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arXiv:quant-ph/0404081

Local control theory for unitary transformations: Application to quantum computing without leakage

Shlomo E. Sklarz, David J. Tannor

Published Apr 13, 2004• 2 citations
arXivHTMLPDF

Abstract

We present a local optimal control strategy to produce desired unitary transformations. Unitary transformations are central to all quantum computational algorithms. Many realizations of quantum computation use a submanifold of states, comprising the quantum register, coupled by an external driving field to a collection of additional mediating excited states. Previous attempts to apply control theory to induce unitary transformations on the quantum register, while successful, produced pulses that drive the population out of the computational register at intermediate times. Leakage of population from the register is undesirable since often the states outside the register are prone to decay and decoherence, and populating them causes a decrease in the final fidelity. In this work we devise a local optimal control method for achieving target unitary transformations on a quantum register, while avoiding intermediate leakage out of the computational submanifold. The technique exploits a phase locking of the field to the system such as to eliminate the undesirable excitation. This method is then applied to produce an SU(6) Fourier transform on the vibrational levels of the ground electronic state of the Na$_2$ molecule. The emerging mechanism uses two photon resonances to create a transformation on the quantum register while blocking one photon resonances to excited states.

DOI
10.48550/arxiv.quant-ph/0404081

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We present a local optimal control strategy to produce desired unitary transformations. Unitary transformations are central to all quantum computational algorithms. Many realizations of quantum computation use a submanifold of states, comprising the quantum register, coupled by an external driving field to a collection of additional mediating excited states. Previous attempts to apply control theory to induce unitary transformations on the quantum register, while successful, produced pulses that drive the population out of the computational register at intermediate times. Leakage of population from the register is undesirable since often the states outside the register are prone to decay and decoherence, and populating them causes a decrease in the final fidelity. In this work we devise a local optimal control method for achieving target unitary transformations on a quantum register, while avoiding intermediate leakage out of the computational submanifold. The technique exploits a phase locking of the field to the system such as to eliminate the undesirable excitation. This method is then applied to produce an SU(6) Fourier transform on the vibrational levels of the ground electronic state of the Na2_22​ molecule. The emerging mechanism uses two photon resonances to create a transformation on the quantum register while blocking one photon resonances to excited states.
The dynamics of a quantum system driven by an external field is well described by a unitary transformation generated by a time-dependent Hamiltonian. The inverse problem of finding the field that generates a specific unitary transformation is the subject of study. The unitary transformation which can represent an algorithm in a quantum computation is imposed on a subset of quantum states embedded in a larger Hilbert space. Optimal control theory is used to solve the inversion problem irrespective of the initial input state. A unified formalism based on the Krotov method is developed leading to a different scheme. The schemes are compared for the inversion of a two-qubit Fourier transform using as registers the vibrational levels of the $X{}^{1}{\ensuremath{\Sigma}}_{g}^{+}$ electronic state of Na2.{\mathrm{Na}}_{2}.Na2​. Raman-like transitions through the $A{}^{1}{\ensuremath{\Sigma}}_{u}^{+}$ electronic state induce the transitions. Light fields are found that are able to implement the Fourier transform within a picosecond time scale. Such fields can be obtained by pulse-shaping techniques of a femtosecond pulse. Of the schemes studied, the square modulus scheme converges fastest. A study of the implementation of the Q qubit Fourier transform in the Na2{\mathrm{Na}}_{2}Na2​ molecule was carried out for up to five qubits. The classical computation effort required to obtain the algorithm with a given fidelity is estimated to scale exponentially with the number of levels. The observed moderate scaling of the pulse intensity with the number of qubits in the transformation is rationalized.
Quantum optimal control theory is applied to find optimal pulses for controlling the motion of an ion and a molecule for two different applications. Those optimal pulses enable the control of the dynamics of the system by driving the atom or the molecule from an initial state to desired states.The evolution equations obtained by means of the quantum optimal control theory are resolved iteratively using a monotonic convergent algorithm. A number of simulation parameters are varied in order to get the optimal pulses including the duration of the pulses, the time step of the time grid, a penalty factor that limits the maximal intensity of the fields, and a guess pulse which is used to start the optimal control.The optimal pulses obtained for each application are analyzed by Fourier transform, and also by looking at the time evolution of the populations that they generate in the system.The first application is the preparation of specific vibrational states of acetylene that are usually not reachable from the ground state, and that would remain unpopulated by usual spectroscopy. Relevant state energies and transition dipole moments are extracted from the experimental literature and especially from the global acetylene Hamiltonian conferring an uncommon precision to the control simulation. The control starts from the ground state. The target states belongs to the polyad Ns=1, Nr=5 of acetylene which includes two vibrational dark states and one vibrational bright state. First, the simulation is performed with the Schrödinger equation and in a second step, with the Liouville--von Neumann equation, as mixed states are prepared. Indeed, the control starts from a Boltzmann distribution of population in the rotational levels of the vibrational ground state chosen in order to simulate an experimental condition. But the distribution is truncated to limit the computational effort. One of the dark states appears to be a potential target for a realistic experimental investigation because the average population of the Rabi oscillation remains high and decoherence is expected to be weak. The optimal pulses obtained have a high fidelity, have a spectrum with well-resolved peak frequencies, and their experimental feasibility seems achievable within the current abilities of experimental laboratories.The second application is to propose an experimental realization of a microscopic physical device able to simulate quantum dynamics. The idea is to use the motional states of a Cd^+ ion trapped in an anharmonic potential to realize a quantum dynamics simulator of a single-particle Schrödinger equation. In this way, the motional states store the information and the optimal pulse manipulates this information to realize operations. In the present case, the simulated dynamics was the propagation of a wave packet in a harmonic potential. Starting from an initial quantum state, the pulse acts on the system to modify the motional states of the ion in such a way that the final superposition of motional states corresponds to the results of the dynamics. This simulation is performed with the Liouville--von Neumann equation and also with the Lindblad equation as dissipation is included to test the robustness of the pulse against perturbations of the potential. The optimal pulses that are obtained have a high fidelity which shows that the ion trap system has correctly realized the quantum dynamics simulation. The optimal pulses are valid for any initial condition if the potential of the simulation or the mass of the propagated wave packet is unchanged.
A quantum gate is realized by specific unitary transformations operating on states representing qubits. Considering a quantum system employed as an element in a quantum computing scheme, the task is therefore to enforce the pre-specified unitary transformation. This task is carried out by an external time dependent field. Optimal control theory has been suggested as a method to compute the external field which alters the evolution of the system such that it performs the desire unitary transformation. This study compares two recent implementations of optimal control theory to find the field that induces a quantum gate. The first approach is based on the equation of motion of the unitary transformation. The second approach generalizes the state to state formulation of optimal control theory. This work highlight the formal relation between the two approaches.
Coherent quantum control shows promise in preparing and manipulating internal states within a quantum system coherently. Specifically, it aims to smoothly transition the system from an initial state to a desired target state using external electromagnetic fields. This transfer of population between ground states is crucial in various quantum processes, including applications in photochemistry for controlling reactions with fast laser fields, enhancing Nuclear Magnetic Resonance images, quantum optics and quantum information science. Quantum information processing manipulates data with unitary operators called quantum logic gates, which control state transitions in qubits. These gates are vital for future quantum computers, requiring high efficiency and robustness despite experimental limitations and errors, like inconsistent control fields. The primary goal in quantum computing is to transfer population from the ground state to the target state with a high fidelity , good robustness and minimum interaction time between the electromagnetic fields and the system. Currently, creating a control method that meets all three criteria is a significant challenge in quantum information. However, by merging reverse engineering using the Lewis-Riesenfeld dynamic invariant with optimal control theory, we've shown that designing such a control field can be simplified into a Lagrange optimization problem with integral constraints. This hybrid method, called RIO (Robust Inverse Optimization), is adaptable and applicable to diverse quantum control challenges.
We consider the control problem of generating unitary transformations, which is especially relevant to current research in quantum information processing and computing, in contrast to the usual state-to-state or the more general observable expectation value control problems. A previous analysis of optimal control landscapes for unitary transformations from a kinematic perspective in the finite-dimensional unitary matrices is extended to a dynamical one in the infinite-dimensional function space of the time-dependent external field. The underlying dynamical landscape is defined as the Frobenius square norm of the difference between the control unitary matrix and the target matrix. A nonsingular adaptation matrix is introduced to provide additional freedom for exploring and manipulating key features, specifically the slope and curvature, of the control landscapes. The dynamical analysis reveals many essential geometric features of optimal control landscapes for unitary transformations, including bounds on the local landscape slope and curvature. Close examination of the curvatures at the critical points shows that the unitary transformation control landscapes are free of local traps and proper choices of the adaptation matrix may facilitate the search for optimal control fields producing desired unitary transformations, in particular, in the neighborhood of the global extrema.
Fast control of long-lived quantum registers often relies on an auxiliary degree of freedom, a mediator, but real mediator excitation can introduce decoherence, leakage, or loss when the mediator is not the coherence-optimized component. We introduce dispersive interaction via analytical linear inversion (DIAL), an analytical control construction for excitation-suppressed interaction engineering in mediator-coupled quantum registers. In the dispersive regime, we show that off-resonant driving of an effective two-level mediator transition yields a diagonal Hamiltonian whose Pauli-ZZZ interaction rates depend linearly on the drive intensities through a transfer matrix determined by the register-resolved transition spectrum. This establishes a direct linear map from experimentally accessible controls to induced multi-qubit interactions, enabling control design without iterative propagation of the full driven dynamics. We turn this structure into the target-aware DIAL algorithm, which selects favorable off-resonant tones and determines their drive intensities for a prescribed target interaction. We benchmark the resulting controls across 100 register realizations for each register size n=2,3,4,5n=2,3,4,5n=2,3,4,5, targeting the highest-order interaction Z1⋯ZnZ_1\cdots Z_nZ1​⋯Zn​ with an interaction phase of magnitude π/4π/4π/4, which generates an nnn-qubit GHZ state up to local equivalence. Propagation under the full driven mediator-register dynamics yields typical gate fidelities above 0.9970.9970.997, while the maximum transient mediator excitation remains near 1%1\%1% on average, far below what is typically possible with resonant Rabi drives. The manuscript is accompanied by an open-source Python package implementing the target-aware DIAL algorithm and the associated numerical validation workflow.