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arXiv:0904.4861 · Physical Review Letters

How Long Can a Quantum Memory Withstand Depolarizing Noise?

Fernando Pastawski, Alastair Kay, Norbert Schuch, J. I. Cirac

Published Aug 20, 2009• 22 citations
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Abstract

We investigate the possibilities and limitations of passive Hamiltonian protection of a quantum memory against depolarizing noise. Without protection, the lifetime of a single qubit is independent of N, the number of qubits composing the memory. In the presence of a protecting Hamiltonian, the lifetime increases at most logarithmically with N. We construct an explicit time-independent Hamiltonian which saturates this bound, exploiting the noise itself to achieve the protection.

DOI
10.1103/physrevlett.103.080501

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We investigate the possibilities and limitations of passive Hamiltonian protection of a quantum memory against depolarizing noise. Without protection, the lifetime of a single qubit is independent of N, the number of qubits composing the memory. In the presence of a protecting Hamiltonian, the lifetime increases at most logarithmically with N. We construct an explicit time-independent Hamiltonian which saturates this bound, exploiting the noise itself to achieve the protection.
This thesis is devoted to the study of coherent storage of quantum information as well as its potential applications.Quantum memories are crucial to harnessing the potential of quantum physics for information processing tasks.They are required for almost all quantum computation proposals.However, despite the large arsenal of theoretical techniques and proposals dedicated to their implementation, the realization of long-lived quantum memories remains an elusive task.Encoding information in quantum states associated to many-body topological phases of matter and protecting them by means of a static Hamiltonian is one of the leading proposals to achieve quantum memories.While many genuine and well publicized virtues have been demonstrated for this approach, equally real limitations were widely disregarded.In the first two projects of this thesis, we study limitations of passive Hamiltonian protection of quantum information under two different noise models.Chapter 2 deals with arbitrary passive Hamiltonian protection for a many body system under the effect of local depolarizing noise.It is shown that for both constant and time dependent Hamiltonians, the optimal enhancement over the natural single-particle memory time is logarithmic in the number of particles composing the system.The main argument involves a monotonic increase of entropy against which a Hamiltonian can provide little protection.Chapter 3 considers the recoverability of quantum information when it is encoded in a many-body state and evolved under a Hamiltonian composed of known geometrically local interactions and a weak yet unknown Hamiltonian perturbation.We obtain some generic criteria which must be fulfilled by the encoding of information.For specific proposals of protecting Hamiltonian and encodings such as Kitaev's toric code and a subsystem code proposed by Bacon, we additionally provide example perturbations capable of destroying the memory which imply upper bounds for the provable memory times.vi
The ability to protect quantum information from the effect of noise is one of the major goals of quantum information processing. In this article, we study limitations on the asymptotic stability of quantum information stored in passive N-qubit systems. We consider the effect of small imperfections in the implementation of the protecting Hamiltonian in the form of perturbations or weak coupling to a ground state environment. We prove that, regardless of the protecting Hamiltonian, there exists a perturbed evolution that necessitates a final error correcting step when the state of the memory is read. Such an error correction step is shown to require a finite error threshold, the lack thereof being exemplified by the 3D compass model. We go on to present explicit weak Hamiltonian perturbations which destroy the logical information stored in the 2D toric code in a time O(log(N)).
Fresh qubits, not size, set how long a noisy quantum system can remember. With kkk input qubits reinitialized per time step, the longest memory any reservoir can attain under depolarizing noise is exp⁡[Θ(k)]\exp[Θ(k)]exp[Θ(k)] time steps, whatever its size, dynamics, and input encoding. A universal upper bound and an explicit reservoir that attains the exponential scaling establish this optimal resource law. The bound depends on the dynamics only through the entropy that fresh qubits remove per time step, so it holds for any Markovian noisy dynamics without feedback, including measurement-free error correction.
The ability to protect quantum information from the effect of noise is one of the major goals of quantum information processing. In this article, we study limitations on the asymptotic stability of quantum information stored in passive NNN-qubit systems. We consider the effect of small imperfections in the implementation of the protecting Hamiltonian in the form of perturbations or weak coupling to a ground state environment. We thus depart from the usual Markovian approximation for a thermal bath by conce ntrating on models for which part of the evolution can be calculated exactly. We prove that, regardless of the protecting Hamiltonian, there exists a perturbed evolution that necessitates a final error correcting step for the state of the memory to be read. Such an error correction step is shown to require a finite error threshold, the lack thereof being exemplified by the 3D XZ-compass model \cite{bacon-2006}. We go on to present explicit weak Hamiltonian perturbations which destroy the logical information stored in the 2D toric code in a time O(log⁡(N))O(\log(N))O(log(N)).
We demonstrate a decoherence-free quantum memory of one qubit. By encoding the qubit into the decoherence-free subspace (DFS) of a pair of trapped 9Be+ ions, we protect the qubit from environment-induced dephasing that limits the storage time of a qubit composed of a single ion. We measured the storage time under ambient conditions and under interaction with an engineered noisy environment and observed that encoding into the DFS increases the storage time by up to an order of magnitude. The encoding reversibly transfers an arbitrary qubit stored in a single ion to the DFS of two ions.
We derive ultimate precision bounds for estimating parameters encoded in \emph{time-dependent} Hamiltonians in the presence of general Markovian noise, allowing for arbitrary adaptive protocols with fast controls and noiseless ancillas. Extending the minimization-over-purifications framework to time-varying continuous channels, we obtain a differential upper bound on the achievable quantum Fisher information (QFI) that can be evaluated at all times via semidefinite programming. For parameter-independent noise, we prove a universal long-time scaling law: if the coherent (noiseless) dynamics yields Qcoh(T)∼T2kQ_{\mathrm{coh}}(T)\sim T^{2k}Qcoh​(T)∼T2k, then under Markovian noise the QFI scales at most as Q(T)∼T2kQ(T)\sim T^{2k}Q(T)∼T2k in the DHNLS regime, whereas in the DHLS regime it is fundamentally limited to Q(T)∼T2k−1Q(T)\sim T^{2k-1}Q(T)∼T2k−1. We illustrate these behaviors on paradigmatic driven-qubit sensors, exhibiting T4T^{4}T4 and T3T^{3}T3 scalings under dephasing and spontaneous emission, respectively. Finally, we provide explicit continuous exact and approximate quantum error correction constructions -- supplemented by spin-squeezed probes -- that asymptotically saturate the bounds, establishing their tightness.