Trace-norm universality of the Lloyd–Braunstein polynomial-Hamiltonian gate set for continuous-variable quantum computation, and a non-Gaussian Solovay–Kitaev theorem

The standard model of continuous-variable (CV) quantum computation, introduced by Lloyd and Braunstein, uses gates generated by Hamiltonians that are polynomial functions of the position and momentum operators of bosonic modes. Although this gate set is routinely called 'universal' in the literature, it is not known whether it is universal even in the very weak controllability sense that any state can be reached from any other to arbitrary precision in trace norm — already for a single mode. The problem asks to settle this: characterize the trace-norm closure of the set of states reachable from the vacuum under Gaussian gates plus a single non-Gaussian polynomial gate such as the cubic phase gate, and decide whether that closure is the full pure state space; and, as the quantitative efficiency refinement on the same gate set, to prove or refute a Solovay–Kitaev-type theorem giving efficient gate decompositions of non-Gaussian unitaries under the energy-constrained diamond norm, generalizing the known Gaussian-only result.

Background

A single bosonic mode (e.g. one mode of the quantum electromagnetic field) is described on the infinite-dimensional Hilbert space of square-integrable functions, spanned by the number states $|0\rangle, |1\rangle, |2\rangle, \dots$, with quadrature (position and momentum) operators $\hat{X}$ and $\hat{P}$ satisfying the canonical commutation relation $[\hat{X}, \hat{P}] = i$. Unlike the finite-dimensional registers of discrete-variable (DV) quantum computing, these operators are unbounded, which is the source of the analytical subtleties in this problem.

A unitary gate $e^{itH}$ is called Gaussian if $H$ is a polynomial of degree at most two in $\hat{X}$ and $\hat{P}$ (displacements, rotations, shear, and squeezing); Gaussian operations map Gaussian states to Gaussian states and are efficiently classically simulable. The CV analogue of the standard universal gate model, introduced by Lloyd and Braunstein (Physical Review Letters 82, 1784 (1999)), augments the Gaussian gates with any one gate generated by a Hamiltonian of degree three or more — the canonical example being the cubic phase gate $e^{it\hat{X}^3}$. By commutator-closure (Lie-algebraic) arguments, Lloyd and Braunstein argued that this set can simulate the evolution generated by any polynomial Hamiltonian, and subsequent work developed exact and approximate gate decompositions along these lines (Sefi and van Loock, Physical Review Letters 107, 170501 (2011); Kalajdzievski and Quesada, Quantum 5, 394 (2021)). However, these are formal approximation arguments on the level of generators, not theorems about what states can actually be reached: Wu, Tarn, and Li (Physical Review A 73, 012719 (2006)) made the statement precise only in the sense of smooth controllability on the common invariant analytic domain of the unbounded generators, and Vilela Mendes and Man'ko (Journal of Physics A 44, 135302 (2011)) exhibited a different infinite-dimensional gate set that does have the reachability property, showing the question is nonvacuous.

The gap was highlighted by Chabaud and Mehraban in 'Holomorphic representation of quantum computations' (Quantum 6, 831 (2022)), a study of bosonic quantum computation in the Segal–Bargmann holomorphic representation, in which a quantum state is represented by an entire holomorphic function whose zeros encode non-Gaussianity (the number of zeros is the stellar rank, a monotone under Gaussian operations). That paper notes that although the polynomial gate set 'is frequently referred to as universal in the literature, it is not known whether these gates actually form a universal gate set, even in the very weak controllability sense that any state can be reached up to arbitrary precision from any other state using sequences of these gates', and poses as an open problem to characterize the trace-norm closure of the states reachable from the vacuum — already in the single-mode setting. Here the trace norm $\|\rho - \sigma\|_1 = \mathrm{Tr}\sqrt{(\rho-\sigma)^\dagger(\rho-\sigma)}$ is the operational distance for state discrimination, and universality in this sense is exactly the hypothesis underlying any claim that a CV gate model can prepare arbitrary states.

Two companion notions complete the picture. First, in DV quantum computing the Solovay–Kitaev theorem (Dawson and Nielsen, arXiv:quant-ph/0505030 (2005)) states that any gate in a universal gate set can approximate any target unitary to precision $\epsilon$ using only polylogarithmically many gates in $1/\epsilon$, so the computational power is independent of the chosen universal set; in infinite dimension no single norm makes all unitaries close, but Becker, Datta, Lami, and Rouzé (Physical Review Letters 126, 190504 (2021)) proved a Gaussian Solovay–Kitaev theorem under the energy-constrained diamond norm (the diamond norm of a channel, restricted to input states of bounded mean energy, which is the physically meaningful topology for oscillator computations). Whether this extends to gate sets containing non-Gaussian gates is open. Second, the same holomorphic-representation paper and its successor literature connect these questions to the foundations of a CV quantum complexity theory: the recently defined complexity classes for bosonic computations (with BQP lower and PSPACE/EXPSPACE upper bounds — BQP being bounded-error quantum polynomial time, the standard class of decision problems efficiently solvable by a quantum computer, and PSPACE/EXPSPACE the classical classes solvable with polynomial/exponential memory) presuppose an understanding of what polynomial-gate circuits can reach and how efficiently they can be compiled.

Problem Statement

Fix a single bosonic mode with quadrature operators $\hat{X}$, $\hat{P}$ on the Hilbert space of square-integrable functions. Let $\mathcal{G}$ be the Lloyd–Braunstein gate set consisting of all Gaussian unitaries $e^{itH}$ with $H$ a polynomial of degree at most two in $\hat{X}$ and $\hat{P}$, together with one non-Gaussian gate generated by a Hamiltonian of degree three or more, e.g. the cubic phase gate $e^{it\hat{X}^3}$; let $\mathcal{R}$ be the orbit of the vacuum state $|0\rangle$ under $\mathcal{G}$, i.e. the set of states obtainable from $|0\rangle$ by finite sequences of these gates, with all products defined on the common invariant analytic domain of the generators. Since the gates are unitary, $\mathcal{R}$ consists of pure states; note also that the set of pure states is closed in trace norm, so the universality question below concerns pure states.

(a) Characterize the trace-norm closure $\overline{\mathcal{R}}$ of the reachable states. In particular, decide whether $\overline{\mathcal{R}}$ is the set of all pure normalizable states — universality in the weak controllability sense that any pure state can be reached up to arbitrary trace-norm precision from any other pure state (equivalently, from the vacuum) — or exhibit a normalizable pure state at strictly positive trace distance from $\overline{\mathcal{R}}$, together with the invariant or obstruction that separates it. The core question is posed for a single mode with the canonical cubic phase generator; multimode settings and other non-Gaussian generators of degree three or higher are generalizations beyond this single-mode core and are not required.

(b) As the quantitative efficiency refinement on the same gate set: prove or refute a continuous-variable Solovay–Kitaev theorem for non-Gaussian gate sets — that arbitrary polynomial-Hamiltonian unitaries can be approximated to precision $\epsilon$ under the energy-constrained diamond norm using gate sequences of length polylogarithmic in $1/\epsilon$ — generalizing the Gaussian-only theorem of Becker, Datta, Lami, and Rouzé.

Parts (a) and (b) are independent, separately refutable deliverables of one scientific objective: what the standard CV gate set can actually reach and how efficiently it can be compiled, i.e. whether the 'universal' label attached to the Lloyd–Braunstein model is justified in the operational sense presupposed by continuous-variable quantum complexity theory. A complete resolution requires an answer to both (a) and (b); an answer to one of them does not resolve the other. An answer to (a) may be a proof of density, an explicit characterization of $\overline{\mathcal{R}}$, or a separating invariant; an answer to (b) is a quantitative approximation theorem or a counterexample to efficient approximation.

References

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