Classical Coding Approaches to Quantum Applications
arXiv:2004.06834
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. In deep-space optical communications, current receivers for the pure-state classical-quantum channel first measure each qubit channel output and then classically post-process the measurements. This approach is sub-optimal. In this dissertation 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 -bit code. We show that the algorithm is optimal for 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. This suggests a near-term quantum communication advantage over the aforementioned sub-optimal scheme. Quantum error correction is vital to building a universal fault-tolerant quantum computer. We propose an efficient algorithm that can translate a given logical Clifford operation on a stabilizer code into all (equivalence classes of) physical Clifford circuits that realize that operation. In order to achieve universality, one also needs to implement at least one non-Clifford logical operation. So, we develop a mathematical framework for a large subset of diagonal operations in the Clifford hierarchy, which we call Quadratic Form Diagonal (QFD) gates. Then we use the QFD formalism to characterize all stabilizer codes whose code spaces are preserved under the transversal action of the non-Clifford gates on the physical qubits. We also discuss a few purely-classical coding problems motivated by transversal gates. A conscious effort has been made to keep this dissertation self-contained, by including necessary background material on quantum information and computation.
arXiv commentPh.D. Dissertation, Duke University, 260 pages, double-spaced. Links to GitHub code included in text. Defense talk: https://youtu.be/cvAcaujp7Wo. Comments welcome!
References in corpus (63)
- Quantum Computing in the NISQ era and beyond
- Surface codes: Towards practical large-scale quantum computation
- Good Quantum Error-Correcting Codes Exist
- Quantum Teleportation is a Universal Computational Primitive
- Improved Simulation of Stabilizer Circuits
- Universal Quantum Computation with ideal Clifford gates and noisy ancillas
- A Theory of Fault-Tolerant Quantum Computation
- Quantum Error Correction and Orthogonal Geometry
- Quantum computation with optical coherent states
- A meet-in-the-middle algorithm for fast synthesis of depth-optimal quantum circuits
- Simple Quantum Error Correcting Codes
- Experimental Comparison of Two Quantum Computing Architectures
- Operator Quantum Error Correcting Subsystems for Self-Correcting Quantum Memories
- Correcting Quantum Errors with Entanglement
- Instantaneous non-local computation of low T-depth quantum circuits
- The Heisenberg Representation of Quantum Computers
- Characterizing Quantum Gates via Randomized Benchmarking
- Quantum advantage with shallow circuits
- Demonstration of qubit operations below a rigorous fault tolerance threshold with gate set tomography
- Quantum computing with nearest neighbor interactions and error rates over 1%
- Magic state distillation with low overhead
- Methodology for quantum logic gate constructions
- Quantum Error Correction
- Quantum error correction with only two extra qubits
- Ultrahigh Error Threshold for Surface Codes with Biased Noise
- Graph-theoretic Simplification of Quantum Circuits with the ZX-calculus
- The Clifford group, stabilizer states, and linear and quadratic operations over GF(2)
- Fault-tolerant quantum error detection
- Robust two-qubit gates in a linear ion crystal using a frequency-modulated driving force
- Universal transversal gates with color codes - a simplified approach
- Fault-Tolerant Logical Gates in the IBM Quantum Experience
- Leveraging Secondary Storage to Simulate Deep 54-qubit Sycamore Circuits
- Fault-tolerant quantum computation with few qubits
- Asymptotically optimal approximation of single qubit unitaries by Clifford and T circuits using a constant number of ancillary qubits
- Optimized Compilation of Aggregated Instructions for Realistic Quantum Computers
- Catalytic quantum error correction
- Polar codes for classical-quantum channels
- Quantum Belief Propagation
- Threshold Accuracy for Quantum Computation
- Threshold Saturation for Spatially-Coupled LDPC and LDGM Codes on BMS Channels
- Using concatenated quantum codes for universal fault-tolerant quantum gates
- A unified framework for magic state distillation and multi-qubit gate-synthesis with reduced resource cost
- On Universal and Fault-Tolerant Quantum Computing
- Diagonal gates in the Clifford hierarchy
- Polar coding to achieve the Holevo capacity of a pure-loss optical channel
- Unifying gate-synthesis and magic state distillation
- On Optimality of CSS Codes for Transversal
- On the logical operators of quantum codes
- Flag fault-tolerant error correction, measurement, and quantum computation for cyclic CSS codes
- Fault-tolerant gates on hypergraph product codes
- Ground-state energy estimation of the water molecule on a trapped ion quantum computer
- Optimising Clifford Circuits with Quantomatic
- Formal Constraint-based Compilation for Noisy Intermediate-Scale Quantum Systems
- Optimal Measurements for Symmetric Quantum States with Applications to Optical Communication
- Unifying the Clifford Hierarchy via Symmetric Matrices over Rings
- Complex instruction set computing architecture for performing accurate quantum rotations with less magic
- Noise-Adaptive Compiler Mappings for Noisy Intermediate-Scale Quantum Computers
- Universal fault-tolerant quantum computation with Bacon-Shor codes
- Order 3 Symmetry in the Clifford Hierarchy
- Introduction to UniversalQCompiler
- Leveraging Automorphisms of Quantum Codes for Fault-Tolerant Quantum Computation
- Magic state distillation with punctured polar codes
- A Message-Passing Algorithm for Graph Isomorphism