Under mild conditions the experimentally accessible Hamiltonians in ensembles of N coupled, nondegenerate spins-½ are shown to generate the entire Lie group SU(2 N ). This is the case e.g., in nuclear magnetic resonance (NMR) spectroscopy [1], where quantum computing gates [2] are implemented by unitary propagators, viz the ‘rf-pulse sequences’. The maximum achievable coherence transfer amplitude under unitary Hamiltonian dynamics of ensembles translates into a minimum Euclidean distance: the minimal distance between the unitary orbit of some given initial state represented by a density operator (or its signal-relevant components collected in a matrix A) and a given final state or observable (or its components C †). This distance relates to the C-numerical range of A, W C(A) := {tr(UAU -1 C)|U ∈ SU(2 N )}, and its largest absolute value, the C-numerical radius of A, $$ r_C \left( A \right) : = \mathop {{\rm{max}}}\limits_U |tr\left\{ {U AU^{ - 1} C} \right\}|\left[ {\rm{3}} \right] $$ [3]. The geometry of the C-numerical range includes as limiting cases the notions of ensembles comprising entangled as well as classically mixed components. In the latter, entropy and relative entropy relate to Euclidean distances. Given two arbitrary matrices A,C ∈ Mat n (C), the first gradient-flow-based computer algorithm [4-6] for minimising the Euclidean distance between the unitary orbit of A and C † (or rather C) in the general case is presented.
Under mild conditions the experimentally accessible Hamiltonians in ensembles of N coupled, nondegenerate spins-½ are shown to generate the entire Lie group SU(2 N ). This is the case e.g., in nuclear magnetic resonance (NMR) spectroscopy [1], where quantum computing gates [2] are implemented by unitary propagators, viz the ‘rf-pulse sequences’. The maximum achievable coherence transfer amplitude under unitary Hamiltonian dynamics of ensembles translates into a minimum Euclidean distance: the minimal distance between the unitary orbit of some given initial state represented by a density operator (or its signal-relevant components collected in a matrix A) and a given final state or observable (or its components C †). This distance relates to the C-numerical range of A, W C(A) := {tr(UAU -1 C)|U ∈ SU(2 N )}, and its largest absolute value, the C-numerical radius of A, rC(A):=Umax∣tr{UAU−1C}∣[3] [3]. The geometry of the C-numerical range includes as limiting cases the notions of ensembles comprising entangled as well as classically mixed components. In the latter, entropy and relative entropy relate to Euclidean distances. Given two arbitrary matrices A,C ∈ Mat n (C), the first gradient-flow-based computer algorithm [4-6] for minimising the Euclidean distance between the unitary orbit of A and C † (or rather C) in the general case is presented.
Abstract. Quantum simulation is a promising near term application for quantum information processors with the potential to solve computationally intractable problems using just a few dozen interacting qubits [1, 2, 3]. A range of experimental platforms have recently demonstrated the basic functionality of quantum simulation applied to quantum magnetism, quantum phase transitions, and relativistic quantum mechanics [4, 5, 6, 7, 8, 9, 10, 11]. However, in all cases, the physics of the underlying hardware restricts the achievable inter-particle interactions and forms a serious constraint on the versatility of the simulators. To broaden the scope of these analog devices, we develop a suite of pulse sequences that permit a user to efficiently realize average Hamiltonians that are beyond the native interactions of the system [12, 13]. Specifically, this approach permits the generation of all symmetrically coupled translation-invariant two-body Hamiltonians with homogeneous on-site terms, a class which includes all spin-1/2 XYZ chains, but generalized to include long-range couplings. Our work builds on previous work proving that universal simulation is possible using both entangling gates and single-qubit unitaries [14, 15, 16]. We show that determining the appropriate “program ” of unitary pulse sequences which implements an arbitrary Hamiltonian transformation can be formulated as a linear program over functions defined by those pulse sequences, running in polynomial time and scaling efficiently in hardware resources. Our analysis extends from circuit model quantum information to adiabatic quantum evolutions, representing an important and broad-based success in applying functional analysis to the field of quantum information. ar X iv
Quantum computers are able to operate on coherent superpositions of states, and to isolate a single global property of the set of computed quantities via interference. In principle, this permits them to solve certain problems exponentially faster thana classical computer, but not one has yet succeeded in implementing a true quantum computer on more than two quantum bits. Recently, however, it has been found that an ensemble of independent and identical quantum computers can perform mots of the same feats than a single quantum computer could, while at the same time bringing massive classical parallelism to bear on its computations. Such an ensemble quantum computer can be realized, to a limited extent, by nuclear magnetic resonance (NMR) spectroscopy on ordinary liquids at room temperature and pressure. This simple implementation depends on special kinds of mixed states, called pseudo-pure states, whose preparation entails a loss of signal that is exponential in the number of spins. While this would appear to limit such an implementation to ca. 8-10 spins in the foreseeable future, NMR spectroscopy has now permitted the first experimental demonstrations of all the basic features of quantum computing. We claim, moreover, that the product operator formalism, on which the theory of NM R spectroscopy is based, provides an efficient framework within which to analyze algorithms and decoherence effects in quantum computing more generally. This is illustrated by presenting our recent experimentally implementation of a quantum error correcting code.
We present a general method which expresses a unitary operator by the product of operators allowed by the Hamiltonian of spin-1/2 systems. In this method, the generator of an operator is found first, and then the generator is expanded by the base operators of the product operator formalism. Finally, the base operators disallowed by the Hamiltonian, including more than two-body interaction operators, are replaced by allowed ones by the axes transformation and coupling order reduction technique. This method directly provides pulse sequences for the nuclear magnetic resonance quantum computer, and can be generally applied to other systems.
Quantum simulation is a promising near term application for quantum information processors with the potential to solve computationally intractable problems using just a few dozen interacting qubits. A range of experimental platforms have recently demonstrated the basic functionality of quantum simulation applied to quantum magnetism, quantum phase transitions and relativistic quantum mechanics. However, in all cases, the physics of the underlying hardware restricts the achievable inter-particle interactions and forms a serious constraint on the versatility of the simulators. To broaden the scope of these analog devices, we develop a suite of pulse sequences that permit a user to efficiently realize average Hamiltonians that are beyond the native interactions of the system. Specifically, this approach permits the generation of all symmetrically coupled translation-invariant two-body Hamiltonians with homogeneous on-site terms, a class which includes all spin- XYZ chains, but generalized to include long-range couplings. Our work builds on previous work proving that universal simulation is possible using both entangling gates and single-qubit unitaries. We show that determining the appropriate 'program' of unitary pulse sequences which implements an arbitrary Hamiltonian transformation can be formulated as a linear program over functions defined by those pulse sequences, running in polynomial time and scaling efficiently in hardware resources. Our analysis extends from circuit model quantum information to adiabatic quantum evolutions, representing an important and broad-based success in applying functional analysis to the field of quantum information.
This article shows how C-numerical-range related new strucures may arise from practical problems in quantum control – and vice versa, how an understanding of these structures helps to tackle hot topics in quantum information. We start out with an overview on the role of C-numerical ranges in current research problems in quantum theory: the quantum mechanical task of maximising the projection of a point on the unitary orbit of an initial state onto a target state C relates to the C-numerical radius of A via maximising the trace function . In quantum control of n qubits one may be interested (i) in having U∈SU(2 n ) for the entire dynamics, or (ii) in restricting the dynamics to local operations on each qubit, i.e., to the n-fold tensor product . Interestingly, the latter then leads to a novel entity, the local C-numerical range W loc(C,A), whose intricate geometry is neither star-shaped nor simply connected in contrast to the conventional C-numerical range. This is shown in the accompanying article on Relative C-Numerical Ranges for Application in Quantum Control and Quantum Information [Dirr, G., Helmke, U., Kleinsteuber, M. and Schulte-Herbrüggen, T., 2008 Dirr, G, Helmke, U, Kleinsteuber, M and Schulte-Herbrüggen, T. 2008. Relative C-numerical ranges for application in quantum control and quantum information. Linear and Multilinear Algebra, 56: 27–51. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar], Linear and Multilinear Algebra, 56, 27–51]. We present novel applications of the C-numerical range in quantum control assisted by gradient flows on the local unitary group: (1) they serve as powerful tools for deciding whether a quantum interaction can be inverted in time (in a sense generalising Hahn's famous spin echo); (2) they allow for optimising witnesses of quantum entanglement. (3) We conclude by connecting the relative C-numerical range to problems of constrained quantum optimisation, for which we also give Lagrange-type gradient-flow algorithms.
Experiments in coherent magnetic resonance, microwave, and optical spectroscopy control quantum-mechanical ensembles by guiding them from initial states toward target states by unitary transformation. Often, the coherences detected as signals are represented by a non-Hermitian operator. Hence, spectroscopic experiments, such as those used in nuclear magnetic resonance, correspond to unitary transformations between operators that in general are not Hermitian. A gradient-based systematic procedure for optimizing these transformations is described that finds the largest projection of a transformed initial operator onto the target operator and, thus, the maximum spectroscopic signal. This method can also be used in applied mathematics and control theory.