scholarly journals Quantum Computing, Seifert Surfaces, and Singular Fibers

2019 ◽  
Vol 1 (1) ◽  
pp. 12-22 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Quantum Computer ◽  
Complete Information ◽  
Singular Fiber ◽  
Dynkin Diagrams ◽  
Singular Fibers ◽  
Seifert Surfaces ◽  
Trefoil Knot ◽  
Universal Quantum

The fundamental group π 1 ( L ) of a knot or link L may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the L of such a quantum computer model and computes their braid-induced Seifert surfaces and the corresponding Alexander polynomial. In particular, some d-fold coverings of the trefoil knot, with d = 3 , 4, 6, or 12, define appropriate links L, and the latter two cases connect to the Dynkin diagrams of E 6 and D 4 , respectively. In this new context, one finds that this correspondence continues with Kodaira’s classification of elliptic singular fibers. The Seifert fibered toroidal manifold Σ ′ , at the boundary of the singular fiber E 8 ˜ , allows possible models of quantum computing.

scholarly journals Benchmarking an 11-qubit quantum computer

2019 ◽  
Vol 10 (1) ◽  
Author(s):  
K. Wright ◽  
K. M. Beck ◽  
S. Debnath ◽  
J. M. Amini ◽  
Y. Nam ◽  
...  
Keyword(s):  
Quantum Computing ◽  
Quantum Physics ◽  
Quantum Computer ◽  
Success Rates ◽  
Single Qubit ◽  
The Past ◽  
Large Numbers ◽  
Universal Quantum ◽  
Qubit Gate ◽  
Fully Connected

AbstractThe field of quantum computing has grown from concept to demonstration devices over the past 20 years. Universal quantum computing offers efficiency in approaching problems of scientific and commercial interest, such as factoring large numbers, searching databases, simulating intractable models from quantum physics, and optimizing complex cost functions. Here, we present an 11-qubit fully-connected, programmable quantum computer in a trapped ion system composed of 13 171Yb+ ions. We demonstrate average single-qubit gate fidelities of 99.5$$\%$$%, average two-qubit-gate fidelities of 97.5$$\%$$%, and SPAM errors of 0.7$$\%$$%. To illustrate the capabilities of this universal platform and provide a basis for comparison with similarly-sized devices, we compile the Bernstein-Vazirani and Hidden Shift algorithms into our native gates and execute them on the hardware with average success rates of 78$$\%$$% and 35$$\%$$%, respectively. These algorithms serve as excellent benchmarks for any type of quantum hardware, and show that our system outperforms all other currently available hardware.

scholarly journals Quantum computing thanks to Bianchi groups

2019 ◽  
Vol 198 ◽  
pp. 00012 ◽  
Author(s):  
Michel Planat
Keyword(s):  
Quantum Computing ◽  
Trefoil Knot ◽  
Integer Ring ◽  
Universal Quantum ◽  
A Chain ◽  
Low Dimensional ◽  
Free Subgroups ◽  
Knots And Links ◽  
Good Agreement ◽  
Positive Operator Valued Measure

It has been shown that the concept of a magic state (in universal quantum computing: uqc) and that of a minimal informationally complete positive operator valued measure: MIC-POVMs (in quantum measurements) are in good agreement when such a magic state is selected in the set of non-stabilizer eigenstates of permutation gates with the Pauli group acting on it [1]. Further work observed that most found low-dimensional MICs may be built from subgroups of the modular group PS L(2, Z) [2] and that this can be understood from the picture of the trefoil knot and related 3-manifolds [3]. Here one concentrates on Bianchi groups PS L(2, O10) (with O10 the integer ring over the imaginary quadratic field) whose torsion-free subgroups define the appropriate knots and links leading to MICs and the related uqc. One finds a chain of Bianchi congruence n-cusped links playing a significant role [4].

scholarly journals Universal Quantum Computing and Three-Manifolds

2018 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Modular Group ◽  
Bloch Sphere ◽  
Borromean Rings ◽  
Single Qubit ◽  
Whitehead Link ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Dehn Fillings ◽  
Knots And Links

A single qubit may be represented on the Bloch sphere or similarly on the $3$-sphere $S^3$. Our goal is to dress this correspondence by converting the language of universal quantum computing (uqc) to that of $3$-manifolds. A magic state and the Pauli group acting on it define a model of uqc as a POVM that one recognizes to be a $3$-manifold $M^3$. E. g., the $d$-dimensional POVMs defined from subgroups of finite index of the modular group $PSL(2,\mathbb{Z})$ correspond to $d$-fold $M^3$- coverings over the trefoil knot. In this paper, one also investigates quantum information on a few \lq universal' knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on SnapPy. Further connections between POVMs based uqc and $M^3$'s obtained from Dehn fillings are explored.

scholarly journals The Magic of Universal Quantum Computing with Permutations

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Michel Planat ◽  
Rukhsan Ul Haq
Keyword(s):  
Quantum Computing ◽  
Wigner Function ◽  
Low Dimensions ◽  
State Dependent ◽  
Universal Quantum ◽  
Discrete Wigner Function

The role of permutation gates for universal quantum computing is investigated. The “magic” of computation is clarified in the permutation gates, their eigenstates, the Wootters discrete Wigner function, and state-dependent contextuality (following many contributions on this subject). A first classification of a few types of resulting magic states in low dimensions d≤9 is performed.

scholarly journals Group Geometrical Axioms for Magic States of Quantum Computing

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 948 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo M. Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Fundamental Group ◽  
Free Group ◽  
Normal Closure ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Nontrivial Subgroup ◽  
Low Dimensional ◽  
Figure Eight Knot ◽  
Stabilizer States

Let H be a nontrivial subgroup of index d of a free group G and N be the normal closure of H in G. The coset organization in a subgroup H of G provides a group P of permutation gates whose common eigenstates are either stabilizer states of the Pauli group or magic states for universal quantum computing. A subset of magic states consists of states associated to minimal informationally complete measurements, called MIC states. It is shown that, in most cases, the existence of a MIC state entails the two conditions (i) N = G and (ii) no geometry (a triple of cosets cannot produce equal pairwise stabilizer subgroups) or that these conditions are both not satisfied. Our claim is verified by defining the low dimensional MIC states from subgroups of the fundamental group G = π 1 ( M ) of some manifolds encountered in our recent papers, e.g., the 3-manifolds attached to the trefoil knot and the figure-eight knot, and the 4-manifolds defined by 0-surgery of them. Exceptions to the aforementioned rule are classified in terms of geometric contextuality (which occurs when cosets on a line of the geometry do not all mutually commute).

scholarly journals Universal Quantum Computing and Three-Manifolds

2018 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Bloch Sphere ◽  
Borromean Rings ◽  
Single Qubit ◽  
Whitehead Link ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Dehn Fillings ◽  
Knots And Links ◽  
Positive Operator Valued Measure

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S3. Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold M3. More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group PSL(2, Z) correspond to d-fold M3- coverings over the trefoil knot. In this paper, one also investigates quantum information on a few ‘universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and M3’s obtained from Dehn fillings are explored.

QUANTUM DOT COMPUTING GATES

2006 ◽  
Vol 04 (02) ◽  
pp. 233-296 ◽  
Author(s):  
GOONG CHEN ◽  
ZIJIAN DIAO ◽  
JONG U. KIM ◽  
ARUP NEOGI ◽  
KERIM URTEKIN ◽  
...  
Keyword(s):  
Quantum Dots ◽  
Quantum Computing ◽  
Quantum Dot ◽  
Quantum Computer ◽  
Semiconductor Quantum Dots ◽  
Quantum Gates ◽  
Promising Candidate ◽  
Universal Quantum ◽  
Physical Attributes

Semiconductor quantum dots are a promising candidate for future quantum computer devices. Presently, there are three major proposals for designing quantum computing gates based on quantum dot technology: (i) electrons trapped in microcavity; (ii) spintronics; (iii) biexcitons. We survey these designs and show mathematically how, in principle, they will generate 1-bit rotation gates as well as 2-bit entanglement and, thus, provide a class of universal quantum gates. Some physical attributes and issues related to their limitations, decoherence and measurement are also discussed.

scholarly journals Universal Quantum Computing and Three-Manifolds

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 773 ◽  
Author(s):  
Michel Planat ◽  
Raymond Aschheim ◽  
Marcelo Amaral ◽  
Klee Irwin
Keyword(s):  
Quantum Computing ◽  
Bloch Sphere ◽  
Borromean Rings ◽  
Single Qubit ◽  
Whitehead Link ◽  
Trefoil Knot ◽  
Universal Quantum ◽  
Dehn Fillings ◽  
Knots And Links ◽  
Positive Operator Valued Measure

A single qubit may be represented on the Bloch sphere or similarly on the 3-sphere S 3 . Our goal is to dress this correspondence by converting the language of universal quantum computing (UQC) to that of 3-manifolds. A magic state and the Pauli group acting on it define a model of UQC as a positive operator-valued measure (POVM) that one recognizes to be a 3-manifold M 3 . More precisely, the d-dimensional POVMs defined from subgroups of finite index of the modular group P S L ( 2 , Z ) correspond to d-fold M 3 - coverings over the trefoil knot. In this paper, we also investigate quantum information on a few ‘universal’ knots and links such as the figure-of-eight knot, the Whitehead link and Borromean rings, making use of the catalog of platonic manifolds available on the software SnapPy. Further connections between POVMs based UQC and M 3 ’s obtained from Dehn fillings are explored.

scholarly journals Universal quantum computation via quantum controlled classical operations

2021 ◽  
Author(s):  
Sebastian Horvat ◽  
Xiaoqin Gao ◽  
Borivoje Dakic
Keyword(s):  
Quantum Computing ◽  
Control System ◽  
Quantum Computation ◽  
Quantum Control ◽  
Quantum Computer ◽  
Affirmative Answer ◽  
Quantum Gates ◽  
Processing Power ◽  
Universal Quantum ◽  
Hadamard Gate

Abstract A universal set of gates for (classical or quantum) computation is a set of gates that can be used to approximate any other operation. It is well known that a universal set for classical computation augmented with the Hadamard gate results in universal quantum computing. Motivated by the latter, we pose the following question: can one perform universal quantum computation by supplementing a set of classical gates with a quantum control, and a set of quantum gates operating solely on the latter? In this work we provide an affirmative answer to this question by considering a computational model that consists of 2n target bits together with a set of classical gates controlled by log(2n + 1) ancillary qubits. We show that this model is equivalent to a quantum computer operating on n qubits. Furthermore, we show that even a primitive computer that is capable of implementing only SWAP gates, can be lifted to universal quantum computing, if aided with an appropriate quantum control of logarithmic size. Our results thus exemplify the information processing power brought forth by the quantum control system.

Universal quantum computing in linear nearest neighbor architectures

2011 ◽  
Vol 11 (3&4) ◽  
pp. 300-312
Author(s):  
Preethika Kumar ◽  
Steven R. Skinner
Keyword(s):  
Quantum Computing ◽  
Solid State ◽  
Control Parameter ◽  
Nearest Neighbor ◽  
Quantum Computer ◽  
Gate Operation ◽  
One Dimensional ◽  
Single Qubit ◽  
Universal Quantum ◽  
Target Qubit

We introduce a scheme for realizing universal quantum computing in a linear nearest neighbor architecture with fixed couplings. We first show how to realize a controlled-NOT gate operation between two adjacent qubits without having to isolate the two qubits from qubits adjacent to them. The gate operation is implemented by applying two consecutive pulses of equal duration, but varying amplitudes, on the target qubit. Since only a single control parameter is required in implementing our scheme, it is very efficient. We next show how our scheme can be used to realize single qubit rotations and two-qubit controlled-unitary operations. As most proposals for solid state implementations of a quantum computer use a one-dimensional line of qubits, the schemes presented here will be extremely useful.

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