arbitrary single-qutrit gates, combining directly addressable SU(2) rotations with auxiliary-state-mediated phase operations. To support this, we derive a multilevel pulse-area theorem that provides an explicit analytic mapping between gate parameters and control fields, enabling systematic design of high-fidelity microwave pulse sequences. Numerical simulations with 1,2-propanediol confirm the robustness of our approach, achieving Walsh-Hadamard gates with minimal leakage from the computational subspace. We further examine four SU(2) decomposition strategies and find that phase-error sensitivity depends on the decomposition sequence, while amplitude errors propagate along specific coherence pathways. Our results establish asymmetric-top molecules as a viable platform for qutrit-based quantum operations and offer an analytical method for precise quantum control of complex multilevel systems.

quant-pharXiv:2605.03468•May 6, 2026•0 citations
Several scenarios are proposed and analyzed for engineering of arbitrary preselected SU(3) transformations of laser-driven atomic qutrits. Two of the most natural implementations of qutrits are considered, in which the three qutrit states are coupled to each other via two-photon transitions through either (i) a common state in a tripod linkage pattern or (ii) two upper states in an M-shaped linkage chain. The SU(3) transformation for the tripod qutrit can be realized by 3 Givens SU(2) rotations, which require 9 consecutive interaction steps. Alternatively, because under certain conditions the propagator of the tripod system reduces to the Householder reflection operator, any SU(3) transformation can be constructed physically by 3 Householder reflections, each of which is implemented in a single interaction step. As an example, the discrete Fourier transform can be synthesized by 7 consecutive interaction steps with Givens rotations or, alternatively, by only a single Householder reflection and a phase gate. For the M-qutrit, the propagator is given by coupled Householder reflections and it cannot be reduced to Givens rotations or independent Householder reflections. By using these coupled Householder reflections it is shown that an arbitrary SU(3) transformation of the M qutrit can be realized with just two fields in at most 3 interaction steps; the discrete Fourier transform, in particular, requires only 2 interaction steps.
We propose a simple physical implementation of the quantum Householder reflection (QHR) $\mathbf{M}(v)=\mathbf{I}\ensuremath{-}2\ensuremath{\mid}v⟩⟨v\ensuremath{\mid}$ in a quantum system of NNN degenerate states (forming a qunit) coupled simultaneously to an ancillary (excited) state by NNN resonant or nearly resonant pulsed external fields. We also introduce the generalized QHR $\mathbf{M}(v;\ensuremath{\varphi})=\mathbf{I}+({e}^{i\ensuremath{\varphi}}\ensuremath{-}1)\ensuremath{\mid}v⟩⟨v\ensuremath{\mid}$, which can be produced in the same NNN-pod system when the fields are appropriately detuned from resonance with the excited state. We use these two operators as building blocks in constructing arbitrary preselected unitary transformations. We show that the most general U(N)\mathrm{U}(N)U(N) transformation can be factorized (and thereby produced) by either $N\ensuremath{-}1$ standard QHRs and an NNN-dimensional phase gate, or $N\ensuremath{-}1$ generalized QHRs and a one-dimensional phase gate. Viewed mathematically, these QHR factorizations provide parametrizations of the U(N)\mathrm{U}(N)U(N) group. As an example, we propose a recipe for constructing the quantum Fourier transform (QFT) by at most NNN interaction steps. For example, the QFT requires a single QHR for N=2N=2N=2, and only two QHRs for N=3N=3N=3 and 4.
We derive an adiabatic technique that implements the most general SU(ddd) transformation in a quantum system of ddd degenerate states, featuring a qudit. This technique is based on the factorization of the SU(ddd) transformation into ddd generalized quantum Householder reflections, each of which is implemented by a two-shot stimulated Raman adiabatic passage with appropriate static phases. The energy of the lasers needed to synthesize a single Householder reflection is shown to be remarkably constant as a function of ddd. This technique is directly applicable to a linear trapped ion system with d+1d+1d+1 ions. We implement the quantum Fourier transform numerically in a qudit with d=4d=4d=4 (defined as a quartit) as an example.
Physical quantum systems are commonly composed of more than two levels and offer the capacity to encode information in higher-dimensional spaces beyond the qubit, starting with the three-level qutrit. Here, we encode neutral-atom qutrits in an ensemble of ultracold 87Rb{}^{87}\mathrm{Rb}87Rb and demonstrate arbitrary single-qutrit SU(3) gates. We generate a full set of gates using only two resonant microwave tones, including synthesizing a gate that effects a direct coupling between the two disconnected levels in the three-level $\mathrm{\ensuremath{\Lambda}}$-scheme. Using two different gate sets, we implement and characterize the Walsh-Hadamard Fourier transform, and find similar final-state fidelity and purity from both approaches. This work establishes the ultracold neutral-atom qutrit as a promising platform for qutrit-based quantum information processing, extensions to ddd-dimensional qudits, and explorations in multilevel quantum state manipulations with nontrivial geometric phases.
We propose an efficient technique for creation and manipulation of entangled states and for construction of arbitrary many-particle unitary transformations in a linear ion trap. We show that, under suitable conditions on the laser pulse areas and detunings, the propagator within the single-excitation manifold has the form of a Householder reflection (HR). We demonstrate that a single HR is sufficient to create an entangled WWW-type state or to transfer one entangled state into another. An arbitrary NNN-dimensional unitary transformation can be constructed by at most $N\ensuremath{-}1$ HRs, i.e., significantly faster than O(N2)O({N}^{2})O(N2) steps in previous proposals, which makes this technique less sensitive to heating and other types of decoherence. We present examples of synthesis of quantum Fourier transforms for four, five, and six ions, which are constructed using only two, three, and four HRs, respectively.
We present a theoretical framework for universal single-qutrit control in asymmetric-top molecules, advancing molecular quantum information processing. In this approach, the qutrit is encoded in three rotational eigenstates, with an auxiliary state providing independent phase control within the computational manifold. We explore an analytic protocol for arbitrary single-qutrit gates, combining directly addressable SU(2) rotations with auxiliary-state-mediated phase operations. To support this, we derive a multilevel pulse-area theorem that provides an explicit analytic mapping between gate parameters and control fields, enabling systematic design of high-fidelity microwave pulse sequences. Numerical simulations with 1,2-propanediol confirm the robustness of our approach, achieving Walsh-Hadamard gates with minimal leakage from the computational subspace. We further examine four SU(2) decomposition strategies and find that phase-error sensitivity depends on the decomposition sequence, while amplitude errors propagate along specific coherence pathways. Our results establish asymmetric-top molecules as a viable platform for qutrit-based quantum operations and offer an analytical method for precise quantum control of complex multilevel systems.