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 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 High-fidelity single-qubit gates using non-adiabatic rapid passage

2007 ◽  
Vol 7 (7) ◽  
pp. 594-608
Author(s):  
R. Li ◽  
M. Hoover ◽  
F. Gaitan
Keyword(s):  
Quantum Interference ◽  
Fault Tolerant ◽  
Quantum Gates ◽  
High Fidelity ◽  
Interference Effects ◽  
Adiabatic Rapid Passage ◽  
Single Qubit ◽  
The One ◽  
Error Probabilities ◽  
Rapid Passage

Numerical simulation results are presented which suggest that a class of non-adiabatic rapid passage sweeps first realized experimentally in 1991 should be capable of implementing a set of quantum gates that is universal for one-qubit unitary operations and whose elements operate with error probabilities $P_{e}<10^{-4}$. The sweeps are non-composite and generate controllable quantum interference effects which allow the one-qubit gates produced to operate non-adiabatically while maintaining high accuracy. The simulations suggest that the one-qubit gates produced by these sweeps show promise as possible elements of a fault-tolerant scheme for quantum computing.

QUANTUM PRECISION LIMITS FOR ANY IMPLEMENTATION OF SINGLE QUBIT GATES UNDER CONSERVATION LAWS

2008 ◽  
Vol 06 (supp01) ◽  
pp. 701-706
Author(s):  
TOKISHIRO KARASAWA ◽  
MASANAO OZAWA ◽  
JULIO GEA-BANACLOCHE ◽  
KAE NEMOTO
Keyword(s):  
Lower Bound ◽  
Conservation Laws ◽  
Lower Bounds ◽  
Conservation Law ◽  
Conserved Quantity ◽  
Quantum Gate ◽  
Single Qubit ◽  
Geometrical Relation ◽  
Precision Limit ◽  
Qubit Gate

A quantum gate is implemented by control interactions between qubits and their ancilla system. It has been shown that the control interactions have possibilities to induce the dynamical decoherence on the qubits if an additive conservation law is assumed in the interactions and the ancilla system is finite. This decoherece put the precision limit on the gate, which cannot be removed from the qubit by optimizing the interaction and the initialization of the ancilla system. In this paper, we give the outline of investigating the precision limit which is formulated by the lower bound of the gate infidelity, one minus the squared fidelity, for an arbitrary self-adjoint gate on a single qubit. We show rigorous lower bounds in terms of the variance of the conserved quantity and a simple geometrical relation between the conservation law to be assumed and the gates to be implemented. We also comment on another approach to provide the precision limit for an arbitrary single qubit gate under a conservation law.

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.

scholarly journals Galvanic Phase Coupling of Superconducting Flux Qubits

2021 ◽  
Vol 11 (23) ◽  
pp. 11309
Author(s):  
Mun Dae Kim
Keyword(s):  
Coupling Strength ◽  
Fault Tolerant ◽  
Coupled System ◽  
Inductive Coupling ◽  
Galvanic Coupling ◽  
Flux Qubits ◽  
Superconducting Flux Qubits ◽  
Qubit Gate ◽  
Central Loop ◽  
Fundamental Boundary

We investigate the galvanic coupling schemes of superconducting flux qubits. From the fundamental boundary conditions, we obtain the effective potential of the coupled system of two or three flux qubits to provide the exact Lagrangian of the system. While usually the two-qubit gate has been investigated approximately, in this study we derive the exact inductive coupling strength between two flux qubits coupled directly and coupled through a connecting central loop. We observe that the inductive coupling strength needs to be included exactly to satisfy the criteria of fault-tolerant quantum computing.

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.

scholarly journals Exact dynamics of single-qubit-gate fidelities under the measurement-based quantum computation scheme

2012 ◽  
Vol 86 (4) ◽  
Author(s):  
L. G. E. Arruda ◽  
F. F. Fanchini ◽  
R. d. J. Napolitano ◽  
J. E. M. Hornos ◽  
A. O. Caldeira
Keyword(s):  
Quantum Computation ◽  
Single Qubit ◽  
Computation Scheme ◽  
Qubit Gate

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.

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