# Browsing by Author "Calderbank, Robert"

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Item Open Access Designing Quantum Channels Induced by Diagonal Gates(2023) Hu, JingzhenThe challenge of quantum computing is to combine error resilience with universal computation. Diagonal gates such as the transversal T gate play an important role in implementing a universal set of quantum operations. We introduce a framework that describes the process of preparing a code state, applying a diagonal physical gate, measuring a code syndrome, and applying a Pauli correction that may depend on the measured syndrome (the average logical channel induced by an arbitrary diagonal gate). The framework describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. The interaction of code states and diagonal gates depends on the signs of Z-stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. We derive necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provide an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate, the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find applications in magic state distillation and elsewhere. When all the signs are positive, we characterize all possible CSS codes, invariant under transversal Z-rotation through π/2^l, that are constructed from classical Reed-Muller codes by deriving the necessary and sufficient constraints on the level l. According to the divisibility conditions, we construct new families of CSS codes using cosets of the first order Reed-Muller code defined by quadratic forms. The generator coefficient framework extends to arbitrary stabilizer codes but the more general class of non-degenerate stabilizer codes does not bring advantages when designing the code parameters.

Relying on the generator coefficient framework, we introduce a method of synthesizing CSS codes that realizes a target logical diagonal gate at some level l in the Clifford hierarchy. The method combines three basic operations: concatenation, removal of Z-stabilizers, and addition of X-stabilizers. It explicitly tracks the logical gate induced by a diagonal physical gate that preserves a CSS code. The first step is concatenation, where the input is a CSS code and a physical diagonal gate at level l inducing a logical diagonal gate at the same level. The output is a new code for which a physical diagonal gate at level l+1 induces the original logical gate. The next step is judicious removal of Z-stabilizers to increase the level of the induced logical operator. We identify three ways of climbing the logical Clifford hierarchy from level l to level l+1, each built on a recursive relation on the Pauli coefficients of the induced logical operators. Removal of Z-stabilizers may reduce distance, and the purpose of the third basic operation, addition of X-stabilizers, is to compensate for such losses. Our approach to logical gate synthesis is demonstrated by two proofs of concept: the [[2^(l+1) − 2, 2, 2]] triorthogonal code family, and the [[2^m, (m choose r) , 2^(min{r, m-r})]] quantum Reed-Muller code family.

Item Open Access Exploration and Application of Dimensionality Reduction and Clustering Techniques to Diabetes Patient Health Records(2017-05-24) Gopinath, SidharthThis research examines various data dimensionality reduction techniques and clustering methods. The goal was to apply these ideas to a test dataset and a healthcare dataset to see how they practically work and what conclusions we could draw from their application. Specifically, we hoped to identify similar clusters of diabetes patients and develop hypotheses of risk for adverse events for further research into sub-populations of diabetes patients. Upon further research and application, it became apparent that the data dimensionality reduction and clustering methods are sensitive to the parameter settings and must be fine-tuned carefully to be successful. Additionally, we saw several statistically significant differences in outcomes for the clusters identified with these data. We focused on coronary artery disease and kidney disease. Focusing on these clusters, we found a high proportion of patients taking medications for heart or kidney conditions Based on these findings, we were able to decide on future paths building upon this research that could lead to more actionable conclusions.Item Open Access Learning from Geometry(2016) Huang, JiajiSubspaces and manifolds are two powerful models for high dimensional signals. Subspaces model linear correlation and are a good fit to signals generated by physical systems, such as frontal images of human faces and multiple sources impinging at an antenna array. Manifolds model sources that are not linearly correlated, but where signals are determined by a small number of parameters. Examples are images of human faces under different poses or expressions, and handwritten digits with varying styles. However, there will always be some degree of model mismatch between the subspace or manifold model and the true statistics of the source. This dissertation exploits subspace and manifold models as prior information in various signal processing and machine learning tasks.

A near-low-rank Gaussian mixture model measures proximity to a union of linear or affine subspaces. This simple model can effectively capture the signal distribution when each class is near a subspace. This dissertation studies how the pairwise geometry between these subspaces affects classification performance. When model mismatch is vanishingly small, the probability of misclassification is determined by the product of the sines of the principal angles between subspaces. When the model mismatch is more significant, the probability of misclassification is determined by the sum of the squares of the sines of the principal angles. Reliability of classification is derived in terms of the distribution of signal energy across principal vectors. Larger principal angles lead to smaller classification error, motivating a linear transform that optimizes principal angles. This linear transformation, termed TRAIT, also preserves some specific features in each class, being complementary to a recently developed Low Rank Transform (LRT). Moreover, when the model mismatch is more significant, TRAIT shows superior performance compared to LRT.

The manifold model enforces a constraint on the freedom of data variation. Learning features that are robust to data variation is very important, especially when the size of the training set is small. A learning machine with large numbers of parameters, e.g., deep neural network, can well describe a very complicated data distribution. However, it is also more likely to be sensitive to small perturbations of the data, and to suffer from suffer from degraded performance when generalizing to unseen (test) data.

From the perspective of complexity of function classes, such a learning machine has a huge capacity (complexity), which tends to overfit. The manifold model provides us with a way of regularizing the learning machine, so as to reduce the generalization error, therefore mitigate overfiting. Two different overfiting-preventing approaches are proposed, one from the perspective of data variation, the other from capacity/complexity control. In the first approach, the learning machine is encouraged to make decisions that vary smoothly for data points in local neighborhoods on the manifold. In the second approach, a graph adjacency matrix is derived for the manifold, and the learned features are encouraged to be aligned with the principal components of this adjacency matrix. Experimental results on benchmark datasets are demonstrated, showing an obvious advantage of the proposed approaches when the training set is small.

Stochastic optimization makes it possible to track a slowly varying subspace underlying streaming data. By approximating local neighborhoods using affine subspaces, a slowly varying manifold can be efficiently tracked as well, even with corrupted and noisy data. The more the local neighborhoods, the better the approximation, but the higher the computational complexity. A multiscale approximation scheme is proposed, where the local approximating subspaces are organized in a tree structure. Splitting and merging of the tree nodes then allows efficient control of the number of neighbourhoods. Deviation (of each datum) from the learned model is estimated, yielding a series of statistics for anomaly detection. This framework extends the classical {\em changepoint detection} technique, which only works for one dimensional signals. Simulations and experiments highlight the robustness and efficacy of the proposed approach in detecting an abrupt change in an otherwise slowly varying low-dimensional manifold.

Item Open Access MITIGATING COHERENT NOISE(2023) Liang, QingzhongStochastic errors in quantum systems occur randomly but coherent errors may be more damaging since they can accumulate in a particular direction. We develop a framework for designing decoherence free subspaces (DFS), that are unperturbed by coherent noise. We consider a particular form of coherent $Z$-errors and construct stabilizer codes that form DFS for such noise (``Z-DFS''). More precisely, we develop conditions for transversal $\exp(\imath \theta \sigma_Z)$ to preserve a stabilizer code subspace for all $\theta$. If the code is error-detecting, then this implies a trivial action on the logical qubits. These conditions require the existence of a large number of weight-$2$ $Z$-stabilizers, and together, these weight-$2$ $Z$-stabilizers generate a direct product of single-parity-check codes.

By adjusting the size of these components, we are able to construct a constant rate family of CSS Z-DFS codes. Invariance under transversal $\exp(\frac{\imath \pi}{2^l} \sigma_Z)$ translates to a trigonometric equation satisfied by $\tan\frac{2\pi}{2^l}$, and for every non-zero $X$-component of a stabilizer, there is a trigonometric equation that must be satisfied. The $Z$-stabilizers supported on this non-zero $X$-component form a classical binary code C, and the trigonometric constraint connects signs of $Z$-stabilizers to divisibility of weights in $C^{\perp}$. This construction may be of independent interest to classical coding theorists who have long been interested in codes $C$ with the property that all weights are divisible by some integer $d$. If we require that transversal $\exp(\frac{\imath \pi}{2^l} \sigma_Z)$ preserves the code space only up to some finite level $l$ in the Clifford hierarchy, then we can construct higher level gates necessary for universal quantum computation. The aforesaid code $C$ contains a self-dual code and the classical Gleason's theorem constrains its weight enumerator.

The trigonometric conditions corresponding to higher values of $l$ lead to generalizations of Gleason's theorem that may be of independent interest to classical coding theorists. The $[[16, 4, 2]]$ Reed-Muller code and the family of $[[4L^2, 1, 2L]]$ Shor codes are included in our general framework.

Item Open Access Understanding Operator Reed-Muller Codes Through the Weyl Transform(2018-04-25) Wang, WeiyaoThis paper expands the framework on the multidimensional generalizations of binary Reed-Muller code, operator Reed-Muller codes, where the codewords are projection operators through the Weyl Transform. The Weyl Transform of these operator Reed- Muller codes maps the operators to vectors, and it is isometric. This nice property gives new proofs for some known results and produce a simpler decoding algorithm. In particular, the property provides a different framework to analyze the distance spectrum of second operator Reed-Muller codes without using the Dickson’s Theorem.