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arXiv:2107.11914 · Scientific Reports

Stabilizer codes for open quantum systems

Francisco Revson F. Pereira, Stefano Mancini, Giuliano G. La Guardia

Published Jun 29, 2023• 6 citations
arXivHTMLPDF

Abstract

The Lindblad master equation describes the evolution of a large variety of open quantum systems. An important property of some open quantum systems is the existence of decoherence-free subspaces. A quantum state from a decoherence-free subspace will evolve unitarily. However, there is no procedural and optimal method for constructing a decoherence-free subspace. In this paper, we develop tools for constructing decoherence-free stabilizer codes for open quantum systems governed by the Lindblad master equation. This is done by pursuing an extension of the stabilizer formalism beyond the celebrated group structure of Pauli error operators. We then show how to utilize decoherence-free stabilizer codes in quantum metrology in order to attain the Heisenberg limit scaling with low computational complexity.

DOI
10.1038/s41598-023-37434-0

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The Lindblad master equation describes the evolution of a large variety of open quantum systems. An important property of some open quantum systems is the existence of decoherence-free subspaces. A quantum state from a decoherence-free subspace will evolve unitarily. However, there is no procedural and optimal method for constructing a decoherence-free subspace. In this paper, we develop tools for constructing decoherence-free stabilizer codes for open quantum systems governed by the Lindblad master equation. This is done by pursuing an extension of the stabilizer formalism beyond the celebrated group structure of Pauli error operators. We then show how to utilize decoherence-free stabilizer codes in quantum metrology in order to attain the Heisenberg limit scaling with low computational complexity.
In quantum computing, nice error bases as generalization of the Pauli basis were introduced by Knill. These bases are known to be projective representations of finite groups. In this paper, we propose a group representation approach to the study of quantum stabilizer codes. We utilize this approach to define decoherence-free subspaces (DFSs). Unlike previous studies of DFSs, this type of DFSs does not involve any spatial symmetry assumptions on the system-environment interaction. Thus, it can be used to construct quantum error-avoiding codes (QEACs) that are fault tolerant automatically. We also propose a new simple construction of QEACs and subsequently develop several classes of QEACs. Finally, we present numerical simulation results encoding the logical error rate over physical error rate on the fidelity performance of these QEACs. Our study demonstrates that DFSs-based QEACs are capable of providing a generalized and unified framework for error-avoiding methods.
It was shown recently [D. A. Lidar et al., Phys. Rev. Lett. 81, 2594 (1998)] that within the framework of the semigroup Markovian master equation, decoherence-free (DF) subspaces exist which are stable to first order in time to a perturbation. Here this result is extended to the non-Markovian regime and generalized. In particular, it is shown that within both the semigroup and the non-Markovian operator sum representation, DF subspaces are stable to all orders in time to a symmetry-breaking perturbation. DF subspaces are thus ideal for quantum memory applications. For quantum computation, however, the stability result does not extend beyond the first order. Thus, to perform robust quantum computation in DF subspaces, they must be supplemented with quantum error correcting codes.
We develop a novel framework to engineer persistent oscillatory modes in Markovian open quantum systems governed by a time-independent Lindblad master equation. We show that oscillatory modes can be created when the Hamiltonian and jump operator can be expressed in the same block-diagonal form. A key feature of the framework is that the dissipator of the Lindblad master equation are generally non-zero. We identify the weak and strong conditions, where the onset of the oscillatory modes is dependent and independent of the parameters of the system, respectively. Our method extends beyond the typical decoherence-free subspace approach, in which the dissipator is zero. We demonstrate the applicability of this framework using various models, showing how carefully tailored system-environment interactions can produce sustained coherent oscillations.
Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this stabilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computation which always preserves the DFS (natural fault-tolerant computation) can be performed using only two-body interactions. This is in marked contrast to standard error correcting codes, where all known constructions using one- or two-body interactions must leave the code space during the on-time of the fault-tolerant gates. A further consequence of our universality construction is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many quantum systems, is thus asymptotically universal.
Decoherence free subspaces are a steady-state structure of the open quantum system which preserves quantum coherence between the states lying with in it and thus has found a variety of applications throughout quantum information science and technology. In this paper we study the computational complexity of deciding whether an open quantum system admits a decoherence free subspace or not. More specifically we study this problem with in the context of Markovian open quantum systems, governed by the time-independent Lindblad master equation. Along the way we introduce the kkk-Local Lindbladian problem, which captures the difficulty of computing purity decay rates under Lindbladian dynamics. We show that both problems are hard for the complexity class Quantum Merlin Arthur (QMA) when the locality k≥5k \geq 5k≥5, with the first under perfect completeness and the second being complete for QMA. Our hardness construction generalizes Kitaev's clock Hamiltonian construction to the open quantum system setting by encoding the execution of a quantum circuit into the steady subspace of a Lindbladian containing both pure and mixed history states. This subspace is then mixed depending on the output of the encoded circuit. Our results suggest that deciding whether a generic Markovian open quantum system admits a decoherence free subspace is intractable even for quantum computation.
Known as decoherence, the unavoidable interaction of a quantum system with its surrounding environment is usually considered to be detrimental for quantum information processing. In this thesis the coherent, open loop control of such open systems is studied. Concepts from quantum control theory and the theory of open quantum system are adopted in order to fight decoherence and implement quantum gates in a noiseless manner. In particular, Lie algebraic methods and numerical optimization tools are used to investigate the control properties of a single spin interacting with a spin environment. We show that, independent of the size of the environment, every unitary transformation can be implemented on the system spin through a single control field. We proceed by investigating dynamical decoupling, a method to suppress the interactions with the environment, for finite- and for in- finite dimensional systems. We prove that every finite dimensional system can be protected from decoherence, even if the environment is infinite dimensional, whereas for noise described by a Lindblad master equation dynamical decoupling will never succeed. This will lead to a new method to distinguish decoherence from intrinsic noise terms. We further prove that not every infinite dimensional system can be protected from decoherence through dynamical decoupling. Afterwards we investigate dynamical decoupling of systems that are described by quadratic Hamiltonians, showing that such interactions can always be suppressed with two simple operations. In the last part we investigate the coherent control of a Lindblad master equation. We show that a strong noise process exhibiting a decoherence free subspace can substantially increase the number of unitary operations that can be implemented, allowing us to fully control parts of the system. Afterwards we develop a scheme to make Hamiltonians and Lindbladians commutative by adding an auxiliary system. The old, possibly non-commutative dynamics, is recovered through a non-selective measurement.