scholarly journals Universal Qudit Hamiltonians

2021 ◽  
Author(s):  
Stephen Piddock ◽  
Ashley Montanaro
Keyword(s):  
State Energy ◽  
Entangled State ◽  
Quantum Computers ◽  
List Type ◽  
Universal Family ◽  
Finite Dimensional ◽  
Universal Quantum ◽  
Aklt Model ◽  
Almost All ◽  
Natural Way

AbstractA family of quantum Hamiltonians is said to be universal if any other finite-dimensional Hamiltonian can be approximately encoded within the low-energy space of a Hamiltonian from that family. If the encoding is efficient, universal families of Hamiltonians can be used as universal analogue quantum simulators and universal quantum computers, and the problem of approximately determining the ground-state energy of a Hamiltonian from a universal family is QMA-complete. One natural way to categorise Hamiltonians into families is in terms of the interactions they are built from. Here we prove universality of some important classes of interactions on qudits (d-level systems): We completely characterise the k-qudit interactions which are universal, if augmented with arbitrary Hermitian 1-local terms. We find that, for all $$k \geqslant 2$$ k ⩾ 2 and all local dimensions $$d \geqslant 2$$ d ⩾ 2 , almost all such interactions are universal aside from a simple stoquastic class. We prove universality of generalisations of the Heisenberg model that are ubiquitous in condensed-matter physics, even if free 1-local terms are not provided. We show that the SU(d) and SU(2) Heisenberg interactions are universal for all local dimensions $$d \geqslant 2$$ d ⩾ 2 (spin $$\geqslant 1/2$$ ⩾ 1 / 2 ), implying that a quantum variant of the Max-d-Cut problem is QMA-complete. We also show that for $$d=3$$ d = 3 all bilinear-biquadratic Heisenberg interactions are universal. One example is the general AKLT model. We prove universality of any interaction proportional to the projector onto a pure entangled state.

scholarly journals A UNIVERSAL QUANTUM ESTIMATOR

2005 ◽  
Vol 03 (supp01) ◽  
pp. 123-132
Author(s):  
L. C. KWEK ◽  
K. W. CHOO ◽  
JIANGFENG DU ◽  
ARTUR K. EKERT ◽  
CAROLINA MOURA ALVES ◽  
...  
Keyword(s):  
Building Blocks ◽  
Quantum Gates ◽  
Quantum Computers ◽  
Quantum Networks ◽  
Modern Computer ◽  
Universal Quantum ◽  
Almost All

Almost all computational tasks in the modern computer can be designed from basic building blocks. These building blocks provide a powerful and efficient language for describing algorithms. In quantum computers, the basic building blocks are the quantum gates. In this tutorial, we will look at quantum gates that act on one and two qubits and briefly discuss how these gates can be used in quantum networks.

Recent results in trapped-ion quantum computing at NIST

2001 ◽  
Vol 1 (Special) ◽  
pp. 113-123
Author(s):  
D. Kielpinski ◽  
A. Ben-Kish ◽  
J. Britton ◽  
V. Meyer ◽  
M.A. Rowe ◽  
...  
Keyword(s):  
The State ◽  
Entangled State ◽  
Bell’S Inequality ◽  
Ambient Conditions ◽  
Bell's Inequality ◽  
Quantum Register ◽  
Sigma Level ◽  
Universal Quantum ◽  
Encoding Method ◽  
First Time

We review recent experiments on entanglement, Bell's inequality, and decoherence-free subspaces in a quantum register of trapped {9Be+} ions. We have demonstrated entanglement of up to four ions using the technique of Molmer and Sorensen. This method produces the state ({|\uparrow\uparrow\rangle}+{|\downarrow\downarrow\rangle})/\sqrt{2} for two ions and the state ({\downarrow}{\downarrow}{\downarrow}{\downarrow} \rangle + | {\uparrow}{\uparrow}{\uparrow}{\uparrow} \rangle)/\sqrt{2} for four ions. We generate the entanglement deterministically in each shot of the experiment. Measurements on the two-ion entangled state violates Bell's inequality at the 8\sigma level. Because of the high detector efficiency of our apparatus, this experiment closes the detector loophole for Bell's inequality measurements for the first time. This measurement is also the first violation of Bell's inequality by massive particles that does not implicitly assume results from quantum mechanics. Finally, we have demonstrated reversible encoding of an arbitrary qubit, originally contained in one ion, into a decoherence-free subspace (DFS) of two ions. The DFS-encoded qubit resists applied collective dephasing noise and retains coherence under ambient conditions 3.6 times longer than does an unencoded qubit. The encoding method, which uses single-ion gates and the two-ion entangling gate, demonstrates all the elements required for two-qubit universal quantum logic.

scholarly journals Asymptotic velocity for four celestial bodies

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170426
Author(s):  
Andreas Knauf
Keyword(s):  
Celestial Mechanics ◽  
Integrable Systems ◽  
Energy Surface ◽  
Initial Conditions ◽  
Theme Issue ◽  
Asymptotic Velocity ◽  
Pair Potentials ◽  
Finite Dimensional ◽  
Celestial Bodies ◽  
Almost All

Asymptotic velocity is defined as the Cesàro limit of velocity. As such, its existence has been proved for bounded interaction potentials. This is known to be wrong in celestial mechanics with four or more bodies. Here, we show for a class of pair potentials including the homogeneous ones of degree − α for α ∈(0, 2), that asymptotic velocities exist for up to four bodies, dimension three or larger, for any energy and almost all initial conditions on the energy surface. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

Derivations Tangential to Compact Group Actions: Spectral Conditions in the Weak Closure

1985 ◽  
Vol 37 (1) ◽  
pp. 160-192 ◽  
Author(s):  
Ola Bratteli ◽  
Frederick M. Goodman
Keyword(s):  
Finite Elements ◽  
Common Core ◽  
Dimensional Space ◽  
List Type ◽  
Finite Dimensional Space ◽  
Type Number ◽  
Compact Lie Group ◽  
Parameter Group ◽  
Finite Dimensional ◽  
Image Position

Let G be a compact Lie group and a an action of G on a C*-algebra as *-automorphisms. Let denote the set of G-finite elements for this action, i.e., the set of those such that the orbit {αg(x):g ∊ G} spans a finite dimensional space. is a common core for all the *-derivations generating one-parameter subgroups of the action α. Now let δ be a *-derivation with domain such that Let us pose the following two problems:Is δ closable, and is the closure of δ the generator of a strongly continuous one-parameter group of *-automorphisms?If is simple or prime, under what conditions does δ have a decompositionwhere is the generator of a one-parameter subgroup of α(G) and is a bounded, or approximately bounded derivation?

A Remark on Disintegrations With Almost All Components Non -Additive

1979 ◽  
Vol 31 (4) ◽  
pp. 786-788 ◽  
Author(s):  
Nghiem Dang-Ngoc
Keyword(s):  
Real Function ◽  
Measure Space ◽  
Gibbs States ◽  
List Type ◽  
Additive Measure ◽  
Finitely Additive Measure ◽  
Coin Tossing ◽  
Definition Of ◽  
Almost All

We extend a theorem of L. E. Dubins on “purely finitely additive disintegrations” of measures (cf. [4]) and apply this result to the disintegrations of extremal Gibbs states with respect to the asymptotic algebra enlarging another result of L. E. Dubins on the symmetric coin tossing game.We recall the following definition of L. E. Dubins (cf. [3], [4]): Let (X , , μ) be a measure space, a sub σ-algebra of . A real function σx (A), is called a measurable-disintegration of μ if:(i) ∀x ∊ X , σx(.) is a finitely additive measure .(ii) ∀A ∊ , σ. (A) is constant on each -atom.(iii) For each A ∊ , σ. (A) is measurable with respect to the completion of by μ and (iv)σx(B) = 1 if x ∊ B ∊ .

Groups with Finite Dimensional Irreducible Multiplier Representations

1985 ◽  
Vol 37 (4) ◽  
pp. 635-643 ◽  
Author(s):  
A. K. Holzherr
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Sufficient Conditions ◽  
Locally Compact Group ◽  
Identity Operator ◽  
Central Projection ◽  
List Type ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Necessary And Sufficient

Let G be a locally compact group and ω a normalized multiplier on G. Denote by V(G) (respectively by V(G, ω)) the von Neumann algebra generated by the regular representation (respectively co-regular representation) of G. Kaniuth [6] and Taylor [14] have characterized those G for which the maximal type I finite central projection in V(G) is non-zero (respectively the identity operator in V(G)).In this paper we determine necessary and sufficient conditions on G and ω such that the maximal type / finite central projection in V(G, ω) is non-zero (respectively the identity operator in V(G, ω)) and construct this projection explicitly as a convolution operator on L2(G). As a consequence we prove the following statements are equivalent,(i) V(G, ω) is type I finite,(ii) all irreducible multiplier representations of G are finite dimensional,(iii) Gω (the central extension of G) is a Moore group, that is all its irreducible (ordinary) representations are finite dimensional.

scholarly journals Assessing the quantumness of the annealing dynamics via Leggett Garg’s inequalities: a weak measurement approach

2019 ◽  
Vol 9 (1) ◽  
Author(s):  
V. Vitale ◽  
G. De Filippis ◽  
A. de Candia ◽  
A. Tagliacozzo ◽  
V. Cataudella ◽  
...  
Keyword(s):  
Quantum Computation ◽  
Quantum Computers ◽  
Quantum Annealing ◽  
Adiabatic Quantum Computation ◽  
Universal Quantum ◽  
Full Quantum ◽  
Measurement Approach ◽  
Coherent Behavior ◽  
Sizable Number

Abstract Adiabatic quantum computation (AQC) is a promising counterpart of universal quantum computation, based on the key concept of quantum annealing (QA). QA is claimed to be at the basis of commercial quantum computers and benefits from the fact that the detrimental role of decoherence and dephasing seems to have poor impact on the annealing towards the ground state. While many papers show interesting optimization results with a sizable number of qubits, a clear evidence of a full quantum coherent behavior during the whole annealing procedure is still lacking. In this paper we show that quantum non-demolition (weak) measurements of Leggett Garg inequalities can be used to efficiently assess the quantumness of the QA procedure. Numerical simulations based on a weak coupling Lindblad approach are compared with classical Langevin simulations to support our statements.

scholarly journals QUANTUM ENTANGLEMENT TRANSFER BETWEEN SPIN PAIRS

2010 ◽  
Vol 08 (07) ◽  
pp. 1111-1120 ◽  
Author(s):  
QING-YOU MENG ◽  
FU-LIN ZHANG ◽  
JING-LING CHEN
Keyword(s):  
Quantum Entanglement ◽  
Entangled State ◽  
Specific Time ◽  
Initial State ◽  
Maximally Entangled State ◽  
Initial States ◽  
Entanglement Transfer ◽  
Spin Pairs ◽  
Almost All

The transfer of entanglement from source particles (SPs) to target particles (TPs) via the Heisenberg interaction H = s1 ⋅ s2 has been investigated. In our research, TPs are two qubits and SPs are two qubits or qutrits. When TPs are two qubits, we find that no matter what state the TPs are initially prepared in, at the specific time t = π the quantity of entanglement of the TPs can attain 1 after interaction with the SPs which stay on the maximally entangled state. When TPs are two qutrits, the maximal quantity of entanglement of the TPs is proportional to the quantity of entanglement of the initial state of the TPs and cannot attain 1 for almost all the initial states of the TPs. Here we propose an iterated operation which can make the TPs go to the maximal entangled state.

scholarly journals Complete slices and homological properties of tilted algebras

1994 ◽  
Vol 36 (3) ◽  
pp. 347-354 ◽  
Author(s):  
Ibrahim Assem ◽  
Flávio Ulhoa Coelho
Keyword(s):  
Representation Theory ◽  
Global Dimension ◽  
Indecomposable Modules ◽  
Tilted Algebra ◽  
Finite Dimensional ◽  
Tilted Algebras ◽  
Homological Dimensions ◽  
The Subject ◽  
Left And Right ◽  
Almost All

It is reasonable to expect that the representation theory of an algebra (finite dimensional over a field, basic and connected) can be used to study its homological properties. In particular, much is known about the structure of the Auslander-Reiten quiver of an algebra, which records most of the information we have on its module category. We ask whether one can predict the homological dimensions of a module from its position in the Auslander-Reiten quiver. We are particularly interested in the case where the algebra is a tilted algebra. This class of algebras of global dimension two, introduced by Happel and Ringel in [7], has since then been the subject of many investigations, and its representation theory is well understood by now (see, for instance, [1], [7], [8], [9], [11], [13]).In this case, the most striking feature of the Auslander-Reiten quiver is the existence of complete slices, which reproduce the quiver of the hereditary algebra from which the tilted algebra arises. It follows from well-known results that any indecomposable successor (or predecessor) of a complete slice has injective (or projective, respectively) dimension at most one, from which one deduces that a tilted algebra is representation-finite if and only if both the projective and the injective dimensions of almost all (that is, all but at most finitely many non-isomorphic) indecomposable modules equal two (see (3.1) and (3.2)). On the other hand, the authors have shown in [2, (3.4)] that a representation-infinite algebra is concealed if and only if both the projective and the injective dimensions of almost all indecomposable modules equal one (see also [14]). This leads us to consider, for tilted algebras which are not concealed, the case when the projective (or injective) dimension of almost all indecomposable successors (or predecessors, respectively) of a complete slice equal two. In order to answer this question, we define the notions of left and right type of a tilted algebra, then those of reduced left and right types (see (2.2) and (3.4) for the definitions).

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