# Classical Coding Approaches to Quantum Applications

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2020

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## Abstract

Quantum information science strives to leverage the quantum-mechanical nature of our universe in order to achieve large improvements in certain information processing tasks. Such tasks include quantum communications and fault-tolerant quantum computation. In this dissertation, we make contributions to both of these applications.

In deep-space optical communications, the mathematical abstraction of the binary phase shift keying (BPSK) modulated pure-loss optical channel is called the pure-state channel. It takes classical inputs and delivers quantum outputs that are pure (qubit) states. To achieve optimal information transmission, if classical error-correcting codes are employed over this channel, then one needs to develop receivers that collectively measure all output qubits in order to optimally identify the transmitted message. In general, it is hard to determine these optimal collective measurements and even harder to realize them in practice. So, current receivers first measure each qubit channel output and then classically post-process the measurements. This approach is sub-optimal. We investigate a recently proposed quantum algorithm for this task, which is inspired by classical belief-propagation algorithms, and analyze its performance on a simple $5$-bit code. We show that the algorithm makes optimal decisions for the value of each bit and it appears to achieve optimal performance when deciding the full transmitted message. We also provide explicit circuits for the algorithm in terms of standard gates. For deep-space optical communications, this suggests a near-term quantum advantage over the aforementioned sub-optimal scheme. Such a communication advantage appears to be the first of its kind.

Quantum error correction is vital to building a universal fault-tolerant quantum computer. An $[\![ n,k,d ]\!]$ quantum error-correcting code (QECC) protects $k$ information (or logical) qubits by encoding them into quantum states of $n > k$ physical qubits such that any undetectable error must affect at least $d$ physical qubits. In this dissertation we focus on stabilizer QECCs, which are the most widely used type of QECCs. Since we would like to perform universal (i.e., arbitrary) quantum computation on the $k$ logical qubits, an important problem is to determine fault-tolerant $n$-qubit physical operations that induce the desired logical operations. Our first contribution here is a systematic algorithm that can translate a given logical Clifford operation on a stabilizer QECC into all (equivalence classes of) physical Clifford circuits that realize that operation. We exploit binary symplectic matrices to make this translation efficient and call this procedure the Logical Clifford Synthesis (LCS) algorithm.

In order to achieve universality, a quantum computer also needs to implement at least one non-Clifford logical operation. We develop a mathematical framework for a large subset of diagonal (unitary) operations in the Clifford hierarchy, and we refer to these as Quadratic Form Diagonal (QFD) gates. We show that all $1$- and $2$-local diagonal gates in the hierarchy are QFD, and we rigorously derive their action on Pauli matrices. This framework of QFD gates includes many non-Clifford gates and could be of independent interest. Subsequently, we use the QFD formalism to characterize all $[\![ n,k,d ]\!]$ stabilizer codes whose code subspaces are preserved under the transversal action of $T$ and $T^{-1}$ gates on the $n$ physical qubits. The $T$ and $T^{-1}$ gates are among the simplest non-Clifford gates to engineer in the lab. By employing a ``reverse LCS'' strategy, we also discuss the logical operations induced by these physical gates. We discuss some important corollaries related to triorthogonal codes and the optimality of CSS codes with respect to $T$ and $T^{-1}$ gates. We also describe a few purely-classical coding problems motivated by physical constraints arising from fault-tolerance. Finally, we discuss several examples of codes and determine the logical operation induced by physical $Z$-rotations on a family of quantum Reed-Muller codes. A conscious effort has been made to keep this dissertation self-contained, by including necessary background material on quantum information and computation.

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Rengaswamy, Narayanan (2020). *Classical Coding Approaches to Quantum Applications*. Dissertation, Duke University. Retrieved from https://hdl.handle.net/10161/20902.

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