Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Programming and Computer Software

Some algorithms for calculating unitary matrices for quantum circuits

Vladimir P. Gerdt, Alexander N. Prokopenya

Published Mar 1, 2010• 8 citations
Publisher

Abstract

Algorithms for calculating unitary matrices determined by quantum circuits are discussed. The algorithms are used in the program QuantumCircuit designed for modeling quantum circuits. Practical implementations of the algorithms as functions written in the language of the Mathematica system are suggested.
DOI
10.1134/s036176881002009x

Related papers

A mathematica program for constructing quantum circuits and computing their unitary matrices

Vladimir P. Gerdt, Robert Kragler, Alexander N. Prokopenya

In this paper we briefly describe a Mathematica program for simulation of quantum circuits and illustrate some of its facilities by simple examples. Unlike other Mathematica-based quantum simulators, our program provides a user-friendly graphical interface for generating quantum circuits and computing the circuit unitary matrices. In addition to standard linear algebra-based tools, our program implements special computer-algebra technique for constructing the multivariate polynomial system that, for a circuit composed from the Toffoli and Hadamard gates, uniquely defines the circuit matrix.

Physics of Particles and Nuclei Letters•Nov 19, 2009•12 citations

Simulation of quantum error correction by means of QuantumCircuit package

Vladimir P. Gerdt, Alexander N. Prokopenya

In the paper, the problem of simulation of quantum error correction by means of error correcting codes is discussed. Examples of error correction by means of quantum circuits constructed with the help of the QuantumCircuit package written in the language of the computer algebra system Mathematica are presented.

Programming and Computer Software•May 1, 2013•8 citations

Explicit Quantum Circuits for Block Encodings of Certain Sparse Matrices

Daan Camps, Lin Lin, Roel Van Beeumen +1 more

Abstract. Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest [Formula: see text] in a larger unitary transformation [Formula: see text] that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such a gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of [Formula: see text], which is difficult in general, and not trivial even for well structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well structured sparse matrices and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.

arXiv:2203.10236•SIAM Journal on Matrix Analysis and Applications•Mar 11, 2024•78 citations

Quantum circuit design methods and applications

Ammar Daskin

Simulating quantum mechanical evolutions in general is difficult on classical computers because the size of the Hilbert space grows exponentially with the number of particles involved in the simulations. However, a quantum computer, which obeys the laws of quantum mechanics, is believed to perform such simulations more efficiently. The time evolution operator, a unitary matrix, of a quantum system acts as a quantum gate changing the state of the system in time. Therefore, any computation on quantum computers can be interpreted as a unitary transformation on quantum state vectors. Generally, quantum gates are sufficiently simple operations so as to avoid difficulty in the physical implementation. The implementation of a general given computation represented by a unitary matrix on quantum computers requires finding an array of elementary quantum gates describing the desired computation. This describes a matrix decomposition problem known as the quantum circuit design problem. In this dissertation, we address the following questions concerning quantum circuit designs: How hard is it to find suitable quantum circuits for a given computation? What are the possible circuit design methods? How do we apply these to simulations of quantum problems, in particular quantum chemistry problems? To answer these questions, we categorize methods for designing quantum circuits as evolutionary algorithms and deterministic methods. We apply our developed group leaders optimization algorithm, an evolutionary algorithm, to find quantum circuits for known quantum algorithms and the simulation of molecular Hamiltonians. Then, we present a deterministic circuit design approach to produce universal circuits, which can be programmable. Moreover, we show how to use these programmable circuit designs within the quantum phase estimation algorithm to find eigenvalues of nonunitary matrices and study resonances in quantum systems. Finally, the quantum phase estimation algorithm is employed to efficiently solve ranking problems: in particular, multiple network alignment is considered to find the similarities between protein-protein interaction networks and molecular networks.

•Jan 1, 2014•2 citations

Simulation of a quantum algorithm for phase estimation

Alexander N. Prokopenya

A quantum algorithm for estimating the phase, which determines the eigenvalue of a unitary operator, is discussed. It is assumed that the eigenvector of this operator and the corresponding quantum circuit are given. The memory register where the approximate phase value is stored consists of n qubits, which makes it possible to determine the phase accurate to 2 −n with the probability greater than 8/π 2 . By way of example, computations for the case of the quantum phase shift operator are performed. The simulation of the quantum algorithm and the computation of the eigenvalue are performed using the QuantumCircuit package written in the Wolfram Mathematica computer algebra system. This system is also used to perform all the computations and visualize the results.

Programming and Computer Software•Mar 1, 2015•7 citations

An Extended Approach for Generating Unitary Matrices for Quantum Circuits

Zhiqiang Li, Wei Zhang, Gaoman Zhang +4 more

In this paper, we do research on generating unitary matrices for quantum circuits automatically. We consider that quantum circuits are divided into six types, and the unitary operator expressions for each type are offered... | Find, read and cite all the research you need on Tech Science Press

Computers, materials & continua/Computers, materials & continua (Print)•Jan 1, 2020•5 citations
In this paper we briefly describe a Mathematica program for simulation of quantum circuits and illustrate some of its facilities by simple examples. Unlike other Mathematica-based quantum simulators, our program provides a user-friendly graphical interface for generating quantum circuits and computing the circuit unitary matrices. In addition to standard linear algebra-based tools, our program implements special computer-algebra technique for constructing the multivariate polynomial system that, for a circuit composed from the Toffoli and Hadamard gates, uniquely defines the circuit matrix.
Abstract. Many standard linear algebra problems can be solved on a quantum computer by using recently developed quantum linear algebra algorithms that make use of block encodings and quantum eigenvalue/singular value transformations. A block encoding embeds a properly scaled matrix of interest [Formula: see text] in a larger unitary transformation [Formula: see text] that can be decomposed into a product of simpler unitaries and implemented efficiently on a quantum computer. Although quantum algorithms can potentially achieve exponential speedup in solving linear algebra problems compared to the best classical algorithm, such a gain in efficiency ultimately hinges on our ability to construct an efficient quantum circuit for the block encoding of [Formula: see text], which is difficult in general, and not trivial even for well structured sparse matrices. In this paper, we give a few examples on how efficient quantum circuits can be explicitly constructed for some well structured sparse matrices and discuss a few strategies used in these constructions. We also provide implementations of these quantum circuits in MATLAB.
Simulating quantum mechanical evolutions in general is difficult on classical computers because the size of the Hilbert space grows exponentially with the number of particles involved in the simulations. However, a quantum computer, which obeys the laws of quantum mechanics, is believed to perform such simulations more efficiently. The time evolution operator, a unitary matrix, of a quantum system acts as a quantum gate changing the state of the system in time. Therefore, any computation on quantum computers can be interpreted as a unitary transformation on quantum state vectors. Generally, quantum gates are sufficiently simple operations so as to avoid difficulty in the physical implementation. The implementation of a general given computation represented by a unitary matrix on quantum computers requires finding an array of elementary quantum gates describing the desired computation. This describes a matrix decomposition problem known as the quantum circuit design problem. In this dissertation, we address the following questions concerning quantum circuit designs: How hard is it to find suitable quantum circuits for a given computation? What are the possible circuit design methods? How do we apply these to simulations of quantum problems, in particular quantum chemistry problems? To answer these questions, we categorize methods for designing quantum circuits as evolutionary algorithms and deterministic methods. We apply our developed group leaders optimization algorithm, an evolutionary algorithm, to find quantum circuits for known quantum algorithms and the simulation of molecular Hamiltonians. Then, we present a deterministic circuit design approach to produce universal circuits, which can be programmable. Moreover, we show how to use these programmable circuit designs within the quantum phase estimation algorithm to find eigenvalues of nonunitary matrices and study resonances in quantum systems. Finally, the quantum phase estimation algorithm is employed to efficiently solve ranking problems: in particular, multiple network alignment is considered to find the similarities between protein-protein interaction networks and molecular networks.
A quantum algorithm for estimating the phase, which determines the eigenvalue of a unitary operator, is discussed. It is assumed that the eigenvector of this operator and the corresponding quantum circuit are given. The memory register where the approximate phase value is stored consists of n qubits, which makes it possible to determine the phase accurate to 2 −n with the probability greater than 8/π 2 . By way of example, computations for the case of the quantum phase shift operator are performed. The simulation of the quantum algorithm and the computation of the eigenvalue are performed using the QuantumCircuit package written in the Wolfram Mathematica computer algebra system. This system is also used to perform all the computations and visualize the results.