Universality of arbitrary self-adjoint polynomial Hamiltonians
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Problem
Can every physical bosonic unitary be approximated by a single self-adjoint polynomial Hamiltonian evolution at any finite input energy? Fix \(n\geq1\) and a unitary \(U\) on \(L^2(\mathbb R^n)\) with \(U\mathcal S(\mathbb R^n)\subseteq\mathcal S(\mathbb R^n)\). Here \(\mathcal S\) is the Schwartz space of smooth, rapidly decreasing functions. The canonical operators obey \([q_j,p_k]=i\delta_{jk}\). Define the total photon number and the energy-constrained channel distance by
In Eq. (1), \(\mathcal U(\rho)=U\rho U^\dagger\) and \(R\) is any auxiliary reference system. For every finite \(E\geq0\) and \(\varepsilon>0\), does there exist a real Weyl-ordered polynomial \(P(q_1,p_1,\ldots,q_n,p_n)\) with a self-adjoint realization such that
The polynomial and its degree in Eq. (2) may depend on \(U,E,\varepsilon\).
Source
Arzani, Booth, and Chabaud establish polynomial universality in Theorems 1–3, with the finite-mode finite-block construction in arXiv Appendix B [ABC25]. The self-adjoint realization required here is supplied by the explicit completion below. That completion is an editorial deduction, not a claim that the paper proves self-adjointness of its unmodified polynomial.
Progress
Theorem 2 constructs a symmetric polynomial realizing any Hermitian matrix on a finite Fock block. The block is invariant. Theorems 1 and 3 combine finite-dimensional unitary approximation with this construction [ABC25].
To complete the domain argument, let \(B\) be a symmetric polynomial of degree \(d\) realizing a chosen finite block. Let \(K\) bound the total photon numbers in that block. On the finite Fock span, \(\|B\psi\|\leq C\|(N+1)^{d/2}\psi\|\). Choose an integer \(r\) with \(2r(K+1)>d/2\) and set
\begin{equation} Q(N)=\left[\prod_{j=0}^{K}(N-j)\right]^{2r}, \qquad P=B+Q(N). \tag{3} \end{equation}The operator \(Q(N)\) in Eq. (3) is self-adjoint on its spectral domain. It vanishes on the chosen block and dominates \((N+1)^{d/2}\) at large photon number. Thus \(B\) has relative bound zero with respect to \(Q(N)\). The Kato–Rellich theorem gives a self-adjoint closure of \(P\) with the same finite block [Tes09].
For a cutoff at total photon number \(L\), the discarded input probability is at most \(E/(L+1)\). This bound also holds with a reference system. Finite-dimensional approximation of the images of the retained basis vectors gives an approximating unitary on a larger finite Fock block. The completion in Eq. (3) realizes that block exactly. Taking \(L\) and then the output block large enough proves Eq. (2) for every finite number of modes.
Comment
The finite-block universality and approximation results are peer-reviewed. The self-adjoint completion above applies the standard Kato–Rellich theorem to remove the complement-domain issue. It can increase the polynomial degree, so no degree-\(3d\) claim is made for the completed Hamiltonian. This result permits a different polynomial for each target and accuracy. It does not prove universality of an arbitrary fixed non-Gaussian generator with Gaussian controls.
References
- [ABC25]
- F. Arzani, R. I. Booth, and U. Chabaud, "Effective Descriptions of Bosonic Systems Can Be Considered Complete," Nature Communications 16, 9744 (2025).DOIarXiv
- [Tes09]
- G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, Graduate Studies in Mathematics 99, American Mathematical Society (2009), Theorem 6.4, p. 135. Author-hosted full text.link