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arXiv:quant-ph/0210057 · Physical Review A

Exploring noiseless subsystems via nuclear magnetic resonance

Evan M. Fortunato, Lorenza Viola, Marco A. Pravia, Emanuel H. Knill, Raymond Laflamme, Timothy F. Havel, David G. Cory

Published Jun 19, 2003• 39 citations
arXivHTMLPDF

Abstract

Noiseless subsystems offer a general and efficient method for protecting quantum information in the presence of noise that has symmetry properties. A paradigmatic class of error models displaying nontrivial symmetries emerges under collective noise behavior, which implies a permutationally invariant interaction between the system and the environment. We expand our previous investigation of the noiseless subsystem idea [L. Viola et al., Science 293, 2059 (2001)] by reporting and analyzing NMR experiments that demonstrate the preservation of a qubit encoded in a three-qubit noiseless subsystem for general collective noise. A complete set of input states is used to determine the superoperator for the implemented one-qubit process and to confirm that the fidelity of entanglement is improved for a large, noncommutative set of engineered errors. To date, this is the largest set of error operators that has been successfully corrected for by any quantum code.

DOI
10.1103/physreva.67.062303

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Noiseless subsystems offer a general and efficient method for protecting quantum information in the presence of noise that has symmetry properties. A paradigmatic class of error models displaying nontrivial symmetries emerges under collective noise behavior, which implies a permutationally invariant interaction between the system and the environment. We expand our previous investigation of the noiseless subsystem idea [L. Viola et al., Science 293, 2059 (2001)] by reporting and analyzing NMR experiments that demonstrate the preservation of a qubit encoded in a three-qubit noiseless subsystem for general collective noise. A complete set of input states is used to determine the superoperator for the implemented one-qubit process and to confirm that the fidelity of entanglement is improved for a large, noncommutative set of engineered errors. To date, this is the largest set of error operators that has been successfully corrected for by any quantum code.
We demonstrate the protection of one bit of quantum information against all collective noise in three nuclear spins. Because no subspace of states offers this protection, the quantum bit was encoded in a proper noiseless subsystem. We therefore realize a general and efficient method for protecting quantum information. Robustness was verified for a full set of noise operators that do not distinguish the spins. Verification relied on the most complete exploration of engineered decoherence to date. The achieved fidelities show improved information storage for a large, noncommutative set of errors.
We show a way to correct collective error through a three-qubit quantum code. The encoding and decoding (i.e., error recovery) operations for such a code, which protects one qubit of quantum information perfectly (with infinite ``distance'') against collective decoherence, are presented. The code is actually a manifestation of a noiseless subsystem first reported by Knill, Laflamme, and Viola [Phys. Rev. Lett. 84, 2525 (2000)], and provides a good illustration of the relationship between noiseless subsystems and error-correcting codes already noted by these authors. We also show in detail how this code can be used to preserve an arbitrary joint state (including entangled states) of two qubits, by pairing each of them with two other qubits, even in the case in which the two resulting clusters interact in nonequivalent ways with a common environment.
Quantum error correction has been successfully applied to enhance the precision of parameter estimation in the presence of noise. Nonetheless, existing methods require a number of noiseless, controllable ancillae and lack efficient encoding and decoding procedures. In this Letter, we demonstrate that subsystem error correction provides a new direction that can substantially simplify the metrological protocol. We derive general conditions under which subsystem stabilizer codes achieve the Heisenberg limit and show that, for broad classes of noise, this can be realized by syndrome-free protocols using at most a single ancilla qubit. Furthermore, we extend this framework to dynamical error correction and show that Floquet codes can protect time-dependent metrological signals in reaching the Heisenberg limit.
The development of a quantum computer presents one of the greatest challenges in science and engineering to date. The promise of more efficient computing based on entangled quantum states and the superposition principle has led to a worldwide explosion of interest in the fields of quantum information and computation. Among the number of hurdles which must first be cleared before we witness a physical realization are problems associated with environment-induced decoherence and noise more generally. However, the discovery of quantum error correction and the establishment of the accuracy threshold theorem provide us with the hope of someday harnessing the potential power a functioning fault-tolerant quantum information processor has to offer. This dissertation contributes to this effort by investigating a particular class of quantum error correcting codes, namely noiseless subsystem encodings. The passive approach to error correction taken by these encodings provides an efficient means of protection from symmetrically coupled system-environment interactions. Here I will present methods for determining the subsystem-preserving evolutions for noiseless subsystem encodings supported by arbitrary-dimensional physical quantum systems. Implications for universal, collective decoherence-free quantum computation using the derived operations are discussed. Moreover, I will present a proposal for an optical device which is capable of preparing a variety of these noiseless subsystem encodings through a postselection strategy.
We introduced an open quantum system model to describe the statistical fluctuations of a spin ensemble in NMR. The model considers an ensemble measurement where the detection coil does not distinguish spins, and accounts for the state update rule. The analysis brings clarity and accuracy in describing the notion of spin noise and derives a correct statistical distribution and correlation function for the spin noise signal. \n \nWe propose a proof-of-principle experiment to encode one logical qubit in the noise protected subspace of three identical spins in a methyl group. We use a symmetry analysis to derive the dipole moment allowed transitions, which enable us to access the noiseless subsystem. We fur- ther analyze the symmetry of the hetronuclear dipolar relaxation, which is one of the responsible mechanisms for observing a noise protected state. Our analytical calculations predict features of of the NMR peaks that are in agreement with the experimental observations. \n \nWe propose a quantum key distribution protocol that simplifies the task of classical data processing in a trusted relay network. A new announcement strategy is proposed which leads to reassigning the task of error correction and privacy amplification from the intermediate user to the end-node users. We examine the security of the proposed protocol analytically, derive the key rate for two well-known examples of BB84 and 6-state protocols numerically, and consider a few imperfections arising in practical QKD.
The most significant achievements of this project were: (1) Development and validation of the hardware and software needed to implement "strongly modulating pulses", by which high precision quantum gates can be obtain in realistic systems of up to 12 qubits; (2) A detailed analysis and evaluation of a 3-qubit quantum Fourier transform via full quantum process tomography; (3) An extensive set of mathematical and computational techniques to accomplish these goals, including Hadamard products, the real density matrix, and methods of fitting superoperators to experimental data; (4) Creation of a 3-qubit "noiseless subsystem", of a Bell state on two 2-qubit decoherence-free subspaces (DFS), and the invention of "partial" pseudopure states which will enable us to demonstrate robust methods for controlling multi-DFS-qubit systems by NMR; (5) Implementation of several quantum chaotic maps, and the discovery that these provide a scalable approach to determining the magnitude and kind of errors present in complex quantum computations; (6) The invention of a "spin amplifier", by which entanglement can be used to enable single spin measurement, and a small-scale demonstration by NMR; (7) Experiments demonstrating that the foregoing advances enable implementation of complex entangling unitary and decoherent operations, culminating in creation of a 12-qubit CAT state.