From Magic State Distillation to Dynamical Systems

Abstract

Magic State Distillation (MSD) has been a research focus for fault-tolerant quantum computing due to the need for non-Clifford resource in gaining quantum advantage. Although many of the MSD protocols so far are based on stabilizer codes with transversal $T$ gates, there exists quite several protocols that don't fall into this class. Here we propose a method to map MSD protocols to iterative dynamical systems under the framework of stabilizer reduction. With the proposed mapping, we are able to analyze the performance of MSD protocols using techniques from dynamical systems theory, easily simulate the distillation process of input states under arbitrary noise and visualize it using flow diagram. We apply our mapping to common MSD protocols for $|T⟩$ state and find some interesting properties: The $[[15,1,3]]$ code may distill states corresponding to $\sqrt{T}$ gate and the $[[5,1,3]]$ code can distill the magic state corresponding to the $T$ gate. Besides, we examine the exotic MSD protocols that may distill into other magic states proposed in [Eur. Phys. J. D 70, 55 (2016)] and identify the condition for distillable magic states. We also study new MSD protocols generated by concatenating different codes and numerically demonstrate that concatenation can generate MSD protocols with various magic states. By concatenating efficient codes with exotic codes, we can reduce the overhead of the exotic MSD protocols. We believe our proposed method will be a useful tool for simulating and visualization MSD protocols for canonical MSD protocols on $|T⟩$ as well as other unexplored MSD protocols for other states.

to be appeared in Quantum

References in corpus (23)

• Surface codes: Towards practical large-scale quantum computation
• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• Quantum error correction below the surface code threshold
• Restrictions on Transversal Encoded Quantum Gate Sets
• Magic state distillation with low overhead
• Magic state distillation in all prime dimensions using quantum Reed-Muller codes
• Improved magic states distillation for quantum universality
• Multilevel distillation of magic states for quantum computing
• Encoding a magic state with beyond break-even fidelity
• A magic state's fidelity can be superior to the operations that created it
• The cost of universality: A comparative study of the overhead of state distillation and code switching with color codes
• Fault-tolerant magic state preparation with flag qubits
• Coherence in quantum error-correcting codes
• Practical approximation of single-qubit unitaries by single-qubit quantum Clifford and T circuits
• Towards low overhead magic state distillation
• Distillation with sublogarithmic overhead
• 2-D color code quantum computation
• A State Distillation Protocol to Implement Arbitrary Single-qubit Rotations
• Efficient Magic State Distillation by Zero-Level Distillation
• Optimal local unitary encoding circuits for the surface code
• Towers of generalized divisible quantum codes
• Code switching revisited: Low-overhead magic state preparation using color codes
• Small Codes for Magic State Distillation

Cited by in corpus (2)

• Efficient simulation of logical magic state preparation protocols
• Classical representation of local Clifford operators