Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms

Khaneja, Navin; Reiss, Timo O.; Kehlet, Cindie; Schulte‐Herbrüggen, Thomas; Glaser, Steffen J.

doi:10.1016/j.jmr.2004.11.004

Cited by 1,629 publications

(1,805 citation statements)

References 40 publications

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“…The most common approach is to rely on open-loop quantum control techniques, including optimal control algorithms, based on analytical [1] or numerical [2] solutions, Lyapunov design [3] and Hamiltonian engineering [4]. An alternative strategy, inspired by the success of classical control, is feedback control [5].…”

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confidence: 99%

Coherent feedback control of a single qubit in diamond

Hirose

Cappellaro

2016

Nature

103

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Engineering desired operations on qubits subjected to the deleterious effects of their environment is a critical task in quantum information processing, quantum simulation and sensing. The most common approach is to rely on open-loop quantum control techniques, including optimal control algorithms, based on analytical [1] or numerical [2] solutions, Lyapunov design [3] and Hamiltonian engineering [4]. An alternative strategy, inspired by the success of classical control, is feedback control [5]. Because of the complications introduced by quantum measurement [6], closed-loop control is less pervasive in the quantum settings and, with exceptions [7,8], its experimental implementations have been mainly limited to quantum optics experiments. Here we implement a feedback control algorithm with a solid-state spin qubit system associated with the Nitrogen Vacancy (NV) centre in diamond, using coherent feedback [9] to overcome limitations of measurement-based feedback, and show that it can protect the qubit against intrinsic dephasing noise for milliseconds. In coherent feedback, the quantum system is connected to an auxiliary quantum controller (ancilla) that acquires information about the system's output state (by an entangling operation) and performs an appropriate feedback action (by a conditional gate). In contrast to open-loop dynamical decoupling (DD) techniques [10], feedback control can protect the qubit even against Markovian noise and for an arbitrary period of time (limited only by the ancilla coherence time), while allowing gate operations. It is thus more closely related to Quantum Error Correction schemes [11][12][13][14], which however require larger and increasing qubit overheads. Increasing the number of fresh ancillas allows protection even beyond their coherence time. We can further evaluate the robustness of the feedback protocol, which could be applied to quantum computation and sensing, by exploring an interesting tradeoff between information gain and decoherence protection, as measurement of the ancilla-qubit correlation after the feedback algorithm voids the protection, even if the rest of the dynamics is unchanged.To demonstrate coherent feedback with spin qubits, we choose two of the most common tasks for qubits, implementing the no-operation (NOOP) and NOT gates, while cancelling the effects of noise. A simple, measurementbased feedback scheme, exploiting one ancillary qubit, was proposed in [16]. The correction protocol (Fig. 1a) works by entangling the qubit-ancilla system before the desired gate operation. By selecting an entangling operation U c appropriate for the type of bath acting on the system, information about the noise action is encoded in the ancilla state. After undoing the entangling operation, the qubit coherence can be restored by a feedback action, Hadamard gates prepare and read out a superposition state of the qubit, |φ q = 1 √ 2 (|0 + |1 ). Amid entangling gates between qubit and ancilla, the qubit is subjected to noise (and possibly unitary gates U ). We assume the ancil...

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confidence: 99%

Coherent feedback control of a single qubit in diamond

Hirose

Cappellaro

2016

Nature

103

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“…To overcome this problem, optimized pulse engineering techniques inspired by optimal control theory for NMR quantum computing have been developed in recent years. Here we primarily focus on the GRadient Ascent Pulse Engineering (GRAPE) techniques [17].…”

Section: Universal Gatesmentioning

confidence: 99%

“…This can be decomposed into two CNOT gates and three single qubit Hadamard gates. A GRAPE pulse [17] with the length 1.5ms was designed to implement this encoding operation, and was rectified by pulse fixing in experiment. The estimation of the average fidelity before and after the rectification is 86.3% and 97.3%, respectively.…”

Section: Characterization Of Clifford Gatesmentioning

confidence: 99%

“…To compensate, the reference spectrum was measured using a similar experiment with the identity instead of the controlled unitary. The other sources of error were gate fidelities, these were maximized using GRAPE [17]. The algorithm was implemented for 18 different braids that had to be divided into six different groups corresponding to the six distinct values of the Jones polynomials.…”

Section: Dqc1mentioning

confidence: 99%

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NMR Quantum Information Processing

Sarthour¹,

Bonagamba²,

Azevêdo³

et al. 2007

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“…We find it advantageous to supplement these with a modified version [23] of the numerically efficient GRAPE algorithm for the calculation of gradients [24]. With this we find optimization of individual waveforms to be straightforward on a desktop computer, and the design of large numbers of waveforms to be feasible on a high-performance cluster [25].…”

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confidence: 99%

Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System

et al. 2015

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Quantum control in large dimensional Hilbert spaces is essential for realizing the power of quantum information processing. For closed quantum systems the relevant inputoutput maps are unitary transformations, and the fundamental challenge becomes how to implement these with high fidelity in the presence of experimental imperfections and decoherence. For two-level systems (qubits) most aspects of unitary control are well understood, but for systems with Hilbert space dimension d>2 (qudits), many questions remain regarding the optimal design of control Hamiltonians 1 and the feasibility of robust implementation 2,3 . Here we show that arbitrary, randomly chosen unitary transformations can be efficiently designed and implemented in a large dimensional Hilbert space (d=16) associated with the electronic ground state of atomic 133 Cs, 4 achieving fidelities above 0.98 as measured by randomized benchmarking 5 . Generalizing the concepts of inhomogeneous control 6 and dynamical decoupling 7 to d>2 systems, we further demonstrate that these qudit unitary maps can be made robust to both static and dynamic perturbations. Potential applications include improved fault-tolerance in universal quantum computation 8 , nonclassical state preparation for high-precision metrology 9 , implementation of quantum simulations 10 , and the study of fundamental physics related to open quantum systems and quantum chaos 11 .The goal of quantum control is to perform a desired transformation through dynamical evolution driven by a control Hamiltonian H C (t) . For example, one common objective is to evolve the system from a known initial state to a desired final state. If the control task is simple or special symmetries are present, it is sometimes possible to find a high-performing control Hamiltonian through intuition, or to construct one using group theoretic methods 12 . In this letter we explore the use of "optimal control" 1 to design control Hamiltonians for tasks of varying complexity, from state-to-state maps to unitary maps on the entire accessible Hilbert space. The basic procedure is well established: the Hamiltonian H C (t) is parameterized by a set of control variables, and a numerical search is performed to find values that optimize the fidelity with which the control objective is achieved. The application of optimal control to quantum systems originated in NMR 13 and physical chemistry 1 , and has since expanded to include, e. g., ultrafast physics 14 , cold atoms 15,16 , biological molecules 17 , spins in condensed matter 18 , and superconducting circuits 19 .We study the efficacy of numerical design and the performance of the resulting control Hamiltonians using a well developed testbed consisting of the electron and nuclear spins of individual 133 Cs atoms driven by radiofrequency (rf) and microwave (µw) magnetic fields (Fig. 1) 16 . Our experiments show that the optimal control strategy is adaptable to a wide range of control tasks, and that it can generate control Hamiltonians with excellent performance even in the prese...

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Optimal control of coupled spin dynamics: design of NMR pulse sequences by gradient ascent algorithms

Cited by 1,629 publications

References 40 publications

Coherent feedback control of a single qubit in diamond

Coherent feedback control of a single qubit in diamond

NMR Quantum Information Processing

Accurate and Robust Unitary Transformations of a High-Dimensional Quantum System

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