[polyopt] Killer Queen: certify truncated-boson bulk-gap bounds

tracks/polyopt/solutions/killer-queen-92/.gitignore

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+# Research inputs are cited by URL/revision and are not redistributed.
+ref/*.pdf

tracks/polyopt/solutions/killer-queen-92/ALGORITHM.md

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+# Algorithm note: certified bulk-gap upper bounds
+
+**Scope.** This note explains the algorithm requested in
+[Quantum Harness issue #92](https://github.com/QuantumBFS/quantum.harness/issues/92).
+It is not the usual moment-SDP lower bound on a finite Hamiltonian's ground-state
+energy. It is a feasibility hierarchy for excluding a hypothesized gap of an
+infinite lattice directly in the thermodynamic limit.
+
+The primary source is Xu *et al.*, [*The bulk spectral gap is semi-decidable: a
+convergent family of certified upper bounds*](https://arxiv.org/html/2606.03836),
+especially Supplementary §2. The authors' proof-of-principle Julia code is
+[`wangjie212/SpectralGap`](https://github.com/wangjie212/SpectralGap). A public
+copy of Jie Wang's 28 July 2026 talk slides was not found in searches of his
+website, GitHub, or the QuantumBFS repository as of 28 July 2026. The closest
+short software tutorial found is the
+[NCTSSOS state-polynomial example](https://wangjie212.github.io/NCTSSOS/dev/state/).
+
+## 1. The distinction that matters
+
+An ordinary noncommutative ground-energy relaxation has the schematic form
+
+\[
+  \min_{\mathcal L}\;\mathcal L(H),\qquad
+  \mathcal L(1)=1,\qquad M_d(\mathcal L)\succeq0.
+\]
+
+Every physical state supplies feasible moments, so the minimum is a **lower
+bound** on the ground-state energy of the problem being modeled. This is a
+linear optimization over moments.
+
+The bulk-gap algorithm asks a different question:
+
+> For a fixed number \(\gamma\geq0\), can there exist an infinite-volume KMS
+> ground state whose locally non-degenerate bulk gap is at least \(\gamma\)?
+
+At each finite relaxation level \((L,d)\), the answer is relaxed to an SDP
+feasibility test. If that relaxation is **infeasible**, the hypothesized state
+cannot exist, so \(\gamma\) is rigorously excluded. Define
+
+\[
+  \Gamma_{L,d}=\sup\{\gamma:\text{the level-}(L,d)\text{ relaxation is feasible}\}.
+\]
+
+Increasing the local window \(L\) or polynomial degree \(d\) adds necessary
+conditions. Hence the feasible sets shrink and
+
+\[
+  \Gamma_{L,d}\searrow\Delta_{\mathrm{bulk}}.
+\]
+
+For a fixed \(\gamma\), minimizing and maximizing a local observable over the
+same relaxation gives outer bounds which tighten monotonically. This is the
+content of [Theorem 2.5](https://arxiv.org/html/2606.03836#S2.SS2.SSS0.Px1) of
+the paper.
+
+## 2. Why the construction is already in the thermodynamic limit
+
+Let \(\mathcal A_{\mathrm{loc}}\) be the algebra of operators with finite support
+on the infinite graph. The formal sum \(H=\sum_X h_X\) is not itself a bounded
+local operator, but for every local excitation \(a\),
+
+\[
+  [H,a]=\sum_{X:X\cap\operatorname{supp}(a)\ne\varnothing}[h_X,a]
+\]
+
+is a finite, well-defined sum. A state \(\omega\) is a KMS ground state when no
+local excitation lowers its energy:
+
+\[
+  \omega\!\left(a^\dagger[H,a]\right)\geq0
+  \quad\text{for all }a\in\mathcal A_{\mathrm{loc}}.
+\]
+
+The state has a locally non-degenerate bulk gap at least \(\gamma\) when
+
+\[
+  \omega\!\left(a^\dagger[H,a]\right)
+  \geq \gamma\left(\omega(a^\dagger a)-|\omega(a)|^2\right)
+  \quad\text{for every local }a. \tag{1}
+\]
+
+The left side is the excitation energy. The right side is \(\gamma\) times the
+squared component of \(a|\Omega_\omega\rangle\) orthogonal to the ground-state
+vector in the GNS representation.
+
+The finite region \(\Lambda(L)\) is therefore a **local test window**, not a
+finite sample whose spectrum is extrapolated. No boundary condition is chosen.
+For nearest-neighbor interactions, the tested excitations must lie at least one
+graph layer inside the Hamiltonian window: use \(a\) supported on
+\(\Lambda(L-1)\) while the commutator is evaluated with \(H^{\Lambda(L)}\).
+This buffer makes the commutator identical to the infinite-lattice derivation.
+
+This is the logical reason that finite-level infeasibility implies a
+thermodynamic-limit statement. Open-boundary exact diagonalization does not
+have that implication and will be used in this project only as a baseline.
+
+## 3. Why an ordinary moment matrix is insufficient
+
+Equation (1) contains \(|\omega(a)|^2\). The map \(\omega\mapsto|\omega(a)|^2\)
+is nonlinear, and the set of states satisfying the assumed-gap condition is in
+general nonconvex. Merely adding (1) to an ordinary NPA/moment relaxation does
+not give an SDP.
+
+State-polynomial optimization resolves this by introducing a commuting formal
+symbol \(\varsigma(w)\) for every operator word \(w\). It represents
+\(\omega(w)\), while products such as
+\(\varsigma(a^\dagger)\varsigma(a)\) are now ordinary commutative monomials.
+A second linear functional \(\mathcal L\) acts on these state polynomials.
+Nonlinear expressions in \(\omega\) therefore become linear entries in lifted
+pseudo-moments of \(\mathcal L\).
+
+For state-polynomial monomials \(s,t\) of degree at most \(d\), the positivity
+moment matrix is
+
+\[
+  [M_d(\mathcal L)]_{s,t}
+  =\mathcal L\!\left(\varsigma(s^\dagger t)\right). \tag{2}
+\]
+
+The assumed-gap localizing matrix is
+
+\[
+\begin{aligned}
+  [M^{\mathrm{gap},\gamma}(\mathcal L)]_{s,t}
+  =\mathcal L\!\Big[&\tfrac12\varsigma\!\left(
+  s^\dagger[H^{\Lambda(L)},t]
+  -[H^{\Lambda(L)},s^\dagger]t\right)\\
+  &-\gamma\big(\varsigma(s^\dagger t)
+  -\varsigma(s^\dagger)\varsigma(t)\big)\Big]. \tag{3}
+\end{aligned}
+\]
+
+The symmetrization makes the matrix Hermitian. At a **fixed** \(\gamma\), every
+entry in (2) and (3) is affine in the lifted pseudo-moments, so the constraints
+are SDPs. Treating \(\gamma\) as a simultaneous decision variable would create
+bilinear products; in practice one solves a sequence of fixed-\(\gamma\)
+feasibility problems, normally by bisection.
+
+The complete level-\((L,d)\) test is
+
+\[
+\begin{array}{ll}
+\text{find} & \mathcal L\\
+\text{such that}
+ & \mathcal L(1)=1,\\
+ & M_d(\mathcal L)\succeq0,\\
+ & \mathcal L\!\left(\varsigma([H^{\Lambda(L)},w])\right)=0
+   \quad\text{for all allowed interior }w,\\
+ & M^{\mathrm{gap},\gamma}(\mathcal L)\succeq0.
+\end{array} \tag{4}
+\]
+
+The stationarity equations are essential. Positivity plus the gap localizer at
+\(\gamma=0\) expresses the KMS ground-state condition; for \(\gamma>0\), the
+gap localizer imposes the stronger energy-to-variance ratio.
+
+## 4. Completeness and the direction of the certificate
+
+Every genuine infinite-volume state with bulk gap at least \(\gamma\) restricts
+to a feasible point at every \((L,d)\). Therefore:
+
+- **feasible** at one finite level means only “not excluded yet”;
+- **infeasible** at one finite level excludes that \(\gamma\);
+- maximizing the last feasible \(\gamma\) gives an **upper**, not lower, gap
+  bound;
+- increasing \(L,d\) can only improve the upper bound downward.
+
+Conversely, if every level remains feasible, compactness and an Archimedean
+state-polynomial representation theorem construct an infinite-volume KMS state
+obeying (1). This yields convergence, not merely a collection of necessary
+tests. The finite-dimensional local matrix algebras are crucial: their
+generators are bounded, which supplies the Archimedean bounds used in the
+proof.
+
+## 5. Correct algebra for occupation-truncated bosons
+
+For cutoff \(n_{\max}\), one site is the matrix algebra
+\(M_D(\mathbb C)\), \(D=n_{\max}+1\). With matrix units
+\(E_{rs}=|r\rangle\langle s|\),
+
+\[
+  b=\sum_{r=1}^{n_{\max}}\sqrt r\,E_{r-1,r},\qquad
+  b^\dagger=\sum_{r=0}^{n_{\max}-1}\sqrt{r+1}\,E_{r+1,r},\qquad
+  n=\sum_{r=0}^{n_{\max}}rE_{rr}. \tag{5}
+\]
+
+The local multiplication and adjoint rules are exact:
+
+\[
+  E_{rs}E_{uv}=\delta_{su}E_{rv},\qquad
+  E_{rs}^\dagger=E_{sr},\qquad
+  \sum_rE_{rr}=I. \tag{6}
+\]
+
+Operators on distinct sites commute. A particularly important boundary
+identity is
+
+\[
+  [b,b^\dagger]=I-D\,E_{n_{\max},n_{\max}}, \tag{7}
+\]
+
+not the infinite-dimensional canonical commutation relation
+\([b,b^\dagger]=I\). Consequently, selecting a package's built-in “bosonic/CCR”
+algebra would model the wrong problem. The safe implementation choices are:
+
+1. canonicalize words with the finite matrix-unit rules (6); or
+2. use a complete Hermitian basis of \(M_D(\mathbb C)\) and its multiplication
+   table.
+
+The second choice also lowers the apparent Hamiltonian degree. In a local
+matrix basis, on-site terms are linear and edge hopping terms are degree two;
+leaving \(n(n-1)\) as a raw word in \(b,b^\dagger\) makes it look degree four
+and needlessly enlarges the relaxation.
+
+For the issue's Bose–Hubbard model,
+
+\[
+  H=-t\sum_{\langle i,j\rangle}(b_i^\dagger b_j+b_j^\dagger b_i)
+  +\frac12\sum_i n_i(n_i-1)-\mu\sum_i n_i, \tag{8}
+\]
+
+the three graphs have coordination \(z=3\) for \(\{8,3\}\) and \(z=4\) for
+\(\{12,4\}\) and the line graph \(L(G_{8,3})\).
+
+## 6. Symmetry: useful, but it changes the quantified state set
+
+The Hamiltonian is invariant under global \(U(1)\) particle-number rotations.
+Words carry charge “number of creations minus annihilations.” Imposing an
+invariant state sets nonzero-charge moments to zero and block-diagonalizes the
+moment matrices by charge. This can be a large reduction.
+
+It must be reported as a **symmetry-restricted certificate**. The result then
+excludes a gap only among \(U(1)\)-invariant KMS ground states; it is not an
+unrestricted statement if spontaneous symmetry breaking is allowed. We will
+keep unrestricted and symmetry-restricted results separate.
+
+Graph automorphisms fixing the reference site can likewise identify moment
+orbits without changing the physical question when they are used as an exact
+block decomposition. Requiring the state itself to be invariant is a stronger
+assumption and must be labeled.
+
+## 7. Concrete algorithm for issue #92
+
+For a graph \(G\), cutoff \(n_{\max}\), and relaxation level \((L,d)\):
+
+1. Build the rooted ball \(\Lambda(L)\) and its induced edges. For a
+   nearest-neighbor Hamiltonian reserve the outer shell as a commutator buffer.
+2. Construct the finite local algebra using (6), then assemble
+   \(H^{\Lambda(L)}\) from (8).
+3. Enumerate a selected basis of operator and state-polynomial monomials up to
+   total degree \(d\), canonicalizing after every multiplication.
+4. Optionally orbit-reduce by rooted graph symmetry and block by \(U(1)\)
+   charge; record whether this restricts the state.
+5. Create one scalar SDP variable for every independent lifted pseudo-moment.
+6. Assemble normalization, the positivity matrix (2), stationarity equations,
+   and the gap matrix (3).
+7. At fixed \(\gamma\), solve (4). Check primal/dual residuals and the minimum
+   eigenvalues of returned PSD blocks.
+8. Bracket and bisect \(\gamma\). An infeasibility certificate above the final
+   feasible endpoint gives \(\Gamma_{L,d}\).
+9. At fixed feasible \(\gamma\), minimize and maximize
+   \(\rho_0=\omega(n_0)\),
+   \(F_0=\omega((n_0-1)^2)\), and
+   \(K_0=z^{-1}\sum_{j\sim0}\omega(b_0^\dagger b_j+b_j^\dagger b_0)\)
+   over the same SDP.
+10. Repeat in nested \(L,d\). Gap upper bounds must move downward, observable
+    lower bounds upward, and observable upper bounds downward.
+
+Pseudocode:
+
+```text
+for geometry, cutoff, (t, mu), symmetry_mode:
+    algebra = truncated_matrix_algebra(cutoff)
+    for (L, d) in nested_relaxations:
+        patch = rooted_ball(geometry, L, buffer=1)
+        model = bose_hubbard(patch, algebra, U=1, t=t, mu=mu)
+        sdp_template = state_polynomial_relaxation(model, d, symmetry_mode)
+
+        lo, hi = bracket_gap(sdp_template)
+        while hi - lo > gamma_tolerance:
+            gamma = (lo + hi) / 2
+            status, certificate = solve_feasibility(sdp_template, gamma)
+            if reliably_feasible(status): lo = gamma
+            elif reliably_infeasible(status, certificate): hi = gamma
+            else: mark_unresolved_and_stop()
+
+        report Gamma_Ld in [lo, hi]
+        for gamma in selected_assumed_gaps_below_hi:
+            optimize rho0, F0, K0 in both directions
+```
+
+## 8. Validation ladder before claiming new lattice results
+
+1. **Algebra tests.** Verify (5)–(7) exactly for every cutoff and verify
+   different sites commute.
+2. **Atomic limit.** At \(t=0,U=1,\mu=0.5\), recover
+   \(\Delta_{\mathrm{bulk}}=0.5\), \(\rho_0=1\), \(F_0=K_0=0\) for
+   \(n_{\max}\ge2\). The result must not depend on the graph.
+3. **Reference implementation.** Reproduce one small transverse-field Ising
+   value from the released `SpectralGap.jl` code before trusting an algebra
+   extension.
+4. **Tiny finite-patch cross-check.** Use exact diagonalization only to catch
+   signs, factors, and observable definitions. Do not call its gap a bulk
+   certificate.
+5. **Nestedness.** Check the mathematically required monotonic directions in
+   \(L,d\).
+6. **Numerical certificate.** A floating-point status string alone is not a
+   rigorous proof. Preserve residuals and a dual ray/certificate; use verified
+   or rational post-processing for final claims.
+
+The paper itself notes that its reported numerical SDPs used floating-point
+arithmetic without accounting for numerical errors. Thus “certified” describes
+the mathematical implication of exact SDP infeasibility; a machine-checkable
+numerical certificate requires an additional verification step.
+
+## 9. Software conclusion before implementation
+
+- `SpectralGap.jl` implements this specific hierarchy and is therefore the
+  closest reference, but its public API is specialized to Pauli Ising and
+  Heisenberg examples and uses JuMP with Mosek. It is research code, not a
+  general truncated-boson package.
+- NCTSSOS and its successor NCTSSoS.jl support state-polynomial optimization,
+  but their built-in bosonic algebra is CCR. They are useful SDP generators only
+  if the finite matrix algebra is encoded correctly.
+- ncpol2sdpa, NCAlgebra, and generic CVXPY/JuMP tools can represent ordinary
+  noncommutative or custom SDPs, but none supplies issue #92's complete
+  thermodynamic bulk-gap construction as a ready-made function.
+- `hypertiling` is a suitable mature graph generator for the regular
+  \(\{p,q\}\) tilings; its output still needs rooted-ball canonicalization and
+  the line-graph transformation.
+
+The practical research route is therefore to preserve the paper's
+state-polynomial hierarchy and replace only its Pauli word algebra with an
+exact finite-matrix-algebra backend. Finite-patch diagonalization is a valuable
+independent baseline, not a substitute for that work.
+
+## Sources
+
+1. X. Xu *et al.*, [bulk-gap paper and Supplementary
+   Information](https://arxiv.org/html/2606.03836), 2026.
+2. I. Klep *et al.*, [*State polynomials: positivity, optimization and nonlinear
+   Bell inequalities*](https://arxiv.org/abs/2301.12513), *Mathematical
+   Programming* 2024.
+3. J. Wang, [`SpectralGap.jl` reference code](https://github.com/wangjie212/SpectralGap).
+4. NCTSSOS, [state-polynomial optimization
+   tutorial](https://wangjie212.github.io/NCTSSOS/dev/state/).
+5. NCTSSoS.jl, [current package repository](https://github.com/QuantumSOS/NCTSSoS.jl).
+6. M. Schrauth *et al.*, [`hypertiling` paper](https://arxiv.org/abs/2309.10844)
+   and [package](https://pypi.org/project/hypertiling/).