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arXiv:quant-ph/0604203 · The Journal of Chemical Physics

Principles of control for decoherence-free subsystems

Paola Cappellaro, J. S. Hodges, Timothy F. Havel, David G. Cory

Published Jul 27, 2006• 44 citations
arXivHTMLPDF

Abstract

Decoherence-free subsystems (DFSs) are a powerful means of protecting quantum information against noise with known symmetry properties. Although Hamiltonians that can implement a universal set of logic gates on DFS encoded qubits without ever leaving the protected subsystem theoretically exist, the natural Hamiltonians that are available in specific implementations do not necessarily have this property. Here we describe some of the principles that can be used in such cases to operate on encoded qubits without losing the protection offered by the DFSs. In particular, we show how dynamical decoupling can be used to control decoherence during the unavoidable excursions outside of the DFS. By means of cumulant expansions, we show how the fidelity of quantum gates implemented by this method on a simple two physical qubit DFS depends on the correlation time of the noise responsible for decoherence. We further show by means of numerical simulations how our previously introduced "strongly modulating pulses" for NMR quantum information processing can permit high-fidelity operations on multiple DFS encoded qubits in practice, provided that the rate at which the system can be modulated is fast compared to the correlation time of the noise. The principles thereby illustrated are expected to be broadly applicable to many implementations of quantum information processors based on DFS encoded qubits.

DOI
10.1063/1.2216702

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Decoherence-free subsystems (DFSs) are a powerful means of protecting quantum information against noise with known symmetry properties. Although Hamiltonians that can implement a universal set of logic gates on DFS encoded qubits without ever leaving the protected subsystem theoretically exist, the natural Hamiltonians that are available in specific implementations do not necessarily have this property. Here we describe some of the principles that can be used in such cases to operate on encoded qubits without losing the protection offered by the DFSs. In particular, we show how dynamical decoupling can be used to control decoherence during the unavoidable excursions outside of the DFS. By means of cumulant expansions, we show how the fidelity of quantum gates implemented by this method on a simple two physical qubit DFS depends on the correlation time of the noise responsible for decoherence. We further show by means of numerical simulations how our previously introduced "strongly modulating pulses" for NMR quantum information processing can permit high-fidelity operations on multiple DFS encoded qubits in practice, provided that the rate at which the system can be modulated is fast compared to the correlation time of the noise. The principles thereby illustrated are expected to be broadly applicable to many implementations of quantum information processors based on DFS encoded qubits.
Decoherence-Free Subsystems (DFS) are a powerful means of protecting quantum information against noise with known symmetry properties. Although Hamiltonians theoretically exist that can implement a universal set of logic gates on DFS encoded qubits without ever leaving the protected subsystem, the natural Hamiltonians that are available in specific implementations do not necessarily have this property. Here we describe some of the principles that can be used in such cases to operate on encoded qubits without losing the protection offered by the DFS. In particular, we show how dynamical decoupling can be used to control decoherence during the unavoidable excursions outside of the DFS. By means of cumulant expansions, we show how the fidelity of quantum gates implemented by this method on a simple two-physical-qubit DFS depends on the correlation time of the noise responsible for decoherence. We further show by means of numerical simulations how our previously introduced "strongly modulating pulses" for NMR quantum information processing can permit high-fidelity operations on multiple DFS encoded qubits in practice, provided that the rate at which the system can be modulated is fast compared to the correlation time of the noise. The principles thereby illustrated are expected to be broadly applicable to many implementations of quantum information processors based on DFS encoded qubits.
Decoherence-free subspaces protect quantum information from the effects of noise that is correlated across the physical qubits used to implement them. Given the ability to impose suitable Hamiltonians upon such a multi-qubit system, one can also implement a set of logical gates which enables universal computation on this information without compromising this protection. Real physical systems, however, seldom come with the correct Hamiltonians built-in, let alone the ability to turn them off and on at will. In the course of our development of quantum information processing devices based on liquid-state NMR, we have found the task of operating on quantum information encoded in decoherence-free subspaces rather more challenging than is commonly assumed. This contribution presents an overview of these challenges and the methods we have developed for overcoming them in practice. These methods promise to be broadly applicable to many of the physical systems proposed for the implementation of quantum information processing devices.
Decoherence-free subspaces and subsystems (DFS) preserve quantum information by encoding it into symmetry-protected states unaffected by decoherence. An inherent DFS of a given experimental system may not exist; however, through the use of dynamical decoupling (DD), one can induce symmetries that support DFSs. Here, we provide the first experimental demonstration of DD-generated decoherence-free subsystem logical qubits. Utilizing IBM Quantum superconducting processors, we investigate two and three-qubit DFS codes comprising up to six and seven noninteracting logical qubits, respectively. Through a combination of DD and error detection, we show that DFS logical qubits can achieve up to a 23% improvement in state preservation fidelity over physical qubits subject to DD alone. This constitutes a beyond-breakeven fidelity improvement for DFS-encoded qubits. Our results showcase the potential utility of DFS codes as a pathway toward enhanced computational accuracy via logical encoding on quantum processors.
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.
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.
The Uhrig dynamical decoupling sequence achieves high-order decoupling of a single system qubit from its dephasing bath through the use of bang–bang Pauli pulses at appropriately timed intervals. High-order decoupling of single and multiple qubit systems from baths causing both dephasing and relaxation can also be achieved through the nested application of Uhrig sequences, again using single-qubit Pauli pulses. For the three-qubit decoherence free subsystem (DFS) and related subsystem encodings, Pauli pulses are not naturally available operations; instead, exchange interactions provide all required encoded operations. Here we demonstrate that exchange interactions alone can achieve high-order decoupling against general noise in the three-qubit DFS. We present decoupling sequences for a three-qubit DFS coupled to classical and quantum baths and evaluate the performance of the sequences through numerical simulations.