The ability to characterise a Hamiltonian with high precision is crucial for the implementation of quantum technologies. In addition to the well-developed approaches utilising optimal probe states and optimal measurements, the method of optimal control can be used to identify time-dependent pulses applied to the system to achieve higher precision in the estimation of Hamiltonian parameters, especially in the presence of noise. Here, we extend optimally controlled estimation schemes for single qubits to non-commuting dynamics as well as two interacting qubits, demonstrating improvements in terms of maximal precision, time-stability, as well as robustness over uncontrolled protocols.
Understanding commuting local Hamiltonians (CLHs) is at the heart of many questions in quantum computational complexity and quantum physics: quantum error correcting codes, quantum NP, the PCP conjecture, topological order and more.
We consider the class of spin Hamiltonians on a 1D chain with periodic boundary conditions that are (i) translational invariant, (ii) commuting and (iii) scale invariant, where by the latter we mean that the ground state degeneracy is independent of the system size. We correspond a directed graph to a Hamiltonian of this form and show that the structure of its ground space can be read from the cycles of the graph. We show that the ground state degeneracy is the only parameter that distinguishes the phases of these Hamiltonians. Our main tool in this paper is the idea of Bravyi and Vyalyi (2005…
We consider the computational complexity of Hamiltonians which are sums of commuting terms acting on plaquettes in a square lattice of qubits, and we show that deciding whether the ground state minimizes the energy of each local term individually is in the complexity class NP. That is, if the ground states has this property, this can be proven using a classical certificate which can be efficiently verified on a classical computer. Different to previous results on commuting Hamiltonians, our certificate proves the existence of such a state without giving instructions on how to prepare it.
Physical constraints make it challenging to implement and control many-body interactions. For this reason, designing quantum information processes with Hamiltonians consisting of only one- and two-local terms is a worthwhile challenge. Enabling error suppression with two-local Hamiltonians is particularly challenging. A no-go theorem of Marvian and Lidar [Physical Review Letters 113(26), 260504 (2014)] demonstrates that, even allowing particles with high Hilbert-space dimension, it is impossible to protect quantum information from single-site errors by encoding in the ground subspace of any Ha…
Given a Hamiltonian that is a sum of commuting few-body terms, the commuting Hamiltonian problem is to determine if there exists a quantum state that is the simultaneous eigenstate of all of these terms that minimizes each term individually. This problem is known to be in the complexity class quantum Merlin-Arthur, but is widely thought to not be complete for this class. Here we show that a limited form of this problem when the individual terms are all made up of tensor products of Pauli matrices is efficiently solvable on a classical computer and thus in the complexity class P. The problem ca…
We consider whether or not Hamiltonians which are sums of commuting projectors have "trivial" ground states which can be constructed by a local quantum circuit of bounded depth and range acting on a product state. While the toric code only has nontrivial ground states, commuting projector Hamiltonians which are sums of two-body interactions have trivial ground states. We define an "interaction complex" for a Hamiltonian, generalizing the interaction graph, and we show that if this complex can be continuously mapped to a 1-complex using a map with bounded diameter of pre-images then the Hamilto…
We propose to use a quantum adiabatic and simulated-annealing framework to compute theground state of small molecules. The initial Hamiltonian of our algorithms is taken to be themaximum commuting Hamiltonian that consists of a maximal set of commuting terms in the fullHamiltonian of molecules in the Pauli basis. We consider two variants. In the first method, weperform the adiabatic evolution on the obtained time- or path-dependent Hamiltonian with theinitial state as the ground state of the maximum commuting Hamiltonian. However, this methoddoes suffer from the usual problems of adiabatic qua…
An algebraic method has been developed which allows one to engineer several energy levels including the low-energy subspace of interacting spin systems. By introducing ancillary qubits, this approach allows k-body interactions to be captured exactly using 2-body Hamiltonians. Our method works when all terms in the Hamiltonian share the same basis and has no dependence on perturbation theory or the associated large spectral gap. Our methods allow problem instance solutions to be embedded into the ground energy state of Ising spin systems. Adiabatic evolution might then be used to place a comput…
We consider error suppression schemes in which quantum information is encoded into the ground subspace of a Hamiltonian comprising a sum of commuting terms. Since such Hamiltonians are gapped they are considered natural candidates for protection of quantum information and topological or adiabatic quantum computation. However, we prove that they cannot be used to this end in the 2-local case. By making the favorable assumption that the gap is infinite we show that single-site perturbations can generate a degeneracy splitting in the ground subspace of this type of Hamiltonians which is of the sa…
We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians…
In this work we consider the ground space connectivity problem for commuting local Hamiltonians. The ground space connectivity problem asks whether it is possible to go from one (efficiently preparable) state to another by applying a polynomial length sequence of 2-qubit unitaries while remaining at all times in a state with low energy for a given Hamiltonian $H$. It was shown in [Gharibian and Sikora, ICALP'15] that this problem is QCMA-complete for general local Hamiltonians, where QCMA is defined as QMA with a classical witness and BQP verifier. Here we show that the commuting version of th…
Variational Quantum Eigensolver (VQE) is a promising algorithm for near-term quantum machines. It can be used to estimate the ground state energy of a molecule by performing separate measurements of $O(N^4)$ terms. Several recent papers observed that this scaling may be reducible to $O(N^3)$ by partitioning the terms into linear-sized commuting families that can be measured simultaneously. We confirm these empirical observations by studying the MIN-COMMUTING-PARTITION problem at the level of the fermionic Hamiltonian and its encoding into qubits. Moreover, we provide a fast, pre-computable pro…
The local Hamiltonian problem plays the equivalent role of SAT in quantum complexity theory. Understanding the complexity of the intermediate case in which the constraints are quantum but all local terms in the Hamiltonian commute, is of importance for conceptual, physical and computational complexity reasons. Bravyi and Vyalyi showed in 2003, using a clever application of the representation theory of C*-algebras, that if the terms in the Hamiltonian are all two-local, the problem is in NP, and the entanglement in the ground states is local. The general case remained open since then. In this p…
Bipartite Gaussian Boson Sampling for Hamiltonian Cycles in Directed Graphs
College of Computer Science and Technology, National University of Defense Technology（国防科技大学计算机学院）；College of Advanced Interdisciplinary Studies, National University of Defense Technology（国防科技大学前沿交叉学院）
Estimation of expectation values of incompatible observables is an essential practical task in quantum computing, especially for approximating energies of chemical and other many-body quantum systems. In this work we introduce a method for this purpose based on performing a single joint measurement that can be implemented locally and whose marginals yield noisy (unsharp) versions of the target set of non-commuting Pauli observables. We derive bounds on the number of experimental repetitions required to estimate energies up to a certain precision. We compare this strategy to the classical shado…
Every choice of an orthonormal frame in the d-dimensional Hilbert space of a system corresponds to one set of all mutually commuting density matrices or, equivalently, a classical statistical state space of the system; the quantum state space itself can thus be profitably viewed as an SU(d) orbit of classical state spaces, one for each orthonormal frame. We exploit this connection to study the relative volume of separable states of a bipartite quantum system. While the two-qubit case is studied in considerable analytic detail, for higher dimensional systems we fall back on Monte Carlo. Several…
Many applications of practical interest rely on time evolution of Hamiltonians that are given by a sum of Pauli operators. Quantum circuits for exact time evolution of single Pauli operators are well known, and can be extended trivially to sums of commuting Paulis by concatenating the circuits of individual terms. In this paper we reduce the circuit complexity of Hamiltonian simulation by partitioning the Pauli operators into mutually commuting clusters and exponentiating the elements within each cluster after applying simultaneous diagonalization. We provide a practical algorithm for partitio…
Multiple quantum coherences are typically characterised by their coherence number and the number of spins that make up the state, though only the coherence number is normally measured. We present a simple set of measurements that extend our knowledge of the multiple quantum state by recording the coherences in both the $x$ basis and the usual $z$ basis. The coherences in the two bases are related by a similarity transformation. We characterize the growth of the multiple quantum coherences via measurements in the two bases, and show that the rate varies with the coefficient of the driving term …
Quantum Markov networks are a generalization of quantum Markov chains to arbitrary graphs. They provide a powerful classification of correlations in quantum many-body systems---complementing the area law at finite temperature---and are therefore useful to understand the powers and limitations of certain classes of simulation algorithms. Here, we extend the characterization of quantum Markov networks and in particular prove the equivalence of positive quantum Markov networks and Gibbs states of Hamiltonians that are the sum of local commuting terms on graphs containing no triangles. For more ge…
Consider two quantum systems A and B interacting according to a product Hamiltonian H = H_A x H_B. We show that any two such Hamiltonians can be used to simulate each other reversibly (i.e., without efficiency losses) with the help of local unitary operations and local ancillas. Accordingly, all non-local features of a product Hamiltonian -- including the rate at which it can be used to produce entanglement, transmit classical or quantum information, or simulate other Hamiltonians -- depend only upon a single parameter. We identify this parameter and use it to obtain an explicit expression for…
We discuss upper bounds on the rate at which unitary evolution governed by a non-local Hamiltonian can generate entanglement in a bipartite system. Given a bipartite Hamiltonian H coupling two finite dimensional particles A and B, the entangling rate is shown to be upper bounded by c*log(d)*norm(H), where d is the smallest dimension of the interacting particles, norm(H) is the operator norm of H, and c is a constant close to 1. Under certain restrictions on the initial state we prove analogous upper bound for the ancilla-assisted entangling rate with a constant c that does not depend upon dime…
The local Hamiltonian problem is famously complete for the class QMA, the quantum analogue of NP. The complexity of its semi-classical version, in which the terms of the Hamiltonian are required to commute (the CLH problem), has attracted considerable attention recently due to its intriguing nature, as well as in relation to growing interest in the qPCP conjecture. We show here that if the underlying bipartite interaction graph of the CLH instance is a good locally-expanding graph, namely, the expansion of any constant-size set is e-close to optimal, then approximating its ground energy to wit…
In this paper, we introduce a discrete quantum walk model called bipartite walks. Bipartite walks include many known discrete quantum walk models, like arc-reversal walks, vertex-face walks. For the transition matrix of a quantum walk, there is a Hamiltonian associated with it. We will study the Hamiltonians of the bipartite walks. Let $S$ be a skew-symmetric matrix. We are mainly interested in the Hamiltonians of the form $iS$. We show that the Hamiltonian can be written as $iS$ if and only if the adjacency matrix of the bipartite graph is invertible. We show that arc-reversal walks and verte…
Azar Aytash Co.（阿扎尔·阿伊塔什公司）；University of Tabriz（大不里士大学）
Graphical Models have various applications in science and engineering which include physics, bioinformatics, telecommunication and etc. Usage of graphical models needs complex computations in order to evaluation of marginal functions,so there are some powerful methods including mean field approximation, belief propagation algorithm and etc. Quantum graphical models have been recently developed in context of quantum information and computation, and quantum statistical physics, which is possible by generalization of classical probability theory to quantum theory. The main goal of this paper is p…
We consider interactions as bidirectional channels. We investigate the capacities for interaction Hamiltonians and nonlocal unitary gates to generate entanglement and transmit classical information. We give analytic expressions for the entanglement generating capacity and entanglement-assisted one-way classical communication capacity of interactions, and show that these quantities are additive, so that the asymptotic capacities equal the corresponding 1-shot capacities. We give general bounds on other capacities, discuss some examples, and conclude with some open questions.
In this article we investigate the unitary dynamics of squashed entanglement and concurrence measures in Werner state and maximally entangled mixed states (MEMS) under two different Hamiltonians. The aim of the present study is two fold. The first part of the study deals with the dynamics under Heisenberg Hamiltonian and the second part deals under bi-linear bi-quadratic Hamiltonian which is the extension of first Hamiltonian. In both the parts we investigate the dynamical trade off and balancing points for squashed entanglement and concurrence. During the study, we also found the results of e…
Inspired by a recently constructed commuting-projector Hamiltonian for a two-dimensional (2D) time-reversal-invariant topological superconductor [Wang et al., Phys. Rev. B 98, 094502 (2018)], we introduce a commuting-projector model that describes an interacting yet exactly solvable 2D topological insulator. We explicitly show that both the gapped and gapless boundaries of our model are consistent with those of band-theoretic, weakly interacting topological insulators. Interestingly, on certain lattices our time-reversal-symmetric models also enjoy $\mathcal{CP}$ symmetry, leading to intuitive…
We show that two interacting physical systems may admit entangled pure or non separable mixed states evolving in time as if the mutual interaction hamiltonian were absent. In this paper we define these states Interaction Free Evolving (IFE) states and characterize their existence for a generic binary system described by a time independent Hamiltonian. A comparison between IFE subspace and the decoherence free subspace is reported. The set of all pure IFE states is explicitly constructed for a non homogeneous spin star system model.
We study unfrustrated spin Hamiltonians that consist of commuting tensor products of Pauli matrices. Assuming translation-invariance, a family of Hamiltonians that belong to the same phase of matter is described by a map between modules over the translation-group algebra, so homological methods are applicable. In any dimension every point-like charge appears as a vertex of a fractal operator, and can be isolated with energy barrier at most logarithmic in the separation distance. For a topologically ordered system in three dimensions, there must exist a point-like nontrivial charge. If the grou…
Modified logarithmic Sobolev inequality for 1D non-commuting Hamiltonians
University of Cambridge（剑桥大学）；Universität Tübingen（蒂宾根大学）；Universidad Complutense de Madrid（马德里康普顿斯大学）；Instituto de Ciencias Matemáticas (ICMAT)（数学科学研究所）；Instituto de Física Teórica UAM/CSIC（UAM/CSIC理论物理研究所）
In bipartite quantum systems commutation relations between the Hamiltonian of each subsystem and the interaction impose fundamental constraints on the dynamics of each partition. Here we investigate work, heat and entropy production in bipartite systems characterized by particular commutators between their local Hamiltonians and the interaction operator. We consider the formalism of [Weimer, EPL, 83:30008, 2008], in which heat (work) is identified with energy changes that (do not) alter the local von Neumann entropy, as observed in an effective local measurement basis. We demonstrate the conse…
We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group $\langle αI, X,S\rangle$, where $α=e^{iπ/4}$ and $S=\operatorname{diag}(1,i)$. We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians etc. We also highligh…
We consider multi-qubit systems and relate quantitatively the problems of generating cluster states with high value of concurrence of assistance, and that of generating states with maximal bipartite entanglement. We prove an upper bound for the concurrence of assistance. We consider dynamics of spin-1/2 systems that model qubits, with different couplings and possible presence of magnetic field to investigate the appearance of the discussed entanglement properties. We find that states with maximal bipartite entanglement can be generated by an XY Hamiltonian, and their generation can be controll…
The notion of capacity of entanglement is the quantum information theoretic counterpart of the heat capacity which is defined as the second cumulant of the entanglement spectrum. Given any bipartite pure state, we can define the capacity of entanglement as the variance of the modular Hamiltonian in the reduced state of any of the subsystems. Here, we study the dynamics of this quantity under non-local Hamiltonian. Specifically, we address the question: Given an arbitrary non-local Hamiltonian what is the capacity of entanglement that the system can possess? As an useful application, we show th…
Bipartite separability of one parameter families of states using conditional quantum relative Tsallis entropy
Kuvempu University（库韦姆普大学）；Inspire Institute Inc.（Inspire研究所）；Harish-Chandra Research Institute（哈里什-钱德拉研究所）；Institute of Mathematical Sciences（数学科学研究所）；Bangalore University（班加罗尔大学）
We analyze some features of alternative Hermitian and quasi-Hermitian quantum descriptions of simple and bipartite compound systems. We show that alternative descriptions of two interacting subsystems are possible if and only if the metric operator of the compound system can be obtained as tensor product of positive operators on component spaces. Some examples also show that such property could be strictly connected with symmetry properties of the non-Hermitian Hamiltonian.
We investigate a two-level system in resonant contact with a larger environment. The environment typically is in a canonical state with a given temperature initially. Depending on the precise spectral structure of the environment and the type of coupling between both systems, the smaller part may relax to a canonical state with the same temperature as the environment (i.e. thermal relaxation) or to some other quasi equilibrium state (non thermal relaxation). The type of the (quasi) equilibrium state can be related to the distribution of certain properties of the energy eigenvectors of the tota…
What could happen if we pinned a single qubit of a system and fixed it in a particular state? First, we show that this can greatly increase the complexity of static questions -- ground state properties of local Hamiltonian problems with restricted types of terms. In particular, we show that the Pinned commuting and Pinned Stoquastic Local Hamiltonian problems are QMA complete. Second, we show that pinning a single qubit via often repeated measurements also results in universal quantum computation already with commuting and stoquastic Hamiltonians. Finally, we discuss variants of the Ground Sta…
Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians
Max-Planck-Institute of Quantum Optics（马克斯·普朗克量子光学研究所）；Inria, Télécom Paris - LTCI, Institut Polytechnique de Paris（法国国家信息与自动化研究所，巴黎高等电信学校-通信与图像处理实验室，巴黎理工学院）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家研究委员会理论物理研究所）；Fachbereich Mathematik, Eberhard Karls Universität Tübingen（蒂宾根大学数学系）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学和理论物理系）；Department of Mathematics, Technische Universität München（慕尼黑工业大学数学系）
