Catalysis and activation of magic states in fault tolerant architectures

• Stabilizer states are mixtures of pure states defined by commuting Pauli operators, while Clifford unitaries preserve the set of pure stabilizer states.Source: §I · p. 2
• A Clifford computer performs Clifford unitaries, stabilizer-state preparation, classical feedforward and randomness, and Pauli measurements.Source: §I · p. 2
• Clifford transformations are deterministic when they always map input ρ to output ρ′, denoted ρ →D ρ′.Source: §I · p. 2
• Probabilistic Clifford transformations produce the target state with nonzero probability, denoted ρ →P ρ′.Source: §I · p. 2
• Magic catalysis concerns deterministic transformations, whereas magic activation concerns probabilistic transformations.Source: §I · p. 2

II. MAGIC CATALYSIS

• |ϕ⟩|H0⟩→D|H0⟩|H0⟩, although |ϕ⟩↛D|H0⟩, establishing magic catalysis with an unchanged Hadamard eigenstate catalyst.Source: §II · p. 3
• The protocol prepares |ϕ⟩ on qubits A, B, C and |H0⟩ on D, measures Y_CY_D and Z_CZ_D, applies conditional corrections, then keeps A and B.Source: §II · p. 3
• Singlet projection acts as an odd-parity projector in any shared basis and, in the Hadamard basis, yields the desired |H0,0⟩ output after correction.Source: §II · p. 3
• Two random measurements are followed by conditional Clifford unitaries, ensuring the same output for every measurement branch.Source: Figure 1 · p. 3, §II · p. 3
• The uncatalyzed transformation is impossible because stabilizer projections permit only certain rational amplitude ratios, whereas |H0⟩ requires the irrational ratio tan^2(π/8) = √2 − 1.Source: §II · p. 4
• The same catalytic idea extends to a mixed state σϕ because it is Clifford-equivalent to |ϕ⟩, while the broader structure of magic-assisted Clifford operations remains open.Source: §II · p. 4

III. MAGIC ACTIVATION

• The single-shot protocol prepares σ(q) and τ(f), measures Y_BY_C and Z_BZ_C, and postselects the −1,−1 outcome before discarding B and C.Source: Figure 2 · p. 4, §III · p. 5
• For fixed finite n, τ(f)^⊗n is bound below f ≤ f_st + ϵ_n, so multiple copies cannot probabilistically distill states unavailable from one copy.Source: §III · p. 4
• All noisy T-states outside the stabilizer set become useful for some task when combined with an activator, even though neither resource alone can produce the target state.Source: §III · p. 5
• Singlet projection selects odd-parity terms, producing τ(f′) with improved fidelity after renormalization.Source: §III · p. 5
• When 1 > q > 1/2, f′ > f, while optimal measurements on σ(q) alone cannot reach the activated output, proving genuine activation for one copy.Source: §III · p. 5
• The protocol’s improvement cannot currently be shown to iterate with more bound-resource copies, although a stronger later protocol approaches unit fidelity asymptotically.Source: §III · p. 5

• INS states are nonstabilizer states from which no single-qubit nonstabilizer state can be probabilistically extracted.Source: §IV · p. 5
• The family σINS(q, n) satisfies the INS conditions when qmin < q ≤qmax.Source: §IV · pp. 5–6
• The allowed region is empty for n = 1, 2 and appears for n ≥3, so this construction yields no two-qubit INS states.Source: Figure 3 · p. 6, §IV · p. 6
• The construction uses the stabilizer-norm as a witness that sufficiently pure states are nonstabilizer states.Source: §IV · p. 6
• The upper weighting bound follows by showing stabilizer-code projections cannot produce sufficiently pure outputs when q ≤qmax.Source: §IV · p. 6

• This protocol combines an INS state with bound magic states and can asymptotically approach unit fidelity, motivating the term asymptotic activation.Source: §V · pp. 6–7
• Theorem 4 shows that σINS(n) combined with τ(f)⊗n−1, for f > fst, probabilistically produces τ(f′) with f′ →1 as n → ∞.Source: §V · pp. 6–7
• The protocol pairs n−1 noisy T-states with qubits of the INS activator and postselects singlet outcomes for every pair.Source: Figure 4 · p. 7, §V · p. 7
• The result requires noisy T-states above the stabilizer threshold, since the output fidelity approaches unity only when f exceeds fst.Source: §V · pp. 6–7
• Because the number of copies varies with n, existing fixed-n no-go results do not establish that the transformation is impossible without the INS state.Source: §V · p. 7

VI. DISCUSSION AND CONCLUSIONS

• Three protocols demonstrate counterintuitive phenomena from jointly exploiting two different species of nonstabilizer states.This extends prior work focused mainly on many copies of one state.Source: §VI · p. 7
• Magic catalysis enables a deterministic transformation impossible with Clifford operations alone while returning the Hadamard eigenstate catalyst.The protocol depends on specific symmetries of that catalyst.Source: §VI · p. 7
• Magic activation combines a noisy T-state with an activator to probabilistically produce a single-qubit state unavailable from either resource alone.This establishes usefulness for every noisy T-state outside the stabilizer set, while remaining a single-round protocol.Source: §VI · p. 8
• Asymptotic activation uses n−1 noisy T-states and an irreducible n-qubit nonstabilizer resource to approach unit-fidelity magic states.No single-qubit nonstabilizer state can be probabilistically extracted from the special n-qubit resource alone.Source: §VI · p. 8
• Neither magic activation protocol reproduces all three defining features of the cited entanglement-activation example.The paper leaves open whether stronger analogues possessing all three features exist.Source: §VI · p. 8
• Singlet projection verifies that two qubits are nonidentical but projects them into a stabilizer state, so it requires multi-qubit correlations to prepare nonstabilizer states.The paper also reports a many-copy strategy based on this mechanism.Source: §VI · p. 8

Appendix A

• The amplitude ratio in Eq. (7) is restricted to rational values, so the required irrational target ratio cannot be achieved.The finite feasible set also limits approximation quality.Source: Appendix A · p. 8
• A pseudo-stabilizer state is a pure state admitting a Clifford-unitary representation of the specified computational-basis form.The definition introduces the complexity measure based on the smallest valid support size.Source: Appendix A · p. 9
• Genuine stabilizer states have complexity P=1, while the constructed state |ϕ′⟩ has complexity at most 4 and H-states are not pseudo-stabilizer states.These structural facts provide the complexity bound used in the obstruction.Source: Appendix A · pp. 8–9
• Theorem 5 states that any pure single-qubit output obtainable from a pseudo-stabilizer state must have an amplitude ratio in the feasible set R_P.Targets outside R_P are therefore impossible under probabilistic Clifford transformations.Source: Appendix A · p. 9
• For |ϕ⟩, exact production of an H-state is impossible because all feasible ratios are rational, with 1/5 the closest searched value to tan(π/8)^2.The numerical gap from the target exceeds 0.028.Source: Appendix A · p. 9
• The resulting theorem rules out many Clifford transformations but is substantially less general than entanglement-theory majorization criteria.The authors identify minimizing computational-basis support under Clifford unitaries as a possible route toward a broader framework.Source: Appendix A · p. 9

Appendix B

• Finite n copies of |ϕ⟩ cannot probabilistically Clifford-transform into an exact |H0⟩.The obstruction follows because |ϕ⟩^⊗n remains pseudo-stabilizer with finite complexity at most 4n.Source: Appendix B · p. 9
• As n increases, many copies of |ϕ⟩ can approximate |H0⟩ with fidelity asymptotically approaching unity.Thus exact impossibility at finite n is compatible with asymptotic computational usefulness.Source: Appendix B · p. 9

Appendix C

• T-rotation symmetry reduces the analysis to the two parity projectors Π±=(I I±Z_AZ_B)/2.The symmetry applies to arbitrary two-qubit Pauli projections after suitable T rotations.Source: Appendix C · p. 10
• For pure inputs q=1, two T-states can be probabilistically converted into a pure state on the Bloch-sphere equator.This equatorization is used as an intermediate step toward implementing non-Clifford gates.Source: Appendix C · p. 10
• The even-parity projection maps both components of σ(q) to the same stabilizer state, and therefore does so for their mixture.The projected state is proportional to cos(β)sin(β)(|0,0⟩−i|1,1⟩).Source: Appendix C · p. 10
• The decoded projected states lie on the Bloch-sphere equator, with the positive-octant geometry selecting the closest decoding to |T0⟩.This identifies the relevant single-qubit output for the comparison.Source: Appendix C · p. 10
• The two-qubit projection fidelity f′′′ is always smaller than the single-qubit fidelity f′′, so no two-qubit stabilizer projection outperforms the single-qubit projection.This is the appendix's final comparison.Source: Appendix C · p. 10

Graph-code reduction

• The maximum projection probability is bounded by analyzing stabilizer-subspace weights after local-Clifford conversion to a graph code.Local Cliffords preserve Pauli weights, allowing the proof to replace the original stabilizer with a locally equivalent graph-code stabilizer.Source: Appendix D · pp. 10–11

Graph-code representation

• Graph-code stabilizers are expressed through a graph and a binary linear code, with the relevant projector written in terms of the stabilizer.The code structure introduces the dual code C⊥ and separates fixed and code-dependent contributions to stabilizer weights.Source: Appendix D · p. 10

Magic-state decomposition

• The proof also uses Pauli decompositions of multiple T-magic states and bounds their projection through stabilizer signs and weights.The sign-dependent expression is upper-bounded by taking all signs positive before switching to the locally equivalent graph code.Source: Appendix D · p. 11

Weight bound

• The upper bound is maximized when the variable weight contribution w(y′′) vanishes after diagonalizing the generator matrix.The resulting inequality depends only on the classical linear code C⊥, while the fixed contribution comes from the diagonalized part.Source: Appendix D · p. 11

Saturating protocol

• Measuring m qubits in the computational basis and postselecting on “+1” outcomes saturates the projection-probability upper bound.This construction realizes the maximal value identified by the weight analysis.Source: Appendix D · p. 11

Resource family

• The input family consists of separable correlated states, except at specific r values where σ(q, r) becomes a product state.Although the family is not completely general, arbitrary states can be brought into this form by a suitable twirling procedure.Source: Appendix E · p. 11

Daisy chain protocol

• The daisy chain protocol uses many copies of σ(q, r), chaining postselected measurements so that the final unmeasured qubit has fidelity approaching 1 as r →0.The protocol prepares paired copies, measures the first qubit, performs correlated X and Z measurements between successive pairs, and discards measured qubits.Source: Appendix E · pp. 11–12

Asymptotic behavior

• For large n, the protocol converges to a limiting fidelity determined by the dominant eigenvalue of its recurrence matrix whenever the eigenvalues have different magnitudes.Product states are exceptional because their eigenvalues are identical; otherwise one eigenvalue determines the limiting fidelity.Source: Appendix E · p. 12

Universality regime

• Convergence to the limiting fidelity is exponentially fast, with finite-size deviations scaling as (µ2/µ1)^n.The protocol can be combined with standard 5-qubit distillation when its achieved fidelity exceeds the required threshold for universality.Source: Appendix E · p. 12, Figure 5 · p. 12

Scope and limitation

• The protocol purifies all nonproduct states toward some state, but that state may remain below the five-qubit-code threshold, leaving regimes without known universal-computation methods.The authors therefore describe the current bound-state definition as limited, with achievable purification apparently constrained by the raw resource’s correlations.Source: Appendix E · p. 12

Generalization

• A similar protocol and analysis applies to two-qubit correlated states in arbitrary bases, not only the T-basis.Source: Appendix E · p. 12

Twirling protocol

• The twirling protocol independently applies random operators from {1, T, T^2} to both qubits and then applies a random two-qubit operation.These steps transform an arbitrary state toward the σ(q, r) family defined in Eq. (E1).Source: Appendix F · p. 12

Effect of twirling

• The first two randomization steps diagonalize the state in the |T⟩ basis, while the third mixes the symmetric terms.Source: Appendix F · p. 12