Quantum information theory is concerned with identifying how quantum mechanical resources, such as entangled quantum states, can be utilized for a number of information processing tasks, including data storage, computation, communication, and cryptography. Efficient quantum algorithms and protocols have been developed for performing some tasks (e.g., factoring large numbers, securely communicating over a public channel, and simulating quantum mechanical systems) that appear to be very difficult with just classical resources. In addition to identifying the separation between classical and quantum computational power, much of the theoretical focus in this field over the last decade has been concerned with finding novel ways of encoding quantum information that are robust against errors, which is an important step toward building practical quantum information processing devices. In this thesis I present some results on the quantum error-correcting properties of oscillator codes (also described as symplectic lattice codes) and toric codes. Any harmonic oscillator system, such as a mode of light, can be encoded with quantum information via symplectic lattice codes that are robust against shifts in the system's continuous quantum variables. I show the existence of lattice codes whose achievable rates match the one-shot coherent information over the Gaussian quantum channel. Also, I construct a family of symplectic self-dual lattices and search for optimal encodings of quantum information distributed between several oscillators. Toric codes provide encodings of quantum information into two-dimensional spin lattices that are robust against local clusters of errors and which require only local quantum operations for error correction. Numerical simulations of this system under realistic error models provide a calculation of the accuracy threshold for quantum memory using toric codes, which can be related to phase transitions in particular condensed matter models. I also present a local classical processing scheme for correcting errors on toric codes, which demonstrates that quantum information can be maintained in two dimensions by purely local quantum and classical resources.
We numerically study the effects of two forms of quenched disorder on the anyons of the toric code. First, a class of codes based on random lattices of stabilizer operators is presented and shown to be superior to the standard square-lattice toric code for certain forms of biased noise. It is further argued that these codes are close to optimal, in that they tightly reach the upper bound of error thresholds beyond which no correctable Calderbank-Shore-Steane codes can exist. Additionally, we study the classical motion of anyons in toric codes with randomly distributed on-site potentials. In the presence of repulsive long-range interaction between the anyons, a surprising increase in the lifetime of encoded states with disorder strength is reported and explained by an entirely incoherent mechanism. Finally, the coherent transport of the anyons in the presence of both forms of disorder is investigated and a significant suppression of the anyon motion is found.
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 ${\ensuremath{\sigma}}_{0}\ensuremath{\approx}0.243$ 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.