Hermitian polynomials in the quadrature operators $x$ and $p$ are the basic ingredients of continuous-variable quantum information, and increasingly exotic ones are now routinely engineered in superconducting circuits and trapped ions. However, such a polynomial need not define unique quantum dynamics: it may define exactly one, none, or an infinite family of physically distinct ones. Such obstructions are invisible to algebraic arguments and can compromise finite-dimensional truncations, which may fail to converge or silently converge to the wrong dynamics. We develop a method, implemented as an algorithm, which decides which case occurs, and how large the family is, from the coefficients alone, with no functional-analytic input from the user. For Gaussian operators the answer is always one; for non-Gaussian ones, a regime continuous-variable quantum computing cannot do without, the other outcomes occur already for simple and physically natural Hamiltonians. We apply the algorithm to operators drawn from the recent literature.
Hermitian polynomials in the quadrature operators x and p are the basic ingredients of continuous-variable quantum information, and increasingly exotic ones are now routinely engineered in superconducting circuits and trapped ions. However, such a polynomial need not define unique quantum dynamics: it may define exactly one, none, or an infinite family of physically distinct ones. Such obstructions are invisible to algebraic arguments and can compromise finite-dimensional truncations, which may fail to converge or silently converge to the wrong dynamics. We develop a method, implemented as an algorithm, which decides which case occurs, and how large the family is, from the coefficients alone, with no functional-analytic input from the user. For Gaussian operators the answer is always one; for non-Gaussian ones, a regime continuous-variable quantum computing cannot do without, the other outcomes occur already for simple and physically natural Hamiltonians. We apply the algorithm to operators drawn from the recent literature.
Quantum computing involving physical systems with continuous degrees of freedom, such as the quantum states of light, has recently attracted significant interest. However, a well-defined quantum complexity theory for these bosonic computations over infinite-dimensional Hilbert spaces is missing. In this work, we lay foundations for such a research program. We introduce natural complexity classes and problems based on bosonic generalizations of BQP, the local Hamiltonian problem, and QMA. We uncover several relationships and subtle differences between standard Boolean classical and discrete variable quantum complexity classes and identify outstanding open problems. In particular: 1. We show that the power of quadratic (Gaussian) quantum dynamics is equivalent to the class BQL. More generally, we define classes of continuous-variable quantum polynomial time computations with a bounded probability of error based on higher-degree gates. Due to the infinite dimensional Hilbert space, it is not a priori clear whether a decidable upper bound can be obtained for these classes. We identify complete problems for these classes and demonstrate a BQP lower and EXPSPACE upper bound. We further show that the problem of computing expectation values of polynomial bosonic observables is in PSPACE. 2. We prove that the problem of deciding the boundedness of the spectrum of a bosonic Hamiltonian is co-NP-hard. Furthermore, we show that the problem of finding the minimum energy of a bosonic Hamiltonian critically depends on the non-Gaussian stellar rank of the family of energy-constrained states one optimizes over: for constant stellar rank, it is NP-complete; for polynomially-bounded rank, it is in QMA; for unbounded rank, it is undecidable.
Since the early days of quantum information science, diverse strategies have been proposed to encode qubits and implement quantum gates across various physical platforms. Modes and bosonic states offer a flexible framework for quantum information processing, from single-photon optics to superconducting circuits, combining massless and massive particles. Universal gate sets have been established for specific encodings—such as those based on Fock states or the field’s quadratures. However, a general framework clarifying how different bosonic resources—such as mode structure, particle-number statistics, Gaussian and non-Gaussian states and operations—combine to enable universality and potential quantum advantage irrespectively of the encoding remains lacking. A main difficulty lies in the coexistence of distinct paradigms that appear to combine loosely in bosonic quantum information: discrete-variable schemes—where information is encoded in qubits or effective finite-dimensional systems—and continuous-variable schemes—which rely on infinite-dimensional Hilbert spaces associated with the field’s quadratures. Despite the success of each approach, they remain seemingly incompatible in many aspects—in particular in terms of physical resource analysis for universality—due to their underlying mathematical differences. In this work, we apply a superselection-rule-compliant framework for bosonic quantum information that unifies these paradigms. This representation explicitly incorporates the phase reference and enforces total particle number conservation, allowing for a consistent treatment of bosonic quantum information protocols and resources across both discrete and continuous regimes. We derive necessary and sufficient conditions expressed in terms of Gaussian and non-Gaussian resources for universality that apply to any encoding, without resorting to specific mappings. Our results show that continuous variable encodings are not fundamentally distinct from discrete ones, but rather emerge as subspaces within a general bosonic structure. Importantly, in such a widely studied framework, non-Gaussian properties of bosonic states and operations can be washed out, providing evidence that the continuous variables regime is not the most suitable to fully assess the quantum properties of bosons.
We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra A^1, defined as the real subalgebra of the Weyl algebra A1 consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for A^1, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in A^1 generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in A^1 that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in A^1? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of A^1 that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in A^1, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.
Bosonic statistics give rise to remarkable phenomena, from the Hong-Ou-Mandel effect to Bose-Einstein condensation, with applications spanning fundamental science to quantum technologies. Modeling bosonic systems relies heavily on effective descriptions: typical examples include truncating their infinite-dimensional state space and restricting their dynamics to a simple class of Hamiltonians, such as polynomials of canonical operators, which are used to define quantum computing over bosonic modes. However, many natural bosonic Hamiltonians do not belong to this simple class, and some quantum effects harnessed by bosonic computers inherently require infinite-dimensional spaces, questioning the validity of such effective descriptions of bosonic systems. How can we trust results obtained with such simplifying assumptions to capture real effects? Driven by the increasing importance of bosonic systems for quantum technologies, we solve this outstanding problem by showing that these effective descriptions do in fact capture the relevant physics of bosonic systems. Our technical contribution is twofold: firstly, we prove that any physical, bosonic unitary evolution can be strongly approximated by a finite-dimensional unitary evolution; secondly, we show that any finite-dimensional unitary evolution can be generated exactly by a bosonic Hamiltonian that is a polynomial of canonical operators. Beyond their fundamental significance, our results have implications for classical and quantum simulations of bosonic systems, they provide universal methods for engineering bosonic quantum states and Hamiltonians, they show that polynomial Hamiltonians do generate universal gate sets for quantum computing over bosonic modes, and they lead to an infinite-dimensional Solovay--Kitaev theorem.
Classical simulability is ultimately determined by both the dynamics of a quantum system and the observables being evaluated. Lie-algebraic simulation exploits the latter to make exact polynomial-time classical simulations by propagating observables through low-dimensional invariant operator spaces. However, its conventional formulation in terms of polynomial-dimensional dynamical Lie algebras does not directly accommodate bosonic systems as their algebras are neither compact nor semisimple. In this contribution, we overcome this limitation, making bosonic systems accessible to the Lie-algebraic programme of exact polynomial-time classical simulation. We prove that expectation values, fixed-order correlation functions, including multi-time correlators and out-of-time-ordered correlators, and gradients are efficiently computable whenever their operator modules have polynomial dimension. This recovers Gaussian quantum optics and extends it to non-Gaussian input states, while identifying exact polynomial regimes of interacting non-Gaussian dynamics including bounded-photon Kerr and pair-hopping Hamiltonians and nilpotent polynomial phase dynamics. We show that unlike in the finite-dimensional spin and fermionic setting treated previously, a finite-dimensional bosonic generator algebra alone does not guarantee finite observable dynamics. We further derive a controlled perturbative hierarchy for squeezing beyond exact sector confinement and confirm the predicted error orders numerically. We also evaluate operator spreading on interacting chains of up to 400 modes and connect a topological doublon band with flux-reversed edge motion. These results provide a unified formalism for classifying, discovering, and systematically approximating tractable bosonic quantum dynamics with classical polynomial-time simulation.
Bosonic statistics give rise to remarkable phenomena, from the Hong-Ou-Mandel effect to Bose-Einstein condensation, with applications spanning fundamental science to quantum technologies. Modeling bosonic systems relies heavily on effective descriptions: typically, truncating their infinite-dimensional state space or restricting their dynamics to a simple class of Hamiltonians, such as polynomials of canonical operators. However, many natural bosonic Hamiltonians do not belong to these simple classes, and some quantum effects harnessed by bosonic computers inherently require infinite-dimensional spaces. Can we trust that results obtained with such simplifying assumptions capture real effects? We solve this outstanding problem, showing that these effective descriptions do correctly capture the physics of bosonic systems. Our technical contributions are twofold: firstly, we prove that any physical bosonic unitary evolution can be accurately approximated by a finite-dimensional unitary evolution; secondly, we show that any finite-dimensional unitary evolution can be generated exactly by a bosonic Hamiltonian that is a polynomial of canonical operators. Beyond their fundamental significance, our results have implications for classical and quantum simulations of bosonic systems, provide universal methods for engineering bosonic quantum states and Hamiltonians, show that polynomial Hamiltonians generate universal gate sets for quantum computing over bosonic modes, and lead to a bosonic Solovay-Kitaev theorem.