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arXiv:1410.3891

Accurate and robust unitary transformation of a high-dimensional quantum system

Brian E. Anderson, Hector Sosa Martinez, Carlos A. Riofrío, Ivan Deutsch, Poul Jessen

Published Oct 14, 2014• 8 citations
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Abstract

Quantum control in large dimensional Hilbert spaces is essential for realizing the power of quantum information processing. For closed quantum systems the relevant input/output 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 and the feasibility of robust implementation. 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 133Cs, achieving fidelities above 0.98 as measured by randomized benchmarking. Generalizing the concepts of inhomogeneous control and dynamical decoupling 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, nonclassical state preparation for high-precision metrology, implementation of quantum simulations, and the study of fundamental physics related to open quantum systems and quantum chaos.

DOI
10.48550/arxiv.1410.3891

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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 input/output 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 and the feasibility of robust implementation. 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 133Cs, achieving fidelities above 0.98 as measured by randomized benchmarking. Generalizing the concepts of inhomogeneous control and dynamical decoupling 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, nonclassical state preparation for high-precision metrology, implementation of quantum simulations, and the study of fundamental physics related to open quantum systems and quantum chaos.
Unitary transformations are the most general input-output maps available in closed quantum systems. Good control protocols have been developed for qubits, but questions remain about the use of optimal control theory to design unitary maps in high-dimensional Hilbert spaces, and about the feasibility of their robust implementation in the laboratory. Here we design and implement unitary maps in a 16-dimensional Hilbert space associated with the 6S(1/2) ground state of (133)Cs, achieving fidelities >0.98 with built-in robustness to static and dynamic perturbations. Our work has relevance for quantum information processing and provides a template for similar advances on other physical platforms.
Neutral atom systems are an appealing platform for the development and testing of quantum control and measurement techniques. This dissertation presents experimental investigations of control and measurement tools using as a testbed the 16-dimensional hyperfine manifold associated with the electronic ground state of 133^{133}133Cs atoms. On the control side, we present an experimental realization of a protocol to implement robust unitary transformations in the presence of static and dynamic perturbations. We also present an experimental realization of inhomogeneous quantum control. Specifically, we demonstrate our ability to perform two different unitary transformations on atoms that see different light shifts from an optical addressing field. On the measurement side, we present experimental realizations of quantum state and process tomography. The state tomography project encompasses a comprehensive evaluation of several measurement strategies and state estimation algorithms. Our experimental results show that in the presence of experimental imperfections, there is a clear tradeoff between accuracy, efficiency and robustness in the reconstruction. The process tomography project involves an experimental demonstration of efficient reconstruction by using a set of intelligent probe states. Experimental results show that we are able to reconstruct unitary maps in Hilbert spaces with dimension ranging from d=4d = 4d=4 to d=16d = 16d=16. To the best of our knowledge, this is the first time that a unitary process in d=16d = 16d=16 is successfully reconstructed in the laboratory.
Neutral atom systems are an appealing platform for the development and testing of quantum control and measurement techniques. This dissertation presents experimental investigations of control and measurement tools using as a testbed the 16-dimensional hyperfine manifold associated with the electronic ground state of 133^{133}133Cs atoms. On the control side, we present an experimental realization of a protocol to implement robust unitary transformations in the presence of static and dynamic perturbations. We also present an experimental realization of inhomogeneous quantum control. Specifically, we demonstrate our ability to perform two different unitary transformations on atoms that see different light shifts from an optical addressing field. On the measurement side, we present experimental realizations of quantum state and process tomography. The state tomography project encompasses a comprehensive evaluation of several measurement strategies and state estimation algorithms. Our experimental results show that in the presence of experimental imperfections, there is a clear tradeoff between accuracy, efficiency and robustness in the reconstruction. The process tomography project involves an experimental demonstration of efficient reconstruction by using a set of intelligent probe states. Experimental results show that we are able to reconstruct unitary maps in Hilbert spaces with dimension ranging from d=4d = 4d=4 to d=16d = 16d=16. To the best of our knowledge, this is the first time that a unitary process in d=16d = 16d=16 is successfully reconstructed in the laboratory.
As uncorrected quantum processors grow in sophistication, many are being used for analog quantum simulation (AQS) of novel physics beyond the capabilities of classical devices. However, without the guarantee of bounded errors achieved by digitizing the computation, we do not know when and how much to trust an AQS result in the presence of inevitable imperfections. Here, we present the results of using our exper- iment as a testbed for studying the impact of errors on AQS. We have developed a small but highly-accurate processor over a 16-dimensional Hilbert space comprising the combined electron-nuclear spin of a single 133Cs atom in the electronic ground state. Advances in Eigenvalue-Only (EVO) optimal control enable us to repeatedly (>100) perform arbitrary unitary transformations in this space while maintaining accuracy on the device. We use this capability to simulate model systems known to exhibit features of interest, such as chaos and hypersensitivity (quantum kicked top) and quantum phase transitions (the transverse Ising and Lipkin-Meshkov-Glick mod- els). Experimental simulations show high fidelity of the quantum state and dynamical features in the presence of native imperfections. We discuss these results in the con- text of a new framework for quantifying the average sensitivity of simulator outcomes to errors. These theoretical and experimental results support the idea that AQS of ‘macroscopic’ observables (e.g., total magnetization) can be robust in the presence of errors, while observables tied to a specific quantum state (e.g., the fidelity) are not.
Unitary designs are unitary ensembles that emulate Haar-random unitary statistics. They provide a vital tool for studying quantum randomness and have found broad applications in quantum technologies. However, existing research has focused on discrete ensembles, despite that many physical processes, such as in quantum chaos, thermalization, and control, naturally involve continuous ensembles generated from continuous time-evolution. Here we initial the study of continuous unitary designs, addressing fundamental questions about their construction and practical utility. For single-qubit system, we construct explicit unitary 1-design paths from spherical 2-design curves and Hopf fibration theory. For arbitrary dimensions, we develop two systematic construction frameworks, one based on topological bundle theory of the unitary group and the other based on the Heisenberg-Weyl group. On the practical front, our unitary design paths provide analytical solutions to universally robust quantum control. Simulations show they outperform conventional pulse techniques in mitigating arbitrary unknown static noises, demonstrating immediate utility for quantum engineering. Extending unitary designs to the continuous domain not only introduces powerful geometric and topological tools that complement conventional combinatorial and group-theoretic methods, but also enhances experimental feasibility over discrete counterparts which usually involve instantaneous pulses. As an outlook, we anticipate that this work will pave the way for using continuous unitary designs to explore complex quantum dynamics and devise quantum information protocols.
The optimal control of unitary transformations is a fundamental problem in quantum control theory and quantum information processing. The feasibility of performing such optimizations is determined by the computational and control resources required, particularly for systems with large Hilbert spaces. Prior work on unitary transformation control indicates that (i) for controllable systems, local extrema in the search landscape for optimal control of quantum gates have null measure, facilitating the convergence of local search algorithms, but (ii) the required time for convergence to optimal controls can scale exponentially with the Hilbert space dimension. Depending on the control-system Hamiltonian, the landscape structure and scaling may vary. This work introduces methods for quantifying Hamiltonian-dependent and kinematic effects on control optimization dynamics in order to classify quantum systems according to the search effort and control resources required to implement arbitrary unitary transformations.