scholarly journals Solovay–Kitaev Approximations of Special Orthogonal Matrices

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Anuradha Mahasinghe ◽  
Sachiththa Bandaranayake ◽  
Kaushika De Silva
Keyword(s):  
Quantum Computing ◽  
Computational Experience ◽  
Quantum Gate ◽  
Unitary Matrix ◽  
Short Sequence ◽  
Efficient Computation ◽  
Orthogonal Matrices ◽  
Exact Bounds ◽  
Arbitrary Quantum

The circuit-gate framework of quantum computing relies on the fact that an arbitrary quantum gate in the form of a unitary matrix of unit determinant can be approximated to a desired accuracy by a fairly short sequence of basic gates, of which the exact bounds are provided by the Solovay–Kitaev theorem. In this work, we show that a version of this theorem is applicable to orthogonal matrices with unit determinant as well, indicating the possibility of using orthogonal matrices for efficient computation. We further develop a version of the Solovay–Kitaev algorithm and discuss the computational experience.

scholarly journals DECOMPOSITION OF UNITARY MATRICES AND QUANTUM GATES

2013 ◽  
Vol 11 (01) ◽  
pp. 1350015 ◽  
Author(s):  
CHI-KWONG LI ◽  
REBECCA ROBERTS ◽  
XIAOYAN YIN
Keyword(s):  
General Scheme ◽  
Quantum Gate ◽  
Quantum Gates ◽  
Unitary Matrix ◽  
Additional Structure ◽  
Unitary Matrices ◽  
Decomposition Scheme ◽  
Single Qubit ◽  
Matlab Program

A general scheme is presented to decompose a d-by-d unitary matrix as the product of two-level unitary matrices with additional structure and prescribed determinants. In particular, the decomposition can be done by using two-level matrices in d - 1 classes, where each class is isomorphic to the group of 2 × 2 unitary matrices. The proposed scheme is easy to apply, and useful in treating problems with the additional structural restrictions. A Matlab program is written to implement the scheme, and the result is used to deduce the fact that every quantum gate acting on n-qubit registers can be expressed as no more than 2n-1(2n-1) fully controlled single-qubit gates chosen from 2n-1 classes, where the quantum gates in each class share the same n - 1 control qubits. Moreover, it is shown that one can easily adjust the proposed decomposition scheme to take advantage of additional structure evolving in the process.

scholarly journals Quantum learning with noise and decoherence: a robust quantum neural network

2019 ◽  
Author(s):  
Elizabeth Behrman ◽  
Nam Nguyen ◽  
James Steck
Keyword(s):  
Neural Network ◽  
Quantum Computing ◽  
Large Scale ◽  
Learning Approaches ◽  
Quantum Neural Network ◽  
Quantum Learning ◽  
Learning With Noise ◽  
Robustness To Noise ◽  
Robust To Noise ◽  
Arbitrary Quantum

<p>Noise and decoherence are two major obstacles to the implementation of large-scale quantum computing. Because of the no-cloning theorem, which says we cannot make an exact copy of an arbitrary quantum state, simple redundancy will not work in a quantum context, and unwanted interactions with the environment can destroy coherence and thus the quantum nature of the computation. Because of the parallel and distributed nature of classical neural networks, they have long been successfully used to deal with incomplete or damaged data. In this work, we show that our model of a quantum neural network (QNN) is similarly robust to noise, and that, in addition, it is robust to decoherence. Moreover, robustness to noise and decoherence is not only maintained but improved as the size of the system is increased. Noise and decoherence may even be of advantage in training, as it helps correct for overfitting. We demonstrate the robustness using entanglement as a means for pattern storage in a qubit array. Our results provide evidence that machine learning approaches can obviate otherwise recalcitrant problems in quantum computing. </p> <p> </p>

The Essence of Quantum Computing

2018 ◽  
Author(s):  
Rajendra K. Bera
Keyword(s):  
Quantum Computing ◽  
Fourier Series ◽  
Linear Algebra ◽  
Quantum Algorithms ◽  
Quantum States ◽  
Efficient Computation

In Part I we laid the foundation on which quantum algorithms are built. In this part we harness such exotic aspects as superposition, entanglement and collapse of quantum states of that foundation to show how powerful quantum algorithms can be constructed for efficient computation. Appendixes A and B are provided to jog the memory of those who are recently out of touch with linear algebra and Fourier series.

scholarly journals Towards optimization of quantum circuits

Open Physics ◽  
2008 ◽  
Vol 6 (1) ◽  
Author(s):  
Michal Sedlák ◽  
Martin Plesch
Keyword(s):  
Quantum Information ◽  
Quantum Information Processing ◽  
Quantum Circuit ◽  
Quantum Gate ◽  
Quantum Circuits ◽  
Quantum Gates ◽  
Unitary Operation ◽  
Short Sequence ◽  
Toffoli Gates ◽  
Universal Procedure

AbstractAny unitary operation in quantum information processing can be implemented via a sequence of simpler steps — quantum gates. However, actual implementation of a quantum gate is always imperfect and takes a finite time. Therefore, searching for a short sequence of gates — efficient quantum circuit for a given operation, is an important task. We contribute to this issue by proposing optimization of the well-known universal procedure proposed by Barenco et al. [Phys. Rev. A 52, 3457 (1995)]. We also created a computer program which realizes both Barenco’s decomposition and the proposed optimization. Furthermore, our optimization can be applied to any quantum circuit containing generalized Toffoli gates, including basic quantum gate circuits.

Transport of quantum states and separation of ions in a dual RF ion trap

2002 ◽  
Vol 2 (4) ◽  
pp. 257-271 ◽  
Author(s):  
M.A. Rowe ◽  
A. Ben-Kish ◽  
B. DeMarco ◽  
D. Leibfried ◽  
V. Meyer ◽  
...  
Keyword(s):  
Quantum Computing ◽  
Building Block ◽  
Ion Trap ◽  
Internal State ◽  
Ion Traps ◽  
Quantum States ◽  
Unit Cell ◽  
Linear Ion Trap ◽  
Ion Dynamics ◽  
Arbitrary Quantum

We have investigated ion dynamics associated with a dual linear ion trap where ions can be stored in and moved between two distinct locations. Such a trap is a building block for a system to engineer arbitrary quantum states of ion ensembles. Specifically, this trap is the unit cell in a strategy for scalable quantum computing using a series of interconnected ion traps. We have transferred an ion between trap locations 1.2 mm apart in 50 $\mu$s with near unit efficiency ($> 10^{6}$ consecutive transfers) and negligible motional heating, while maintaining internal-state coherence. In addition, we have separated two ions held in a common trap into two distinct traps.

scholarly journals Preparation of an arbitrary two-qubit quantum gate on two spins with an anisotropic Heisenberg interaction

2018 ◽  
Vol 16 (05) ◽  
pp. 1850044 ◽  
Author(s):  
A. R. Kuzmak
Keyword(s):  
Magnetic Field ◽  
Physical System ◽  
Optical Lattice ◽  
Ultracold Atoms ◽  
Quantum Gate ◽  
Pulsed Magnetic Field ◽  
Second Step ◽  
Step Method ◽  
Arbitrary Quantum

We consider the two-step method [A. R. Kuzmak and V. M. Tkachuk, Phys. Lett. A 378 (2014) 1469] for preparation of an arbitrary quantum gate on two spins with anisotropic Heisenberg interaction. At the first step, the system evolves during some period of time. At the second step, we apply pulsed magnetic field individually to each spin. We obtain the conditions for realization of SWAP, iSWAP, [Formula: see text] and entangled gates. Finally, we consider the implementation of this method on the physical system of ultracold atoms in optical lattice.

The Essence of Quantum Computing

2018 ◽  
Author(s):  
Rajendra K. Bera
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Quantum Computation ◽  
Quantum Algorithms ◽  
Underlying Mechanism ◽  
Quantum States ◽  
Quantum Computers ◽  
Efficient Computation

In Part I we laid the foundation on which quantum algorithms are built. In part II we harnessed such exotic aspects of quantum mechanics as superposition, entanglement and collapse of quantum states to show how powerful quantum algorithms can be constructed for efficient computation. In Part III (the concluding part) we discuss two aspects of quantum computation: (1) the problem of correcting errors that inevitably plague physical quantum computers during computations, by algorithmic means; and (2) a possible underlying mechanism for the collapse of the wave function during measurement.

UNITARY MATRICES AND COMMUTATORS

2011 ◽  
Vol 08 (07) ◽  
pp. 1583-1592
Author(s):  
Y. HARDY ◽  
W.-H. STEEB
Keyword(s):  
Quantum Computing ◽  
Complete Solution ◽  
Quantum Gates ◽  
Unitary Matrix ◽  
Kronecker Products ◽  
Unitary Matrices ◽  
Direct Sums

Unitary matrices are the quantum gates in quantum computing. We study the question under which conditions is the commutator of two unitary matrices again a unitary matrix. We consider the general case and a complete solution for the 2 × 2 unitary matrices. We also consider Kronecker products and direct sums of unitary matrices.

scholarly journals Scaling a Unitary Matrix

2014 ◽  
Vol 21 (04) ◽  
pp. 1450013 ◽  
Author(s):  
Alexis De Vos ◽  
Stijn De Baerdemacker
Keyword(s):  
Iterative Method ◽  
Quantum Circuit ◽  
Unitary Matrix ◽  
Real Matrix ◽  
Doubly Stochastic ◽  
Arbitrary Quantum

The iterative method of Sinkhorn allows, starting from an arbitrary real matrix with non-negative entries, to find a so-called ‘scaled matrix’ which is doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and with all line sums equal to 1. We conjecture that a similar procedure exists, which allows, starting from an arbitrary unitary matrix, to find a scaled matrix which is unitary and has all line sums equal to 1. The existence of such algorithm guarantees a powerful decomposition of an arbitrary quantum circuit.

Sign in / Sign up

Export Citation Format

Share Document