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 Number-Theoretic Characterizations of Some Restricted Clifford+T Circuits

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 252
Author(s):  
Matthew Amy ◽  
Andrew N. Glaudell ◽  
Neil J. Ross
Keyword(s):  
Unitary Matrix ◽  
Unitary Matrices ◽  
Single Qubit ◽  
Phase Gate ◽  
Hadamard Gate ◽  
Universal Gate

Kliuchnikov, Maslov, and Mosca proved in 2012 that a 2×2 unitary matrix V can be exactly represented by a single-qubit Clifford+T circuit if and only if the entries of V belong to the ring Z[1/2,i]. Later that year, Giles and Selinger showed that the same restriction applies to matrices that can be exactly represented by a multi-qubit Clifford+T circuit. These number-theoretic characterizations shed new light upon the structure of Clifford+T circuits and led to remarkable developments in the field of quantum compiling. In the present paper, we provide number-theoretic characterizations for certain restricted Clifford+T circuits by considering unitary matrices over subrings of Z[1/2,i]. We focus on the subrings Z[1/2], Z[1/2], Z[1/i2], and Z[1/2,i], and we prove that unitary matrices with entries in these rings correspond to circuits over well-known universal gate sets. In each case, the desired gate set is obtained by extending the set of classical reversible gates {X,CX,CCX} with an analogue of the Hadamard gate and an optional phase gate.

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.

The Solovay-Kitaev algorithm

2006 ◽  
Vol 6 (1) ◽  
pp. 81-95
Author(s):  
C.M. Dawson ◽  
M.A. Nielsen
Keyword(s):  
Fault Tolerant ◽  
Quantum Gate ◽  
Quantum Gates ◽  
Single Qubit ◽  
Classical Algorithm ◽  
Shor's Algorithm ◽  
Finite Set ◽  
Qubit Gate

This pedagogical review presents the proof of the Solovay-Kitaev theorem in the form of an efficient classical algorithm for compiling an arbitrary single-qubit gate into a sequence of gates from a fixed and finite set. The algorithm can be used, for example, to compile Shor's algorithm, which uses rotations of $\pi / 2^k$, into an efficient fault-tolerant form using only Hadamard, controlled-{\sc not}, and $\pi / 8$ gates. The algorithm runs in $O(\log^{2.71}(1/\epsilon))$ time, and produces as output a sequence of $O(\log^{3.97}(1/\epsilon))$ quantum gates which is guaranteed to approximate the desired quantum gate to an accuracy within $\epsilon > 0$. We also explain how the algorithm can be generalized to apply to multi-qubit gates and to gates from SU(d).

scholarly journals Efficient quantum gates and algorithms in an engineered optical lattice

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
A. H. Homid ◽  
M. Abdel-Aty ◽  
M. Qasymeh ◽  
H. Eleuch
Keyword(s):  
Analytical Model ◽  
Open System ◽  
Optical Lattice ◽  
Ultracold Atoms ◽  
Processing Time ◽  
Quantum Algorithms ◽  
Black Box ◽  
Quantum Gate ◽  
Quantum Gates

AbstractIn this work, trapped ultracold atoms are proposed as a platform for efficient quantum gate circuits and algorithms. We also develop and evaluate quantum algorithms, including those for the Simon problem and the black-box string-finding problem. Our analytical model describes an open system with non-Hermitian Hamiltonian. It is shown that our proposed scheme offers better performance (in terms of the number of required gates and the processing time) for realizing the quantum gates and algorithms compared to previously reported approaches.

The implementation of universal quantum memory and gates based on large-scale diamond surface

2016 ◽  
Vol 14 (05) ◽  
pp. 1650026
Author(s):  
Xiao-Ning Qi ◽  
Yong Zhang
Keyword(s):  
Quantum State ◽  
Large Scale ◽  
Quantum Memory ◽  
Quantum Gate ◽  
Nitrogen Vacancy ◽  
Diamond Surface ◽  
Single Qubit ◽  
Model Based ◽  
Universal Quantum ◽  
Jc Model

Nitrogen-vacancy (NV) centers implanted beneath the diamond surface have been demonstrated to be effective in the processing of controlling and reading-out. In this paper, NV center entangled with the fluorine nuclei collective ensemble is simplified to Jaynes–Cummings (JC) model. Based on this system, we discussed the implementation of quantum state storage and single-qubit quantum gate.

scholarly journals Quantum gate teleportation between separated qubits in a trapped-ion processor

Science ◽  
2019 ◽  
Vol 364 (6443) ◽  
pp. 875-878 ◽  
Author(s):  
Yong Wan ◽  
Daniel Kienzler ◽  
Stephen D. Erickson ◽  
Karl H. Mayer ◽  
Ting Rei Tan ◽  
...  
Keyword(s):  
Confidence Level ◽  
Large Scale ◽  
Ion Trap ◽  
Quantum Gate ◽  
Quantum Computers ◽  
Mixed Species ◽  
Cnot Gate ◽  
Classical Communication ◽  
Trapped Ion ◽  
Single Qubit

Large-scale quantum computers will require quantum gate operations between widely separated qubits. A method for implementing such operations, known as quantum gate teleportation (QGT), requires only local operations, classical communication, and shared entanglement. We demonstrate QGT in a scalable architecture by deterministically teleporting a controlled-NOT (CNOT) gate between two qubits in spatially separated locations in an ion trap. The entanglement fidelity of our teleported CNOT is in the interval (0.845, 0.872) at the 95% confidence level. The implementation combines ion shuttling with individually addressed single-qubit rotations and detections, same- and mixed-species two-qubit gates, and real-time conditional operations, thereby demonstrating essential tools for scaling trapped-ion quantum computers combined in a single device.

scholarly journals Measuring the Tangle of Three-Qubit States

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 436 ◽  
Author(s):  
Adrián Pérez-Salinas ◽  
Diego García-Martín ◽  
Carlos Bravo-Prieto ◽  
José Latorre
Keyword(s):  
Direct Measurement ◽  
Canonical Form ◽  
Quantum Circuit ◽  
Quantum Gates ◽  
Tripartite Entanglement ◽  
Single Qubit ◽  
Optimal Values ◽  
Random States ◽  
Qubit States ◽  
Computational Basis

We present a quantum circuit that transforms an unknown three-qubit state into its canonical form, up to relative phases, given many copies of the original state. The circuit is made of three single-qubit parametrized quantum gates, and the optimal values for the parameters are learned in a variational fashion. Once this transformation is achieved, direct measurement of outcome probabilities in the computational basis provides an estimate of the tangle, which quantifies genuine tripartite entanglement. We perform simulations on a set of random states under different noise conditions to asses the validity of the method.

Experimental demonstration of optimized quantum process tomography on the IBM quantum experience

2020 ◽  
pp. 2040004
Author(s):  
Akshay Gaikwad ◽  
Krishna Shende ◽  
Kavita Dorai
Keyword(s):  
Quantum Circuit ◽  
Quantum Gates ◽  
Experimental Demonstration ◽  
Superconducting Qubit ◽  
Quantum Process ◽  
Single Qubit ◽  
Quantum Process Tomography ◽  
Process Tomography ◽  
Quantum Processor ◽  
Least Squares Optimization

We experimentally performed complete and optimized quantum process tomography of quantum gates implemented on superconducting qubit-based IBM QX2 quantum processor via two constrained convex optimization (CCO) techniques: least squares optimization and compressed sensing optimization. We studied the performance of these methods by comparing the experimental complexity involved and the experimental fidelities obtained. We experimentally characterized several two-qubit quantum gates: identity gate, a controlled-NOT gate, and a SWAP gate. The general quantum circuit is efficient in the sense that the data needed to perform CCO-based process tomography can be directly acquired by measuring only a single qubit. The quantum circuit can be extended to higher dimensions and is also valid for other experimental platforms.

scholarly journals DIAGONAL-UNITARY 2-DESIGN AND THEIR IMPLEMENTATIONS BY QUANTUM CIRCUITS

2013 ◽  
Vol 11 (07) ◽  
pp. 1350062 ◽  
Author(s):  
YOSHIFUMI NAKATA ◽  
MIO MURAO
Keyword(s):  
Algebraic Structure ◽  
Quantum Circuits ◽  
Random Choice ◽  
Unitary Matrices ◽  
Single Qubit ◽  
Classical Procedure ◽  
Random States ◽  
Computational Basis

We study efficient generations of random diagonal-unitary matrices, an ensemble of unitary matrices diagonal in a given basis with randomly distributed phases for their eigenvalues. Despite the simple algebraic structure, they cannot be achieved by quantum circuits composed of a few-qubit diagonal gates. We introduce diagonal-unitaryt-designs and present two quantum circuits that implement diagonal-unitary 2-design with the computational basis in N-qubit systems. One is composed of single-qubit diagonal gates and controlled-phase gates with randomized phases, which achieves an exact diagonal-unitary 2-design after applying the gates on all pairs of qubits. The number of required gates is N(N - 1)/2. If the controlled-Z gates are used instead of the controlled-phase gates, the circuit cannot achieve an exact 2-design, but achieves an ϵ-approximate 2-design by applying gates on randomly selected pairs of qubits. Due to the random choice of pairs, the circuit obtains extra randomness and the required number of gates is at most O(N2(N + log 1/∊)). We also provide an application of the circuits, a protocol of generating an exact 2-design of random states by combining the circuits with a simple classical procedure requiring O(N) random classical bits.

Adiabatic holonomic quantum gates for a single qubit

2014 ◽  
Vol T160 ◽  
pp. 014029 ◽  
Author(s):  
Vladimir S Malinovsky ◽  
Sergey Rudin
Keyword(s):  
Quantum Gates ◽  
Single Qubit
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