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arXiv:2609.40226

A physical and universal model of bosonic computations with Solovay-Kitaev theorem

Dorian Rudolph, Arsalan Motamedi, Dhruva Sambrani, Hamid Reza Naeij, Ulysse Chabaud, Sevag Gharibian, Saeed Mehraban

quant-phcs.CCSubmitted Sep 30, 2026• Announced Oct 1, 2026• 0 citations
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Abstract

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.

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Energy, Bosons and Computational Complexity

Rudolph, Dorian, Arsalan Motamedi, Dhruva Sambrani +4 more

We investigate the role of energy, i.e. average photon number, as a resource in the computational complexity of bosonic systems. We show three sets of results: (1. Energy growth rates) There exist bosonic gate sets which increase energy incredibly rapidly, obtaining e.g. infinite energy in finite/constant time. We prove these high energies can make computing properties of bosonic computations, such as deciding whether a given computation will attain infinite energy, extremely difficult, formally undecidable. (2. Lower bounds on computational power) More energy ``='' more computational power. For example, certain gate sets allow poly-time bosonic computations to simulate PTOWER, the set of deterministic computations whose runtime scales as a tower of exponentials with polynomial height. Even just exponential energy and $O(1)$ modes suffice to simulate NP, which, importantly, is a setup similar to that of the recent bosonic factoring algorithm of [Brenner, Caha, Coiteux-Roy and Koenig (2024)]. For simpler gate sets, we show an energy hierarchy theorem. (3. Upper bounds on computational power) Bosonic computations with polynomial energy can be simulated in BQP, ``physical'' bosonic computations with arbitrary finite energy are decidable, and the gate set consisting of Gaussian gates and the cubic phase gate can be simulated in PP, with exponential bound on energy, improving upon the previous PSPACE upper bound. Finally, combining upper and lower bounds yields no-go theorems for a continuous-variable Solovay--Kitaev theorem for gate sets such as the Gaussian and cubic phase gates.

arXiv:2510.08545•Oct 9, 2025•1 citation

Can effective descriptions of bosonic systems 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: 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.

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Coherent-State Propagation: A Computational Framework for Simulating Bosonic Quantum Systems

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We introduce coherent-state propagation, a computational framework for simulating bosonic systems. We focus on bosonic circuits composed of displaced linear optics augmented by Kerr nonlinearities, a universal model of bosonic quantum computation that is also physically motivated by driven Bose-Hubbard dynamics. The method works in the Schrödinger picture representing the evolving state as a sparse superposition of coherent states. We develop approximation strategies that keep the simulation cost tractable in physically relevant regimes, notably when the number of Kerr gates is small or the Kerr nonlinearities are weak, and prove rigorous guarantees for both observable estimation and sampling. In particular, bosonic circuits with logarithmically many Kerr gates admit quasi-polynomial-time classical simulation at exponentially small error in trace distance. We further identify a weak-nonlinearity regime in which the runtime is polynomial for arbitrarily small constant precision. We complement these results with numerical benchmarks on the Bose-Hubbard model with all-to-all connectivity. The method reproduces Fock-basis and matrix-product-state reference data, suggesting that it offers a useful route to the classical simulation of bosonic systems.

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Effective descriptions of bosonic systems can be considered complete

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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.

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Qutrit-Based Orthogonal Approximations with Inverse-Free Quantum Gate Sets

Anuradha Mahasinghe, Dulmi Fernando, Kaushika De Silva

The efficient compiling of arbitrary single-qubit gates into a sequence of gates from a finite gate set is of fundamental importance in quantum computation. The exact bounds of this compilation are given by the Solovay–Kitaev theorem, which serves as a powerful tool in compiling quantum algorithms that require many qubits. However, the inverse closure condition it imposes on the gate set adds a certain complexity to the experimental compilation, making the process less efficient. This was recently resolved by a version of the Solovay–Kitaev theorem for inverse-free gate sets, yielding a significant gain. Considering the recent progress in the field of three-level quantum systems, in which qubits are replaced by qutrits, it is possible to achieve the quantum speedup guaranteed by the Solovay–Kitaev theorem simply from orthogonal gates. Nevertheless, it has not been investigated previously whether the condition of inverse closure can be relaxed for these qutrit-based orthogonal compilations as well. In this work, we answer this positively, by obtaining improved Solovay–Kitaev approximations to an arbitrary orthogonal qutrit gate, with an accuracy ε from a sequence of O(log8.62(1/ε)) orthogonal gates taken from an inverse-free set.

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Universal computation with magic Hamiltonians

Marius Junge, Jason Pollack, Luke Visser

In the conventional theory of quantum computation, universality is discussed in terms of properties of unitary gate sets. In many experimental setups, however, we instead have access to a parametrized set of Hamiltonians, which can be exponentiated for any desired time. Accordingly, we formulate a continuous analogue of universality, where adding one expensive Hamiltonian control to a cheap set of local control operators allows for universal computation. We find several concrete sets of 1- or 2-local Pauli generators $S$ and $n$-qubit ``magic" Hamiltonians $H$ that together generate the full Lie algebra $\mathfrak{su}(2^n)$, with a special focus on the Ising Hamiltonian $J$. We propose a method to directly implement any unitary by exponentiating these generator sets, and give bounds on the needed time of application of the magic Hamiltonian $H$. We find that the number of applications of $H$ needed to approximate any unitary up to accuracy $ε$ scales exponentially in the number of qubits as expected from the standard application of the Solovay-Kitaev theorem. We also find phase transitions in the size of the generated Lie algebras as the parameters of the Ising Hamiltonian are varied. We find an algorithmic application of the Chow/Rashevskii technique to implement unitaries corresponding to iterated commutators.

quant-pharXiv:2609.14757•Sep 15, 2026•0 citations
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.
We investigate the role of energy, i.e. average photon number, as a resource in the computational complexity of bosonic systems. We show three sets of results: (1. Energy growth rates) There exist bosonic gate sets which increase energy incredibly rapidly, obtaining e.g. infinite energy in finite/constant time. We prove these high energies can make computing properties of bosonic computations, such as deciding whether a given computation will attain infinite energy, extremely difficult, formally undecidable. (2. Lower bounds on computational power) More energy ``='' more computational power. For example, certain gate sets allow poly-time bosonic computations to simulate PTOWER, the set of deterministic computations whose runtime scales as a tower of exponentials with polynomial height. Even just exponential energy and O(1)O(1)O(1) modes suffice to simulate NP, which, importantly, is a setup similar to that of the recent bosonic factoring algorithm of [Brenner, Caha, Coiteux-Roy and Koenig (2024)]. For simpler gate sets, we show an energy hierarchy theorem. (3. Upper bounds on computational power) Bosonic computations with polynomial energy can be simulated in BQP, ``physical'' bosonic computations with arbitrary finite energy are decidable, and the gate set consisting of Gaussian gates and the cubic phase gate can be simulated in PP, with exponential bound on energy, improving upon the previous PSPACE upper bound. Finally, combining upper and lower bounds yields no-go theorems for a continuous-variable Solovay--Kitaev theorem for gate sets such as the Gaussian and cubic phase gates.
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.
We introduce coherent-state propagation, a computational framework for simulating bosonic systems. We focus on bosonic circuits composed of displaced linear optics augmented by Kerr nonlinearities, a universal model of bosonic quantum computation that is also physically motivated by driven Bose-Hubbard dynamics. The method works in the Schrödinger picture representing the evolving state as a sparse superposition of coherent states. We develop approximation strategies that keep the simulation cost tractable in physically relevant regimes, notably when the number of Kerr gates is small or the Kerr nonlinearities are weak, and prove rigorous guarantees for both observable estimation and sampling. In particular, bosonic circuits with logarithmically many Kerr gates admit quasi-polynomial-time classical simulation at exponentially small error in trace distance. We further identify a weak-nonlinearity regime in which the runtime is polynomial for arbitrarily small constant precision. We complement these results with numerical benchmarks on the Bose-Hubbard model with all-to-all connectivity. The method reproduces Fock-basis and matrix-product-state reference data, suggesting that it offers a useful route to the classical simulation of bosonic systems.
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.
The efficient compiling of arbitrary single-qubit gates into a sequence of gates from a finite gate set is of fundamental importance in quantum computation. The exact bounds of this compilation are given by the Solovay–Kitaev theorem, which serves as a powerful tool in compiling quantum algorithms that require many qubits. However, the inverse closure condition it imposes on the gate set adds a certain complexity to the experimental compilation, making the process less efficient. This was recently resolved by a version of the Solovay–Kitaev theorem for inverse-free gate sets, yielding a significant gain. Considering the recent progress in the field of three-level quantum systems, in which qubits are replaced by qutrits, it is possible to achieve the quantum speedup guaranteed by the Solovay–Kitaev theorem simply from orthogonal gates. Nevertheless, it has not been investigated previously whether the condition of inverse closure can be relaxed for these qutrit-based orthogonal compilations as well. In this work, we answer this positively, by obtaining improved Solovay–Kitaev approximations to an arbitrary orthogonal qutrit gate, with an accuracy ε from a sequence of O(log8.62(1/ε)) orthogonal gates taken from an inverse-free set.
In the conventional theory of quantum computation, universality is discussed in terms of properties of unitary gate sets. In many experimental setups, however, we instead have access to a parametrized set of Hamiltonians, which can be exponentiated for any desired time. Accordingly, we formulate a continuous analogue of universality, where adding one expensive Hamiltonian control to a cheap set of local control operators allows for universal computation. We find several concrete sets of 1- or 2-local Pauli generators SSS and nnn-qubit ``magic" Hamiltonians HHH that together generate the full Lie algebra su(2n)\mathfrak{su}(2^n)su(2n), with a special focus on the Ising Hamiltonian JJJ. We propose a method to directly implement any unitary by exponentiating these generator sets, and give bounds on the needed time of application of the magic Hamiltonian HHH. We find that the number of applications of HHH needed to approximate any unitary up to accuracy εεε scales exponentially in the number of qubits as expected from the standard application of the Solovay-Kitaev theorem. We also find phase transitions in the size of the generated Lie algebras as the parameters of the Ising Hamiltonian are varied. We find an algorithmic application of the Chow/Rashevskii technique to implement unitaries corresponding to iterated commutators.