Universality of every fixed non-Gaussian polynomial generator
- Field
- Topics
Problem
Does every fixed non-Gaussian polynomial Hamiltonian become universal when all Gaussian controls are available? Fix \(n\geq1\). On \(L^2(\mathbb R^n)\), let \(q_j,p_j\) satisfy \([q_j,p_k]=i\delta_{jk}\). Let \(H_*\) be any real Weyl-ordered polynomial in these operators of degree greater than two. Assume that \(H_*\) is essentially self-adjoint on the Schwartz space \(\mathcal S(\mathbb R^n)\) of smooth, rapidly decreasing functions. Use its unique self-adjoint closure to define \(e^{-itH_*}\). Allowed gates are \(e^{-itH_*}\) for arbitrary real \(t\), together with all Gaussian unitaries. Gaussian unitaries are generated by real polynomials of degree at most two in the canonical operators.
Let \(U\) be any unitary satisfying \(U\mathcal S(\mathbb R^n)\subseteq\mathcal S(\mathbb R^n)\). Fix finite \(E\geq0\) and \(\varepsilon>0\). Does some finite product \(V\) of allowed gates satisfy
In Eq. (1), \(\mathcal U(\rho)=U\rho U^\dagger\), \(\mathcal V(\rho)=V\rho V^\dagger\), and \(R\) is an arbitrary reference system. The fixed \(H_*\) must work for every target \(U\), energy \(E\), and accuracy \(\varepsilon\).
Source
The Discussion of Arzani, Booth, and Chabaud explicitly identifies the missing rigorous equivalence between Gaussian controls plus any fixed higher-degree polynomial and arbitrary polynomial evolutions [ABC25]. This formulation specifies the operator domain and energy-constrained approximation target.
Progress
Arzani, Booth, and Chabaud prove universality when the polynomial Hamiltonian can be chosen for each target. Their Theorem 2 realizes arbitrary finite Fock blocks; Theorem 3 supplies unitary approximation [ABC25]. A polynomial in total photon number can be added outside the retained block to obtain a self-adjoint realization. These generators depend on the desired finite-dimensional action.
The same paper’s Theorem 4 gives a Solovay–Kitaev result for finite-block gate sets constructed from Theorem 2. Its hypotheses do not cover every preassigned higher-degree \(H_*\). The Discussion expressly leaves the fixed-generator equivalence open [ABC25].
Comment
The gap is a proof of the approximation in Eq. (1) for every essentially self-adjoint polynomial \(H_*\) of degree greater than two, or a counterexample. Formal Lie-algebra generation alone does not justify products of unbounded evolutions or their convergence. Arbitrary polynomial universality is therefore a related solved problem with a different resource model.