7 research outputs found
Quantum Error Correction with the Toric-GKP Code
We examine the performance of the single-mode GKP code and its concatenation
with the toric code for a noise model of Gaussian shifts, or displacement
errors. We show how one can optimize the tracking of errors in repeated noisy
error correction for the GKP code. We do this by examining the
maximum-likelihood problem for this setting and its mapping onto a 1D Euclidean
path-integral modeling a particle in a random cosine potential. We demonstrate
the efficiency of a minimum-energy decoding strategy as a proxy for the path
integral evaluation. In the second part of this paper, we analyze and
numerically assess the concatenation of the GKP code with the toric code. When
toric code measurements and GKP error correction measurements are perfect, we
find that by using GKP error information the toric code threshold improves from
to . When only the GKP error correction measurements are perfect
we observe a threshold at . In the more realistic setting when all error
information is noisy, we show how to represent the maximum likelihood decoding
problem for the toric-GKP code as a 3D compact QED model in the presence of a
quenched random gauge field, an extension of the random-plaquette gauge model
for the toric code. We present a new decoder for this problem which shows the
existence of a noise threshold at shift-error standard deviation for toric code measurements, data errors and GKP ancilla errors.
If the errors only come from having imperfect GKP states, this corresponds to
states with just 4 photons or more. Our last result is a no-go result for
linear oscillator codes, encoding oscillators into oscillators. For the
Gaussian displacement error model, we prove that encoding corresponds to
squeezing the shift errors. This shows that linear oscillator codes are useless
for quantum information protection against Gaussian shift errors.Comment: 50 pages, 14 figure
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Equilibration of Edge States in the Quantum Hall State at Filling Fraction ν = 5/2
Among the most interesting approaches in building fault--tolerant quantum computation is utilizing non--Abelian topological phases of matter. These phases of matter are capable of storing information that is less susceptible to loss due to the interactions between the system and its environment. Furthermore, their non--Abelian characteristics allows for reliable performance of logical operations on the stored information. One of the leading physical systems that can realize such non--Abelian topological phases is the quantum Hall system at filling fraction . Since the discovery of this state, the nature of the ground state of this system has been the subject of debate. An important development was made recently when the thermal Hall conductance of this state was measured to be [M. Banerjee \emph{et al.}, Nature {\bf 559}, 205 (2018)]. Taken at face value, this result points to the PH--Pfaffian state as the true ground state of the quantum Hall system at filling fraction . It is the consequence of the assumption that all the modes running along the edge of this system are well--equilibrated with each other. However, as has been pointed out by other authors, this assumption may not be completely justified. In particular, the measured thermal Hall conductance could also be consistent with the anti--Pfaffian state under some experimental conditions. In this thesis, we study those conditions in detail. To achieve this, we propose new fixed point theories that describe the low--temperature physics of the anti--Pfaffian state. We demonstrate that these proposed theories could be consistent with the parameters describing the experimental conditions. For each of these fixed points, we identify the effective low--temperature edge modes and study the effect of strong short--range Coulomb interaction and an approximate spin symmetry on the interactions between them. We derive the kinetic equations that describe the hydrodynamic transport of charge and heat in a general quantum Hall state. This is the expansion of the previous studies and includes the description of transport of strongly coupled edge modes. We use these kinetic equations to describe the hydrodynamic transport of heat and charge in our proposed fixed point theories. We estimate the values of physical parameters in our theory based on the previous experimental studies. This enables us to make meaningful comparisons between our theoretical predictions and the experimental measurements of thermal conductance. We show that there exists an experimentally realistic range of parameters that the anti--Pfaffian state is consistent with the thermal Hall conductance . We identify these ranges of parameters and based upon them, make predictions on the electrical and thermal Hall conductance of the anti--Pfaffian state for a range of temperatures
Recommended from our members
Equilibration of Edge States in the Quantum Hall State at Filling Fraction ν = 5/2
Among the most interesting approaches in building fault--tolerant quantum computation is utilizing non--Abelian topological phases of matter. These phases of matter are capable of storing information that is less susceptible to loss due to the interactions between the system and its environment. Furthermore, their non--Abelian characteristics allows for reliable performance of logical operations on the stored information. One of the leading physical systems that can realize such non--Abelian topological phases is the quantum Hall system at filling fraction . Since the discovery of this state, the nature of the ground state of this system has been the subject of debate. An important development was made recently when the thermal Hall conductance of this state was measured to be [M. Banerjee \emph{et al.}, Nature {\bf 559}, 205 (2018)]. Taken at face value, this result points to the PH--Pfaffian state as the true ground state of the quantum Hall system at filling fraction . It is the consequence of the assumption that all the modes running along the edge of this system are well--equilibrated with each other. However, as has been pointed out by other authors, this assumption may not be completely justified. In particular, the measured thermal Hall conductance could also be consistent with the anti--Pfaffian state under some experimental conditions. In this thesis, we study those conditions in detail. To achieve this, we propose new fixed point theories that describe the low--temperature physics of the anti--Pfaffian state. We demonstrate that these proposed theories could be consistent with the parameters describing the experimental conditions. For each of these fixed points, we identify the effective low--temperature edge modes and study the effect of strong short--range Coulomb interaction and an approximate spin symmetry on the interactions between them. We derive the kinetic equations that describe the hydrodynamic transport of charge and heat in a general quantum Hall state. This is the expansion of the previous studies and includes the description of transport of strongly coupled edge modes. We use these kinetic equations to describe the hydrodynamic transport of heat and charge in our proposed fixed point theories. We estimate the values of physical parameters in our theory based on the previous experimental studies. This enables us to make meaningful comparisons between our theoretical predictions and the experimental measurements of thermal conductance. We show that there exists an experimentally realistic range of parameters that the anti--Pfaffian state is consistent with the thermal Hall conductance . We identify these ranges of parameters and based upon them, make predictions on the electrical and thermal Hall conductance of the anti--Pfaffian state for a range of temperatures
Quantum error correction with the toric Gottesman-Kitaev-Preskill code
We examine the performance of the single-mode Gottesman-Kitaev-Preskill (GKP) code and its concatenation with the toric code for a noise model of Gaussian shifts, or displacement errors. We show how one can optimize the tracking of errors in repeated noisy error correction for the GKP code. We do this by examining the maximum-likelihood problem for this setting and its mapping onto a 1D Euclidean path-integral modeling a particle in a random cosine potential. We demonstrate the efficiency of a minimum-energy decoding strategy as a proxy for the path integral evaluation. In the second part of this paper, we analyze and numerically assess the concatenation of the GKP code with the toric code. When toric code measurements and GKP error correction measurements are perfect, we find that by using GKP error information the toric code threshold improves from 10% to 14%. When only the GKP error correction measurements are perfect we observe a threshold at 6%. In the more realistic setting when all error information is noisy, we show how to represent the maximum likelihood decoding problem for the toric-GKP code as a 3D compact QED model in the presence of a quenched random gauge field, an extension of the random-plaquette gauge model for the toric code. We present a decoder for this problem which shows the existence of a noise threshold at shift-error standard deviation σ0 ≈ 0.243 for toric code measurements, data errors and GKP ancilla errors. If the errors only come from having imperfect GKP states, then this corresponds to states with just four photons or more. Our last result is a no-go result for linear oscillator codes, encoding oscillators into oscillators. For the Gaussian displacement error model, we prove that encoding corresponds to squeezing the shift errors. This shows that linear oscillator codes are useless for quantum information protection against Gaussian shift errors.QCD/Terhal GroupQuantum Computin