Bosonic quantum systems are among the leading architectures for quantum information processing, offering continuous-variable degrees of freedom with strong error-correction capabilities. However, standard bosonic quantum computation models such as the Lloyd-Braunstein [Lloyd and Braunstein, 1999] and hybrid oscillator-qubit models [Brenner, Dias, and Koenig, 2025; Liu et al., 2026] permit dramatic energy growth, leading to unphysical computational power and the breakdown of fundamental algorithmic tools such as universal and efficient compilation [Brenner et al., 2026; Rudolph et al., 2025]. To address this, we introduce a new model of bosonic quantum computation, Bosonic Energy-Preserving Quantum Computation (BEQC), in which energy is treated as a computational resource. Namely, energy is supplied solely through input coherent states, and all gates are generated by energy-preserving Hamiltonians. Thus, by construction, dramatic energy growth is impossible, making the model physically grounded. We next show that BEQC is a computationally robust and universal model in many respects, including: (1. Computational power) BEQC efficiently simulates all polynomial-energy computations in existing models, and exactly recovers BQP in the polynomial energy setting. It further admits several complexity-theoretic upper bounds when varying the energy, precision, and space parameters of the model. (2. Universal gate sets and state synthesis) BEQC has natural universal gate sets based on linear optics and Kerr interactions. In particular, we obtain a Solovay-Kitaev theorem which circumvents previous no-go results. We give various applications, including (a) a protocol for engineering GKP states with rigorous preparation guarantees, (b) Fock state preparation to exponential precision, and (c) native Fock space simulation of any qubit-based unitary.Bosonic quantum systems are among the leading architectures for quantum information processing, offering continuous-variable degrees of freedom with strong error-correction capabilities. However, standard bosonic quantum computation models such as the Lloyd-Braunstein [Lloyd and Braunstein, 1999] and hybrid oscillator-qubit models [Brenner, Dias, and Koenig, 2025; Liu et al., 2026] permit dramatic energy growth, leading to unphysical computational power and the breakdown of fundamental algorithmic tools such as universal and efficient compilation [Brenner et al., 2026; Rudolph et al., 2025]. To address this, we introduce a new model of bosonic quantum computation, Bosonic Energy-Preserving Quantum Computation (BEQC), in which energy is treated as a computational resource. Namely, energy is supplied solely through input coherent states, and all gates are generated by energy-preserving Hamiltonians. Thus, by construction, dramatic energy growth is impossible, making the model physically grounded. We next show that BEQC is a computationally robust and universal model in many respects, including: (1. Computational power) BEQC efficiently simulates all polynomial-energy computations in existing models, and exactly recovers BQP in the polynomial energy setting. It further admits several complexity-theoretic upper bounds when varying the energy, precision, and space parameters of the model. (2. Universal gate sets and state synthesis) BEQC has natural universal gate sets based on linear optics and Kerr interactions. In particular, we obtain a Solovay-Kitaev theorem which circumvents previous no-go results. We give various applications, including (a) a protocol for engineering GKP states with rigorous preparation guarantees, (b) Fock state preparation to exponential precision, and (c) native Fock space simulation of any qubit-based unitary.

quant-pharXiv:2609.40226Oct 1, 20260 citations

Effective descriptions of bosonic systems can be considered complete

Francesco Arzani, Robert I. Booth, Ulysse Chabaud

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.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.

Nature CommunicationsNov 6, 20254 citations

Exponentially-improved effective descriptions of physical bosonic systems

Varun Upreti, Nicolás Quesada, Ulysse Chabaud

The effective description of a bosonic quantum system identifies the minimum finite dimension required to capture its essential dynamics. This effective dimension plays an important role in the complexity of classical and quantum algorithms for learning and simulating bosonic systems. While generic bosonic states require a dimension scaling as $1/ε^2$ for a precision of approximation $ε$ , here we identify a natural energy condition which allows us to improve this scaling exponentially to $\log(1/ε)$ . We then prove that most bosonic quantum states satisfy this condition, and in particular those produced by combining Gaussian dynamics with generic energy-preserving dynamics, which include the output states of universal bosonic quantum circuits. We apply this finding to enhance learning algorithms for bosonic quantum states and we further obtain new classical simulation algorithms for a large class of bosonic systems. Finally, using efficient decompositions of Kerr gates as sums of Gaussian gates, we significantly refine these classical simulation algorithms for universal bosonic quantum circuits. Our results demonstrate that physical bosonic systems are significantly more well-behaved than previously assumed, allowing for efficient descriptions even at high precision.The effective description of a bosonic quantum system identifies the minimum finite dimension required to capture its essential dynamics. This effective dimension plays an important role in the complexity of classical and quantum algorithms for learning and simulating bosonic systems. While generic bosonic states require a dimension scaling as $1/ε^2$ for a precision of approximation $ε$, here we identify a natural energy condition which allows us to improve this scaling exponentially to $\log(1/ε)$. We then prove that most bosonic quantum states satisfy this condition, and in particular those produced by combining Gaussian dynamics with generic energy-preserving dynamics, which include the output states of universal bosonic quantum circuits. We apply this finding to enhance learning algorithms for bosonic quantum states and we further obtain new classical simulation algorithms for a large class of bosonic systems. Finally, using efficient decompositions of Kerr gates as sums of Gaussian gates, we significantly refine these classical simulation algorithms for universal bosonic quantum circuits. Our results demonstrate that physical bosonic systems are significantly more well-behaved than previously assumed, allowing for efficient descriptions even at high precision.

quant-pharXiv:2604.18720Apr 22, 20260 citations

Lie-Algebraic Classical Simulation of Bosonic Systems Beyond Gaussian Dynamics

Adelina Bärligea, Timothy Heightman, Jakob S. Kottmann +1 more

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.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.

quant-pharXiv:2608.17094Aug 19, 20260 citations

Bosonic Quantum Computational Complexity

Ulysse Chabaud, Michael Joseph, Saeed Adel Mehraban +1 more

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.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.

arXiv:2410.04274QuantumMay 20, 20261 citation

When is a bosonic operator a Hamiltonian?

Felix Fischer, Hermann Schulz-Baldes, Daniel Burgarth +1 more

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.

quant-pharXiv:2610.33545Oct 27, 20260 citations

1/ε^2

for a precision of approximation

ε

, here we identify a natural energy condition which allows us to improve this scaling exponentially to

\log(1/ε)

. We then prove that most bosonic quantum states satisfy this condition, and in particular those produced by combining Gaussian dynamics with generic energy-preserving dynamics, which include the output states of universal bosonic quantum circuits. We apply this finding to enhance learning algorithms for bosonic quantum states and we further obtain new classical simulation algorithms for a large class of bosonic systems. Finally, using efficient decompositions of Kerr gates as sums of Gaussian gates, we significantly refine these classical simulation algorithms for universal bosonic quantum circuits. Our results demonstrate that physical bosonic systems are significantly more well-behaved than previously assumed, allowing for efficient descriptions even at high precision.

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.

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.