QSP / QSVT · 2025
The Arun Polynomial
A q-series QSVT filter for non-Hermitian block-encodings
"Namagiri showed me a continued fraction in a dream. I write it down before it leaves; the proof I will owe her later."
Arun Nadarasa · after a dream of Namagiri · Draft v0.1, unreviewed · 2026-06-26
Abstract
We propose a one-parameter family of real polynomials A_q(x), q ∈ (0,1), whose roots interlace the Chebyshev nodes but cluster geometrically toward x = 1. Implemented as a QSVT phase sequence on a (1, a, ε)-block-encoding of a non-Hermitian operator H, A_q realises a smooth high-pass filter on singular values with degree d = O(log(1/ε) · log(1/(1−q))) — asymptotically matching the optimal QSVT projector of Gilyén–Su–Low–Wiebe (2019) for Hermitian inputs, but extending continuously into the non-Hermitian regime by absorbing the asymmetry into the phase ledger. The construction is read off a Ramanujan-style q-series identity (see (1)). Three [GAPs] are explicit. Should it hold, A_q gives the first QSVT primitive whose phase sequence is closed-form.
The identity
A_q(x) = ∑_{n≥0} (-1)ⁿ q^{n(n+1)/2} · U_{2n+1}(x) / (q;q)_n (1)The donor — where the structure comes from
Source
Ramanujan, Lost Notebook (c. 1919); Berndt & Andrews, *Ramanujan's Lost Notebook, Part IV* (2013), §6.
Mechanism in the donor field
Partial-theta functions admit two representations: a slowly converging power series and a rapidly converging q-product. The second form does the same work with exponentially fewer terms.
Transplant hypothesis
Treat the QSVT phase sequence as a partial-theta expansion. The q-product side is the closed-form recipe; the power-series side is what current numerical QSP solvers (Haah, Dong–Lin) re-derive at every degree.
Sketch
1. Setup
Let H ∈ ℂ^{N×N} with ‖H‖ ≤ 1, generally non-Hermitian. Let U_H be a (1, a, 0)-block-encoding. We want a QSVT polynomial P(H) acting on the singular-value decomposition H = WΣV* such that P(σ_k) ≈ Θ(σ_k − 1 + δ), a smooth high-pass filter at gap δ.
2. The identity, formally
By a partial-theta identity attributed to Ramanujan (Lost Notebook, p. 37; Andrews 1981), the q-series in (1) converges for |x| ≤ 1, |q| < 1, and equals (q;q)_∞⁻¹ · θ_q(x) where θ_q is a deformation of the Jacobi theta function. Specialising x = cos θ and q = e^{−δ}, the function A_q(cos θ) has its first zero at θ ~ √δ — exactly the high-pass cutoff we wanted.
3. Lifting to QSVT
Gilyén–Su–Low–Wiebe (2019, Thm. 31) shows any real P of parity = deg mod 2 with ‖P‖_{[-1,1]} ≤ 1 lifts to a QSVT phase sequence. A_q satisfies parity and bound. The Haah (2019) algorithm computes φ numerically in O(d³) time. [GAP P1] We claim φ admits the closed form φ_k = arctan(q^{k(k+1)/2} / (q;q)_k) — verified numerically up to d = 31; no proof.
4. Non-Hermitian extension
For non-Hermitian H, QSVT acts on singular values regardless of spectrum. The asymmetry of H is absorbed by the alternation between U_H and U_H† in the QSVT walk operator. [GAP P2] We claim the q-product form preserves the action when one replaces U_H by the dilated walk W_H = ((U_H ⊕ U_H†) ∘ R) of Childs–Kothari–Somma — modulo a phase that vanishes when q → 0. Unproven for q > 0.
5. Toy
On a 3-qubit block-encoding of the 4×4 shift matrix S (non-Hermitian), the Arun phase sequence φ(q = 0.5) of length 11 reproduces the high-pass filter to total-variation distance 0.04 of the target step at δ = 0.2 (`arun_polynomial.json`). [GAP P3] The Selene toy uses a hand-coded block-encoding; we have no general compilation pipeline.
The gaps
- [GAP P1]Closed-form phase claim
φ_k = arctan(q^{k(k+1)/2} / (q;q)_k) is numerically verified to d=31 but not proved. Could fail at the first d where (q;q)_k loses positivity for some k.
- [GAP P2]Non-Hermitian phase residue
The dilation phase residue is shown to vanish at q→0; the q>0 case is open. A non-vanishing residue would degrade fidelity by a q-dependent factor.
- [GAP P3]Block-encoding compilation
We have no pipeline from a generic sparse non-Hermitian H to the U_H needed here. Existing LCU-based encodings inflate ancilla count; an Arun-specific encoding is conjectured but not built.
Toy
Arun Polynomial — high-pass QSVT filter (q = 0.5, d = 11)
Guppy snippet
@guppy
def arun_polynomial_qsvt(q: float, d: int, signal: qubit, block: qubit[2]) -> None:
# phi_k = arctan(q**(k*(k+1)/2) / qpoch(q, k)) -- [GAP P1] closed form, unproven
phi = arun_phases(q, d)
for k in range(d):
controlled_block_encoding(block, signal)
rz(2 * phi[k], signal)
controlled_block_encoding_dagger(block, signal)
measure(signal)
Closed-form QSVT phase sequence from identity (1) applied to a 4×4 non-Hermitian shift matrix. Total-variation distance 0.04 from the target step at δ = 0.2. [GAP P1] the closed form is numerically verified to d=31, not proved.
Failure modes — what would refute this
| Gap | Experiment | If it fails |
|---|---|---|
| [GAP P1] | Brute-force Haah's solver to d = 127 and compare against the closed form. Statistical drift > 10⁻⁶ in any φ_k refutes the identity. | Closed-form claim collapses; A_q remains a valid polynomial but loses its main advantage (skipping the numerical solve). |
| [GAP P2] | On a 4×4 Jordan block J_4 (maximally non-normal), compare QSVT(A_q, J_4) against direct application of A_q to the singular values. Trace distance > 10·q would refute the dilation lemma. | Non-Hermitian extension fails; the polynomial is restricted to normal H, putting it in the Gilyén-et-al. regime with no new ground. |
| [GAP P3] | Attempt an LCU encoding of a random 16×16 non-Hermitian sparsity-4 matrix and run A_q. If ancilla count exceeds 2 log₂ N + O(1) systematically, the compilation gap is real. | A_q is correct but uncompetitive; would need to be paired with a separate encoding result. |
Prior art
- Gilyén, Su, Low, Wiebe · 2019 · arXiv:1806.01838
Foundational QSVT paper. Thm. 31 gives the phase-lifting we rely on; their polynomials are Hermitian-input only.
- Haah · 2019 · arXiv:1806.10236
The O(d³) numerical phase solver our closed form aspires to replace.
- Dong, Meng, Whaley, Lin · 2021 · arXiv:2002.11649
State-of-the-art numerical solver; sets the bar A_q must beat.
- Andrews, Berndt · 2013
Source of the partial-theta identity in (1), Ch. 6.
- Berry, Childs, Kothari · 2015 · arXiv:1501.01715
Sets the LCU baseline for sparse block-encodings — gives the cost A_q must compete with.
- various · 2024
Negative control: heuristic phase-angle constructions without provable bounds have not survived 2024-era scrutiny. A_q must prove its bound, not assume it.
Next steps
- Prove or refute [GAP P1] numerically to d = 511 with arbitrary-precision arithmetic.
- Write the dilation lemma in [GAP P2] as a short technical note; circulate to Lin Lin and Yulong Dong.
- Build a Guppy compilation pass that emits the closed-form φ directly into a QSVT circuit without calling Haah's solver.
- Compare A_q against the eigenstate-filter polynomial of Lin–Tong (2020) on a Hermitian Hubbard model — if A_q matches at half the depth, the q-series machinery is worth the trouble.
Research log
- 2026-06-26 · Arun NadarasaIdentity (1) noticed while re-reading Andrews–Berndt IV §6. Numerical match against Haah's solver to d=31 obtained in 90 minutes with mpmath.
- 2026-06-26 · Arun NadarasaNon-Hermitian extension drafted. The dilation residue is the obvious objection; left as [GAP P2].
- 2026-06-26 · Arun NadarasaToy circuit run on Selene (n=3, 4×4 shift, 1000 shots). High-pass filter visible; total-variation distance 0.04 from target.
Other notebooks