Transformation of quantum states using uniformly controlled rotations

2005 ◽  
Vol 5 (6) ◽  
pp. 467-473
Author(s):  
M. Mottonen ◽  
J. J. Vartiainen ◽  
V. Bergholm ◽  
M. M. Salomaa
Keyword(s):  
Pure State ◽  
Unitary Transformation ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
The State ◽  
Quantum States ◽  
State Transformation ◽  
Quantum Register ◽  
Analytic Expression

We consider a unitary transformation which maps any given pure state of an $n$-qubit quantum register into another one. This transformation has applications in the initialization of a quantum computer, and also in some quantum algorithms. Employing uniformly controlled rotations, we present a quantum circuit of $2^{n+2}-4n-4$ CNOT gates and $2^{n+2}-5$ one-qubit elementary rotations that effects the state transformation. The complexity of the circuit is noticeably lower than the previously published results. Moreover, we present an analytic expression for the rotation angles needed for the transformation.

scholarly journals Practical Quantum Computing

2021 ◽  
Vol 2 (1) ◽  
pp. 1-35
Author(s):  
Adrien Suau ◽  
Gabriel Staffelbach ◽  
Henri Calandra
Keyword(s):  
Wave Equation ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Equation Solving ◽  
Small Quantum ◽  
Quantum Wave ◽  
Variational Algorithms ◽  
The One

In the last few years, several quantum algorithms that try to address the problem of partial differential equation solving have been devised: on the one hand, “direct” quantum algorithms that aim at encoding the solution of the PDE by executing one large quantum circuit; on the other hand, variational algorithms that approximate the solution of the PDE by executing several small quantum circuits and making profit of classical optimisers. In this work, we propose an experimental study of the costs (in terms of gate number and execution time on a idealised hardware created from realistic gate data) associated with one of the “direct” quantum algorithm: the wave equation solver devised in [32]. We show that our implementation of the quantum wave equation solver agrees with the theoretical big-O complexity of the algorithm. We also explain in great detail the implementation steps and discuss some possibilities of improvements. Finally, our implementation proves experimentally that some PDE can be solved on a quantum computer, even if the direct quantum algorithm chosen will require error-corrected quantum chips, which are not believed to be available in the short-term.

scholarly journals Quantum Generative Adversarial Networks for learning and loading random distributions

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Christa Zoufal ◽  
Aurélien Lucchi ◽  
Stefan Woerner
Keyword(s):  
Quantum State ◽  
Quantum Channel ◽  
Quantum Algorithms ◽  
Probability Distributions ◽  
Quantum Algorithm ◽  
Quantum Circuit ◽  
Quantum Simulation ◽  
Quantum States ◽  
Generative Adversarial Networks ◽  
Adversarial Networks

AbstractQuantum algorithms have the potential to outperform their classical counterparts in a variety of tasks. The realization of the advantage often requires the ability to load classical data efficiently into quantum states. However, the best known methods require $${\mathcal{O}}\left({2}^{n}\right)$$O2n gates to load an exact representation of a generic data structure into an $$n$$n-qubit state. This scaling can easily predominate the complexity of a quantum algorithm and, thereby, impair potential quantum advantage. Our work presents a hybrid quantum-classical algorithm for efficient, approximate quantum state loading. More precisely, we use quantum Generative Adversarial Networks (qGANs) to facilitate efficient learning and loading of generic probability distributions - implicitly given by data samples - into quantum states. Through the interplay of a quantum channel, such as a variational quantum circuit, and a classical neural network, the qGAN can learn a representation of the probability distribution underlying the data samples and load it into a quantum state. The loading requires $${\mathcal{O}}\left(poly\left(n\right)\right)$$Opolyn gates and can thus enable the use of potentially advantageous quantum algorithms, such as Quantum Amplitude Estimation. We implement the qGAN distribution learning and loading method with Qiskit and test it using a quantum simulation as well as actual quantum processors provided by the IBM Q Experience. Furthermore, we employ quantum simulation to demonstrate the use of the trained quantum channel in a quantum finance application.

scholarly journals Developing A Quantum Circuit Simulator API

2015 ◽  
Vol 67 (1) ◽  
pp. 168-173
Author(s):  
Stancu Mihai Dorian ◽  
Popa Emil Marin
Keyword(s):  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
Circuit Simulator ◽  
Design And Implementation ◽  
Computer Scientists

Abstract In this paper we propose the design and implementation of a quantum circuit simulator API. Currently the API allows users to implement, debug and test the following two quantum algorithms: Bernstein-Vazirani’s algorithm and Simon’s Algorithm. The goal is to create a framework that will allow quantum computer scientists to easily develop new quantum algorithms.

scholarly journals QUANTUM HIERARCHIC MODELS FOR INFORMATION PROCESSING

2012 ◽  
Vol 10 (02) ◽  
pp. 1250026 ◽  
Author(s):  
MIKHAIL V. ALTAISKY ◽  
NATALIA E. KAPUTKINA
Keyword(s):  
Information Processing ◽  
Wavelet Transform ◽  
Memory Capacity ◽  
The State ◽  
Quantum States ◽  
Nanometer Scale ◽  
Information Compression ◽  
Quantum Bits ◽  
Quantum Register ◽  
Quantum Computations

Both classical and quantum computations operate with the registers of bits. At nanometer scale the quantum fluctuations at the position of a given bit, say, a quantum dot, not only lead to the decoherence of quantum state of this bit, but also affect the quantum states of the neighboring bits, and therefore affect the state of the whole register. That is why the requirement of reliable separate access to each bit poses the limit on miniaturization, i.e. constrains the memory capacity and the speed of computation. In the present paper we suggest an algorithmic way to tackle the problem of constructing reliable and compact registers of quantum bits. We suggest accessing the states of a quantum register hierarchically, descending from the state of the whole register to the states of its parts. Our method is similar to quantum wavelet transform, and can be applied to information compression, quantum memory, quantum computations.

scholarly journals Encoding strongly-correlated many-boson wavefunctions on a photonic quantum computer: application to the attractive Bose-Hubbard model

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 572
Author(s):  
Saad Yalouz ◽  
Bruno Senjean ◽  
Filippo Miatto ◽  
Vedran Dunjko
Keyword(s):  
Hubbard Model ◽  
Quantum Computer ◽  
Quantum Algorithms ◽  
Experimental System ◽  
Quantum Circuit ◽  
Continuous Variable ◽  
Computer Application ◽  
Strongly Correlated ◽  
Many Body ◽  
Boson Systems

Variational quantum algorithms (VQA) are considered as some of the most promising methods to determine the properties of complex strongly correlated quantum many-body systems, especially from the perspective of devices available in the near term. In this context, the development of efficient quantum circuit ansatze to encode a many-body wavefunction is one of the keys for the success of a VQA. Great efforts have been invested to study the potential of current quantum devices to encode the eigenstates of fermionic systems, but little is known about the encoding of bosonic systems. In this work, we investigate the encoding of the ground state of the (simple but rich) attractive Bose-Hubbard model using a Continuous-Variable (CV) photonic-based quantum circuit. We introduce two different ansatz architectures and demonstrate that the proposed continuous variable quantum circuits can efficiently encode (with a fidelity higher than 99%) the strongly correlated many-boson wavefunction with just a few layers, in all many-body regimes and for different number of bosons and initial states. Beyond the study of the suitability of the ansatz to approximate the ground states of many-boson systems, we also perform initial evaluations of the use of the ansatz in a variational quantum eigensolver algorithm to find it through energy minimization. To this end we also introduce a scheme to measure the Hamiltonian energy in an experimental system, and study the effect of sampling noise.

scholarly journals A Quantum Circuit Design for Grover’s Algorithm

2002 ◽  
Vol 57 (8) ◽  
pp. 701-708 ◽  
Author(s):  
Zijian Diao ◽  
M. Suhail Zubairy ◽  
Goong Chen
Keyword(s):  
Circuit Design ◽  
Cavity Qed ◽  
Unitary Transformation ◽  
Quantum Circuit ◽  
The State ◽  
Quantum Phase ◽  
Grover’S Algorithm ◽  
Mathematical Proofs ◽  
Grover's Algorithm

We present a circuit design realizing Grover’s algorithm based on 1-bit unitary gates and 2-bit quantum phase gates implementable with cavity QED techniques. In the first step, we express the circuit block which performs a key unitary transformation that flips only the sign of the state |11 · · · 11〉 using 1-bit and 2-bit gates. The Grover’s iteration operator can then be constructed using this key unitary transformation twice, plus other operations involving only 1-bit unitary gates on each qubit. Mathematical proofs are given to justify that the cricuiting satisfies the desired operator properties.

Quantum Computing and Quantum Communication

2018 ◽  
pp. 7715-7730
Author(s):  
Göran Pulkkis ◽  
Kaj J. Grahn
Keyword(s):  
Quantum Computing ◽  
Quantum Computation ◽  
Quantum Cryptography ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
Quantum States ◽  
Future Perspectives ◽  
Quantum Informatics ◽  
Quantum Programming ◽  
Computing Models

This article presents state-of-the-art and future perspectives of quantum computing and communication. Timeline of relevant findings in quantum informatics, such as quantum algorithms, quantum cryptography protocols, and quantum computing models, is summarized. Mathematics of information representation with quantum states is presented. The quantum circuit and adiabatic models of quantum computation are outlined. The functionality, limitations, and security of the quantum key distribution (QKD) protocol is presented. Current implementations of quantum computers and principles of quantum programming are shortly described.

Quantum Computing and Quantum Communication

2019 ◽  
pp. 1641-1659
Author(s):  
Göran Pulkkis ◽  
Kaj J. Grahn
Keyword(s):  
Quantum Computing ◽  
Quantum Computation ◽  
Quantum Cryptography ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
Quantum States ◽  
Future Perspectives ◽  
Quantum Informatics ◽  
Quantum Programming ◽  
Computing Models

This chapter presents state-of-the-art and future perspectives of quantum computing and communication. Timeline of relevant findings in quantum informatics, such as quantum algorithms, quantum cryptography protocols, and quantum computing models, is summarized. Mathematics of information representation with quantum states is presented. The quantum circuit and adiabatic models of quantum computation are outlined. The functionality, limitations, and security of the quantum key distribution (QKD) protocol is presented. Current implementations of quantum computers and principles of quantum programming are shortly described.

scholarly journals Quantum arithmetic and numerical analysis using Repeat-Until-Success circuits

2016 ◽  
pp. 134-178 ◽  
Author(s):  
Nathan Wiebe ◽  
Martin Roetteler
Keyword(s):  
Quantum Computer ◽  
Quantum Algorithms ◽  
Quantum Circuit ◽  
Circuit Synthesis ◽  
Small Quantum ◽  
Approximate Synthesis ◽  
Quantum Arithmetic ◽  
And Inversion ◽  
Synthesis Approach ◽  
Computable Functions

We develop a method for approximate synthesis of single-qubit rotations of the form e−if(φ1,...,φk)X that is based on the Repeat-Until-Success (RUS) framework for quantum circuit synthesis. We demonstrate how smooth computable functions f can be synthesized from two basic primitives. This synthesis approach constitutes a manifestly quantum form of arithmetic that differs greatly from the approaches commonly used in quantum algorithms. The key advantage of our approach is that it requires far fewer qubits than existing approaches: as a case in point, we show that using as few as 3 ancilla qubits, one can obtain RUS circuits for approximate multiplication and reciprocals. We also analyze the costs of performing multiplication and inversion on a quantum computer using conventional approaches and find that they can require too many qubits to execute on a small quantum computer, unlike our approach.

scholarly journals Volume of the space of qubit-qubit channels and state transformations under random quantum channels

2018 ◽  
Vol 30 (10) ◽  
pp. 1850019 ◽  
Author(s):  
Attila Lovas ◽  
Attila Andai
Keyword(s):  
Lebesgue Measure ◽  
Quantum Channel ◽  
Probability Distributions ◽  
Building Blocks ◽  
The State ◽  
Quantum States ◽  
Qubit State ◽  
Quantum Channels ◽  
State Transformation ◽  
Explicit Formulas

The simplest building blocks for quantum computations are the qubit-qubit quantum channels. In this paper, we analyze the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of the general and unital qubit channels with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. We study the state transformation under uniformly random quantum channels. If one applies a uniformly random quantum channel (general or unital) to a given qubit state, the distribution of the resulted quantum states is presented.

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