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Science

Experimental Realization of Noiseless Subsystems for Quantum Information Processing

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

Published Sep 14, 2001• 235 citations
Publisher

Abstract

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.

DOI
10.1126/science.1064460

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•Nov 30, 2001•67 citations
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.
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.
There are two complementary approaches to realizing quantum information so that it is protected from a given set of error operators. Both involve encoding information by means of subsystems. One is initialization-based error protection, which involves a quantum operation that is applied before error events occur. The other is operator quantum error correction, which uses a recovery operation applied after the errors have occurred. Together, the two approaches make it clear how quantum information can be stored at all stages of a process involving alternating error and quantum operations. In particular, there is always a subsystem that faithfully represents the desired quantum information. We give a definition of faithful realization of quantum information and show that it always involves subsystems. This supports the ``subsystems principle'' for realizing quantum information. In the presence of errors, one can make use of noiseless, (initialization) protectable, or error-correcting subsystems. We give an explicit algorithm for finding optimal noiseless subsystems. Finding optimal protectable or error-correcting subsystems is in general difficult. Verifying that a subsystem is error correcting is known to involve only linear algebra. We discuss the verification problem for protectable subsystems and reduce it to a simpler version of the problem of finding error-detecting codes.
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 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 maintenance of 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.
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 computers exploit quantum mechanical phenomena as resources to process information.They are therefore far more vulnerable to errors than their classical counterparts.The main obstacle to the practical realization of a working large-scale quantum computer is decoherence, i.e., the process by which a quantum system decays to a classical state through interaction with the environment.We herein discuss two techniques to implement robust quantum information processing, and report on their implementation in a liquid state NMR Quantum Computing experiment.We describe a method to implement a two-qubit gate between spins 2 and 3 of a three-spin linear chain molecule by applying a soft pulse to spin 1, with appropriate choices of the rf-field amplitude ω 1 and duration τ [1].The Bloch-Siegert effect for such a soft pulse is negligible in general, which makes NMR pulse programming and quantum gates design much simpler than with conventional hard π-pulses; it also makes this method suitable for use with both homo-and heteronuclear molecules.We then discuss a simple Operator Quantum Error Correction (OQEC) scheme with three qubits [2].This scheme provides the simplest possible Noiseless Subsystem, in terms of the number of CNOT gates, that is robust against (restricted) fully correlated noise.We demonstrated our OQEC scheme experimentally using a three-qubit NMR Quantum Computer, in which the ancillae are in the maximally mixed state.In this experiment, we made practical use of the soft pulse technique [1] to implement the two-qubit gates in the pulse sequences realizing encoding and decoding operations.
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.