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1. A universal variational quantum eigensolver for non-Hermitian systems

   https://doi.org/10.1038/s41598-023-49662-5  

   Zhao, Huanfeng; Zhang, Peng; Wei, Tzu-Chieh (December 2023, Scientific Reports)

   Abstract Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices—Inspired by Schur’s triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver—With additional$$O(log_{2}{N})$$ $O\left(lo{g}_{2}N\right)$quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm’s feasibility. We also undertake a comprehensive parametric study to validate VQUE’s scalability, generality, and performance in realistic applications. 

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2. 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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3. A quantum hamiltonian simulation benchmark

   https://doi.org/10.1038/s41534-022-00636-x  

   Dong, Yulong; Whaley, K. Birgitta; Lin, Lin (December 2022, npj Quantum Information)

   Abstract Hamiltonian simulation is one of the most important problems in quantum computation, and quantum singular value transformation (QSVT) is an efficient way to simulate a general class of Hamiltonians. However, the QSVT circuit typically involves multiple ancilla qubits and multi-qubit control gates. In order to simulate a certain class of n -qubit random Hamiltonians, we propose a drastically simplified quantum circuit that we refer to as the minimal QSVT circuit, which uses only one ancilla qubit and no multi-qubit controlled gates. We formulate a simple metric called the quantum unitary evolution score (QUES), which is a scalable quantum benchmark and can be verified without any need for classical computation. Under the globally depolarized noise model, we demonstrate that QUES is directly related to the circuit fidelity, and the potential classical hardness of an associated quantum circuit sampling problem. Under the same assumption, theoretical analysis suggests there exists an ‘optimal’ simulation time t opt  ≈ 4.81, at which even a noisy quantum device may be sufficient to demonstrate the potential classical hardness. 

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4. Linear growth of quantum circuit complexity

   https://doi.org/10.1038/s41567-022-01539-6  

   Haferkamp, Jonas; Faist, Philippe; Kothakonda, Naga B.; Eisert, Jens; Yunger Halpern, Nicole (May 2022, Nature Physics)

   Abstract The complexity of quantum states has become a key quantity of interest across various subfields of physics, from quantum computing to the theory of black holes. The evolution of generic quantum systems can be modelled by considering a collection of qubits subjected to sequences of random unitary gates. Here we investigate how the complexity of these random quantum circuits increases by considering how to construct a unitary operation from Haar-random two-qubit quantum gates. Implementing the unitary operation exactly requires a minimal number of gates—this is the operation’s exact circuit complexity. We prove a conjecture that this complexity grows linearly, before saturating when the number of applied gates reaches a threshold that grows exponentially with the number of qubits. Our proof overcomes difficulties in establishing lower bounds for the exact circuit complexity by combining differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits. 

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5. Quantum codes and irreducible products of characters

   https://doi.org/10.1007/s10623-025-01599-8  

   Kubischta, Eric; Teixeira, Ian (April 2025, Designs, Codes and Cryptography)

   Abstract In a recent paper, we defined a type of weighted unitary design called a twisted unitary 1-group and showed that such a design automatically induced error-detecting quantum codes. We also showed that twisted unitary 1-groups correspond to irreducible products of characters thereby reducing the problem of code-finding to a computation in the character theory of finite groups. Using a combination of GAP computations and results from the mathematics literature on irreducible products of characters, we identify many new non-trivial quantum codes with unusual transversal gates. Transversal gates are of significant interest to the quantum information community for their central role in fault tolerant quantum computing. Most unitary$$\text {t}$$ $\text{t}$-designs have never been realized as the transversal gate group of a quantum code. We, for the first time, find nontrivial quantum codes realizing nearly every finite group which is a unitary 2-design or better as the transversal gate group of some error-detecting quantum code. 

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