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Enhanced fault-tolerant quantum computing in $d$-level systems
Earl T. Campbell
TL;DR
The paper studies how Mµ gates fit into the Clifford hierarchy and how their Clifford equivalence depends on the finite-field dimension. It shows that these gates are third-level and non-Clifford for odd prime d > 3, with equivalence classes determined by cubic residues, while the appendix also derives finite-field summation rules.
Problem
The paper addresses how the proposed non-Clifford gates can be verified as third-level Clifford-hierarchy operations and distinguished under Clifford equivalence.
Method
It analyzes Pauli conjugation algebraically and uses permutation-Clifford conjugation plus cubic-residue structure to classify Mµ gates.
Results
For odd prime d > 3, Mµ is third-level and non-Clifford; all nonzero gates are equivalent for d = 2 mod 3, whereas d = 1 mod 3 gives three equivalence classes.
Takeaways & Limitations
The gate classification depends on the arithmetic of the prime dimension, with cubic residues determining whether distinct nonzero Mµ parameters remain Clifford-distinct.
Abstract
from arXiv · showhide
Error correcting codes protect quantum information and form the basis of fault tolerant quantum computing. Leading proposals for fault-tolerant quantum computation require codes with an exceedingly rare property, a transverse non-Clifford gate. Codes with the desired property are presented for $d$-level, qudit, systems with prime $d$. The codes use $n=d-1$ qudits and can detect upto $\sim d/3$ errors. We quantify the performance of these codes for one approach to quantum computation, known as magic state distillation. Unlike prior work, we find performance is always enhanced by increasing $d$.
Appendix A: Clifford hierarchy
The appendix verifies that Mµ lies in the third Clifford-hierarchy level while remaining non-Clifford for odd prime dimensions d > 3. The argument uses conjugation of Pauli operators and fails at d = 3 because 3µ vanishes modulo d.
- Third-level membership: Mµ conjugates the Pauli operator X into a Clifford unitary because the cubic terms cancel, leaving a quadratic exponent.The resulting exponent is t = µ(3n^2 + 3n + 1).
- Third-level membership: All Pauli operators can be written, up to phase, as X^mZ^n, so conjugation by Mµ reduces to the established X case and commutation with Z.Since Clifford operators form a group, the conjugated operator remains Clifford.
- Non-Cliffordness: Mµ is non-Clifford when its conjugation of X is not Pauli, which requires the quadratic coefficient 3µ to be nonzero modulo d.This criterion distinguishes non-Cliffordness from merely belonging to the third level.
- Non-Cliffordness: For d = 3, 3µ vanishes modulo d for every µ, whereas odd prime d > 3 permits the required nonzero coefficient.This is identified as a fundamental reason transversal non-Cliffords are simpler to construct in odd dimensions above three.
Appendix B: Clifford equivalence of Mµ gates
The appendix classifies the Clifford equivalence of Mµ gates by cubic residues after basis-permuting Clifford conjugation. All nonzero gates are equivalent when d = 2 mod 3, while d = 1 mod 3 yields three equivalence classes; the analysis has not yet included Hadamard gates.
- Equivalence criterion: Conjugating Mµ by permutation Cliffords rescales its cubic coefficient from µ to β^3µ, so equivalence is determined by cubic residues.Quadratic terms are collected into a Clifford contribution and do not determine the cubic equivalence class.
- d = 2 mod 3: When d = 2 mod 3, the cubic residue contains every nonzero field element, making all nonzero Mµ gates Clifford equivalent.A suitable β satisfies β^3 = µ/µ′.
- d = 1 mod 3: When d = 1 mod 3, the cubic residue contains (d−1)/3 elements and partitions the nonzero field into three equal cosets.Each coset corresponds to one Clifford equivalence class of µ values.
- d = 1 mod 3: For d = 7, the cubic residue is R7 = {1, 6}, with cosets {2, 5} and {3, 4}, giving three Clifford equivalence classes.The residue and its two nontrivial cosets together partition the six nonzero field elements.
- Scope of classification: The classification so far considers only computational-basis permutations and phases, leaving the effect of Hadamard conjugation unresolved.Thus the stated classes are relative to the restricted set of Clifford operations analyzed here.
Appendix C: Evaluating summations
The appendix evaluates finite-field monomial sums by separating exponents divisible by d−1 from those that are not. This yields a simple zero-versus-minus-one rule and extends it to shifted polynomials.
- Monomial reduction: The summation problem is reduced to evaluating monomial sums S(x^m) after decomposing the polynomial into monomials.This reduction provides the building block for the general polynomial case.
- Monomial sums: When m is not congruent to 0 modulo d−1, multiplying the summation variable by a suitable nonzero y shows S(x^m) = 0.The argument uses a y satisfying y^m ≠ 1 and invariance under reordering the field elements.
- Monomial sums: When m is congruent to 0 modulo d−1, every nonzero field element contributes one, so S(x^m) = d−1 = −1.There are d−1 terms in the sum over nonzero field elements.
- Polynomial sums: For an unshifted polynomial H with no x^0 term, the sum vanishes, while the general shifted case gives S(H) = −hρ.The appendix attributes this dependence to the constant-term structure.