Estimation of the energy of quantum many-body systems is a paradigmatic task in various research fields. In particular, efficient energy estimation may be crucial in achieving a quantum advantage for a practically relevant problem. For instance, the measurement effort poses a critical bottleneck for variational quantum algorithms.
We aim to find the optimal strategy with single-qubit measurements that yields the highest provable accuracy given a total measurement budget. As a central tool, we establish new tail bounds for empirical estimators of the energy. They are helpful for identifying measurement settings that improve the energy estimate the most. This task constitutes an NP-hard problem. However, we are able to circumvent this bottleneck and use the tail bounds to develop a practical, efficient estimation strategy, which we call ShadowGrouping. As the name indicates, it combines shadow estimation methods with grouping strategies for Pauli strings. In numerical experiments, we demonstrate that ShadowGrouping improves upon state-of-the-art methods in estimating the electronic ground-state energies of various small molecules, both in provable and practical accuracy benchmarks. Hence, this work provides a promising way, e.g., to tackle the measurement bottleneck associated with quantum many-body Hamiltonians.
Landauer's principle states that it costs at least kTln2 of work to reset one bit in the presence of a heat bath at temperature T. The bound of kTln2 is achieved in the unphysical infinite-time limit. Here we ask what is possible if one is restricted to finite-time protocols. We prove analytically that it is possible to reset a bit with a work cost close to kTln2 in a finite time. We construct an explicit protocol that achieves this, which involves changing the system's Hamiltonian avoiding quantum coherences, and thermalising. Using concepts and techniques pertaining to single-shot statistica…
Variational quantum eigensolver (VQE), which attracts attention as a promising application of noisy intermediate-scale quantum devices, finds a ground state of a given Hamiltonian by variationally optimizing the parameters of quantum circuits called ansatz. Since the difficulty of the optimization depends on the complexity of the problem Hamiltonian and the structure of the ansatz, it has been difficult to analyze the performance of optimizers for the VQE systematically. To resolve this problem, we propose a technique to construct a benchmark problem whose ground state is guaranteed to be achi…
In a pre-selected Hilbert space of quantum states the unitarity of the evolution is usually guaranteed via a pre-selection of the generator (i.e., of the Hamiltonian operator) in self-adjoint form. In fact, the simultaneous use of both of these pre-selections is overrestrictive. One should be allowed to make a given Hamiltonian self-adjoint only after an {\em ad hoc} generalization of Hermitian conjugation. We argue that in the generalized, hidden-Hermiticity scenario with nontrivial metric, the current concept of solvability (meaning, most often, the feasibility of a non-numerical diagonaliza…
There are many ways to decompose the Hilbert space H of a composite quantum system into tensor product subspaces. Different subsystem decompositions generally imply different interaction Hamiltonians V, and therefore different expectation values for subsystem observables. This means that thE uniqueness of physical predictions is not guaranteed, despite the uniqueness of the total Hamiltonian H and the total Hilbert space. Here we use Clausius' version of the second law of thermodynamics (CSL) and standard identifications of thermodynamic quantities to identify possible subsystem decompositions…
Most classical mechanical systems are based on dynamical variables whose values are real numbers. Energy conservation is then guaranteed if the dynamical equations are phrased in terms of a Hamiltonian function, which then leads to differential equations in the time variable. If these real dynamical variables are instead replaced by integers, and also the time variable is restricted to integers, it appears to be hard to enforce energy conservation unless one can also derive a Hamiltonian formalism for that case. We here show how the Hamiltonian formalism works here, and how it may yield the us…
Comprehending the dynamical behaviour of quantum systems driven by time-varying Hamiltonians is particularly difficult. Systems with as little as two energy levels are not yet fully understood as the usual methods including diagonalisation of the Hamiltonian do not work in this setting. In fact, since the inception of Magnus' expansion in 1954, no fundamentally novel mathematical approach capable of solving the quantum equations of motion with a time-varying Hamiltonian has been devised. We report here of an entirely different non-perturbative approach, termed path-sum, which is always guarant…
Numerical Optimization Strategies for the Variational Hamiltonian Ansatz in Noisy Quantum Environments
Gran Sasso Science Institute（格兰萨索科学研究所）；VSB-Technical University of Ostrava（俄斯特拉发理工大学）；IT4Innovations National Supercomputing Center, VSB–Technical University of Ostrava（俄斯特拉发理工大学 IT4创新国家超算中心）；Technical University of Munich（慕尼黑工业大学）；Fraunhofer Institute for Chemical Technology ICT（弗劳恩霍夫化学技术研究所）；Saarland University（萨尔兰大学）；University of L’Aquila（拉奎拉大学）
A broad range of quantum optimisation problems can be phrased as the question whether a specific system has a ground state at zero energy, i.e.\ whether its Hamiltonian is frustration free. Frustration-free Hamiltonians, in turn, play a central role for constructing and understanding new phases of matter in quantum many-body physics. Unfortunately, determining whether this is the case is known to be a complexity-theoretically intractable problem. This makes it highly desirable to search for efficient heuristics and algorithms in order to, at least, partially answer this question. Here we prove…
In conventional Schrödinger representation the unitarity of the evolution of bound states is guaranteed by the Hermiticity of the Hamiltonian. A non-unitary isospectral simplification of the Hamiltonian, $\mathfrak{h} \to H=Ω\,\mathfrak{h}\,Ω\neq H^\dagger$ induces the change ${\cal L} \to {\cal K}$ of the Hilbert space of states, reflected by the loss of the Hermiticity of $H\neq H^\dagger$. In such a reformulation of the theory the introduction of an {\it ad hoc} inner-product metric reconverts ${\cal K}$ into the third, correct physical Hilbert space ${\cal H}$, unitarily equivalent to ${\c…
Nondemolition filtering of an embedded cluster-state scar under continuous local monitoring
School of Physics and Electronic Engineering, Shanxi University（山西大学物理电子工程学院）；Collaborative Innovation Center of Extreme Optics, Shanxi University（山西大学极端光学协同创新中心）；Department of Physics, University of Science and Technology of China（中国科学技术大学物理系）；School of Instrument Science and Opto-Electronics Engineering, Beijing Information Science and Technology University（北京信息科技大学仪器科学与光电工程学院）；School of Physics, Beihang University（北京航空航天大学物理学院）；School of Applied Science, Beijing Information Science and Technology University（北京信息科技大学应用科学学院）；School of Automation (School of Artificial Intelligence), Beijing Information Science and Technology University（北京信息科技大学自动化学院（人工智能学院））；School of Space and Earth Sciences, Beihang University（北京航空航天大学空间与地球科学学院）
One of the defining features of many-body localization is the presence of extensively many quasi-local conserved quantities. These constants of motion constitute a corner-stone to an intuitive understanding of much of the phenomenology of many-body localized systems arising from effective Hamiltonians. They may be seen as local magnetization operators smeared out by a quasi-local unitary. However, accurately identifying such constants of motion remains a challenging problem. Current numerical constructions often capture the conserved operators only approximately restricting a conclusive unders…
A Hierarchy of Spectral Gap Certificates for Frustration-Free Spin Systems
Universiteit Leiden（莱顿大学）；Instituut-Lorentz, Universiteit Leiden（莱顿大学洛伦兹研究所）；University of Vienna（维也纳大学）；Ulm University（乌尔姆大学）；Center for Integrated Quantum Science and Technology (IQST)（集成量子科学与技术中心）；Università di Padova（帕多瓦大学）；Padua Quantum Technologies Research Center, Università degli Studi di Padova（帕多瓦大学帕多瓦量子技术中心）
A "no-knowledge" measurement of an open quantum system yields no information about any system observable; it only returns noise input from the environment. Surprisingly, performing such a no-knowledge measurement can be advantageous. We prove that a system undergoing no-knowledge monitoring has reversible noise, which can be cancelled by directly feeding back the measurement signal. We show how no-knowledge feedback control can be used to cancel decoherence in an arbitrary quantum system coupled to a Markovian reservoir that is being monitored. Since no-knowledge feedback does not depend on th…
The quantum many-body bound-state problem in its computationally successful coupled cluster method (CCM) representation is reconsidered. In conventional practice one factorizes the ground-state wave functions $|Ψ\rangle= e^S |Φ\rangle$ which live in the "physical" Hilbert space ${\cal H}^{(P)}$ using an elementary ansatz for $|Φ\rangle$ plus a formal expansion of $S$ in an operator basis of multi-configurational creation operators. In our paper a reinterpretation of the method is proposed. Using parallels between the CCM and the so called quasi-Hermitian, alias three-Hilbert-space (THS), quant…
The eigenvalue of the hermitic Hamiltonian is real undoubtedly. Actually, The reality can also be guaranteed by the $PT$-symmetry. The hermiticity and the $PT$-symmetric quantum theory both have requirements regarding the boundary condition. There exists a reverse strategy to investigate the quantum problem. Namely, define the eigenvalue as real first, and, meanwhile, open the boundary condition. Then the behaviors of the wave function at the boundary become rich in meaning. This eigenfunction is generally divergent, and the extent and direction of divergence are closely linked to the energy. …
Suppose the postulate of measurement in quantum mechanics can be extended to quantum field theory, then a local projective measurement at some moment on an object locally coupled with a relativistic quantum field will result in a projection or collapse of the wavefunctional of the combined system defined on the whole time-slice associated with the very moment of the measurement, if the relevant degrees of freedom have nonzero correlations. This implies that the wavefunctionals in the same Hamiltonian system but defined in different reference frames would collapse on different time-slices passi…
Quantum thermodynamics and semi-definite optimization
Institute of Natural Sciences, Shanghai Jiao Tong University（上海交通大学自然科学学院）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；School of Electrical and Computer Engineering, Cornell University（康奈尔大学电气与计算机工程学院）；School of Computer Science, Cornell University（康奈尔大学计算机科学学院）
