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:0712.1272 · Physical Review A

Navigation between quantum states by quantum mirrors

Peter A. Ivanov, Boyan T. Torosov, Nikolay V. Vitanov

Published Jan 22, 2007• 24 citations
arXivHTMLPDF

Abstract

We introduce a technique that allows one to connect any two arbitrary (pure or mixed) superposition states of an $N$-state quantum system. The proposed solution to this inverse quantum mechanical problem is analytical, exact, and very compact. The technique uses standard and generalized quantum Householder reflections (QHRs), which require external pulses of precise areas and frequencies. We show that any two pure states can be linked by just a single generalized QHR. The transfer between any two mixed states with the same dynamic invariants (e.g., the same density matrix eigenvalues) requires in general $N$ QHRs. Moreover, we propose recipes for synthesis of arbitrary preselected mixed states using a combination of QHRs and incoherent processes (pure dephasing or spontaneous emission).

DOI
10.1103/physreva.75.012323

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

Engineering arbitrary pure and mixed quantum states

Alexander Nikolaevich Pechen

Controlled manipulation by atomic- and molecular-scale quantum systems has attracted a lot of research attention in recent years. A fundamental problem is to provide deterministic methods for controlled engineering of arbitrary quantum states. This work proposes a deterministic method for engineering arbitrary pure and mixed states of a wide class of quantum systems. The method exploits a special combination of incoherent and coherent controls (incoherent and coherent radiation) and has two properties which are specifically important for manipulating by quantum systems: it realizes the strongest possible degree of their state control, complete density matrix controllability, meaning the ability to steer arbitrary pure and mixed initial states into any desired pure or mixed final state, and it is all-to-one, such that each particular control transfers all initial system states into one target state.

arXiv:1210.2281•Physical Review A•Oct 6, 2011•63 citations

Physical realization of coupled Hilbert-space mirrors for quantum-state engineering

Elica S. Kyoseva, Nikolay V. Vitanov, Bruce W. Shore

Manipulation of superpositions of discrete quantum states has a mathematical counterpart in the motion of a unit-length statevector in an N-dimensional Hilbert space. Any such statevector motion can be regarded as a succession of two-dimensional rotations. But the desired statevector change can also be treated as a succession of reflections, the generalization of Householder transformations. In multidimensional Hilbert space such reflection sequences offer more efficient procedures for statevector manipulation than do sequences of rotations. We here show how such reflections can be designed for a system with two degenerate levels—a generalization of the traditional two-state atom—that allows the construction of propagators for angular momentum states. We use the Morris–Shore transformation to express the propagator in terms of Morris–Shore basis states and Cayley–Klein parameters, which allows us to connect properties of laser pulses to Hilbert-space motion. Under suitable conditions on the couplings and the common detuning, the propagators within each set of degenerate states represent products of generalized Householder reflections, with orthogonal vectors. We propose physical realizations of this novel geometrical object with resonant, near-resonant and far-off-resonant laser pulses. We give several examples of implementations in real atoms or molecules.

arXiv:0802.4246•Journal of Modern Optics•Sep 10, 2007•26 citations

Generation of arbitrary qubit states by adiabatic evolution split by a phase jump

Kaloyan N. Zlatanov, Nikolay V. Vitanov

We propose a technique for the accurate, flexible, and robust generation of arbitrary preselected coherent superpositions of two quantum states. It uses a sequence of two adiabatic pulses split by a phase jump serving as a control parameter. Each pulse has a chirped detuning, which induces a half crossing, and acts approximately as a half-$\ensuremath{\pi}$ pulse in the adiabatic regime. The phase jump is imprinted onto the population ratio of the created superposition state. Of the various possible relations between the two pulses, we select the case when the Rabi frequency and the detuning of the second pulse are mirror images of those of the first pulse, and the two detunings have opposite signs. Then the mixing angle of the created superposition state depends on the phase jump only. For other arrangements, the superposition mixing angle is shifted by the dynamic phases of the propagators, which makes these cases suitable for state tomography. This sandwich setup comes along with the advantage that it reduces the error $\ensuremath{\epsilon}$ of each individual pulse down to $4{\ensuremath{\epsilon}}^{2}$ overall. Therefore, the proposed technique combines the benefits of robustness stemming from adiabatic evolution with accuracy generated by the twin-pulse error suppression, and flexibility of the created superposition state controlled by the value of the phase jump $\ensuremath{\phi}$. In addition to the general analysis, we present a simple exactly soluble trigonometric model in order to illustrate the proposed technique. In this model, when the pulse area $A$ increases, the nonadiabatic oscillations are damped as ${A}^{\ensuremath{-}1}$ for a single pulse and ${A}^{\ensuremath{-}2}$ for the two-pulse sequence. Finally, the proposed technique is iteratively extended to sequences of $N={2}^{n}$ pulses by concatenating half-$\ensuremath{\pi}$ sequences and splitting them by a phase jump, thereby further reducing the nonadiabatic error $\ensuremath{\epsilon}$ to ${(2\ensuremath{\epsilon})}^{N}$. This makes the proposed technique suitable for generating high-fidelity quantum rotation gates even when starting with errant pulses.

arXiv:1909.07769•Physical Review A•Jan 17, 2020•15 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
We introduce a technique that allows one to connect any two arbitrary (pure or mixed) superposition states of an NNN-state quantum system. The proposed solution to this inverse quantum mechanical problem is analytical, exact, and very compact. The technique uses standard and generalized quantum Householder reflections (QHRs), which require external pulses of precise areas and frequencies. We show that any two pure states can be linked by just a single generalized QHR. The transfer between any two mixed states with the same dynamic invariants (e.g., the same density matrix eigenvalues) requires in general NNN QHRs. Moreover, we propose recipes for synthesis of arbitrary preselected mixed states using a combination of QHRs and incoherent processes (pure dephasing or spontaneous emission).
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.
Controlled manipulation by atomic- and molecular-scale quantum systems has attracted a lot of research attention in recent years. A fundamental problem is to provide deterministic methods for controlled engineering of arbitrary quantum states. This work proposes a deterministic method for engineering arbitrary pure and mixed states of a wide class of quantum systems. The method exploits a special combination of incoherent and coherent controls (incoherent and coherent radiation) and has two properties which are specifically important for manipulating by quantum systems: it realizes the strongest possible degree of their state control, complete density matrix controllability, meaning the ability to steer arbitrary pure and mixed initial states into any desired pure or mixed final state, and it is all-to-one, such that each particular control transfers all initial system states into one target state.
Manipulation of superpositions of discrete quantum states has a mathematical counterpart in the motion of a unit-length statevector in an N-dimensional Hilbert space. Any such statevector motion can be regarded as a succession of two-dimensional rotations. But the desired statevector change can also be treated as a succession of reflections, the generalization of Householder transformations. In multidimensional Hilbert space such reflection sequences offer more efficient procedures for statevector manipulation than do sequences of rotations. We here show how such reflections can be designed for a system with two degenerate levels—a generalization of the traditional two-state atom—that allows the construction of propagators for angular momentum states. We use the Morris–Shore transformation to express the propagator in terms of Morris–Shore basis states and Cayley–Klein parameters, which allows us to connect properties of laser pulses to Hilbert-space motion. Under suitable conditions on the couplings and the common detuning, the propagators within each set of degenerate states represent products of generalized Householder reflections, with orthogonal vectors. We propose physical realizations of this novel geometrical object with resonant, near-resonant and far-off-resonant laser pulses. We give several examples of implementations in real atoms or molecules.
We propose a technique for the accurate, flexible, and robust generation of arbitrary preselected coherent superpositions of two quantum states. It uses a sequence of two adiabatic pulses split by a phase jump serving as a control parameter. Each pulse has a chirped detuning, which induces a half crossing, and acts approximately as a half-$\ensuremath{\pi}$ pulse in the adiabatic regime. The phase jump is imprinted onto the population ratio of the created superposition state. Of the various possible relations between the two pulses, we select the case when the Rabi frequency and the detuning of the second pulse are mirror images of those of the first pulse, and the two detunings have opposite signs. Then the mixing angle of the created superposition state depends on the phase jump only. For other arrangements, the superposition mixing angle is shifted by the dynamic phases of the propagators, which makes these cases suitable for state tomography. This sandwich setup comes along with the advantage that it reduces the error $\ensuremath{\epsilon}$ of each individual pulse down to $4{\ensuremath{\epsilon}}^{2}$ overall. Therefore, the proposed technique combines the benefits of robustness stemming from adiabatic evolution with accuracy generated by the twin-pulse error suppression, and flexibility of the created superposition state controlled by the value of the phase jump $\ensuremath{\phi}$. In addition to the general analysis, we present a simple exactly soluble trigonometric model in order to illustrate the proposed technique. In this model, when the pulse area AAA increases, the nonadiabatic oscillations are damped as ${A}^{\ensuremath{-}1}$ for a single pulse and ${A}^{\ensuremath{-}2}$ for the two-pulse sequence. Finally, the proposed technique is iteratively extended to sequences of N=2nN={2}^{n}N=2n pulses by concatenating half-$\ensuremath{\pi}$ sequences and splitting them by a phase jump, thereby further reducing the nonadiabatic error $\ensuremath{\epsilon}$ to ${(2\ensuremath{\epsilon})}^{N}$. This makes the proposed technique suitable for generating high-fidelity quantum rotation gates even when starting with errant pulses.
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.