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Qutrit Magic State Distillation

Hussain Anwar, Earl T. Campbell, Dan E. Browne

arXiv:1202.2326v2quant-ph

TL;DR

Magic state distillation had been developed mainly for qubits, leaving higher-dimensional systems insufficiently studied despite their relevance to fault-tolerant computation. The paper develops a general higher-dimensional framework and applies it to a 5-qutrit stabilizer code, finding two attractor families whose outputs can be converted into phase-states for non-Clifford computation.

  • Problem

    Magic state distillation had been developed mainly for qubits, while whether it generalizes to higher-dimensional systems remained insufficiently studied.

  • Method

    The paper develops a general method for analyzing stabilizer-code distillation in prime dimensions and applies it to a 5-qutrit code with follow-up parity-checking and equatorialization protocols.

  • Results

    The 5-qutrit protocol distills two qutrit magic-state families, with depolarizing-noise thresholds of 23.3% and 34.5%.

  • Takeaways & Limitations

    The distilled outputs can be converted using stabilizer circuits into phase-states suitable for state-injected non-Clifford unitaries.

Abstract

from arXiv · show

Magic state distillation (MSD) is a purification protocol that plays a central role in fault tolerant quantum computation. Repeated iteration of the steps of a MSD protocol, generates pure single non-stabilizer states, or magic states, from multiple copies of a mixed resource state using stabilizer operations only. Thus mixed resource states promote the stabilizer operations to full universality. Magic state distillation was introduced for qubit-based quantum computation, but little has been known concerning MSD in higher dimensional qudit-based computation. Here, we describe a general approach for studying MSD in higher dimensions. We use it to investigate the features of a qutrit MSD protocol based on the 5-qutrit stabilizer code. We show that this protocol distills non-stabilizer magic states, and identify two types of states, that are attractors of this iteration map. Finally, we show how these states may be converted, via stabilizer circuits alone, into a state suitable for state injected implementation of a non-Clifford phase gate, enabling non-Clifford unitary computation.

I. INTRODUCTION

The paper addresses the limited understanding of magic state distillation beyond qubits by developing a higher-dimensional framework and applying it to a 5-qutrit code. It identifies two distillable qutrit state families and shows their outputs can be converted into resources for non-Clifford computation.

  • Stabilizer operations are fault-tolerant but not universal, motivating mixed non-stabilizer states as resources for state-injected non-Clifford gates.
  • The protocol follows the iterative initialization, stabilizer projection, and Clifford decoding structure used by known magic state distillation schemes.
  • Higher-dimensional distillation is important because relevant physical systems may naturally realize Clifford operations in dimensions greater than two, while the generality of qubit-specific features is unclear.
  • The paper presents the first generalization of magic state distillation to higher-dimensional systems and demonstrates it in a qutrit system.
  • The 5-qutrit protocol distills two state families: Hadamard eigenstates up to a 23.3% depolarizing-noise threshold and four Hadamard-squared eigenstates up to 34.5%.
  • Outputs of the 5-qutrit code can be converted by parity-checking and equatorialization into phase-states suitable for state-injected non-Clifford unitaries.

A. Higher Dimensions and Stabilizer Codes

The paper formulates higher-dimensional qudit states using generalized Pauli operators, stabilizer groups, and complex Bloch vectors. This representation supports analysis of physical-state geometry, Pauli orbits, and density-operator constraints.

  • Generalized single-qudit Pauli operators are traceless, non-Hermitian unitaries obeying XZ = ω^-1ZX and X^d = Z^d = I.
  • A stabilizer state is a simultaneous +1 eigenvector of an Abelian Pauli subgroup, whose codewords form the stabilizer-code space.
  • The chosen Pauli basis imposes α*_j,k = α_-j,-k, so only (d^2 − 1)/2 complex Bloch components are independent.
  • Beyond qubits, mixed states are represented by complex Bloch vectors in C^(d^2−1)/2 with an inner product and distance inherited from their components.
  • Within suitable hyperplanes, physical states form a convex polytope with the structure of a standard simplex.
  • Pauli conjugation changes Bloch-component phases, generating orbital equivalence classes under which the later magic states are unique.

C. Qutrits

The qutrit specialization uses a four-parameter complex Bloch representation and analyzes Pauli-orbit phases for qutrit states. These orbit phases characterize states equivalent under qutrit Pauli conjugation.

  • For qutrits, the Pauli basis uses σ_j,k = ω^-jkX^jZ^k with ω = e^(2πi/3), and a state requires four complex independent parameters.
  • The qutrit orbital Bloch phases summarize how conjugation by the nine σ_j,k operators changes the phases of the four independent Bloch components.
  • The magic states identified later are unique up to one of the Pauli-generated orbital Bloch phases.

D. Stabilizer Dynamics and the Clifford Group

The paper uses the stabilizer formalism and Clifford gates to define qutrit dynamics and implement distillation. The protocol prepares noisy copies, postselects stabilizer measurements, decodes the code space, and studies the resulting nonlinear map numerically.

  • The Clifford group is generated by the higher-dimensional Hadamard, phase, and controlled-NOT gates, while Pauli measurements use POVM elements because qudit Paulis are generally non-Hermitian.
  • The higher-dimensional MSD framework formulates the three protocol steps for any stabilizer code in any prime dimension.
  • The protocol prepares identical noisy resource states, measures stabilizer generators, postselects the +1 outcomes, and aborts on nontrivial syndromes.
  • Successful projection places the copies in the code subspace, after which a Clifford decoding operation maps logical operators to a single physical qudit.
  • The output Bloch components are multivariable complex polynomials of order n, and qutrit fixed points are therefore studied numerically because analytic solutions were not found.

IV. THE 5-QUTRIT CODE

The generalized [[5,1,3]]3 code defines a qutrit distillation map, whose behavior depends on the chosen decoding Clifford and separates states into two qualitatively different families.

  • The five-qudit [[5,1,3]]d stabilizers provide the basis for studying the qutrit distillation map.The exact map for four Bloch components is given in the appendix.
  • The decoding is not unique, and an additional Clifford simplifies iteration by avoiding the cycling seen without it.
  • The protocol separates states into Hadamard-plane states and a second family outside that plane.

A. Hadamard-like Distillation

The qutrit Hadamard plane organizes the first distillation family geometrically, with |H+⟩ and |H−⟩ as distillable attractors while |Hi⟩ is not an attractor. The five-qutrit code has asymmetric error suppression, and a seven-qutrit code can enlarge the useful input region.

  • Hadamard-like Distillation: Both |H+⟩ and |H−⟩ are distillable attractors, whereas |Hi⟩ is not; their regions are symmetric because a Clifford maps the two states into each other.
  • Hadamard-like Distillation: The Hadamard plane is represented by an equilateral triangle whose vertices are Hadamard eigenstates, and distillable regions lie inside this state-space geometry.
  • Hadamard-like Distillation: Error suppression is asymmetric: the distillation region attracts more strongly toward |Hi⟩ than |H−⟩ from |H+⟩.
  • Hadamard-like Distillation: The [[5,1,3]]3 code performs worse than the qubit 15-qubit protocol, whose output error scales as ϵout ≈35ϵ3.
  • Hadamard-like Distillation: The seven-qutrit code distills mixed states toward non-stabilizer segments and can enlarge the overall region before the five-qutrit code distills |H±⟩.

B. Hadamard-squared subspace

A second family consists of H2-symmetric states that are attractor fixed points of the five-qutrit protocol. Their numerical error suppression is approximately linear for small depolarizing noise, unlike the analytically characterized Hadamard states.

  • Hadamard-squared subspace: The second magic-state family is an H2 eigenstate within a degenerate eigenspace, so H2 does not uniquely specify the state.
  • Hadamard-squared subspace: The H2-symmetric states are clearly attractor fixed points, although no closed-form analytic expressions are available for them.
  • Hadamard-squared subspace: For small depolarizing noise, the output error follows ϵout ≈ϵn with n ≈1 from the log-log plot.
  • Hadamard-squared subspace: The protocol evaluates syndrome-measurement success probabilities for both |ϕ⟩ and |H±⟩ under depolarizing noise.

V. PROMOTING THE CLIFFORD GROUP

The distilled |H+⟩ and |ϕ⟩ states are not directly known to implement non-Clifford gates, but stabilizer sub-protocols convert them into phase-states suitable for gate injection.

  • The resulting phase-states hold phase information and can be used for injection of a non-Clifford gate.
  • Parity checking converts |H+⟩ and |ϕ⟩ into the plus-state |Ψ+⟩, after which equatorialization produces a desired phase-state.

A. The parity-checker protocol

The protocol prepares noisy inputs for parity checking, exponentially suppresses one noise component while controlling another, and converts purified plus-states into phase-states.

  • A. The parity-checker protocol: ηn vanishes exponentially for δ0 = 0 and η0 < 1/3, while nonzero δ0 eventually becomes problematic after approximately log(δ0) rounds.The protocol therefore suppresses the large-noise component rapidly but has a finite useful iteration window when small noise is present.
  • A. The parity-checker protocol: For ϵ = 10−8, four parity-checking rounds produce a plus-state with total error 2.707 × 10−8.The initial parameters are η0 ∼0.0152 and δ0 = 10−8/3, and total noise decreases during the first three rounds.
  • B. Equatorialization: Two purified plus-state copies are projected with a 2-qutrit stabilizer measurement and decoded to probabilistically produce a phase-state.The projection uses the subspace spanned by the stated logical qutrit states.
  • B. Equatorialization: The resulting phase-state supplies the desired magic-state form after the equatorialization transformation.The conclusion follows from the relation ω + ω2 = −1 in the output expression.

C. Gate injection

The paper describes qutrit gate injection from phase states and connects distilled states to non-Clifford computation through promoted Clifford operations.

  • C. Gate injection: Gate injection uses a phase state together with a second qutrit state, followed by measurement and outcome-dependent decoding.The measurement outcome determines a diagonal unitary acting on the input state.
  • C. Gate injection: For ∣Φ0,π⟩, the generated unitaries form a small group of order 4 up to global phase, so repeated attempts can reach any desired unitary with high probability.The random outcomes correspond to steps of a random walk over phase gates.
  • C. Gate injection: The non-Clifford unitary N generates, together with the Clifford group, an infinite-order single-qutrit group.The paper notes that this structure makes a dense cover of SU(3) plausible, but does not provide a complete proof.
  • C. Gate injection: The five-qutrit protocol distills two families of states, including Hadamard eigenstates with a 23.3% depolarizing-noise threshold and another family reaching 34.4%.The second family consists of eigenstates of a nondegenerate Clifford operator outside the Hadamard plane.
  • C. Gate injection: The outputs can be converted into other magic states that provide a unitary promoting the Clifford group, with good but inconclusive evidence for universal quantum computation.The paper identifies universality as an open question rather than a proved result.

Appendix A: Clifford equivalences and cycling behaviour

The appendix explains how decoding choices affect the qutrit distillation map, including cycling among equivalent states and a corrected map used to identify fixed points numerically.

  • Appendix A: Clifford equivalences and cycling behaviour: Using canonical decoding without the additional R rotation preserves purification but causes outputs to cycle between different states.Hadamard eigenstates oscillate between ∣H±⟩, while the ∣ϕ⟩ state follows a four-cycle related by Clifford operations.
  • Appendix A: Clifford equivalences and cycling behaviour: The cycling also occurs for depolarized states, so every state in the four-cycle is distilled by the five-qutrit code.Applying the corrective transformation removes the cycling behaviour from the iteration map.
  • Appendix A: Clifford equivalences and cycling behaviour: The protocol starts with five identical qutrit copies, projects onto the stabilizer-code subspace, and computes output Bloch components from traces involving the input state and projector.The stabilizer generators and logical operators define the projector used in the distillation calculation.
  • Appendix A: Clifford equivalences and cycling behaviour: The four independent Bloch components evolve through coupled fifth-order polynomial expressions, with the remaining components given by complex conjugation.The corrected expressions incorporate the additional Clifford correction after each iteration.
  • Appendix A: Clifford equivalences and cycling behaviour: Because the corrected fixed-point equations are general simultaneous complex polynomials of order five, the fixed points are found numerically rather than algebraically.The authors invoke the Abel–Ruffini theorem to explain why a general closed-form solution is unavailable.

Appendix C: The size of the promoted Clifford group

The appendix establishes that the promoted single-qutrit Clifford-plus-non-Clifford group has infinitely many distinct unitaries, while leaving approximate universality unresolved.

  • Appendix C: The size of the promoted Clifford group: An alternating sequence of Hadamard gates and the non-Clifford gate is used to construct a unitary K for analyzing the promoted group.The eigenvalues of K are calculated analytically.
  • Appendix C: The size of the promoted Clifford group: Because two eigenvalue parameters are irrational, every power of K is different, proving that the group C+N has infinite order.This establishes infinitely many unitaries but not a complete universality theorem.
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