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arXiv:0705.4119 · Physical Review Letters

Fidelity Enhancement by Logical Qubit Encoding

Michael K. Henry, Chandrasekhar Ramanathan, J. S. Hodges, Colm A. Ryan, Michael J. T. Ditty, Raymond Laflamme, David G. Cory

Published Nov 28, 2007• 15 citations
arXivHTMLPDF

Abstract

We demonstrate coherent control of two logical qubits encoded in a decoherence free subspace (DFS) of four dipolar-coupled protons in an NMR quantum information processor. A pseudopure fiducial state is created in the DFS, and a unitary logical qubit entangling operator evolves the system to a logical Bell state. The four-spin molecule is partially aligned by a liquid crystal solvent, which introduces strong dipolar couplings among the spins. Although the system Hamiltonian is never fully specified, we demonstrate high fidelity control over the logical degrees of freedom. In fact, the DFS encoding leads to higher fidelity control than is available in the full four-spin Hilbert space.

DOI
10.1103/physrevlett.99.220501

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We demonstrate coherent control of two logical qubits encoded in a decoherence free subspace (DFS) of four dipolar-coupled protons in an NMR quantum information processor. A pseudopure fiducial state is created in the DFS, and a unitary logical qubit entangling operator evolves the system to a logical Bell state. The four-spin molecule is partially aligned by a liquid crystal solvent, which introduces strong dipolar couplings among the spins. Although the system Hamiltonian is never fully specified, we demonstrate high fidelity control over the logical degrees of freedom. In fact, the DFS encoding leads to higher fidelity control than is available in the full four-spin Hilbert space.
Liquid-phase NMR is a general-purpose testbed for developing methods of coherent control relevant to quantum information processing. Here we extend these studies to the coherent control of logical qubits and in particular to the unitary gates necessary to create entanglement between logical qubits. We report an experimental implementation of a conditional logical gate between two logical qubits that are each in decoherence-free subspaces that protect the quantum information from fully correlated dephasing.
A critical step in experimental quantum information processing (QIP) is to implement control of quantum systems protected against decoherence via informational encodings, such as quantum error-correcting codes, noiseless subsystems, and decoherence-free subspaces. These encodings lead to the promise of fault-tolerant QIP, but they come at the expense of resource overheads. Part of the challenge in studying control over multiple logical qubits is that QIP testbeds have not had sufficient resources to analyze encodings beyond the simplest ones. The most relevant resources are the number of available qubits and the cost to initialize and control them. Here we demonstrate an encoding of logical information that permits control over multiple logical qubits without full initialization, an issue that is particularly challenging in liquid-state NMR. The method of subsystem pseudopure states will allow the study of decoherence control schemes on up to six logical qubits using liquid-state NMR implementations.
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.
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 report the implementation of an entanglement transfer on a four-qubit liquid-state nuclear-magnetic resonance quantum information processor. This consists of creating an (pseudopure) entangled state among two directly coupled spins, and then transferring this two-spin state to another pair of spins whose direct interactions are negligible. Such transfers are expected to be an essential operation in scalable quantum computer architectures, and provide a useful benchmark for the coherent control available in specific implementations.
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.