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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. Theory of quantum circuits with Abelian symmetries

   https://doi.org/10.1103/PhysRevResearch.6.043292  

   Marvian, Iman (December 2024, Physical Review Research)

   Quantum circuits with gates (local unitaries) respecting a global symmetry have broad applications in quantum information science and related fields, such as condensed-matter theory and quantum thermodynamics. However, despite their widespread use, fundamental properties of such circuits are not well understood. Recently, it was found that generic unitaries respecting a global symmetry cannot be realized, even approximately, using gates that respect the same symmetry. This observation raises important open questions: What unitary transformations can be realized with $k$-local gates that respect a global symmetry? In other words, in the presence of a global symmetry, how does the locality of interactions constrain the possible time evolution of a composite system? In this work, we address these questions for the case of Abelian (commutative) symmetries and develop constructive methods for synthesizing circuits with such symmetries. Remarkably, as a corollary, we find that, while the locality of interactions still imposes additional constraints on realizable unitaries, certain restrictions observed in the case of non-Abelian symmetries do not apply to circuits with Abelian symmetries. For instance, in circuits with a general non-Abelian symmetry such as $\mathrm{SU}\left(d\right)$, the unitary realized in a subspace with one irreducible representation (charge) of the symmetry dictates the realized unitaries in multiple other sectors with inequivalent representations of the symmetry. Furthermore, in certain sectors, rather than all unitaries respecting the symmetry, the realizable unitaries are the symplectic or orthogonal subgroups of this group. We prove that none of these restrictions appears in the case of Abelian symmetries. This result suggests that global non-Abelian symmetries may affect the thermalization of quantum systems in ways not possible under Abelian symmetries. Published by the American Physical Society2024 

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3. 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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4. Unleashed from constrained optimization: quantum computing for quantum chemistry employing generator coordinate inspired method

   https://doi.org/10.1038/s41534-024-00916-8  

   Zheng, Muqing; Peng, Bo; Li, Ang; Yang, Xiu; Kowalski, Karol (December 2024, npj Quantum Information)

   Abstract Hybrid quantum-classical approaches offer potential solutions to quantum chemistry problems, yet they often manifest as constrained optimization problems. Here, we explore the interconnection between constrained optimization and generalized eigenvalue problems through the Unitary Coupled Cluster (UCC) excitation generators. Inspired by the generator coordinate method, we employ these UCC excitation generators to construct non-orthogonal, overcomplete many-body bases, projecting the system Hamiltonian into an effective Hamiltonian, which bypasses issues such as barren plateaus that heuristic numerical minimizers often encountered in standard variational quantum eigensolver (VQE). Diverging from conventional quantum subspace expansion methods, we introduce an adaptive scheme that robustly constructs the many-body basis sets from a pool of the UCC excitation generators. This scheme supports the development of a hierarchical ADAPT quantum-classical strategy, enabling a balanced interplay between subspace expansion and ansatz optimization to address complex, strongly correlated quantum chemical systems cost-effectively, setting the stage for more advanced quantum simulations in chemistry. 

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5. Synthesis of energy-conserving quantum circuits with XY interaction

   https://doi.org/10.1088/2058-9565/ad53fa  

   Bai, Ge; Marvian, Iman (September 2024, Quantum science and technology)

   Abstract We study quantum circuits constructed from $\sqrt{\mathrm{i}\text{SWAP}}$gates and, more generally, from the entangling gates that can be realized with theXX + YYinteraction alone. Such gates preserve the Hamming weight of states in the computational basis, which means they respect the global U(1) symmetry corresponding to rotations around thezaxis. Equivalently, assuming that the intrinsic Hamiltonian of each qubit in the system is the PauliZoperator, they conserve the total energy of the system. We develop efficient methods for synthesizing circuits realizing any desired energy-conserving unitary usingXX + YYinteraction with or without single-qubit rotations around thezaxis. Interestingly, implementing generic energy-conserving unitaries, such as CCZ and Fredkin gates, with two-local energy-conserving gates requires the use of ancilla qubits. When single-qubit rotations around thez-axis are permitted, our scheme requires only a single ancilla qubit, whereas with theXX+YYinteraction alone, it requires two ancilla qubits. In addition to exact realizations, we also consider approximate realizations and show how a general energy-conserving unitary can be synthesized using only a sequence of $\sqrt{\mathrm{i}\text{SWAP}}$gates and two ancillary qubits, with arbitrarily small error, which can be bounded via the Solovay–Kitaev theorem. Our methods are also applicable for synthesizing energy-conserving unitaries when, rather than theXX + YYinteraction, one has access to any other energy-conserving two-body interaction that is not diagonal in the computational basis, such as the Heisenberg exchange interaction. We briefly discuss the applications of these circuits in the context of quantum computing, quantum thermodynamics, and quantum clocks. 

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