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Efficiently computable bounds for magic state distillation

Xin Wang, Mark M. Wilde, Yuan Su

arXiv:1812.10145v2quant-phcs.ITmath-phmath.ITmath.MP

TL;DR

The resource theory of non-stabilizer states needs measures that quantify non-stabilizerness. The paper introduces a family of thauma measures and formulates selected members as convex optimization problems. Max-thauma provides an efficiently computable magic-state-distillation benchmark that can outperform mana, while min-thauma bounds regularized relative entropy of magic.

  • Problem

    The resource theory of non-stabilizer states needs measures that quantify non-stabilizerness.

    Source: Appendix C · p. 9

  • Method

    The paper introduces a family of thauma measures and formulates selected members as convex optimization problems.

    Source: Appendix B · p. 8, Appendix C · p. 9

  • Results

    Max-thauma provides an efficiently computable magic-state-distillation benchmark that can outperform mana, while min-thauma bounds regularized relative entropy of magic.

    Source: Appendix C · p. 9, Appendix E · pp. 11–12

  • Takeaways & Limitations

    The thauma family supplies computable tools for benchmarking distillation and constraining resource conversion in the non-stabilizer resource theory.

    Source: Appendix E · pp. 11–13

Abstract

from arXiv · show

Magic-state distillation (or non-stabilizer state manipulation) is a crucial component in the leading approaches to realizing scalable, fault-tolerant, and universal quantum computation. Related to non-stabilizer state manipulation is the resource theory of non-stabilizer states, for which one of the goals is to characterize and quantify non-stabilizerness of a quantum state. In this paper, we introduce the family of thauma measures to quantify the amount of non-stabilizerness in a quantum state, and we exploit this family of measures to address several open questions in the resource theory of non-stabilizer states. As a first application, we establish the hypothesis testing thauma as an efficiently computable benchmark for the one-shot distillable non-stabilizerness, which in turn leads to a variety of bounds on the rate at which non-stabilizerness can be distilled, as well as on the overhead of magic-state distillation. We then prove that the max-thauma can be used as an efficiently computable tool in benchmarking the efficiency of magic-state distillation and that it can outperform pervious approaches based on mana. Finally, we use the min-thauma to bound a quantity known in the literature as the "regularized relative entropy of magic." As a consequence of this bound, we find that two classes of states with maximal mana, a previously established non-stabilizerness measure, cannot be interconverted in the asymptotic regime at a rate equal to one. This result resolves a basic question in the resource theory of non-stabilizer states and reveals a difference between the resource theory of non-stabilizer states and other resource theories such as entanglement and coherence.

Supplemental Material: Efficiently computable bounds for magic state distillation

The supplemental material gives more detailed analyses and proofs of the main-text results, sometimes reiterating steps for explicitness and self-containment.

Appendix A: The stabilizer formalism and discrete Wigner function

This appendix reviews the stabilizer formalism and discrete Wigner-function framework underlying the resource theory of non-stabilizer states.

Stabilizer formalism

  • Stabilizer operations form the restricted operation set used in most known fault-tolerant schemes.They include computational-basis preparation and measurement plus restricted unitary operations.Source: Appendix A · p. 7

Heisenberg–Weyl operators

  • For prime-dimensional systems, shift and boost operators define the Heisenberg–Weyl operators used by the formalism.The operators are parameterized by elements of Z_d × Z_d and tensor across composite systems.Source: Appendix A · p. 7

Clifford group

  • Clifford operators are unitaries that map Heisenberg–Weyl operators to Heisenberg–Weyl operators under conjugation, up to phases.These operators form the Clifford group.Source: Appendix A · p. 7

Stabilizer states

  • The stabilizer-state set is the convex hull of pure stabilizer states obtained by applying Clifford operators to |0⟩.Source: Appendix A · p. 7

Discrete Wigner function

  • Positive discrete Wigner functions identify states associated with bound magic: some such states cannot yield magic at a strictly positive distillation rate.The Wigner framework is also used to study quasi-probability representations in quantum information.Source: Appendix A · p. 7
  • The discrete Wigner representation assigns phase-space values through phase-space point operators, with an operator representation defined by W_V(u) := 1/d Tr A_uV.The appendix also introduces associated trace and spectral norms.Source: Appendix A · pp. 7–8

Appendix B: Efficient computation of thauma via convex optimization

This appendix shows that thauma can be computed through convex optimization using the relative entropy objective and a convex feasible set.

  • Thauma is defined through a relative-entropy optimization over the relevant set of subnormalized operators.Source: Appendix B · p. 8
  • The optimization is convex because relative entropy is convex in σ and the constraint set is convex.Source: Appendix B · p. 8
  • An efficient quantum-relative-entropy optimization algorithm is used to provide a program for computing θ(ρ).Source: Appendix B · p. 8

Appendix C: Generalized thauma measures

This appendix defines generalized thauma measures from Petz–Rényi and sandwiched Rényi divergences and establishes their monotonicity, computability limits, and ordering relations.

Generalized thauma

  • Generalized thauma measures are constructed from relative-entropy-like divergences that satisfy data processing and suitable positivity conditions.These conditions imply that the resulting thauma is a magic monotone.Source: Appendix C · pp. 8–9

Rényi divergences

  • The Petz–Rényi and sandwiched Rényi relative entropies are defined with support and orthogonality conditions governing finite versus infinite values.Source: Appendix C · pp. 8–9

Petz–Rényi thauma

  • Petz–Rényi thauma is defined over α ranges where the Petz–Rényi relative entropy obeys data processing, including α ∈ [0, 1) ∪ (1, 2].The min-thauma used in the main text equals θ_0(ρ).Source: Appendix C · p. 9

Sandwiched Rényi thauma

  • Sandwiched Rényi thauma is defined over α ranges satisfying data processing, and its α → ∞ limit yields the max-thauma.Source: Appendix C · p. 9

Ordering relations

  • Ordering relations among the underlying Rényi divergences induce corresponding ordering relations among thauma measures.Source: Appendix C · p. 9

Appendix D: SDPs of min- and max-thauma and proof of Proposition 1

This appendix formulates min- and max-thauma as semidefinite programs and proves their additivity. It then establishes their ordering around the regularized thauma.

SDP formulations

  • Min- and max-thauma are represented by primal and dual semidefinite programs.The appendix derives the min-thauma dual SDP and presents the primal and dual SDPs for max-thauma.Source: Appendix D · p. 10

Additivity

  • Proposition 1 establishes additivity of both min-thauma and max-thauma for tensor-product states.Source: Appendix D · p. 10

Ordering

  • The min-thauma is a lower bound and the max-thauma an upper bound for regularized thauma.Source: Appendix D · p. 10

Appendix E: Monotonicity of max-thauma on average under stabilizer operations

This appendix proves that max-thauma is monotone on average under stabilizer operations and evaluates thauma quantities for four named states. The state-specific analyses use SDP bounds whose matching values yield exact equalities.

Monotonicity

  • Max-thauma does not increase on average under general stabilizer protocols.The proof maps feasible SDP solutions through completely positive stabilizer maps and uses the conditional-probability interpretation of stabilizer operations.Source: Appendix E · p. 11

Exact evaluations

  • For the Strange, Norrell, H+, and T states, the appendix evaluates min-thauma, regularized thauma, and max-thauma exactly.These exact evaluations are summarized in Proposition 2.Source: Appendix E · p. 11

Strange state

  • The Strange state satisfies θmin(S) = θ∞(S) = θmax(S) = log2(5/3).Matching lower and upper bounds establish the equality through the ordering θmin(S) ≤ θ∞(S) ≤ θmax(S).Source: Appendix E · p. 12

Norrell state

  • The Norrell state satisfies θmin(N) = θ∞(N) = θmax(N) = log2(3/2).The appendix also relates this common value to the regularized relative entropy of magic using a stabilizer-state argument.Source: Appendix E · pp. 12–13

H+ state

  • The H+ state satisfies θmin(H+) = θ∞(H+) = θmax(H+) = log2(3 − √3).The equality follows by matching SDP lower and upper bounds and applying the min–regularized–max ordering.Source: Appendix E · p. 13

T state

  • The T state’s min-, regularized, and max-thauma values coincide after matching SDP bounds.The upper-bound construction uses a state with nonnegative discrete Wigner function and a positive-semidefinite operator inequality.Source: Appendix E · pp. 13–14

Appendix G: Proof of Proposition 3

The proof of Proposition 3 uses the min-thauma as a bound on overlap with states in the relevant free-state set and applies data processing to hypothesis-testing relative entropy.

Proof strategy

  • The proof bounds distillation performance using the inequality Tr[|φ⟩⟨φ|^⊗nτ] ≤ 2^-nθmin(φ) for free states τ.Source: Appendix G · p. 14
  • Data processing of Dε supplies the third inequality in the bound.Source: Appendix G · p. 14

Conclusion

  • Because k is arbitrary, the argument yields the stated bound, with the proof of the companion result proceeding similarly.Source: Appendix G · p. 14

Appendix H: Second-order converse

The second-order converse uses a minimizer of thauma and the variance term of quantum relative entropy to refine the finite-blocklength bound. Its proof combines an earlier upper bound with the second-order expansion of quantum hypothesis-testing relative entropy.

Converse bound

  • Corollary 4 gives a finite-blocklength converse for n source states and error tolerance ε ∈ (0, 1).Source: Appendix H · p. 15
  • The bound is expressed using a thauma minimizer σ, the inverse normal cumulative distribution Φ^-1, and V(ρ∥σ).The variance term is defined as V(ρ∥σ) = Tr ρ(log2 ρ − log2 σ)^2 − D(ρ∥σ)^2.Source: Appendix H · p. 15

Proof

  • The proof combines the upper bound in Proposition 3 with the second-order expansion of quantum hypothesis-testing relative entropy.Source: Appendix H · p. 15

Appendix I: Bound on efficiency of magic-state distillation

Proposition 5 gives a lower bound on the efficiency of distilling a magic state from independent copies of a resource state, with the proof establishing the bound for all integer input sizes and success probabilities.

  • Proposition 5 lower-bounds the efficiency of distilling a magic state σ from independent copies of a resource state ρ.The proof assumes a stabilizer protocol that outputs σ from ρ^⊗n with probability p.Source: Appendix I · p. 15
  • The proof uses an intermediate state τ together with Proposition 8 and the stabilizer nature of |0⟩.Source: Appendix I · p. 15
  • The bound applies for every integer n ≥1 and success probability p ∈(0, 1].Source: Appendix I · p. 15

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