Synthesis of quantum circuits for $d$-level systems by using cosine-sine

2009 ◽  
Vol 9 (5&6) ◽  
pp. 423-443
Author(s):  
Y. Nakajima ◽  
Y. Kawano ◽  
H. Sekigawa ◽  
M. Nakanishi ◽  
S. Yamashita ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Quantum Circuit ◽  
Previous Method ◽  
Quantum Circuits ◽  
Divide And Conquer ◽  
Synthesis Methods ◽  
Quantum Fourier Transform ◽  
Asymptotically Optimal ◽  
Polynomial Size

We study the problem of designing minimal quantum circuits for any operations on $n$ qudits by means of the cosine-sine decomposition. Our method is based on a divide-and-conquer strategy. In that strategy, the size of the produced quantum circuit depends on whether the partitioning is balanced. We provide a new cosine-sine decomposition based on a balanced partitioning for $d$-level systems. The produced circuit is not asymptotically optimal except when $d$ is a power of two, but, when the number of qudits $n$ is small, our method can produce the smallest quantum circuit compared to the circuits produced by other synthesis methods. For example, when $d=3$ (three-level systems) and $n=2$ (two qudits), then the number of two-qudit operations called CINC, which is a generalized versions of CNOT, is 36 whereas the previous method needs 156 CINC gates. Moreover, we show that our method is useful for designing a polynomial-size quantum circuit for the radix-$d$ quantum Fourier transform.

scholarly journals Quantum arithmetic operations based on quantum fourier transform on signed integers

2020 ◽  
Vol 18 (06) ◽  
pp. 2050035
Author(s):  
Engin Şahin
Keyword(s):  
Fourier Transform ◽  
Quantum Circuits ◽  
Quantum Computers ◽  
Quantum Fourier Transform ◽  
Arithmetic Operations ◽  
Absolute Value ◽  
Addition And Subtraction ◽  
Quantum Arithmetic

The quantum Fourier transform (QFT) brings efficiency in many respects, especially usage of resource, for most operations on quantum computers. In this study, the existing QFT-based and non-QFT-based quantum arithmetic operations are examined. The capabilities of QFT-based addition and multiplication are improved with some modifications. The proposed operations are compared with the nearest quantum arithmetic operations. Furthermore, novel QFT-based subtraction, division and exponentiation operations are presented. The proposed arithmetic operations can perform nonmodular operations on all signed numbers without any limitation by using less resources. In addition, novel quantum circuits of two’s complement, absolute value and comparison operations are also presented by using the proposed QFT-based addition and subtraction operations.

scholarly journals Human-Competitive Evolution of Quantum Computing Artefacts by Genetic Programming

2006 ◽  
Vol 14 (1) ◽  
pp. 21-40 ◽  
Author(s):  
Paul Massey ◽  
John A. Clark ◽  
Susan Stepney
Keyword(s):  
Fourier Transform ◽  
Quantum Computing ◽  
Genetic Programming ◽  
Quantum Algorithms ◽  
Quantum Circuits ◽  
Quantum Fourier Transform ◽  
Competitive Evolution ◽  
Specific Quantum

We show how Genetic Programming (GP) can be used to evolve useful quantum computing artefacts of increasing sophistication and usefulness: firstly specific quantum circuits, then quantum programs, and finally system-independent quantum algorithms. We conclude the paper by presenting a human-competitive Quantum Fourier Transform (QFT) algorithm evolved by GP.

Protocol and quantum circuit for implementing theN-bit discrete quantum Fourier transform in cavity QED

2010 ◽  
Vol 43 (6) ◽  
pp. 065503 ◽  
Author(s):  
Hong-Fu Wang ◽  
Xiao-Qiang Shao ◽  
Yong-Fang Zhao ◽  
Shou Zhang ◽  
Kyu-Hwang Yeon
Keyword(s):  
Fourier Transform ◽  
Cavity Qed ◽  
Quantum Circuit ◽  
Quantum Fourier Transform

The quantum Fourier transform on a linear nearest neighbor architecture

2007 ◽  
Vol 7 (4) ◽  
pp. 383-391
Author(s):  
Y. Takahashi ◽  
N. Kunihiro ◽  
K. Ohta
Keyword(s):  
Fourier Transform ◽  
Nearest Neighbor ◽  
Quantum Circuit ◽  
Quantum Fourier Transform ◽  
Factoring Algorithm

We show how to construct an efficient quantum circuit for computing a good approximation of the quantum Fourier transform on a linear nearest neighbor architecture. The constructed circuit uses no ancillary qubits and its depth and size are $O(n)$ and $O(n\log n)$, respectively, where $n$ is the length of the input. The circuit is useful for decreasing the size of Fowler et al.'s quantum circuit for Shor's factoring algorithm on a linear nearest neighbor architecture.

A new algorithm for producing quantum circuits using KAK decompositions

2006 ◽  
Vol 6 (1) ◽  
pp. 67-80
Author(s):  
M.Y. Nakajima ◽  
Y. Kawano ◽  
H. Sekigawa
Keyword(s):  
Matrix Decomposition ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Unitary Matrix ◽  
Square Root ◽  
Quantum Fourier Transform ◽  
Lie Group Theory ◽  
Special Cases ◽  
Cartan Involution ◽  
The Given

We provide a new algorithm that translates a unitary matrix into a quantum circuit according to the G=KAK theorem in Lie group theory. With our algorithm, any matrix decomposition corresponding to type-AIII KAK decompositions can be derived according to the given Cartan involution. Our algorithm contains, as its special cases, Cosine-Sine decomposition (CSD) and Khaneja-Glaser decomposition (KGD) in the sense that it derives the same quantum circuits as the ones obtained by them if we select suitable Cartan involutions and square root matrices. The selections of Cartan involutions for computing CSD and KGD will be shown explicitly. As an example, we show explicitly that our method can automatically reproduce the well-known efficient quantum circuit for the $n$-qubit quantum Fourier transform.

Hardness of classically simulating quantum circuits with unbounded Toffoli and fan-out gates

2014 ◽  
Vol 14 (13&14) ◽  
pp. 1149-1164
Author(s):  
Yasuhiro Takahashi ◽  
Takeshi Yamazaki ◽  
Kazuyuki Tanaka
Keyword(s):  
Probability Distribution ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Constant Depth ◽  
Toffoli Gate ◽  
Single Qubit ◽  
Polynomial Size ◽  
Toffoli Gates ◽  
Output Probability ◽  
Universal Gates

We study the classical simulatability of constant-depth polynomial-size quantum circuits followed by only one single-qubit measurement, where the circuits consist of universal gates on at most two qubits and additional gates on an unbounded number of qubits. First, we consider unbounded Toffoli gates as additional gates and deal with the weak simulation, i.e., sampling the output probability distribution. We show that there exists a constant-depth quantum circuit with only one unbounded Toffoli gate that is not weakly simulatable, unless $\bqp \subseteq \postbpp \cap \am$. Then, we consider unbounded fan-out gates as additional gates and deal with the strong simulation, i.e., computing the output probability. We show that there exists a constant-depth quantum circuit with only two unbounded fan-out gates that is not strongly simulatable, unless $\p = \pp$. These results are in contrast to the fact that any constant-depth quantum circuit without additional gates on an unbounded number of qubits is strongly and weakly simulatable.

scholarly journals Classical simulations of Abelian-group normalizer circuits with intermediate measurements

2014 ◽  
Vol 14 (3&4) ◽  
pp. 181-216
Author(s):  
Juan Bermejo-Vega ◽  
Maarten Van den Nest
Keyword(s):  
Fourier Transform ◽  
Abelian Group ◽  
Probability Distribution ◽  
Normal Form ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Quadratic Functions ◽  
Quantum Circuits ◽  
Quantum Fourier Transform ◽  
Finite Abelian Groups

Quantum normalizer circuits were recently introduced as generalizations of Clifford circuits: a normalizer circuit over a finite Abelian group G is composed of the quantum Fourier transform (QFT) over G, together with gates which compute quadratic functions and automorphisms. In \cite{VDNest_12_QFTs} it was shown that every normalizer circuit can be simulated efficiently classically. This result provides a nontrivial example of a family of quantum circuits that cannot yield exponential speed-ups in spite of usage of the QFT, the latter being a central quantum algorithmic primitive. Here we extend the aforementioned result in several ways. Most importantly, we show that normalizer circuits supplemented with intermediate measurements can also be simulated efficiently classically, even when the computation proceeds adaptively. This yields a generalization of the Gottesman-Knill theorem (valid for n-qubit Clifford operations) to quantum circuits described by arbitrary finite Abelian groups. Moreover, our simulations are twofold: we present efficient classical algorithms to sample the measurement probability distribution of any adaptive-normalizer computation, as well as to compute the amplitudes of the state vector in every step of it. Finally we develop a generalization of the stabilizer formalism relative to arbitrary finite Abelian groups: for example we characterize how to update stabilizers under generalized Pauli measurements and provide a normal form of the amplitudes of generalized stabilizer states using quadratic functions and subgroup cosets.

scholarly journals A divide-and-conquer algorithm for quantum state preparation

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Israel F. Araujo ◽  
Daniel K. Park ◽  
Francesco Petruccione ◽  
Adenilton J. da Silva
Keyword(s):  
Computational Cost ◽  
Quantum Circuit ◽  
Divide And Conquer ◽  
Computational Time ◽  
Quantum Computers ◽  
Dimensional Vector ◽  
Quantum Devices ◽  
Divide And Conquer Algorithm ◽  
Quantum State Preparation ◽  
Quantum Device

AbstractAdvantages in several fields of research and industry are expected with the rise of quantum computers. However, the computational cost to load classical data in quantum computers can impose restrictions on possible quantum speedups. Known algorithms to create arbitrary quantum states require quantum circuits with depth O(N) to load an N-dimensional vector. Here, we show that it is possible to load an N-dimensional vector with exponential time advantage using a quantum circuit with polylogarithmic depth and entangled information in ancillary qubits. Results show that we can efficiently load data in quantum devices using a divide-and-conquer strategy to exchange computational time for space. We demonstrate a proof of concept on a real quantum device and present two applications for quantum machine learning. We expect that this new loading strategy allows the quantum speedup of tasks that require to load a significant volume of information to quantum devices.

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