arXiv:quant-ph/0111166v1 30 Nov 2001 Implementation of Universal Control on a Decoherence-Free Qubit Evan M. Fortunato 1 ∗ , Lorenza Viola 2 † , Jonathan Hodges 3 , Grum Teklemariam 4 , and David G. Cory 1 1 Department of Nuclear Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 2 Los Alamos National Laboratory, Mail Stop B256, Los Alamos, NM 87545 3 Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027 4 Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139 We demonstrate storage and manipulation of one qubit encoded into a decoherence-free subspace (DFS) of two nuclear spins using liquid state nuclear magnetic resonance (NMR) techniques. The DFS is spanned by states that are unaffected by arbitrary collective phase noise. Encoding and decoding procedures reversibly map an arbitrary qubit state from a single data spin to the DFS and back. The implementation demonstrates the robustness of the DFS memory against engineered dephasing with arbitrary strength as well as a substantial increase in the amount of quantum information retained, relative to an un-encoded qubit, under both engineered and natural noise processes. In addition, a universal set of logical manipulations over the encoded qubit is also realized. Although intrinsic limitations prevent maintaining full noise tolerance during quantum gates, we show how the use of dynamical control methods at the encoded level can ensure that computation is protected with finite distance. We demonstrate noise-tolerant control over a DFS qubit in the presence of engineered phase noise significantly stronger than observed from natural noise sources. I. INTRODUCTION The ability to effectively protect the coherence properties of a quantum information processing (QIP) device against the detrimental effects of environmental interactions is a prerequisite for realizing any potential gain of quantum computation and quantum information theory [1]. Approaches based on noiseless (or “decoherence-free” [2]) coding offer a promising venue for meeting the challenge of noise-tolerant QIP. The theory of decoherence-free subspaces (DFSs) has been the focus of intensive development particularly by Zanardi, Lidar, and coworkers [3–9]. Recently, the DFS idea has been incorporated within the more general approach based on noiseless subsystems (NSs) [10–13], which recover DFSs and their benefits as special instances. The primary motivation behind “passive” noise control strategies relying on either DFSs or NSs is to take advantage of specific symmetries occurring in the noise process to single out subspaces or subsystems of the physical information processor that are inaccessible to noise. Once information is appropriately encoded into such noiseless structures, robust storage is ensured without requiring further active correction – as long as the underlying symmetry dominates. These features, together with their stability against symmetry-perturbing errors [5,6,8] and the consequent potential for concatenation with quantum error-correcting codes [7], make noiseless codes natural candidates as robust quantum memories. To date, experimental implementations include studies of DF states in quantum optical systems [14] , and one-bit quantum memories based on both a DFS of two trapped ions [15] and a NS of three nuclear spins [16]. Achieving robust quantum information storage represents only a first, though indispensable, step toward the goal of reliable QIP. An important advance in this direction came from the identification of universality schemes, which in principle enable DFSs (or NSs) to support universal encoded quantum computation in a way that remains fully protected against noise. Both existential [11,17] and constructive results [18] have been established. While the latter are especially appealing for a class of proposed quantum computing architectures governed by Heisenberg exchange interactions [19], implementations of these schemes remain difficult due to the stringent symmetry and tunability requirements on the control Hamiltonians. Here, we take a first experimental step towards encoded quantum computation by demonstrating universal control over a one-bit DF quantum register of two nuclear spins. A novel key ingredient we use to implement encoded quantum gates is the combination of robust control design with the use of dynamical decoupling methods [20–23] directly on encoded degrees of freedom. Our results suggest that this may serve as a useful strategy for practically coping with the constraints required for DF computation. The paper is organized as follows. In Sect. II we review the collective decoherence model that is relevant to the work, along with the prescriptions from the DFS theory for both protected storage and manipulation of quantum information in a two-qubit system. In Sect. III, we outline our proposed approach to noise-tolerant control of DFS encoded qubits based on concatenating encoded decoupling methods with robust control design. The general principles are developed starting from the physical NMR setting relevant to the experiment. Sect. IV contains an account of the control techniques used in the experiment and the reliability measures adopted to quantify the accuracy of the implementation. In particular, a notion of gate entanglement fidelity, generalizing Schumacher’s definition to allow a desired unitary evolution on the quantum data, is proposed and related to other fidelity metrics relevant to QIP. The experimental results demonstrating protected storage and universal protected quantum logic are presented and discussed in Sect. V and VI, respectively. 1