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:1009.4957 · Journal of Physics A Mathematical and Theoretical

Bang–bang control design for quantum state transfer based on hyperspherical coordinates and optimal time–energy control

Weiwei Zhou, S G Schirmer, Ming Zhang, Hong-Yi Dai

Published Feb 17, 2011• 18 citations
arXivHTMLPDF

Abstract

We present a constructive control scheme for solving quantum state engineering problems based on the parameterization of the state vector in terms of complex hyperspherical coordinates. Unlike many control schemes based on factorization of unitary operators, the scheme gives explicit expressions for all generalized Euler angles in terms of the hyperspherical coordinates of the initial and final states. The factorization, when applicable, has added benefits that phase rotations can be combined and performed concurrently. The control procedure can be realized using simple bang–bang or square-wavefunction controls. Optimal time–energy control is considered to find the optimal control amplitude. The extension of the scheme to implement arbitrary unitary operators is also discussed.

DOI
10.1088/1751-8113/44/10/105303

Related papers

Bang-bang control design for quantum state transfer based on hyperspherical coordinates and optimal time-energy control

Weiwei Zhou, S. G. Schirmer, Ming Zhang

Abstract. We present a constructive control scheme for solving quantum state engineering problems based on a parametrization of the state vector in terms of complex hyperspherical coordinates. Unlike many control schemes based on factorization of unitary operators, the scheme gives explicit expressions for all generalized Euler angles in terms of the hyperspherical coordinates of the initial and final states. The factorization, when applicable, has added benefits that phase rotations can be combined and performed concurrently. The control procedure can be realized using simple bang-bang or square-wave-function controls. Optimal time-energy control is considered to find the optimal control amplitude. The extension of the scheme to implement arbitrary unitary operators is also discussed.

•Jan 1, 2011•18 citations

Time-optimal single-scalar control on a qubit of unitary dynamics

Chungwei Lin, Qi Ding, Petros T. Boufounos +4 more

Optimal control theory is applied to analyze the time-optimal solution with a single scalar control knob in a two-level quantum system without quantum decoherence. Emphasis is placed on the dependence on the maximum control strength ${u}_{\text{max}}$. General constraints on the optimal protocol are derived and used to rigorously parametrize the time-optimal solution. Two concrete problems are investigated. For generic state preparation problems, both multiple bang-bang and bang-singular-bang are legitimate and should be considered. Generally, the optimal is bang-bang for small ${u}_{\text{max}}$, and there exists a state-dependent critical amplitude above which singular control emerges. For the X-gate operation of a qubit, the optimal protocol is exclusively multiple bang-bang. The minimum gate time is about 80% of that based on the resonant Rabi $\ensuremath{\pi}$-pulse over a wide range of control strength; in the ${u}_{\text{max}}\ensuremath{\rightarrow}0$ limit, this ratio is derived to be $\ensuremath{\pi}/4$. To develop practically feasible protocols, we present methods to smooth the abrupt changes in the bang-bang control while preserving perfect gate fidelity. The presence of bang-bang segments in the time-optimal protocol indicates that the high-frequency components and a full calculation (instead of the commonly adopted rotating wave approximation) are essential for the ultimate quantum speed limit.

arXiv:2504.01785•Physical Review A•Apr 2, 2025•2 citations

Constructive control of quantum systems using factorization of unitary operators

S. G. Schirmer, Andrew D. Greentree, Viswanath Ramakrishna +1 more

We demonstrate how structured decompositions of unitary operators can be employed to derive control schemes for finite-level quantum systems that require only sequences of simple control pulses such as square wave pulses with finite rise and decay times or Gaussian wavepackets. To illustrate the technique, it is applied to find control schemes to achieve population transfers for pure-state systems, complete inversions of the ensemble populations for mixed-state systems, create arbitrary superposition states and optimize the ensemble average of dynamic observables.

arXiv:quant-ph/0211042•Journal of Physics A Mathematical and General•Sep 17, 2002•64 citations

Optimal bounded local sine-waveform controls for state transfer in $N$-level quantum systems

Ming Zhang, Weiwei Zhou, Hong‐Yi Dai +2 more

Based on a parametrization of pure quantum states we explicitly construct a sequence of (at most) $4N-5$ local time-continuous waveform controls to achieve a specified state transition for $N$-level quantum systems when sufficient controls of the Hamiltonian are available. The control magnitudes are further optimized in terms of a time-energy performance, which is a generalization of the time performance index. Trajectory-constrained optimal local time-continuous waveform controls, including both local sine-waveforms and $n^{\rm th}$-order-polynomial waveform controls are obtained in terms of time-energy performance. It is demonstrated that constrained optimal local $n^{\rm th}$-order-polynomial waveform controls approach constrained optimal bang-bang controls when $n\rightarrow\infty$.

•Oct 14, 2010•0 citations

Steering quantum dynamics via bang-bang control: Implementing optimal fixed-point quantum search algorithm

Gaurav Bhole, V. S. Anjusha, T. S. Mahesh

A robust control over quantum dynamics is of paramount importance for quantum technologies. Many of the existing control techniques are based on smooth Hamiltonian modulations involving repeated calculations of basic unitaries resulting in time complexities scaling rapidly with the length of the control sequence. Here we show that bang-bang controls need one-time calculation of basic unitaries and hence scale much more efficiently. By employing a global optimization routine such as the genetic algorithm, it is possible to synthesize not only highly intricate unitaries, but also certain nonunitary operations. We demonstrate the unitary control through the implementation of the optimal fixed-point quantum search algorithm in a three-qubit nuclear magnetic resonance (NMR) system. Moreover, by combining the bang-bang pulses with the crusher gradients, we also demonstrate nonunitary transformations of thermal equilibrium states into effective pure states in three- as well as five-qubit NMR systems.

arXiv:1512.08385•Physical Review A•Apr 26, 2016•21 citations

Improving quantum gate fidelities using optimized Euler angles

K. Ch. Chatzisavvas, Goce Chadzitaskos, C. Daskaloyannis +1 more

An explicit algorithm for calculating the optimized Euler angles for both qubit state transfer and gate engineering given two arbitrary fixed Hamiltonians is presented. It is shown how the algorithm enables us to efficiently implement single qubit gates even if the control is severely restricted and the experimentally accessible Hamiltonians are far from orthogonal. It is further shown that using the optimized Euler angles can significantly improve the fidelity of quantum operations even for systems where the experimentally accessible Hamiltonians are nearly orthogonal. Unlike schemes such as composite pulses, the proposed scheme does not significantly increase the number of local operations or gate operation times.

arXiv:0805.2894•Physical Review A•Nov 23, 2009•4 citations
We present a constructive control scheme for solving quantum state engineering problems based on the parameterization of the state vector in terms of complex hyperspherical coordinates. Unlike many control schemes based on factorization of unitary operators, the scheme gives explicit expressions for all generalized Euler angles in terms of the hyperspherical coordinates of the initial and final states. The factorization, when applicable, has added benefits that phase rotations can be combined and performed concurrently. The control procedure can be realized using simple bang–bang or square-wavefunction controls. Optimal time–energy control is considered to find the optimal control amplitude. The extension of the scheme to implement arbitrary unitary operators is also discussed.
Abstract. We present a constructive control scheme for solving quantum state engineering problems based on a parametrization of the state vector in terms of complex hyperspherical coordinates. Unlike many control schemes based on factorization of unitary operators, the scheme gives explicit expressions for all generalized Euler angles in terms of the hyperspherical coordinates of the initial and final states. The factorization, when applicable, has added benefits that phase rotations can be combined and performed concurrently. The control procedure can be realized using simple bang-bang or square-wave-function controls. Optimal time-energy control is considered to find the optimal control amplitude. The extension of the scheme to implement arbitrary unitary operators is also discussed.
Optimal control theory is applied to analyze the time-optimal solution with a single scalar control knob in a two-level quantum system without quantum decoherence. Emphasis is placed on the dependence on the maximum control strength umax{u}_{\text{max}}umax​. General constraints on the optimal protocol are derived and used to rigorously parametrize the time-optimal solution. Two concrete problems are investigated. For generic state preparation problems, both multiple bang-bang and bang-singular-bang are legitimate and should be considered. Generally, the optimal is bang-bang for small umax{u}_{\text{max}}umax​, and there exists a state-dependent critical amplitude above which singular control emerges. For the X-gate operation of a qubit, the optimal protocol is exclusively multiple bang-bang. The minimum gate time is about 80% of that based on the resonant Rabi $\ensuremath{\pi}$-pulse over a wide range of control strength; in the ${u}_{\text{max}}\ensuremath{\rightarrow}0$ limit, this ratio is derived to be $\ensuremath{\pi}/4$. To develop practically feasible protocols, we present methods to smooth the abrupt changes in the bang-bang control while preserving perfect gate fidelity. The presence of bang-bang segments in the time-optimal protocol indicates that the high-frequency components and a full calculation (instead of the commonly adopted rotating wave approximation) are essential for the ultimate quantum speed limit.
We demonstrate how structured decompositions of unitary operators can be employed to derive control schemes for finite-level quantum systems that require only sequences of simple control pulses such as square wave pulses with finite rise and decay times or Gaussian wavepackets. To illustrate the technique, it is applied to find control schemes to achieve population transfers for pure-state systems, complete inversions of the ensemble populations for mixed-state systems, create arbitrary superposition states and optimize the ensemble average of dynamic observables.
Optimal bounded local sine-waveform controls for state transfer in NNN-level quantum systems
Based on a parametrization of pure quantum states we explicitly construct a sequence of (at most) 4N−54N-54N−5 local time-continuous waveform controls to achieve a specified state transition for NNN-level quantum systems when sufficient controls of the Hamiltonian are available. The control magnitudes are further optimized in terms of a time-energy performance, which is a generalization of the time performance index. Trajectory-constrained optimal local time-continuous waveform controls, including both local sine-waveforms and nthn^{\rm th}nth-order-polynomial waveform controls are obtained in terms of time-energy performance. It is demonstrated that constrained optimal local nthn^{\rm th}nth-order-polynomial waveform controls approach constrained optimal bang-bang controls when n→∞n\rightarrow\inftyn→∞.
A robust control over quantum dynamics is of paramount importance for quantum technologies. Many of the existing control techniques are based on smooth Hamiltonian modulations involving repeated calculations of basic unitaries resulting in time complexities scaling rapidly with the length of the control sequence. Here we show that bang-bang controls need one-time calculation of basic unitaries and hence scale much more efficiently. By employing a global optimization routine such as the genetic algorithm, it is possible to synthesize not only highly intricate unitaries, but also certain nonunitary operations. We demonstrate the unitary control through the implementation of the optimal fixed-point quantum search algorithm in a three-qubit nuclear magnetic resonance (NMR) system. Moreover, by combining the bang-bang pulses with the crusher gradients, we also demonstrate nonunitary transformations of thermal equilibrium states into effective pure states in three- as well as five-qubit NMR systems.
An explicit algorithm for calculating the optimized Euler angles for both qubit state transfer and gate engineering given two arbitrary fixed Hamiltonians is presented. It is shown how the algorithm enables us to efficiently implement single qubit gates even if the control is severely restricted and the experimentally accessible Hamiltonians are far from orthogonal. It is further shown that using the optimized Euler angles can significantly improve the fidelity of quantum operations even for systems where the experimentally accessible Hamiltonians are nearly orthogonal. Unlike schemes such as composite pulses, the proposed scheme does not significantly increase the number of local operations or gate operation times.