3D Topological Quantum Memory with a Power-Law Energy Barrier

Michnicki, Kamil

doi:10.1103/physrevlett.113.130501

Cited by 33 publications

(54 citation statements)

References 29 publications

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“…Recent discoveries [24][25][26] show that this tradeoff is not necessary in 3D. Our result extends prior findings [13,15] derived for stabilizer codes to a broader, widely studied class of models that includes quantum double [2], LevinWen [7], and Turaev-Viro [8] models among others.…”

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confidence: 86%

Local Topological Order Inhibits Thermal Stability in 2D

Landon-Cardinal

Poulin

2013

Phys. Rev. Lett.

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We study the robustness of quantum information stored in the degenerate ground space of a local, frustration-free Hamiltonian with commuting terms on a 2D spin lattice. On one hand, a macroscopic energy barrier separating the distinct ground states under local transformations would protect the information from thermal fluctuations. On the other hand, local topological order would shield the ground space from static perturbations. Here we demonstrate that local topological order implies a constant energy barrier, thus inhibiting thermal stability.PACS numbers: 03.67.Pp, 03.65.Ud, 03.67.Ac A self-correcting quantum memory [1] is a physical system whose quantum state can be preserved over a long period of time without the need for any external intervention. The archetypical self-correcting classical memory is the two-dimensional (2D) Ising ferromagnet. The ground state of this system is two-fold degenerate-all-spin up and all-spin down-so it can store one bit of information. If the memory is put into contact with a heat bath after being initialized in one of these ground states, thermal fluctuations will lead to the creation of small error droplets of inverted spins. The boundary of such droplets are domain walls, i.e., one-dimensional excitations whose energy is proportional to the droplet perimeter. If the temperature is below the critical Curie temperature, the Boltzman factor will prevent the creation of macroscopic error droplets. Thus, when the system is cooled down (either physically or algorithmically) after some macroscopic storage time, it will very likely return to its original ground state: the memory is thermally stable.This behaviour contrasts with the 1D Ising ferromagnet whose domain walls are point-like excitations. Therefore, they can freely diffuse on the chain at no energy cost. As a consequence, arbitrarily large error droplets can form, so this 1D memory is thermally unstable.While the 2D Ising ferromagnet features thermal stability, it is vulnerable to static, local perturbations. Indeed, an arbitrarily weak magnetic field breaks the ground state degeneracy and favours one ground state over the other. When this perturbed system is subject to thermal fluctuations, the bulk contribution of the magnetic field overwhelms the boundary tension of the domain wall, so once error droplets reach a critical size, they rapidly expand to corrupt the memory. This type of instability plagues any systems with a local order parameter, so they cannot be robust quantum memories. Indeed, distinct ground states give different values of this order parameter, so a local field coupling to the order parameter lifts degeneracy.In 2D and higher, there exists quantum systems with no local order parameter and whose spectrum is stable under weak, local perturbations. These systems have a degenerate ground state separated from the other energy levels by a constant energy gap, and perturbations only alter these features by an exponentially vanishing amount as a function of the system size. Kitaev's toric code [2] is the bes...

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confidence: 86%

Local Topological Order Inhibits Thermal Stability in 2D

Landon-Cardinal

Poulin

2013

Phys. Rev. Lett.

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“…A different approach to construct stabilizer codes with a macroscopic energy barrier has been proposed by Michnicki [40], who introduced the notion of code welding to construct new codes by combining existing ones. The welding technique leads to a construction of a topological stabilizer code with a polynomially growing energy barrier in three spatial dimensions.…”

Section: B Self-correction and Fault Tolerancementioning

confidence: 99%

Fault-tolerant logical gates in quantum error-correcting codes

2015

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Recently, S. Bravyi and R. König [Phys. Rev. Lett. 110, 170503 (2013)] have shown that there is a trade-off between fault-tolerantly implementable logical gates and geometric locality of stabilizer codes. They consider locality-preserving operations which are implemented by a constant-depth geometrically local circuit and are thus fault tolerant by construction. In particular, they show that, for local stabilizer codes in D spatial dimensions, locality-preserving gates are restricted to a set of unitary gates known as the Dth level of the Clifford hierarchy. In this paper, we explore this idea further by providing several extensions and applications of their characterization to qubit stabilizer and subsystem codes. First, we present a no-go theorem for self-correcting quantum memory. Namely, we prove that a three-dimensional stabilizer Hamiltonian with a locality-preserving implementation of a non-Clifford gate cannot have a macroscopic energy barrier. This result implies that non-Clifford gates do not admit such implementations in Haah's cubic code and Michnicki's welded code. Second, we prove that the code distance of a D-dimensional local stabilizer code with a nontrivial locality-preserving mth-level Clifford logical (2007)], saturate the bound for m = D. Third, we prove that the qubit erasure threshold for codes with a nontrivial transversal mth-level Clifford logical gate is upper bounded by 1/m. This implies that no family of fault-tolerant codes with transversal gates in increasing level of the Clifford hierarchy may exist. This result applies to arbitrary stabilizer and subsystem codes and is not restricted to geometrically local codes. Fourth, we extend the result of Bravyi and König to subsystem codes. Unlike stabilizer codes, the so-called union lemma does not apply to subsystem codes. This problem is avoided by assuming the presence of an error threshold in a subsystem code, and a conclusion analogous to that of Bravyi and König is recovered.

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“…the lifetime of topological quantum memories: long-range interactions between anyons [17][18][19][20], energy [2,[21][22][23][24][25][26], and entropic barriers [27] to suppress anyon propagation, disorder to localize the anyons [28][29][30], and engineered dissipation to remove entropy and excitations from the system [31][32][33] (see Refs. [34][35][36] for recent reviews).…”

Section: Introductionmentioning

confidence: 99%

Exponential lifetime improvement in topological quantum memories

Bardyn

Karzig

2016

Phys. Rev. B

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We propose a simple yet efficient mechanism for passive error correction in topological quantum memories. Our scheme relies on driven-dissipative ancilla systems which couple to local excitations (anyons) and make them "sink" in energy, with no required interaction among ancillae or anyons. Through this process, anyons created by some thermal environment end up trapped in potential "trenches" that they themselves generate, which can be interpreted as a "memory foam" for anyons. This self-trapping mechanism provides an energy barrier for anyon propagation and removes entropy from the memory by favoring anyon recombination over anyon separation (responsible for memory errors). We demonstrate that our scheme leads to an exponential increase of the memory-coherence time with system size L, up to an upper bound L max , which can increase exponentially with /T , where T is the temperature and is some energy scale defined by potential trenches. This results in a double exponential increase of the memory time with /T , which greatly improves over the Arrhenius (single-exponential) scaling found in typical quantum memories.

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3D Topological Quantum Memory with a Power-Law Energy Barrier

Cited by 33 publications

References 29 publications

Local Topological Order Inhibits Thermal Stability in 2D

Local Topological Order Inhibits Thermal Stability in 2D

Fault-tolerant logical gates in quantum error-correcting codes

Exponential lifetime improvement in topological quantum memories

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