We implement arbitrary maps between pure states in the 16-dimensional Hilbert space associated with the ground electronic manifold of ^{133}Cs. This is accomplished by driving atoms with phase modulated radio-frequency and microwave fields, using modulation waveforms found via numerical optimization and designed to work robustly in the presence of imperfections. We evaluate the performance of a sample of randomly chosen state maps by randomized benchmarking, obtaining an average fidelity >99%. Our protocol advances state-of-the-art quantum control and has immediate applications in quantum metrology and tomography.

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...

The need to perform quantum state tomography on ever larger systems has spurred a search for methods that yield good estimates from incomplete data. We study the performance of compressed sensing (CS) and least squares (LS) estimators in a fast protocol based on continuous measurement on an ensemble of cesium atomic spins. Both efficiently reconstruct nearly pure states in the 16dimensional ground manifold, reaching average fidelitiesFCS = 0.92 andFLS = 0.88 using similar amounts of incomplete data. Surprisingly, the main advantage of CS in our protocol is an increased robustness to experimental imperfections.PACS numbers: 03.65. Wj, 42.50.Dv, Recovering a full description of a complex system from limited information is a central problem in science and engineering. In physics one often seeks to estimate an unknown quantum state based on measurement data [1], generally a formidable challenge for large systems given that O(d 2 ) real parameters are needed to describe arbitrary states in a d-dimensional Hilbert space. In quantum information science, however, the states of interest are nearly pure and can be described by O(d) parameters. Algorithms that make use of this prior information to obtain good estimates from a reduced number of measurements fall under the general heading of compressed sensing [2], a family of techniques used in signal processing tasks that range from movie recommendation to earthquake analysis. Gross et al. [3,4] have developed one such algorithm that gives good estimates of nearly pure quantum states in a d-dimensional Hilbert space from the expectation values of O(d log d) orthogonal observables, a substantial saving when d is large. This algorithm was recently benchmarked against a standard maximum likelihood estimator in an experiment with photonic qubits and the two were found to yield similar results [5]. Generalization to process tomography has led to similar improvements when the process is close to unitary [6].In this work we study the laboratory performance of quantum state reconstruction based on compressed sensing (CS) and least-squares [7] (LS) estimators in the context of continuous measurement. Our physical testbed consists of the 16-dimensional hyperfine manifold of magnetic sublevels in the electronic ground state of atomic cesium. The data required for quantum tomography is gathered by performing a weak (nonprojective) continuous measurement on an ensemble of atoms while dynamically evolving their state with known driving fields [8][9][10][11]. This approach differs substantially from conventional quantum tomography in that the measurement record contains information about the expectation values of a continuum of nonorthogonal observables instead FIG. 1.(Color online) Schematic of the experiment. An ensemble of identically prepared cesium atoms is probed with an optical beam and polarimeter to obtain a continuous measurement of the spin observable fz in the f = 3 hyperfine state. The atoms sit at the center of a plexiglass cube that supports coil pairs used to apply bias an...