scholarly journals Quantum universality by state distillation

2009 ◽  
Vol 9 (11&12) ◽  
pp. 1030-1052
Author(s):  
B.W. Reichardt
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Fault Tolerant ◽  
Upper Bounds ◽  
Mixed States ◽  
Single Qubit ◽  
Noise Rate ◽  
Pure States ◽  
Rate Threshold ◽  
Stabilizer States

Quantum universality can be achieved using classically controlled stabilizer operations and repeated preparation of certain ancilla states. Which ancilla states suffice for universality? This ``magic states distillation" question is closely related to quantum fault tolerance. Lower bounds on the noise tolerable on the ancilla help give lower bounds on the tolerable noise rate threshold for fault-tolerant computation. Upper bounds show the limits of threshold upper-bound arguments based on the Gottesman-Knill theorem. We extend the range of single-qubit mixed states that are known to give universality, by using a simple parity-checking operation. For applications to proving threshold lower bounds, certain practical stability characteristics are often required, and we also show a stable distillation procedure.}{No distillation upper bounds are known beyond those given by the Gottesman-Knill theorem. One might ask whether distillation upper bounds reduce to upper bounds for single-qubit ancilla states. For multi-qubit pure states and previously considered two-qubit ancilla states, the answer is yes. However, we exhibit two-qubit mixed states that are not mixtures of stabilizer states, but for which every postselected stabilizer reduction from two qubits to one outputs a mixture of stabilizer states. Distilling such states would require true multi-qubit state distillation methods.

scholarly journals Improved upper bounds on the stabilizer rank of magic states

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 606
Author(s):  
Hammam Qassim ◽  
Hakop Pashayan ◽  
David Gosset
Keyword(s):  
Relative Error ◽  
Upper Bound ◽  
Linear Codes ◽  
Upper Bounds ◽  
Quantum Circuits ◽  
Symmetric Product ◽  
Simulation Algorithm ◽  
Single Qubit ◽  
Classical Algorithm ◽  
Stabilizer States

In this work we improve the runtime of recent classical algorithms for strong simulation of quantum circuits composed of Clifford and T gates. The improvement is obtained by establishing a new upper bound on the stabilizer rank of m copies of the magic state |T⟩=2−1(|0⟩+eiπ/4|1⟩) in the limit of large m. In particular, we show that |T⟩⊗m can be exactly expressed as a superposition of at most O(2αm) stabilizer states, where α≤0.3963, improving on the best previously known bound α≤0.463. This furnishes, via known techniques, a classical algorithm which approximates output probabilities of an n-qubit Clifford + T circuit U with m uses of the T gate to within a given inverse polynomial relative error using a runtime poly(n,m)2αm. We also provide improved upper bounds on the stabilizer rank of symmetric product states |ψ⟩⊗m more generally; as a consequence we obtain a strong simulation algorithm for circuits consisting of Clifford gates and m instances of any (fixed) single-qubit Z-rotation gate with runtime poly(n,m)2m/2. We suggest a method to further improve the upper bounds by constructing linear codes with certain properties.

Quantum cloning of identical mixed qubits

2007 ◽  
Vol 7 (5&6) ◽  
pp. 551-558
Author(s):  
H. Fan ◽  
B.-Y. Liu ◽  
K.-J. Shi
Keyword(s):  
Upper Bound ◽  
Quantum Cloning ◽  
Single Qubit ◽  
Antisymmetric State ◽  
Pure States ◽  
Symmetric States

Quantum cloning of two identical mixed qubits $\rho \otimes \rho$ is studied. We propose the quantum cloning transformations not only for the triplet (symmetric) states but also for the singlet (antisymmetric) state. We can copy these two identical mixed qubits to $M$ ($M\ge 2$) copies. This quantum cloning machine is optimal in the sense that the shrinking factor between the input and the output single qubit achieves the upper bound. The result shows that we can copy two identical mixed qubits with the same quality as that of two identical pure states.

scholarly journals The Essential Dimension of Stacks of Parabolic Vector Bundles over Curves

2012 ◽  
Vol 10 (3) ◽  
pp. 455-488 ◽  
Author(s):  
Indranil Biswas ◽  
Ajneet Dhillon ◽  
Nicole Lemire
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Vector Bundles ◽  
Upper Bounds ◽  
Essential Dimension ◽  
Parabolic Structure ◽  
Moduli Stack

AbstractWe find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our calculations also give lower bounds on the essential dimension of the semistable locus inside the moduli stack of vector bundles of rank r and degree d without parabolic structure.

scholarly journals Construction of genuinely entangled subspacesand the associated bounds on entanglement measures for mixed states

2021 ◽  
Author(s):  
Konstantin Antipin
Keyword(s):  
Information Processing ◽  
Quantum Information ◽  
Lower Bounds ◽  
Quantum Information Processing ◽  
Multipartite Entanglement ◽  
Quantum Channels ◽  
Valuable Resource ◽  
Mixed States ◽  
Entanglement Measures ◽  
Pure States

Abstract Genuine entanglement is the strongest form of multipartite entanglement. Genuinely entangled pure states contain entanglement in every bipartition and as such can be regarded as a valuable resource in the protocols of quantum information processing. A recent direction of research is the construction of genuinely entangled subspaces — the class of subspaces consisting entirely of genuinely entangled pure states. In this paper we present methods of construction of such subspaces including those of maximal possible dimension. The approach is based on the composition of bipartite entangled subspaces and quantum channels of certain types. The examples include maximal subspaces for systems of three qubits, four qubits, three qutrits. We also provide lower bounds on two entanglement measures for mixed states, the concurrence and the convex-roof extended negativity, which are directly connected with the projection on genuinely entangled subspaces.

The Heine-Borel theorem in extended basic logic

1949 ◽  
Vol 14 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Fundamental Theorem ◽  
Upper Bounds ◽  
Basic Logic ◽  
Real Numbers ◽  
Consistent Theory

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.

scholarly journals Class numbers of real cyclotomic fields of composite conductor

2014 ◽  
Vol 17 (A) ◽  
pp. 404-417 ◽  
Author(s):  
John C. Miller
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Upper Bounds ◽  
Class Numbers ◽  
Prime Ideals ◽  
Cyclotomic Fields ◽  
Class Field ◽  
Divisibility Properties ◽  
Composite Conductor ◽  
Hilbert Class Field

AbstractUntil recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds.However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method.In this paper, we study in particular the cyclotomic fields of composite conductor.

Sharp upper bounds on perfect retrieval in the Hopfield model

1999 ◽  
Vol 36 (03) ◽  
pp. 941-950 ◽  
Author(s):  
Anton Bovier
Keyword(s):  
Fixed Points ◽  
Lower Bounds ◽  
Upper Bound ◽  
Random Variables ◽  
Upper Bounds ◽  
Negative Association ◽  
Hopfield Model ◽  
Gradient Dynamics ◽  
One Step ◽  
The One

We prove a sharp upper bound on the number of patterns that can be stored in the Hopfield model if the stored patterns are required to be fixed points of the gradient dynamics. We also show corresponding bounds on the one-step convergence of the sequential gradient dynamics. The bounds coincide with the known lower bounds and confirm the heuristic expectations. The proof is based on a crucial idea of Loukianova (1997) using the negative association properties of some random variables arising in the analysis.

Connections between relative entropy of entanglement and geometric measure of entanglement

2004 ◽  
Vol 4 (4) ◽  
pp. 252-272
Author(s):  
T.-C. Wei ◽  
M. Ericsson ◽  
P.M. Goldbart ◽  
W.J. Munro
Keyword(s):  
Lower Bound ◽  
Relative Entropy ◽  
Upper Bounds ◽  
Mixed States ◽  
Geometric Measure ◽  
Entanglement Of Formation ◽  
Entanglement Measures ◽  
Pure States ◽  
Relative Entropy Of Entanglement ◽  
Geometric Measure Of Entanglement

As two of the most important entanglement measures---the entanglement of formation and the entanglement of distillation---have so far been limited to bipartite settings, the study of other entanglement measures for multipartite systems appears necessary. Here, connections between two other entanglement measures---the relative entropy of entanglement and the geometric measure of entanglement---are investigated. It is found that for arbitrary pure states the latter gives rise to a lower bound on the former. For certain pure states, some bipartite and some multipartite, this lower bound is saturated, and thus their relative entropy of entanglement can be found analytically in terms of their known geometric measure of entanglement. For certain mixed states, upper bounds on the relative entropy of entanglement are also established. Numerical evidence strongly suggests that these upper bounds are tight, i.e., they are actually the relative entropy of entanglement.

Optimization of coherent attacks in generalizations of the BB84 quantum bit commitment protocol

2002 ◽  
Vol 2 (1) ◽  
pp. 66-96
Author(s):  
R.W. Spekkens ◽  
T. Rudolph
Keyword(s):  
State Estimation ◽  
Quantum State ◽  
Upper Bound ◽  
Upper Bounds ◽  
Entangled State ◽  
Quantum State Estimation ◽  
Quantum Bit ◽  
Bit Commitment ◽  
Single Qubit ◽  
Qubit States

It is well known that no quantum bit commitment protocol is unconditionally secure. Nonetheless, there can be non-trivial upper bounds on both Bob's probability of correctly estimating Alice's commitment and Alice's probability of successfully unveiling whatever bit she desires. In this paper, we seek to determine these bounds for generalizations of the BB84 bit commitment protocol. In such protocols, an honest Alice commits to a bit by randomly choosing a state from a specified set and submitting this to Bob, and later unveils the bit to Bob by announcing the chosen state, at which point Bob measures the projector onto the state. Bob's optimal cheating strategy can be easily deduced from well known results in the theory of quantum state estimation. We show how to understand Alice's most general cheating strategy, (which involves her submitting to Bob one half of an entangled state) in terms of a theorem of Hughston, Jozsa and Wootters. We also show how the problem of optimizing Alice's cheating strategy for a fixed submitted state can be mapped onto a problem of state estimation. Finally, using the Bloch ball representation of qubit states, we identify the optimal coherent attack for a class of protocols that can be implemented with just a single qubit. These results provide a tight upper bound on Alice's probability of successfully unveiling whatever bit she desires in the protocol proposed by Aharonov et al., and lead us to identify a qubit protocol with even greater security.

scholarly journals Upper Bounds for Online Ramsey Games in Random Graphs

2009 ◽  
Vol 18 (1-2) ◽  
pp. 259-270 ◽  
Author(s):  
MARTIN MARCINISZYN ◽  
RETO SPÖHEL ◽  
ANGELIKA STEGER
Keyword(s):  
Large Class ◽  
Lower Bounds ◽  
Random Graphs ◽  
Upper Bound ◽  
Upper Bounds ◽  
Arbitrary Size ◽  
Threshold Functions ◽  
Current Graph ◽  
Player Game ◽  
Monochromatic Copy

Consider the following one-player game. Starting with the empty graph onnvertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one ofravailable colours. The player's goal is to avoid creating a monochromatic copy of some fixed graphFfor as long as possible. We prove an upper bound on the typical duration of this game ifFis from a large class of graphs including cliques and cycles of arbitrary size. Together with lower bounds published elsewhere, explicit threshold functions follow.

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