We study the complexity of a problem "Common Eigenspace" -- verifying consistency of eigenvalue equations for composite quantum systems. The input of the problem is a family of pairwise commuting Hermitian operators H_1,...,H_r on a Hilbert space (C^d)^{\otimes n} and a string of real numbers h_1,...,h_r. The problem is to determine whether a common eigenspace specified by equalities (H_a - h_a)|ψ>=0, a=1,...,r, has a positive dimension. We consider two cases: (i) all operators H_a are k-local; (ii) all operators H_a are factorized. It can be easily shown that both problems belong to the class QMA - the quantum analogue of NP, and that some NP-complete problems can be reduced to either (i) or (ii). A non-trivial question is whether the problems (i) or (ii) belong to NP? We show that the answer is positive for some special values of k and d. Also we prove that the problem (ii) can be reduced to its special case, such that all operators H_a are factorized projectors and all h_a=0.
Quantum computing can efficiently simulate Hamiltonian dynamics of many-body quantum physics, a task that is generally intractable with classical computers. The hardness lies at the ubiquitous anti-commutative relations of quantum operators, in corresponding with the notorious negative sign problem in classical simulation. Intuitively, Hamiltonians with more commutative terms are also easier to simulate on a quantum computer, and anti-commutative relations generally cause more errors, such as in the product formula method. Here, we theoretically explore the role of anti-commutative relation in…
We propose a circuit-model quantum algorithm for eigenpath traversal that is based on a combination of concepts from Grover's search and adiabatic quantum computation. Our algorithm deploys a sequence of reflections determined from eigenspaces of instantaneous Hamiltonians defined along an adiabatic schedule in order to prepare a ground state of a target problem Hamiltonian. We provide numerical evidence suggesting that, for combinatorial search problems, our algorithm can find a solution faster, on average, than Grover's search. We demonstrate our findings by applying both algorithms to solvi…
In this article we study two-dimensional Dirac Hamiltonians with non-commutativity both in coordinates and momenta from an algebraic perspective. In order to do so, we consider the graded Lie algebra $\mathfrak{sl}(2|1)$ generated by Hermitian bilinear forms in the non-commutative dynamical variables and the Dirac matrices in $2+1$ dimensions. By further defining a total angular momentum operator, we are able to express a class of Dirac Hamiltonians completely in terms of these operators. In this way, we analyze the energy spectrum of some simple models by constructing and studying the represe…
It is well known that an (in general, non-commutative) set of non-Hermitian operators $Λ_j$ with real eigenvalues need not necessarily represent observables. We describe a specific class of quantum models in which these operators plus the underlying physical Hilbert-space metric $Θ$ are all represented in terms of an auxiliary operator $(N+1)-$plet $Z_k$, $k=0,1,\ldots,N$. Our formalism degenerates to the ${\cal PT}-$symmetric quantum mechanics at $N=2$, with metric $Θ=Z_2Z_1$, parity ${\cal P}=Z_2$, charge ${\cal C}=Z_1$ and Hamiltonian $H=Z_0$.
在最近的工作[D. K. Burgarth等人，国家通信5，5173（2014）]表明，一系列频繁的测量可以将量子系统的动力学投影到动力学可能更复杂的子空间上。在这个子空间中，甚至可以实现完全的可控性，尽管在投影之前系统的可控性非常差，因为控制哈密顿量彼此交换。我们也可以考虑相反的情况：量子系统的任何哈密顿算符，一般来说彼此之间是不可交换的，可以通过将它们嵌入到扩展的希尔伯特空间中而使它们成为可交换的，因此扩展空间中的动力学变得平凡而简单。这种使非对易的哈密尔顿算子成为对易的思想叫做“哈密尔顿纯化"。“通过频繁的测量将系统投影回原始希尔伯特空间，恢复了原始的非对易哈密顿算子。在这里，我们将这个想法推广到开放系统动力学，提出了一个简单的结构，使Lindbladian，以及哈密顿，交换的一个更大的空间与辅助系统。我们表明，原来的动态可以恢复通过经常测量辅助系统在一个非选择性的方式。此外，我们还提供了一个通用的Lindbladian对，它描述了一个“可访问的”开放量子系统的通用系统大小。这使我们能够得出结论，通过一系列频繁的非选择性测量，一个不可访问的开放量子系统通常变得可访问。这进一步阐明了测量反作用对量子系统控制的作用。
Closest Accessible Symmetry reduction: a tool for Hamiltonian interpolation analysis
Qilimanjaro Quantum Tech（基利马尼哈罗量子技术）；Universitat de Barcelona（巴塞罗那大学）；Institut de Ciències del Cosmos, Universitat de Barcelona（巴塞罗那大学宇宙科学研究所）；Barcelona Supercomputing Center（巴塞罗那超级计算中心）
We consider Markovian dynamics of a finitely dimensional open quantum system featuring a weak unitary symmetry, i.e., when the action of a unitary symmetry on the space of density matrices commutes with the master operator governing the dynamics. We show how to encode the weak symmetry in quantum stochastic dynamics of the system by constructing a weakly symmetric representation of the master operator: a symmetric Hamiltonian, and jump operators connecting only the symmetry eigenspaces with a fixed eigenvalue ratio. In turn, this representation simplifies both the construction of the master op…
We propose a wave operator method to calculate eigenvalues and eigenvectors of large parameter-dependent matrices, using an adaptative active subspace. We consider a hamiltonian which depends on external adjustable or adiabatic parameters, using adaptative projectors which follow the successive eigenspaces when the adjustable parameters are modified. The method can also handle non-hermitian hamiltonians. An iterative algorithm is derived and tested through comparisons with a standard wave operator algorithm using a fixed active space and with a standard block-Davidson method. The proposed appr…
Quantum optimal control has numerous important applications ranging from pulse shaping in magnetic-resonance imagining to laser control of chemical reactions and quantum computing. Our objective is to address two major challenges that have limited the success of applications of quantum optimal control so far: non-commutativity inherent in quantum systems and non-convexity of quantum optimal control problems involving more than three quantum levels. Methodologically, we address the non-commutativity of the control Hamiltonian at different times by the use of Magnus expansion. To tackle the non-…
The goal of this work is to build a dynamical theory of defects for quantum spin systems. A kinematic theory for an indefinite number of defects is first introduced exploiting distinguishable Fock space. Dynamics are then incorporated by allowing the defects to become mobile via a microscopic Hamiltonian. This construction is extended to topologically ordered systems by restricting to the ground state eigenspace of Hamiltonians generalizing the golden chain. We illustrate the construction with the example of a spin chain with $\mathbf{Vec}(\mathbb{Z}/2\mathbb{Z})$ fusion rules, employing gener…
Quasi-optimal sampling from Gibbs states via non-commutative optimal transport metrics
Fachbereich Mathematik, Universität Tübingen（蒂宾根大学数学系）；Dep. of App. Mathematics and Theo. Physics, University of Cambridge（剑桥大学应用数学与理论物理系）；Inria, Télécom Paris - LTCI, Institut Polytechnique de Paris（巴黎理工学院电信学院LTCI实验室法国国家信息与自动化研究所）
The Lotka-Volterra (LV) dynamics is investigated in the framework of the Weyl-Wigner (WW) quantum mechanics (QM) extended to one-dimensional Hamiltonian systems, $\mathcal{H}(x,\,k)$, constrained by the $\partial^2 \mathcal{H} / \partial x \, \partial k = 0$ condition. Supported by the Heisenberg-Weyl non-commutative algebra, where $[x,\,k] = i$, the canonical variables $x$ and $k$ are interpreted in terms of the LV variables, $y = e^{-x}$ and $z = e^{-k}$, eventually associated with the number of individuals in a closed competitive dynamics: the so-called prey-predator system. The WW framewor…
