Magic state distillation in all prime dimensions using quantum Reed-Muller codes

Abstract

We propose families of protocols for magic state distillation -- important components of fault tolerance schemes --- for systems of odd prime dimension. Our protocols utilize quantum Reed-Muller codes with transversal non-Clifford gates. We find that, in higher dimensions, small and effective codes can be used that have no direct analogue in qubit (two-dimensional) systems. We present several concrete protocols, including schemes for three-dimensional (qutrit) and five-dimensional (ququint) systems. The five-dimensional protocol is, by many measures, the best magic state distillation scheme yet discovered. It excels both in terms of error threshold with respect to depolarising noise (36.3%) and the efficiency measure know as "yield", where, for a large region of parameters, it outperforms its qubit counterpart by many orders of magnitude.

Updated from V1 to include results on the remarkable d=5 case

References in corpus (37)

• Fault-tolerant quantum computation by anyons
• Mixed State Entanglement and Quantum Error Correction
• Quantum Teleportation is a Universal Computational Primitive
• Universal Quantum Computation with ideal Clifford gates and noisy ancillas
• Quantum Computing with Very Noisy Devices
• A Theory of Fault-Tolerant Quantum Computation
• Distillation of secret key and entanglement from quantum states
• Fault-tolerant quantum computation with high threshold in two dimensions
• Topological fault-tolerance in cluster state quantum computation
• Restrictions on Transversal Encoded Quantum Gate Sets
• Bound entanglement can be activated
• Entanglement-assisted local manipulation of pure quantum states
• Hudson's Theorem for finite-dimensional quantum systems
• Quantum Encodings in Spin Systems and Harmonic Oscillators
• Quantum computation based on d-level cluster states
• Fault tolerant quantum computation with very high threshold for loss errors
• Fault-Tolerant Quantum Computation with Higher-Dimensional Systems
• Improved magic states distillation for quantum universality
• Universal simulation of Hamiltonian dynamics for qudits
• Qudit versions of the qubit "pi-over-eight" gate
• Qutrit Magic State Distillation
• Architectural design for a topological cluster state quantum computer
• Catalysis and activation of magic states in fault tolerant architectures
• Bound States for Magic State Distillation in Fault-Tolerant Quantum Computation
• Efficient Quantum Polar Coding
• Fully fault tolerant quantum computation with non-deterministic gates
• Experimental magic state distillation for fault-tolerant quantum computing
• Subsystem stabilizer codes cannot have a universal set of transversal gates for even one encoded qudit
• Noise Thresholds for Higher Dimensional Systems using the Discrete Wigner Function
• Valence bond solid formalism for d-level one-way quantum computation
• Quantum Online Memory Checking
• On the security and degradability of Gaussian channels
• Tight Noise Thresholds for Quantum Computation with Perfect Stabilizer Operations
• Simulating Hamiltonian dynamics using many-qudit Hamiltonians and local unitary control
• Engineering complex topological memories from simple Abelian models
• Generalised state spaces and non-locality in fault tolerant quantum computing schemes
• Upper bounds on fault tolerance thresholds of noisy Clifford-based quantum computers

Cited by in corpus (125)

• Electromagnetic Force and Momentum
• Electromagnetic Angular Momentum
• Scalable Designs for Quasiparticle-Poisoning-Protected Topological Quantum Computation with Majorana Zero Modes
• Qudits and high-dimensional quantum computing
• The Resource Theory of Stabilizer Computation
• A universal qudit quantum processor with trapped ions
• Quantum Information Scrambling in a Superconducting Qutrit Processor
• Enhanced fault-tolerant quantum computing in $d$-level systems
• Fault-tolerant conversion between the Steane and Reed-Muller quantum codes
• Quantum computation with realistic magic state factories
• Ultracold polar molecules as qudits
• Qutrit randomized benchmarking
• Multilevel distillation of magic states for quantum computing
• Quantifying nonstabilizerness of matrix product states
• Quantifying quantum speedups: improved classical simulation from tighter magic monotones
• Stabilizer entropies and nonstabilizerness monotones
• Practical trapped-ion protocols for universal qudit-based quantum computing
• Factoring with Qutrits: Shor's Algorithm on Ternary and Metaplectic Quantum Architectures
• Resource costs for fault-tolerant linear optical quantum computing
• Experimental high-dimensional Greenberger-Horne-Zeilinger entanglement with superconducting transmon qutrits
• Experimental engineering of arbitrary qudit states with discrete-time quantum walks
• Efficiently computable bounds for magic state distillation
• A unified framework for magic state distillation and multi-qubit gate-synthesis with reduced resource cost
• A magic state's fidelity can be superior to the operations that created it
• Implementation of a Walsh-Hadamard gate in a superconducting qutrit
• Efficient Decoders for Qudit Topological Codes
• Molecular Asymmetry and Optical Cycling: Laser Cooling Asymmetric Top Molecules
• Robust and Efficient High-dimensional Quantum State Tomography
• Quantum Computing with Parafermions
• Towards low overhead magic state distillation
• Magic State Distillation with Low Space Overhead and Optimal Asymptotic Input Count
• Experimental Realization of Two Qutrits Gate with Tunable Coupling in Superconducting Circuits
• A fast fault-tolerant decoder for qubit and qudit surface codes
• Uniqueness of noncontextual models for stabilizer subtheories
• Quantum state engineering using one-dimensional discrete-time quantum walks
• Distilling one-qubit magic states into Toffoli states
• Transversal Clifford gates on folded surface codes
• Arithmetic Circuits for Multilevel Qudits Based on Quantum Fourier Transform
• Qudit Colour Codes and Gauge Colour Codes in All Spatial Dimensions
• Error Thresholds for Abelian Quantum Double Models: Increasing the bit-flip Stability of Topological Quantum Memory
• Normal form for single-qutrit Clifford+T operators and synthesis of single-qutrit gates
• Qutrit Magic State Distillation Tight in Some Directions
• Learning efficient decoders for quasi-chaotic quantum scramblers
• An efficient magic state approach to small angle rotations
• Improved HDRG decoders for qudit and non-Abelian quantum error correction
• Characterization of an operational quantum resource in a critical many-body system
• Qudit entanglers using quantum optimal control
• Qudit Dynamical Decoupling on a Superconducting Quantum Processor
• Hybrid Oscillator-Qubit Quantum Processors: Instruction Set Architectures, Abstract Machine Models, and Applications
• Retrieving information from a black hole using quantum machine learning
• Qutrit and Ququint Magic States
• Maximum nonlocality and minimum uncertainty using magic states
• Quantum Information Processing by Weaving Quantum Talbot Carpets
• Quantifying dynamical magic with completely stabilizer preserving operations as free
• A Practical Introduction to Benchmarking and Characterization of Quantum Computers
• Parafermions in a Kagome lattice of qubits for topological quantum computation
• Randomized benchmarking for qudit Clifford gates
• Entanglement distance for an arbitrary state of M qubits
• Entanglement distance for arbitrary $M$-qudit hybrid systems
• Realizing nonadiabatic holonomic quantum computation beyond the three-level setting
• One-Shot Yield-Cost Relations in General Quantum Resource Theories
• Order 3 Symmetry in the Clifford Hierarchy
• Towers of generalized divisible quantum codes
• Lattice Surgery on the Raussendorf Lattice
• Measurement-only topological quantum computation without forced measurements
• Non-Uniform Code Concatenation for Universal Fault-Tolerant Quantum Computing
• Approximate stabilizer rank and improved weak simulation of Clifford-dominated circuits for qudits
• Preserving multi-level quantum coherence by dynamical decoupling
• Designing the Quantum Channels Induced by Diagonal Gates
• Quantum Pin Codes
• The $2T$-qutrit, a two-mode bosonic qutrit
• The magic of universal quantum computing with permutations
• Multi-frequency control and measurement of a spin-7/2 system encoded in a transmon qudit
• Qutrit Circuits and Algebraic Relations: A Pathway to Efficient Spin-1 Hamiltonian Simulation
• Protecting qudit operations from noise by continuous dynamical decoupling
• Mitigating Coherent Noise by Balancing Weight-2 $Z$-Stabilizers
• Universal quantum computing with parafermions assisted by a half fluxon
• Entangling power of symmetric two-qubit quantum gates
• Local Dimension Invariant Qudit Stabilizer Codes
• Quantum phase transition in the $Z_3$ Kitaev-Potts model
• States that are far from being stabilizer states
• Mixed-state additivity properties of magic monotones based on quantum relative entropies for single-qubit states and beyond
• Extracting randomness from magic quantum states
• Improved Simulation of Quantum Circuits by Fewer Gaussian Eliminations
• Contextual bound states for qudit magic state distillation
• Ancilla-driven quantum computation for qudits and continuous variables
• Pauli-based model of quantum computation with higher-dimensional systems
• Certifying dimension of quantum systems by sequential projective measurements
• Cost of postselection in decision theory
• Limitations of Classically-Simulable Measurements for Quantum State Discrimination
• Broadcasting of quantum correlations in qubit-qudit systems
• Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits
• Qudit Shadow Estimation Based on the Clifford Group and the Power of a Single Magic Gate
• The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems
• Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates
• Randomised benchmarking for universal qudit gates
• Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits
• Building Qutrit Diagonal Gates from Phase Gadgets
• Geometric structure and transversal logic of quantum Reed-Muller codes
• Scaling W state circuits in the qudit Clifford hierarchy
• A Universal Circuit Set Using the $S_3$ Quantum Double
• Counting stabiliser codes for arbitrary dimension
• Classical simulability of Clifford+T circuits with Clifford-augmented matrix product states
• Qudit vs. Qubit: Simulated performance of error correction codes in higher dimensions
• Synthesis and Arithmetic of Single Qutrit Circuits
• A note on many valued quantum computational logics
• Time-dependent Rabi frequencies to protect quantum operations on an atomic qutrit by continuous dynamical decoupling
• Robust Control and Entanglement of Qudits in Neutral Atom Arrays
• Qudit low-density parity-check codes
• A Complete and Natural Rule Set for Multi-Qutrit Clifford Circuits
• From Magic State Distillation to Dynamical Systems
• Low Overhead Qutrit Magic State Distillation
• Finite-Function-Encoding Quantum States
• Quantifying magic via quantum $(\alpha ,\beta )$ Jensen-Shannon divergence
• General entropic constraints on CSS codes within magic distillation protocols
• Noise-Canceling Quantum Feedback: non-Hermitian Dynamics with Applications to State Preparation and Magic State Distillation
• Magic tricycles: Efficient magic state generation with finite block-length quantum LDPC codes
• Entanglement transfer, accumulation and retrieval via quantum-walk-based qubit-qudit dynamics
• Experimental Approach to Demonstrating Contextuality for Qudits
• Universal quantum computation via scalable measurement-free error correction
• Scalable platform for qudit-based quantum computing using polar molecules
• Synthesis of Single Qutrit Circuits from Clifford+R
• Constructing all qutrit controlled Clifford+T gates in Clifford+T
• Gottesman-Knill Limit on One-way Communication Complexity: Tracing the Quantum Advantage down to Magic Resources
• Transversal AND in Quantum Codes