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1. Preparing Hamiltonians for quantum simulation: A computational framework for Cartan decomposition via Lax dynamics

   https://doi.org/10.1090/mcom/4056  

   Chu, Moody (February 2025, Mathematics of Computation)

   Quantum algorithms usually are described via quantum circuits representable as unitary operators. Synthesizing the unitary operators described mathematically in terms of the unitary operators recognizable as quantum circuits is essential. One such a challenge lies in the Hamiltonian simulation problem, where the matrix exponential of a large-scale skew-Hermitian matrix is to be computed. Most current techniques are prone to approximation errors, whereas the parametrization of the underlying Hamiltonian via the Cartan decomposition is more promising. To prepare for such a simulation, this work proposes to tackle the Cartan decomposition by means of the Lax dynamics. The advantages include not only that it is numerically feasible with no matrices involved, but also that this approach offers a genuine unitary synthesis within the integration errors. This work contributes to the theoretic and algorithmic foundations in three aspects: exploiting the quaternary representation of Hamiltonian subalgebras; describing a common mechanism for deriving the Lax dynamics; and providing a mathematical theory of convergence. 

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2. Lax dynamics for Cartan decomposition with applications to Hamiltonian simulation

   https://doi.org/10.1093/imanum/drad018  

   Chu, Moody T (April 2023, IMA Journal of Numerical Analysis)

   Simulating the time evolution of a Hamiltonian system on a classical computer is hard—The computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrate its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge—Assuming that a quantum system composing of spin-½ particles can be manipulated at will, then it is tenable to engineer the interaction between those particles according to the one that is to be simulated, and thus predict the value of physical quantities by simply performing the appropriate measurements on the system. Establishing a linkage between the unitary operators described mathematically as a logic solution and the unitary operators recognizable as quantum circuits for execution, is therefore essential for algorithm design and circuit implementation. Most current techniques are fallible because of truncation errors or the stagnation at local solutions. This work offers an innovative avenue by tackling the Cartan decomposition with the notion of Lax dynamics. Within the integration errors that is controllable, this approach gives rise to a genuine unitary synthesis that not only is numerically feasible, but also can be utilized to gauge the quality of results produced by other means, and extend the knowledge to a wide range of applications. This paper aims at establishing the theoretic and algorithmic foundations by exploiting the geometric properties of Hamiltonian subalgebras and describing a common mechanism for deriving the Lax dynamics. 

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3. Simple and High-Precision Hamiltonian Simulation by Compensating Trotter Error with Linear Combination of Unitary Operations

   https://doi.org/10.1103/PRXQuantum.6.010359  

   Zeng, Pei; Sun, Jinzhao; Jiang, Liang; Zhao, Qi (March 2025, PRX Quantum)

   Trotter and linear combination of unitary (LCU) operations are two popular Hamiltonian simulation methods. The Trotter method is easy to implement and enjoys good system-size dependence endowed by commutator scaling, while the LCU method admits high-accuracy simulation with a smaller gate cost. We propose Hamiltonian simulation algorithms using LCU to compensate Trotter error, which enjoy both of their advantages. By adding few gates after the $K\mathrm{th}$-order Trotter formula, we realize a better time scaling than $2K\mathrm{th}$-order Trotter. Our first algorithm exponentially improves the accuracy scaling of the $K\mathrm{th}$-order Trotter formula. For a generic Hamiltonian, the estimated gate counts of the first algorithm can be 2 orders of magnitude smaller than the best analytical bound of fourth-order Trotter formula. In the second algorithm, we consider the detailed structure of Hamiltonians and construct LCU for Trotter errors with commutator scaling. Consequently, for lattice Hamiltonians, the algorithm enjoys almost linear system-size dependence and quadratically improves the accuracy of the $K\mathrm{th}$-order Trotter. For the lattice system, the second algorithm can achieve 3 to 4 orders of magnitude higher accuracy with the same gate costs as the optimal Trotter algorithm. These algorithms provide an easy-to-implement approach to achieve a low-cost and high-precision Hamiltonian simulation. 

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4. Time-dependent unbounded Hamiltonian simulation with vector norm scaling

   https://doi.org/10.22331/q-2021-05-26-459  

   An, Dong; Fang, Di; Lin, Lin (May 2021, Quantum)

   null (Ed.)

   The accuracy of quantum dynamics simulation is usually measured by the error of the unitary evolution operator in the operator norm, which in turn depends on certain norm of the Hamiltonian. For unbounded operators, after suitable discretization, the norm of the Hamiltonian can be very large, which significantly increases the simulation cost. However, the operator norm measures the worst-case error of the quantum simulation, while practical simulation concerns the error with respect to a given initial vector at hand. We demonstrate that under suitable assumptions of the Hamiltonian and the initial vector, if the error is measured in terms of the vector norm, the computational cost may not increase at all as the norm of the Hamiltonian increases using Trotter type methods. In this sense, our result outperforms all previous error bounds in the quantum simulation literature. Our result extends that of [Jahnke, Lubich, BIT Numer. Math. 2000] to the time-dependent setting. We also clarify the existence and the importance of commutator scalings of Trotter and generalized Trotter methods for time-dependent Hamiltonian simulations. 

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5. General form of effective operators from hidden sectors

   https://doi.org/10.1007/JHEP05(2025)167  

   Ahmed, Aqeel; Chacko, Zackaria; Flood, Ina; Kilic, Can; Najjari, Saereh (May 2025, Journal of High Energy Physics)

   A<sc>bstract</sc> We perform a model-independent analysis of the dimension-six terms that are generated in the low energy effective theory when a hidden sector that communicates with the Standard Model (SM) through a specific portal operator is integrated out. We work within the Standard Model Effective Field Theory (SMEFT) framework and consider the Higgs, neutrino and hypercharge portals. We find that, for each portal, the forms of the leading dimension-six terms in the low-energy effective theory are fixed and independent of the dynamics in the hidden sector. For the Higgs portal, we find that two independent dimension-six terms are generated, one of which has a sign that, under certain conditions, is fixed by the requirement that the dynamics in the hidden sector be causal and unitary. In the case of the neutrino portal, for a single generation of SM fermions and assuming that the hidden sector does not violate lepton number, a unique dimension-six term is generated, which corresponds to a specific linear combination of operators in the Warsaw basis. For the hypercharge portal, a unique dimension-six term is generated, which again corresponds to a specific linear combination of operators in the Warsaw basis. For both the neutrino and hypercharge portals, under certain conditions, the signs of these terms are fixed by the requirement that the hidden sector be causal and unitary. We perform a global fit of these dimension-six terms to electroweak precision observables, Higgs measurements and diboson production data and determine the current bounds on their coefficients. 

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