PaperlayerView exact revision on arXiv

Source-linked AI summary

Lower bounds on the non-Clifford resources for quantum computations

Michael Beverland, Earl Campbell, Mark Howard, Vadym Kliuchnikov

arXiv:1904.01124v2quant-ph

TL;DR

The paper addresses the practical cost of magic-state production by establishing lower bounds for resource conversion, single-qubit unitary synthesis, and computational tasks. It introduces new monotones and a canonical form for post-selected stabilizer circuits, obtaining bounds including a logarithmic T-state requirement for approximate unitary synthesis and near-optimal bounds for key operations.

  • Problem

    Magic-state distillation can dominate the space-time overhead of fault-tolerant quantum computation, motivating lower bounds on the resource states required for important tasks.

  • Method

    The paper uses stabilizer nullity, the dyadic monotone, stabilizer extent, and a canonical form for post-selected stabilizer circuits to derive resource lower bounds.

  • Results

    The paper strengthens lower bounds for resource conversion, unitary synthesis, and computational tasks, including a bound of 1/7·log_2(1/ε) − 4/3 T-states for some approximate single-qubit unitary protocols.

  • Takeaways & Limitations

    Catalysis enables a broad class of resource-state conversions that are impossible without it, while some bounds for modular adders and multiply-controlled operations are close to optimal.

  • Takeaways & Limitations

    The stabilizer extent is not known to be multiplicative for all states, so some inter-conversion bounds do not apply with arbitrary catalysts.

Abstract

from arXiv · show

We establish lower-bounds on the number of resource states, also known as magic states, needed to perform various quantum computing tasks, treating stabilizer operations as free. Our bounds apply to adaptive computations using measurements and an arbitrary number of stabilizer ancillas. We consider (1) resource state conversion, (2) single-qubit unitary synthesis, and (3) computational tasks. To prove our resource conversion bounds we introduce two new monotones, the stabilizer nullity and the dyadic monotone, and make use of the already-known stabilizer extent. We consider conversions that borrow resource states, known as catalyst states, and return them at the end of the algorithm. We show that catalysis is necessary for many conversions and introduce new catalytic conversions, some of which are close to optimal. By finding a canonical form for post-selected stabilizer computations, we show that approximating a single-qubit unitary to within diamond-norm precision $\varepsilon$ requires at least $1/7\cdot\log_2(1/\varepsilon) - 4/3$ $T$-states on average. This is the first lower bound that applies to synthesis protocols using fall-back, mixing techniques, and where the number of ancillas used can depend on $\varepsilon$. Up to multiplicative factors, we optimally lower bound the number of $T$ or $CCZ$ states needed to implement the ubiquitous modular adder and multiply-controlled-$Z$ operations. When the probability of Pauli measurement outcomes is 1/2, some of our bounds become tight to within a small additive constant.

1 Introduction and high-level overview

The paper develops lower bounds on magic-state resources for state conversion, unitary synthesis, and computational tasks, treating stabilizer operations as free. It introduces monotones and catalytic-conversion analysis, then derives bounds for controlled operations, adders, and approximate single-qubit synthesis.

  • Motivation: The paper asks for the minimum copies of a resource state needed for a computational task using arbitrary stabilizer operations.The motivation is that magic-state production can dominate fault-tolerant computation overhead.
  • Methods: The authors introduce stabilizer nullity and dyadic monotones, alongside stabilizer extent, to lower-bound resource-state production and conversion.Stabilizer nullity is non-increasing under stabilizer operations, and its additivity rules out some conversions even with catalysts.
  • Resource-state conversion: Catalysis is necessary for a broad class of resource-state conversions, while several catalytic conversions are shown to be achievable.A catalyst is borrowed during the computation and returned unchanged at the end.
  • Resource-state conversion: The conversion bounds reveal an asymptotic gap: converting n |T⟩ states to |CCZ⟩ states and back yields fewer than n |T⟩ states.The paper also notes that many known conversion algorithms do not match the bounds.
  • Computational tasks: For n ≥3, multiply-controlled-Z requires at least n |T⟩, n/2 |CS⟩, or n/3 |CCZ⟩ states.The best known implementation uses n −2 |CCZ⟩ states.
  • Computational tasks: The modular adder requires at least n −2 |T⟩, (n −2)/2 |CS⟩, or (n −2)/3 |CCZ⟩ states, while the best known implementation uses n −1 |CCZ⟩ states.In the restricted half-probability measurement setting, n −2 |CCZ⟩ states are required for the modular adder.
  • Unitary synthesis: The paper derives lower bounds for approximating arbitrary single-qubit unitaries in diamond norm and applies them to general synthesis strategies involving measurements and ancillas.The analysis uses a canonical form for stabilizer circuits applied to resource states.
  • Measurement with probability 1/2: In the half-probability Pauli-measurement setting, the dyadic monotone shows that the n−2-|CCZ⟩ multiply-controlled-Z circuit is optimal.This setting excludes arbitrary single-qubit Pauli measurements.

2 Some basic techniques

This section develops stabilizer nullity and stabilizer extent as monotones for analyzing resource-state conversions under stabilizer operations, including catalytic settings. It also establishes a field-based no-go theorem and illustrates catalytic conversions and conversion-bound limitations.

  • Stabilizer nullity: The stabilizer nullity is the number of hosted qubits minus the logarithm of the stabilizer size, and it is zero for stabilizer states.It is invariant under Clifford unitaries and stabilizer-state inclusion or removal.
  • Stabilizer nullity: Pauli measurements never increase stabilizer nullity: the stabilizer size either stays unchanged or increases by at least a factor of two.Both measurement outcomes satisfy ν(|φ⟩) ≤ ν(|ψ⟩).
  • Stabilizer extent: The stabilizer extent is another monotone under stabilizer operations, defined by minimizing over complex linear combinations of stabilizer states.It is submultiplicative and multiplicative for specified collections of states satisfying the lemma’s conditions.
  • Catalysis: Catalysts are resource states borrowed during a stabilizer computation and returned unchanged, enabling conversions that are impossible without them.The section gives examples where catalysis is necessary and sufficient, while stabilizer-nullity additivity can rule out some catalyzed conversions.
  • Field-based conversion bounds: Density-matrix entries in a number field are preserved by stabilizer circuits, preventing conversions from states with entries in Q(i) to a T state requiring Q(ζ8).This rules out producing |T⟩ from any number of |CS⟩ or |CCZ⟩ states without additional resources.

3 Conversion between resource states

The paper develops resource-state conversion bounds and catalytic protocols for Clifford magic states, including phase-polynomial, W_n, and adder-based constructions. These results connect monotone-based lower bounds with explicit conversions, while identifying catalytic necessity, conversion gaps, and several near-optimal protocols.

  • Overview: Conversion bounds and algorithms enable resource-cost comparisons across different input magic states.The authors position these results as groundwork for later computational-task bounds and broader comparisons between distillation protocols.
  • Catalytic conversions: Catalysis is necessary for some conversions, and the paper introduces two families of catalytic conversion circuits.The families target Clifford magic states and higher Clifford-hierarchy resource states, including adder-based constructions.
  • Phase polynomial protocols: For diagonal third-level unitaries U, the resource state |U⟩ supports conversion protocols characterized by the minimum T-count τ(U) and stabilizer nullity ν.Theorem 3.1 formulates a general conversion result using τ(U) and ν, with phase-polynomial protocols summarized in Figure 3.
  • Phase polynomial protocols: |W_n⟩ cannot yield T-states without a catalyst, but catalysis converts |W_n⟩ into |T⟩^(n−1), while measurement reduces |W_n⟩ to |W_(n−1)⟩.The reduction measures one qubit and applies a conditional Clifford correction when the outcome is 1.
  • One-bit adder conversion protocols: Adder-based protocols compute Hamming-weight information with a one-qubit adder that can be un-computed using Clifford gates and Pauli measurements.This construction supports catalytic conversions involving higher-level gates and amortizes correction costs through recursion.
  • One-bit adder conversion protocols: 1 − 1/2^(d−1) copies of |CCZ⟩ are asymptotically necessary and sufficient to produce |π_j/2^d⟩ under 50% measurement protocols.Theorem A.16 gives an explicit protocol with catalysts, while Lemma 6.9 supplies the matching lower bound.
  • Conversion bounds: ν(|T⟩)=1 and ν(|CCZ⟩)=3, while stabilizer extent values are ξ(|T⟩)=1.17157 and ξ(|CCZ⟩)=1.77778.These monotones yield r ≥ 3.63356 for producing a |CCZ⟩-related CSS state from |T⟩ states, versus r′ ≤ 3 in the reverse direction.
  • Conversion bounds: Many conversion algorithms do not match the listed monotone bounds, indicating that more efficient algorithms remain to be found.Tables 1 and 2 summarize known algorithms alongside the tightest bounds from stabilizer extent or nullity.

4 Computational task lower bounds

The paper lower bounds resource-state consumption for multiply controlled-Z gates, modular adders, and the quantum Fourier transform by transferring state-production bounds to computational tasks.

  • Resource conversions: Catalytic conversion tables compare best-known algorithms with bounds from stabilizer extent or nullity, but extent bounds require a multiplicativity condition.The stated caveat concerns arbitrary catalysts.
  • Proof strategy: The proof strategy bootstraps lower bounds for producing U|S⟩ or catalytically producing |Φ⟩ into lower bounds for implementing U.Stabilizer nullity supplies the relevant resource accounting, with ν(|T⟩)=1, ν(|CS⟩)=2, and ν(|CCZ⟩)=3.
  • 4.1 Lower bounds for the CnZ gate: n |T⟩, n/2 |CS⟩, or n/3 |CCZ⟩ states are necessary for C^(n−1)Z when n ≥3.The same bounds apply to producing the corresponding multiply controlled-Z state.
  • 4.2 Lower bounds for the modular adder: n + 1 |T⟩, (n + 1)/2 |CS⟩, or (n + 1)/3 |CCZ⟩ states are required by the stronger adder bound.A separate bound gives n −2, (n −2)/2, and (n −2)/3 for the corresponding resource states.
  • 4.2 Lower bounds for the modular adder: n −2 |T⟩ states are necessary for the n-qubit quantum Fourier transform.The bound follows from the stabilizer-nullity calculation used for the adder.

5 Lower bounds for approximate unitary synthesis

The paper develops lower bounds for approximate single-qubit unitary synthesis that remain applicable to adaptive measurement protocols and arbitrary stabilizer ancillas. Its canonical-form and approximation analysis yields logarithmic resource requirements for T, CS, and CCZ states.

  • Scope: The synthesis bounds allow Pauli measurements, measurement-dependent adaptivity, and an arbitrary number of ancillary qubits.They target approximation of arbitrary single-qubit unitaries using Clifford gates and Pauli measurements.
  • Scope boundary: The unitary-synthesis bounds do not hold when catalyst states are allowed.This differs from stabilizer-nullity bounds, which benefit from additivity.
  • Canonical form: Theorem 5.3 reduces post-selected stabilizer circuits to commuting Pauli projectors, a Clifford unitary, and a stabilizer state.The number of independent commuting Paulis is k = ν(|ψin⟩) − ν(|ψout⟩).
  • Adaptive protocols: Measurement outcomes make output quality and resource consumption random variables, so the analysis converts average fidelity into per-run guarantees.With probability at least (C −1)/C, individual outputs achieve fidelity at least 1 − Cδ.
  • CS- and CCZ-state bounds: 1/3·log2(1/ε) − 2/3 CS states and 1/4·log2(1/ε) − 1/2 CCZ states lower bound approximation for some one-qubit states when ε < 1/8.The CS bound includes the same example condition; the CCZ result is stated for an existing state.
  • T-state bounds: 1/3·log2(1/ε) − 2/3 T states suffice as a lower bound for approximating some one-qubit states within trace distance ε < 1/8.The same bound applies to all states satisfying ε < |⟨ψ|0⟩|^2 < 3ε.

6 Tighter lower bounds with measurement probabilities one half

This section develops tighter resource lower bounds when every Pauli measurement outcome has probability one half, using the dyadic monotone alongside stabilizer-state structure. The bounds establish optimality or near-optimality for multiply-controlled-Z gates, modular adders, and selected resource-state conversions.

  • Scope: Half-probability measurements permit stronger lower bounds, although the resulting monotone applies only to circuits in this measurement setting.The section targets circuits where each measurement outcome occurs with probability one half.
  • Dyadic monotone: The dyadic monotone is the maximum power of two in the denominator across Pauli expectations for states with entries in Z[i, 1/2].It is invariant under Clifford unitaries and is non-increasing under the relevant half-probability Pauli measurements.
  • Dyadic monotone: µ2 is nonnegative and equals zero exactly for stabilizer states.This makes the quantity faithful to non-stabilizerness within the stated state class.
  • Multiply-controlled-Z: At least n − 2 |CCZ⟩ states are needed for the n-qubit multiply-controlled-Z gate under half-probability measurements.The well-known construction therefore achieves the lower bound in this setting.
  • Modular adder: At least n − 2 |CCZ⟩ states are needed for the n-qubit modular adder, while the best known construction uses n − 1 states.The bound is one |CCZ⟩ state short of the known construction; the corresponding T-state bound is 2n − 5 for n ≥ 3.
  • Resource-state conversion: Catalytic conversion of |CCZ⟩ states into |πj/2d⟩ states requires asymptotically at least 1 − 1/2d−1 |CCZ⟩ states per output.For odd j and d ≥ 2, this matches the introduced conversion protocols when only 50% Pauli measurements are allowed.

7 Conclusion and open problems

The conclusion summarizes lower bounds and the new stabilizer nullity, dyadic monotone, and canonical post-selected-circuit tools, then identifies extensions and scope boundaries for future work.

  • Conclusion: The paper establishes resource lower bounds for state conversion, unitary synthesis, and computational tasks using new monotones and a canonical form for post-selected stabilizer circuits.The authors anticipate broader applications of these tools.
  • Future applications: Proposed applications include lower bounds for multiply-controlled adders, Hamming-weight state preparation, Hamming-weight computation, and small quantum Fourier-transform circuits.These are presented as prospective applications rather than established results in this section.
  • Open problems and limitations: The multiplicativity of stabilizer extent is unknown for all states, so not all inter-conversion bounds apply with arbitrary catalysts.It is multiplicative for all states used in the paper.
  • Open problems and limitations: Exact inter-conversion of stabilizer states is unavoidably lossy, while extending this result to inexact qubit conversions remains open.The corresponding inexact extension has been achieved for odd-prime qudits but not qubits.
  • Open problems and limitations: Whether more efficient modular-adder and multiply-controlled-Z algorithms exist outside the half-probability measurement setting remains an open question.The questions specifically concern leaving the probability-half restriction.

A.1 Generic circuits for injecting diagonal gates

The appendix gives a generic injection protocol for applying a diagonal unitary using its resource state, CNOTs, measurements, and conditional corrections. For a level-k unitary, the corrections lie at most at level k − 1.

  • Generic injection: The protocol takes a resource state |U⟩ = U|+⟩⊗n and an arbitrary n-qubit input state |α⟩.The resource and input occupy separate n-qubit registers.
  • Generic injection: CNOTs couple the resource and input registers, after which the resource register is measured in the computational basis.The measurement produces outcomes m(1), …, m(n).
  • Conditional corrections: Each measurement outcome controls an X correction followed by a conditional correction UXkU† on the corresponding input qubit.All X corrections are completed before the conditional conjugated corrections are applied.
  • Output: The output register is U|α⟩, while the measured resource register is left in a known computational-basis state.This establishes the protocol’s correctness for diagonal unitaries.
  • Hierarchy structure: If U belongs to level k of the Clifford hierarchy, the conditional corrections belong to at most level k − 1.Conjugating Pauli X operators by U lowers the hierarchy level by one.

A.2 Reducing the cost of unitary synthesis using T gates

This section reduces single-qubit unitary-synthesis cost by exploiting resource-state injection and conversion protocols for T gates. The reported protocols reduce average and worst-case T-gate consumption relative to Clifford-and-T synthesis.

  • T-gate injection: The conversion family uses 3 + 1/(4k) T gates per T gate on average and 3.5 + 1/(2k) in the worst case.The parameter k controls the stated overhead terms.
  • Baseline synthesis: Approximating a single-qubit rotation with Clifford and T gates uses fewer than 3 log2(1/ε) + O(log(log2(1/ε))) T gates typically and fewer than 4 log2(1/ε) + O(1) in the worst case.These are the baseline synthesis costs discussed before the resource-state reduction.
  • Synthesis comparison: Using the broader resource-assisted gate set reaches a break-even point with Clifford-and-T synthesis while retaining the bound 4 log2(1/ε) + O(1).The comparison concerns the number of T gates in the synthesized sequence.
  • T-gate injection: Applying a T gate consumes four T gates in one protocol, while the best protocol found consumes three T gates per application.The section further uses catalysis and conversion protocols to reduce this cost.
  • Synthesis gains: The resulting protocol achieves a 20% average-case reduction and a 10% worst-case reduction in T gates for single-qubit rotation synthesis.The circuit shown uses three T gates per applied T gate on average.

A.3 Explicit circuits for some common resource conversions

This section collects explicit circuits for converting common resource states, including controlled-unitary constructions whose measurements succeed with probability one half.

  • Figure 11 records a four-T-gate circuit for implementing the CCZ gate.
  • Figures 12–15 present explicit resource-state conversion circuits, including conversions involving CCZ, CS, and related states.These constructions are used as subroutines for the paper’s results.
  • With probability one half, measuring the control qubit of |CU⟩ in the Z basis produces |U⟩; otherwise it produces |+⟩^⊗n.This applies when U is diagonal and CU is its controlled version.
  • With probability one half, measuring the first qubit in the Y basis produces the state |C_nS^m⟩ for outcome m = ±1.The cited proposition supplies the correctness proof for this measurement-based conversion.

A.4 Extent values

This section reports extent values for common resource states and outlines a numerical-to-exact procedure, while noting that some multi-qubit values were not rigorously certified.

  • The extent calculation uses approximate primal and dual linear-program solutions, exact-expression reconstruction, and equality of the primal and dual optima.
  • Some multi-qubit extent values were reconstructed from eight-digit numerical approximations rather than rigorously calculated.The paper identifies this as a limitation of the extent calculations in Table 4.
  • Table 4 lists exact expressions for the extent of resource states used in Tables 1 and 2.The tabulated values are reported as accurate to within eight digits of precision.
  • Equal extent does not imply Clifford equivalence, as illustrated by the states discussed in this section.

A.5 Further details on phase polynomial protocols

This section develops phase-polynomial protocols for diagonal third-level unitaries, relating T-count, matrix structure, and catalytic resource conversions.

  • Resource conversion: Theorem 3.1 gives a resource conversion for magic states |U⟩ associated with diagonal third-level unitaries, using stabilizer nullity ν and T-count τ(U).
  • Phase-polynomial representation: A diagonal third-level unitary is represented by a cubic phase polynomial whose terms use Z_2-linear functions and coefficients in Z_8.The nonzero terms are encoded by a binary matrix P and coefficient vector a.
  • T-count decomposition: Odd phase-polynomial coefficients determine the non-Clifford part, while even coefficients contribute a Clifford unitary.The circuit’s T-count equals the number of odd-valued coefficients, and an all-even phase polynomial is Clifford.
  • T-count decomposition: Writing f = g + 2h separates U_f into non-Clifford U_g and Clifford U_2h, with τ(U_g) equal to the minimum number of terms in g.This minimum is found over equivalent phase-polynomial representatives.
  • Conversion protocols: Two equivalent phase polynomials differing by Δ require τ(U_Δ) T states, equal to the number of terms where they differ.
  • Catalytic conversions: If all rows of the phase-polynomial matrix have even Hamming weight, the associated state cannot be converted to |T⟩ without a catalyst.This follows because the unitary is then a product of CS and CCZ gates with entries in Q(i).
  • Catalytic conversions: For W_n, τ(W_n) = n + 1, while its full-rank matrix gives μ(|W_n⟩) = n and a catalytic conversion to T states.W_2 is Clifford-equivalent to CS, and |W_3⟩ is Clifford-equivalent to |CCZ⟩, yielding the stated CCZ conversion.

A.6 Lower bound reduction from the modular adder to the multiply-controlled Z state

This section reduces production of the multiply-controlled-Z state to implementation of a controlled modular increment, and then to the modular adder.

  • Reduction to the modular adder: The reduction proceeds by first producing |C_nZ⟩ from the controlled increment and then implementing that increment with the modular adder.
  • Controlled modular increment: A controlled modular increment maps |j⟩|a⟩ to |j⟩|a + j mod 2^n⟩ for j ∈ {0,1}.The second register stores a using n qubits.
  • Controlled modular increment: Applying the controlled increment to |+⟩^⊗n|0⟩ reduces the circuit to an n-controlled NOT and produces a state Clifford-equivalent to |C_nZ⟩.
  • Reduction to the modular adder: The controlled modular increment can be implemented with a modular adder by using the last qubit of its first input as control and fixing the other input qubits to |0⟩.

A.7 Canonical form for post-selected stabilizer computations

This appendix puts post-selected stabilizer computations into a canonical form: a Clifford unitary followed by commuting Pauli projections, with resource-state stabilizers separated from ancillary stabilizer qubits.

  • A.7 Canonical form for post-selected stabilizer computations: A state with 2^r stabilizers can be Clifford-decomposed into r |0⟩ ancillas tensored with a state having a trivial stabilizer.This separates stabilizer degrees of freedom from the non-stabilizer component.
  • A.7 Canonical form for post-selected stabilizer computations: Post-selected stabilizer circuits reduce to a Clifford unitary and independent commuting Pauli operators, with k = n − m projections for input and output nullities n and m.The output also includes a stabilizer state and the +1 eigenspace projectors of the Pauli operators.
  • A.7 Canonical form for post-selected stabilizer computations: If input and output nullities are equal, the two states are related by a Clifford unitary.Thus, no nontrivial Pauli projections are needed when m = n.
  • A.7 Canonical form for post-selected stabilizer computations: The canonical form requires considering only mutually commuting Pauli measurements when enumerating stabilizer circuits acting on a fixed input state.The construction removes measurements that conflict with input stabilizers by replacing them with Clifford operations or discarding impossible branches.
  • A.7 Canonical form for post-selected stabilizer computations: Measurements anticommuting with an input stabilizer are equivalent to randomly applying Clifford unitaries, with the two measurement outcomes occurring with equal probability.This equivalence allows such measurements to be pushed through later operations and absorbed into the final Clifford.
  • A.7 Canonical form for post-selected stabilizer computations: Commuting measured Paulis can be conjugated through Clifford gates and collected into a subgroup G that remains commuting and excludes −I.The resulting generators can be restricted to the non-ancillary qubits after separating |0⟩ ancillas.
  • A.7.1 Decoupling stabilizer states: Ancillary |0⟩ qubits can be removed from the canonical expression using a decoupling lemma.The lemma reduces the relevant equivalence to states with trivial stabilizer, completing the passage from the general canonical form to Theorem A.7.
  • A.7.1 Decoupling stabilizer states: The decoupling proof constructs a Clifford that commutes with the relevant Pauli Z operators, then uses induction to establish Clifford equivalence after adding |0⟩ ancillas.The construction uses CNOT gates on the ancilla qubits to preserve their |0⟩ state.

A.8 Conversion protocols for dyadic rational powers of T gate √

The appendix develops catalytic protocols for applying dyadic rational phase rotations and producing their resource states, using CCZ gates and 50%-probability stabilizer measurements. It also supplies the dyadic-monotone machinery used to derive approximation lower bounds.

  • Catalytic rotation protocols: The protocols include |T⟩ as the case j = 1, d = 3 and use fewer than one CCZ gate per state asymptotically when producing many identical copies.The construction recursively converts higher-angle resource states using catalysts at lower dyadic angles.
  • Catalytic rotation protocols: 2k + 1 phase rotations exp(iθ|1⟩⟨1|) can be applied in parallel using one such rotation, k CCZ gates, and k doubled-angle rotations.This recursive identity is the basis for the catalytic protocol for dyadic angles.
  • Catalytic rotation protocols: 2k unitaries R(πj/2^d) can be applied in parallel using one catalytic |πj/2^d⟩ state, k CCZ gates, and k + 1 lower-angle rotations.The construction assumes positive integers k, d and odd integer j, with measurement outcomes occurring with probability 50%.
  • Catalytic rotation protocols: ad,k = 2^(d−1)(k−1)+2 rotations can be executed using bd,k = (2^(d−1)−1)(k−1)+d−1 CCZ states and one catalyst at each lower dyadic angle.Asymptotically, producing |πj/2^d⟩ uses 1 − 1/2^(d−1) CCZ states per copy.
  • Dyadic monotone construction: The dyadic monotone extends the 2-adic valuation through norm functions Nd on nested cyclotomic rational sets.Multiplicativity and the identity Nd(x)^2 = Nd+1(x) make the extension additive and consistent across the nested sets.
  • Dyadic monotone construction: The dyadic monotone is positive and vanishes exactly for stabilizer states, based on a valuation condition for elements equal to ±1.This establishes its role as a resource monotone for the restricted measurement setting.
  • Dyadic monotone construction: The valuation satisfies v2(x + y) ≥ min(v2(x), v2(y)), while norm-function properties ensure multiplicativity and well-definedness.These properties support calculations of the dyadic monotone for states with dyadic-rational amplitudes.
  • Approximation lower bounds: 7N_T ≥ 1/7 log2(1/ε) − 1/14 lower bounds T-state resources for certain ε-approximations, with a corresponding unitary-synthesis bound stated for diamond-norm precision.The approximation argument assumes ε < 1/8 for the state result and ε < 1/(28C) for the unitary result.
Loading 1904.01124v2…