Foundations 2

Abstract: 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. Modelling 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 results obtained with such simplifying assumptions to 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: first, we prove that any physical bosonic unitary evolution can be accurately approximated by a finite-dimensional unitary evolution; second, 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.

Abstract: Compiling Bell games under cryptographic assumptions replaces the need for physical separation, allowing nonlocality to be probed with a single untrusted device. While Kalai et al. (STOC'23) showed that this compilation preserves quantum advantages, its quantitative quantum soundness has remained an open problem. We address this gap with two primary contributions. First, we establish the first quantitative quantum soundness bounds for every bipartite compiled Bell game whose optimal quantum strategy is finite-dimensional: any polynomial-time prover's score in the compiled game is negligibly close to the game's ideal quantum value. More generally, for all bipartite games we show that the compiled score cannot significantly exceed the bounds given by a newly formalized convergent sequential Navascués-Pironio-Acín (NPA) hierarchy. Second, we provide a full characterization of this sequential NPA hierarchy, establishing it as a robust numerical tool that is of independent interest. Finally, for games without finite-dimensional optimal strategies, we explore the necessity of NPA approximation error for quantitatively bounding their compiled scores, linking these considerations to the complexity conjecture $\mathrm{MIP}^{\mathrm{co}}=\mathrm{coRE}$ and open challenges such as quantum homomorphic encryption correctness for "weakly commuting" quantum registers.

Abstract: We continue the line of work initiated by Kalai et al. (STOC '23), studying "compiled" nonlocal games played between a classical verifier and a single quantum prover, with cryptography simulating the spatial separation between the players. The central open question in this area is to understand the soundness of this compiler against quantum strategies, and apart from results for specific games, all that is known is the recent "qualitative" result of Kulpe et al. (STOC '25) showing that the success probability of a quantum prover in the compiled game is bounded by the game's quantum commuting-operator value in the limit as the cryptographic security parameter goes to infinity. In this work, we make progress towards a quantitative understanding of quantum soundness for general games, by giving a concrete framework to bound the quantum value of compiled nonlocal games. Building on the result of Kulpe et al. together with the notion of "nice" sum-of-squares certificates, introduced by Natarajan and Zhang (FOCS '23) to bound the value of the compiled CHSH game, we extend the niceness framework and construct a hierarchy of semidefinite programs that searches exclusively over nice certificates. We show that this hierarchy converges to the optimal quantum value of the game. Additionally, we present a transformation to make any degree-1 sum-of-squares certificate nice. This approach provides a systematic method to reproduce all known bounds for special classes of games together with Kulpe et al.'s bound for general games from the same framework.

Abstract: Compiled nonlocal games transfer the power of Bell-type multi-prover tests into a single-device setting by replacing spatial separation with cryptography. Concretely, the KLVY compiler (STOC'23) maps any multi-prover game to an interactive single-prover protocol, using quantum homomorphic encryption. A crucial security property of such compilers is quantum soundness, which ensures a dishonest quantum prover cannot exceed the original game's quantum value. For practical cryptographic implementations, this soundness must be quantitative, providing concrete bounds, rather than merely asymptotic. While quantitative quantum soundness has been established for the KLVY compiler in the bipartite case, it has only been shown asymptotically for multipartite games. This is a significant gap, as multipartite nonlocality exhibits phenomena with no bipartite analogue, and the difficulty of enforcing space-like separation makes single-device compilation especially compelling. This work closes this gap by showing the quantitative quantum soundness of the KLVY compiler for all multipartite nonlocal games. On the way, we introduce an NPA-like hierarchy for quantum instruments and prove its completeness, thereby characterizing correlations from non-signaling sequential strategies. We further develop novel geometric arguments for the decomposition of sequential strategies into their signaling and non-signaling parts, which might be of independent interest.

Abstract: Non-local games are a powerful tool to distinguish between correlations possible in classical and quantum worlds. Kalai et al. (STOC'23) proposed a compiler that converts multipartite non-local games into interactive protocols with a single prover, relying on cryptographic tools to remove the assumption of physical separation of the players. While quantum completeness and classical soundness of the construction have been established for all multipartite games, quantum soundness is known only in the special case of bipartite games. In this paper, we prove that the Kalai \emph{et al.}'s compiler indeed achieves quantum soundness for all multipartite compiled non-local games, by showing that any correlations that can be generated in the asymptotic case correspond to quantum commuting strategies. Our proof uses techniques from the theory of operator algebras, and relies on a characterisation of sequential operationally no-signalling strategies as quantum commuting operator strategies in the multipartite case, thereby generalising several previous results. On the way, we construct universal C*-algebras of sequential PVMs and prove a new chain rule for Radon-Nikodym derivatives of completely positive maps on C*-algebras which may be of independent interest.