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Hamiltonian Eigenvalue Transformation by Tridiagonal Gadgets

Arthur Braida, Joseph Cunningham, Jérémie Roland

arXiv:2610.03295v1quant-phcs.DS

TL;DR

Analog devices implement local Hamiltonians, but the polynomial transformations used in algorithms are generally nonlocal. This paper encodes polynomial actions as a single time-independent local Hamiltonian on an invariant subspace, using weighted ancilla chains. The construction supports marked-state filtering and analog search, with resource costs in ancillas, locality, and energy scale.

  • Problem

    Analog algorithms can require nonlocal polynomial transformations even when the device implements only a local Hamiltonian.

    Source: §1 · p. 1

  • Method

    The paper attaches ancilla chains to the input Hamiltonian and combines their isolated eigenvalue branches in a single local, time-independent Hamiltonian.

    Source: §1 · p. 2

  • Results

    The construction synthesizes polynomial actions on an invariant subspace, and its marked-state filter uses O(log n) ancillas at constant accuracy for analog search.

    Source: §1 · p. 2, §1.2 · p. 5, §6 · p. 25

  • Takeaways & Limitations

    The construction trades sequential circuit calls for ancillas, increased locality, and energy scale while providing a static Hamiltonian action.

    Source: §6 · p. 25

    The full-space analog-search spectrum is numerically supported only up to n = 20; formal validation is left for future work.

    Source: §6 · p. 25

Abstract

from arXiv · show

An analog device implements a local Hamiltonian H, but a polynomial P (H) is in general not local, so the device cannot implement it. We carry P (H), to any prescribed accuracy, as the action of a single time-independent local Hamiltonian on an explicitly described invariant subspace. Short chains of ancilla qubits are attached to H. A chain of 2m sites has a unique isolated eigenvalue that is an analytic function of the input vanishing to order exactly 2m, because the input must cross the chain and come back before it can shift the energy at the far end. Chains of different lengths therefore form a triangular family, and a weighted sum of them reproduces a prescribed polynomial term by term. Even chains give the even part and odd chains the odd part, so any polynomial is reached. We prove uniqueness, analyticity and coefficient bounds uniform in the chain length for inputs H of norm below any r < 1. Where the circuit model pays degree 2l in 2l sequential oracle calls, the result here is one Hamiltonian of locality two more than that of H, which pays the degree in O(l^2 + log2^(1/eps)) ancillas and in energy scale. As an application we filter a marked eigenstate of a local Hamiltonian. Composing a synthesised square along the Chebyshev doubling identity works, but its energy scale grows quasi-polynomially with the degree. Iterating instead the exact eigenvalue branch of the two-site chain k times gives a filter on k + 1 ancilla qubits whose error decays geometrically in k, at an energy scale polynomial in the inverse passband width and independent of k, the locality growing by one per stage. In adiabatic optimisation with a rank-one driver, the construction replaces that non-local driver, by a Hamiltonian with O(log n) ancillas and O(log n)- body terms, at an energy scale polynomial in n. Simulations including every ancilla reproduce the spectrum of the ideal algorithm.

1 Introduction

The paper asks how an analog device can implement a polynomial of a local Hamiltonian, then constructs time-independent local gadgets whose action on an invariant subspace approximates that polynomial. It applies the construction to marked-state filtering and analog search, trading circuit depth for ancillas, locality, and energy scale.

1 Introduction

  • A device can implement a local Hamiltonian with the marked string as its ground state, whereas the rank-one projector used by analog search is nonlocal.The paper frames polynomial transformation as the route between the local input and the desired projector.Source: §1 · pp. 1–2

1.1 The gadget

  • Attaching a chain of alternating input and unit couplings makes its isolated eigenvalue depend on the input only after the input traverses the chain and returns.Chains of different lengths therefore have different leading orders, allowing weighted eigenvalues to reproduce polynomial terms successively.Source: §1 · p. 2, §1.1 · pp. 3–4

1.2 Results

  • A unique analytic chain eigenvalue and uniform coefficient bounds support polynomial synthesis for inputs with norm below any r < 1.The branch for a chain of 2m sites vanishes at order exactly 2m, making the chains a triangular family.Source: §1.2 · p. 4
  • Weighted chains yield a single time-independent (κ + 2)-local Hamiltonian implementing an even polynomial on an invariant subspace; odd chains supply the odd part.The synthesis uses O(L^2) ancillas for L chains and has a geometrically decaying truncation error.Source: §1.2 · p. 4
  • For analog search, iterating the exact two-site branch uses O(log n) ancillas at constant accuracy and yields a (κ + O(log n))-local Hamiltonian.Its normalized energy scale is bounded by a polynomial in σ/d independent of the number of stages.Source: §1.2 · p. 5
  • The marked-state filter can use the exact two-site branch instead of a synthesized Chebyshev square, avoiding the latter route's energy scale growth with degree.The iterated branch has geometrically rather than doubly exponentially decaying error, which the paper says is sufficient for analog search.Source: §1.2 · p. 5

1.3 Related work

  • Unlike perturbative gadgets, which reproduce target interactions through a low-energy effective Hamiltonian, this construction computes an isolated eigenvalue branch exactly.The paper also distinguishes its static local Hamiltonian from circuit-based transformations that sequence operations in time.Source: §1.3 · pp. 5–6

1.4 Organisation

  • The paper develops the chain and its analysis, solves the synthesis problem, applies it to marked-state filtering and analog search, then discusses limitations and open questions.Appendices cover the synthesized-square alternative, longer proofs, and the odd-chain construction.Source: §1.4 · p. 6

2 The even gadget

The even gadget is a tridiagonal chain with alternating input and unit couplings; its characteristic polynomial yields an analytic eigenvalue equation for studying the branch through zero. The construction has a local Hamiltonian realization, and odd-chain variants provide the corresponding odd functions.

2.1 Definition and local realisation

  • Alternating input and unit bonds define the tridiagonal chain, whose final site carries detuning −∆.The matrix becomes an operator when the scalar input is replaced by a Hamiltonian.Source: §2.1 · p. 7
  • For κ-local H, the even gadget is realizable on an invariant subspace using 2m ancilla qubits with locality κ + 2; for m = 1, one ancilla and one extra locality suffice.The one-excitation sector realizes the chain’s hopping terms and boundary projector.Source: §2.1 · p. 8
  • Odd-chain variants have an odd eigenvalue branch with leading order 2n + 1, supplying a triangular basis for odd targets.Even chains produce even functions; combining both parities reaches any polynomial, with constant and linear terms added directly.Source: §2.1 · p. 8, §2.2 · p. 10

2 The even gadget

  • The chain’s characteristic polynomial is computed using a transfer matrix whose powers follow the Chebyshev recurrence, yielding a reduced eigenvalue equation.The reduced equation and its partial-fraction expansion are used to analyze the target eigenvalue.Source: §2 · p. 6, §2.2 · pp. 8–9

2.2 The characteristic polynomial

  • The characteristic polynomial depends on x only through x², so the eigenvalue branch singled out by an invariant condition is even in x.The reduced equation uses a partial-fraction expansion, with poles determined by zeros of the recurrence sequence S_m.Source: §2.2 · pp. 9–10

3 The target eigenvalue

This section establishes a unique analytic eigenvalue branch through zero for each chain and controls its Taylor coefficients uniformly in chain length. The results provide the basis for polynomial synthesis and identify the detuning needed for a chosen input radius.

3 The target eigenvalue

  • For target radius r < 1 and detuning ∆ ≥ ∆⋆(r), each chain has a unique simple eigenvalue near zero, analytic throughout |x| < r, with leading order exactly 2m.The leading orders make chains of different lengths a triangular family; the threshold scales as ∆⋆(r) ∼ (1 − r)^−2.Source: §3 · p. 11

3.1 A two-sided bracket on the real axis

  • The real-axis bracket has relative width O(x^2m), pinning down coefficients below order x^4m and revealing a run of zero coefficients that makes the basis sparse.The branch has a convergent Taylor expansion near x = 0.Source: §3 · p. 10, §3.2 · p. 11

3.3 Coefficient bounds on the complex disc

  • On a complex disc, Rouché’s theorem establishes a unique root near zero; the resulting coefficient bound is uniform in chain length: |c_k^(m)| ≤ C_r ∆ r^−2k.Here C_r = 2/(1 − r²), and the result applies for every m ≥ 1 and k ≥ 0.Source: §3 · p. 11, §3.3 · pp. 12–13

3.4 The working detuning ∆⋆(r)

  • In the analysis, 0 is never used, and ∆min(2, r) may be replaced throughout by the smaller ∆min(q, r).The analysis assumes ∆ > 1; for r > 1/2, ∆ ≥ ∆⋆(r) already implies ∆ > 1.Source: §3.4 · p. 14
  • The detuning threshold is set by the shortest chains: below r ≃ 0.4935, the m = 1 chain requires ∆ > 2r; above it, the m = 2 chain sets the threshold.For r approaching 1, ∆⋆(r) ∼ (1 − r)^−2.Source: §3.4 · p. 14

4 Synthesis of an even polynomial

Chains of different lengths provide a triangular family of eigenvalues whose weighted sum synthesizes an even polynomial. The synthesis converges for inputs with ∥H∥<r and is realised by one local Hamiltonian on an invariant subspace.

4 Synthesis of an even polynomial

  • The section combines chains of different lengths to reproduce a prescribed even polynomial term by term, with the approximation tail controlled by the number of chains.The eigenvalue coefficients are uniformly bounded in chain length, supporting the synthesis across the admissible input range.Source: §4 · p. 14, §4.2 · p. 17

4.1 The triangular system

  • Distinct leading orders make chain eigenvalues a triangular basis, so back-substitution uniquely chooses weights matching the target even polynomial through degree 2L.The constant term p0I is added directly rather than synthesised by the chains.Source: §4 · pp. 14–15, §4.1 · p. 15

4.2 Convergence of the synthesis

  • Sparse back-substitution keeps partial weights quasi-polynomial, so the geometric truncation tail converges for every ∥H∥<r.The convergence rate is set by the margin between the input norm and the analyticity radius.Source: §4.2 · pp. 16–17

4.3 The realised Hamiltonian

  • Disjoint chains attached to H commute, and their weighted sum is a single (κ + 2)-local Hamiltonian whose invariant encoded subspace carries the synthesized polynomial.The construction uses O(L^2) ancilla qubits; its energy-scale cost grows with the target through the synthesis weights.Source: §4.3 · pp. 18–19

5 Application: a filter for a marked state

The section builds a local filter for an isolated marked eigenstate, then uses it in adiabatic search. Iterating the exact two-site-chain transformation avoids the quasi-polynomial energy growth of a synthesized square, while the search evidence remains numerical and limited in scope.

5 Application: a filter for a marked state

  • The marked-state projector is nonlocal, but a local Hamiltonian with a unique marked zero-energy eigenvector supports a polynomial filter that approximates the projector when its stopband amplitude is small.For R(0) = 1 and |R(x)| ≤ η on δ ≤ |x| ≤ σ, the spectral theorem gives projector error at most η.Source: §5 · pp. 19–20

5.1 Why not the polynomial directly

  • Synthesizing the Chebyshev filter directly takes Θ(ℓ) chains, Θ(ℓ^2) ancillas, and exponentially large weights; composing a synthesized square reduces ancillas but leaves quasi-polynomial energy growth.The Chebyshev filter is normalized to one at the marked energy and is small away from it; the polynomial synthesis targets Rℓ − 1 because the gadget branches vanish at zero.Source: §5.1 · pp. 20–21

5.2 One exact block, iterated

  • Iterating the exact two-site-chain eigenvalue map yields a filter on k + 1 ancillas, with one added unit of locality per stage and energy scale independent of k.Its stopband error decays geometrically in k, rather than doubly exponentially as for the Chebyshev filter.Source: §5.2 · p. 22
  • At d = σ/10, ∆ = 4σ, and η = 10^-2, six stages use 7 ancilla qubits at normalized norm 3.4 · 10^4, versus above 10^11 for the synthesized square.At ∆ = 10σ, the stage-count coefficient 2/log2 ϱ is 1.007; larger detuning improves the map toward the Chebyshev map but raises energy cost.Source: §5.2 · p. 22
  • A single long chain after the entry stage keeps locality at κ + 3, but its ancilla count is linear in σ/d for small entry detuning and quadratic when the detuning exceeds σ.The long-chain power gains less marked-point separation than iteration, and its normalized norm can grow exponentially with chain length.Source: §5.2 · pp. 22–23

5.3 Adiabatic search

  • The analog-search gadget uses O(log n) ancillas and O(log n)-local terms for constant accuracy, but the simulated spectrum is for the Hamming-weight cost function.The filter depends on the driver alone; for another cost function, the spectrum is not the one simulated here.Source: §5.3 · pp. 23–24
  • For n ≤ 20, the full gadget spectrum reproduces analog search and its anticrossing-gap scaling within 15%, while non-encoded levels remain separated near the anticrossing.The interpolation is numerically diagonalized with all ancillas; a proof that it works for every n remains open.Source: §5.3 · p. 24

6 Discussion

The discussion presents exact chain eigenvalue branches as a way to encode polynomial transformations of a local Hamiltonian in one time-independent local Hamiltonian. It highlights the filter application’s resource trade-offs and notes that the adiabatic-search validation is numerical rather than proved for all sizes.

6 Discussion

  • A degree-2ℓ polynomial transformation can be carried by a time-independent Hamiltonian that is only two more local than the input, using O(ℓ^2 + log2(1/ε)) ancillas and a larger energy scale.The construction uses weighted ancilla chains and acts on an explicitly described invariant subspace.Source: §6 · pp. 24–25
  • For filtering, iterating the exact two-site-chain map avoids synthesis and radius restrictions, with energy scale independent of stage count but error decreasing geometrically rather than doubly exponentially.The number of stages grows as log(1/η), instead of log log(1/η).Source: §6 · p. 25
  • The marked-state filter approximates a rank-one projector using O(log n) ancillas and O(log n)-body terms at polynomial energy scale; a constant-locality variant uses more ancillas.Source: §6 · p. 25
  • With a noncommuting adiabatic driver, the full Hamiltonian numerically reproduces analog-search spectra and anticrossing-gap scaling through n = 20, but formal validation is left for future work.The discussion identifies perturbative effective-Hamiltonian methods as a natural tool for that validation.Source: §6 · p. 25
  • The discussion proposes exploring other useful gadget transformations and whether exact-spectrum ancillas can realize block encodings of prescribed matrices.Source: §6 · p. 25

A The Chebyshev recursion with a synthesised square

This appendix constructs the Chebyshev filter by recursively composing synthesized approximations to the square map. It controls the accumulated approximation error by keeping each realized stage inside a certified disc, and derives the resulting ancilla and locality costs.

A.1 Construction

  • Each ideal stage is a prefactor times a square plus a constant, arranged by the Chebyshev doubling identity so the target recursion remains bounded by σ on the input interval.The realized stages substitute a synthesized square for the exact square.Source: §A.1 · p. 26
  • If the square-synthesis error and margin conditions hold, every realized stage stays within the certified radius and the normalized encoded action approximates the degree-2ℓ Chebyshev filter.The construction requires σ < r and a margin σ < ¯σ < r.Source: §A.1 · pp. 26–27
  • The encoded filter’s error is bounded by controlling the realized-versus-ideal stage deviation and the Chebyshev stopband on the nonzero input spectrum.The normalized encoded action divides the final realized stage by its marked-point value, which is at least σ/2.Source: §A.1 · pp. 26–27
  • The recursion uses L = O(k + log(1/η)) chains per stage and O(L^2) ancillas overall, adding two to the input locality.Each stage uses fresh disjoint chains; increasing the target degree adds stages rather than changing the synthesis weights.Source: §A.1 · pp. 26–27

A.2 Energy

The synthesized-square recursion achieves the Chebyshev filter but incurs a substantial energy-scale cost. Each stage multiplies the Hamiltonian norm by the synthesis weight, producing quasi-polynomial growth with degree.

A.2 Energy

  • The recursion’s normalized norm grows geometrically with its number of stages, hence quasi-polynomially in ℓ, because each stage feeds its input through weighted square-synthesis chains.At fixed energy budget, this norm ratio stretches evolution times.Source: §A.2 · pp. 27–28
  • The energy bound depends on the total synthesis weight ΛL, whose available bound grows geometrically with L; the authors do not establish a better bound.Their measurements indicate that obvious improvements to this bound are unavailable.Source: §A.2 · p. 28

A.3 Numbers

The reported synthesis error decreases from 9.1·10^-6 at L = 5 to 7.1·10^-8 at L = 8, while the weight sum eventually grows geometrically rather than linearly. The filtering example also shows that meeting a stopband target does not remove the large normalized-norm cost.

A.3 Numbers

  • The synthesis error is 9.1·10^-6 at L = 5 and 7.1·10^-8 at L = 8.The reported radius is r = 0.7562 at ∆ = 10.Source: §A.3 · p. 28
  • ΛL reaches 722 at L = 40 and 1.13·10^8 at L = 80, so no bound of the form ΛL = O(L) is available.The measured ratio |αj+1/αj| falls with ∆ and remains below 1 at the tested detunings.Source: §A.3 · pp. 28–29
  • At k = 6, one chain per stage reaches a realised stopband of 10^-5, while the normalised norm remains above 10^11.The norm is set by each stage's weight ΛL ≥ ∆, not by the number of chains.Source: §A.3 · p. 29

B Deferred proofs

The deferred proofs establish the recurrences and identities used in the main argument, then derive the eigenvalue equation from the chain boundary condition. Together they provide the algebraic tools for characterising the chain's eigenvalue branch.

B.1 Proof of Lemma 2

  • The recurrence for Θ2j rearranges into the matrix relation (6), and iteration from (Θ0, Θ−1) = (1, 0) gives (7).This proves Lemma 2 from the recurrence in (5).Source: §B.1 · p. 29

B.2 Proof of Lemma 3

  • The closed form for Sk follows from recurrence (9) and its characteristic roots, while identity (10) follows by induction using Cayley–Hamilton.The base cases are m = 1 and m = 2.Source: §B.2 · p. 29

B.3 Proof of Lemma 4

  • Rescaling Sk turns its recurrence into the Chebyshev recurrence, yielding roots wj and the factorisation; the proof also bounds root locations and residues.The roots are shown to be outside D, and κj = sin^2 ϑj/m > 0.Source: §B.3 · pp. 29–30

B.4 Proof of Proposition 1

  • Changing the chain boundary modifies only the last diagonal entry, and expanding that row gives the eigenvalue equation (12).The proof uses the recurrence for Θ2m and its vector form.Source: §B.4 · p. 30

B.5 Proof of Proposition 2

The proof establishes that the selected eigenvalue branch is even in the input and that a root exists uniquely within a specified interval. Its bounds scale with the chain-dependent quantity M.

B.5 Proof of Proposition 2

  • The characteristic polynomial depends on x only through x^2, so the ordered eigenvalues agree for x and −x and an invariantly selected branch is even.This establishes the claim for real x.Source: §B.5 · p. 30

B.6 Proof of Proposition 3

  • For 0 ≤ λ ≤ M/∆, the eigenvalue equation has exactly one root, which tends to zero with x.The proof uses the strict increase of λ(λ + ∆) and the non-decreasing property of −x^2Rm.Source: §B.6 · p. 30
  • The root bounds use M = Θ(x^2m), and the bracket in the lower bound has relative width O(x^2m) as x approaches zero.The interval is obtained using the upper bound λ ≤ M/∆.Source: §B.6 · p. 30

B.7 Proof of Theorem 1

These deferred proofs establish analyticity and coefficient properties of the chain eigenvalue branch, bounds on the analytic continuation, and index restrictions used in the composition argument. They also give supporting algebraic and monotonicity results for the stated lemmas and corollaries.

B.7 Proof of Theorem 1

  • The implicit function theorem gives a unique analytic branch through zero, and its series starts at x^2m; the m = 1 case instead has an exact quadratic equation.The branch is even, and the first nonzero coefficient is at order 2m.Source: §B.7 · p. 31

B.8 Proof of Lemma 6

  • Lemma 6 bounds the terms involving wj when |λ| < 1 − r, using separate cases according to whether sin ϑj is at least 1 − r.The proof sums bounds across the m − 1 terms.Source: §B.8 · pp. 31–32

B.9 Proof of Theorem 2

  • Rouché’s theorem gives precisely one simple zero in |λ| < ρ, and the zero continues analytically for every |x| < r.The continuation agrees near x = 0 with the branch from Theorem 1.Source: §B.9 · p. 32

B.10 Proof of Proposition 4

  • For m = 1, the eigenvalue equation is λ^2 + ∆λ − x^2 = 0, and its root vanishing at zero is bounded using |x| < r ≤ ∆/2.Lemma 5 converts the root bound into the stated coefficient bound.Source: §B.10 · p. 32

B.11 Proof of Lemma 7

  • The proof of Lemma 7 reduces the condition to a quadratic inequality whose admissible branch is s ≤ s−.The other branch lies outside s < (1 − r)^2.Source: §B.11 · p. 33

B.12 Proof of Corollary 2

  • The Corollary 2 conditions hold for every m > m0 once they hold at m0, because the relevant sequence am strictly decreases with m.The proof also eliminates ρ from the conditions by setting ρ = 2am/∆.Source: §B.12 · p. 33

B.13 Proof of Proposition 6

  • In the index expansion, contributions vanish for the stated intermediate ranges, while j = 1 always remains in the retained range.The proof treats k = 2 separately because the two steps used for larger k fail there.Source: §B.13 · p. 33

B.14 Proof of Proposition 7

The proof bounds the rescaled coefficients by a recurrence and then uses induction to control their growth. The following proof step reindexes an inner sum and applies the coefficient bound.

B.14 Proof of Proposition 7

  • The recurrence inequality holds for every k > q, including k = 2, and summing it yields a bound in terms of A_⌊L/2⌋.The k = 2 case follows because the general bound has an additional non-negative term there.Source: §B.14 · p. 34
  • Strong induction bounds A_L by νΦ_L, reducing the coefficient-growth estimate to bounding the product Φ_L.The proof establishes the base case at L = q and uses the bound at ⌊L/2⌋ for larger L.Source: §B.14 · p. 34

B.15 Proof of Proposition 8

  • The proof of Proposition 8 reindexes the inner sum using n = k − j and applies the coefficient bound (44).Source: §B.15 · p. 34

B.16 Proof of Proposition 9

The proof controls the iterated eigenvalue branch by tracking its growth and the operator norm across stages. The odd gadget supplies an odd branch whose leading order depends on chain length, with coefficient bounds applying across chain lengths below radius one.

B.16 Proof of Proposition 9

  • Each iteration increases locality by one, while the stopband [d, σ] maps into [−σ, σ] and remains there under subsequent stages.Source: §B.16 · p. 35
  • The marked value grows first geometrically with factor ϱ, then with factor θ up to corrections whose combined cost is less than four.The slow phase lasts until u_i reaches ∆; the fast-phase bound then applies.Source: §B.16 · p. 36
  • The scaling bound E is controlled by parameters depending only on ∆/σ, while its denominator is bounded using u_0 > 2d^2/σ.The resulting bound is stated in terms of d/σ and the exponent β.Source: §B.16 · p. 36

C The odd gadget

  • The odd gadget changes the chain boundary while preserving the interior, allowing the even-family results to transpose with two modifications.Source: C The odd gadget · p. 36

C.1 The chain and its eigenvalue equation

  • The odd chain ends on a unit bond and uses a boundary term ∆o x; its Hamiltonian has locality two greater than that of H.Source: §C.1 · p. 37
  • The chain’s characteristic equation has a parity symmetry: the branch selected through the origin is odd in x.Source: §C.1 · p. 37, §C.2 · p. 38
  • For every chain length n ≥ 1, the relevant bound is uniform on each disk of radius r < 1, so no chain length is excluded there.The zeros of 1 + M(x) lie on the unit circle.Source: §C.1 · pp. 37–38

C.2 The branch

  • The odd branch starts at order 2n + 1 with coefficient ∆o; its sparser expansion has no exceptional shortest chain.The odd family also has an additional run of zero coefficients compared with the even case.Source: §C.2 · p. 38

C.3 Coefficient bounds on the complex disc

The odd branch admits coefficient bounds on disks of radius below one when its boundary coupling is sufficiently small. The odd and even families can then be combined through triangular matching to represent a polynomial on the encoded subspace.

C.3 Coefficient bounds on the complex disc

  • The odd eigenvalue branch is analytic on |x| < r under the stated coefficient-bound condition, with its coefficients bounded as in Theorem 5.Source: §C.3 · p. 39
  • Every n ≥ 1 is admissible at every radius r < 1 when ∆o is small enough, and one bound certifies all n ≥ n0.The shortest chain in the basis sets the constraint.Source: §C.3 · p. 39
  • The odd gadget requires small ∆o; this raises the energy scale through synthesis weights, while ∆o cancels from the error bound.Source: §C.3 · p. 40

C.4 The dictionary

  • Odd-target matching remains lower triangular, and its growth and truncation factors cancel to give the stated coefficient and input-norm bounds.Source: §C.4 · p. 40
  • Combining even and odd chains with direct constant and linear terms realizes a general polynomial on the joint encoded subspace.Source: §C.4 · p. 41
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