scholarly journals Scaling a Unitary Matrix

2014 ◽  
Vol 21 (04) ◽  
pp. 1450013 ◽  
Author(s):  
Alexis De Vos ◽  
Stijn De Baerdemacker
Keyword(s):  
Iterative Method ◽  
Quantum Circuit ◽  
Unitary Matrix ◽  
Real Matrix ◽  
Doubly Stochastic ◽  
Arbitrary Quantum

The iterative method of Sinkhorn allows, starting from an arbitrary real matrix with non-negative entries, to find a so-called ‘scaled matrix’ which is doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and with all line sums equal to 1. We conjecture that a similar procedure exists, which allows, starting from an arbitrary unitary matrix, to find a scaled matrix which is unitary and has all line sums equal to 1. The existence of such algorithm guarantees a powerful decomposition of an arbitrary quantum circuit.

scholarly journals Quick Quantum Circuit Simulation

2021 ◽  
Vol 3 (4) ◽  
Author(s):  
Daniel Evans
Keyword(s):  
Quantum State ◽  
Quantum Computer ◽  
Circuit Simulation ◽  
Matrix Multiplication ◽  
Quantum Circuit ◽  
Machine Language ◽  
Unitary Matrix ◽  
Initial State ◽  
Final State ◽  
Quantum Uncertainty

Quick Quantum Circuit Simulation (QQCS) is a software system for computing the result of a quantum circuit using a notation that derives directly from the circuit, expressed in a single input line. Quantum circuits begin with an initial quantum state of one or more qubits, which are the quantum analog to classical bits. The initial state is modified by a sequence of quantum gates, quantum machine language instructions, to get the final state. Measurements are made of the final state and displayed as a classical binary result. Measurements are postponed to the end of the circuit because a quantum state collapses when measured and produces probabilistic results, a consequence of quantum uncertainty. A circuit may be run many times on a quantum computer to refine the probabilistic result. Mathematically, quantum states are 2n -dimensional vectors over the complex number field, where n is the number of qubits. A gate is a 2n ×2n unitary matrix of complex values. Matrix multiplication models the application of a gate to a quantum state. QQCS is a mathematical rendering of each step of a quantum algorithm represented as a circuit, and as such, can present a trace of the quantum state of the circuit after each gate, compute gate equivalents for each circuit step, and perform measurements at any point in the circuit without state collapse. Output displays are in vector coefficients or Dirac bra-ket notation. It is an easy-to-use educational tool for students new to quantum computing.

scholarly journals Block-ZXZsynthesis of an arbitrary quantum circuit

2016 ◽  
Vol 94 (5) ◽  
Author(s):  
A. De Vos ◽  
S. De Baerdemacker
Keyword(s):  
Quantum Circuit ◽  
Arbitrary Quantum

scholarly journals Modelling of Grover’s quantum search algorithms: implementations of Simple quantum simulators on classical computers

2020 ◽  
pp. 65-128
Author(s):  
Sergey Ulyanov ◽  
Andrey Reshetnikov ◽  
Olga Tyatyushkina
Keyword(s):  
Quantum Computation ◽  
Quantum Computer ◽  
Search Algorithm ◽  
Quantum Circuit ◽  
Unitary Matrix ◽  
Single Qubit ◽  
Simple Quantum ◽  
Grover’S Search Algorithm ◽  
The Cost ◽  
Universal Gate

Models of Grover’s search algorithm is reviewed to build the foundation for the other algorithms. Thereafter, some preliminary modifications of the original algorithms by others are stated, that increases the applicability of the search procedure. A general quantum computation on an isolated system can be represented by a unitary matrix. In order to execute such a computation on a quantum computer, it is common to decompose the unitary into a quantum circuit, i.e., a sequence of quantum gates that can be physically implemented on a given architecture. There are different universal gate sets for quantum computation. Here we choose the universal gate set consisting of CNOT and single-qubit gates. We measure the cost of a circuit by the number of CNOT gates as they are usually more difficult to implement than single qubit gates and since the number of single-qubit gates is bounded by about twice the number of CNOT’s.

scholarly journals QUANTUM SOFTWARE REUSABILITY

2003 ◽  
Vol 14 (05) ◽  
pp. 777-796 ◽  
Author(s):  
ANDREAS KLAPPENECKER ◽  
MARTIN RÖTTELER
Keyword(s):  
Circuit Design ◽  
Unique Feature ◽  
Fourier Transforms ◽  
Design Method ◽  
Quantum Circuit ◽  
Quantum Circuits ◽  
Unitary Matrix ◽  
Mathematical Basis ◽  
Discrete Hartley Transform ◽  
Software Reusability

The design of efficient quantum circuits is an important issue in quantum computing. It is in general a formidable task to find a highly optimized quantum circuit for a given unitary matrix. We propose a quantum circuit design method that has the following unique feature: It allows to construct efficient quantum circuits in a systematic way by reusing and combining a set of highly optimized quantum circuits. Specifically, the method realizes a quantum circuit for a given unitary matrix by implementing a linear combination of representing matrices of a group, which have known fast quantum circuits. We motivate and illustrate this method by deriving extremely efficient quantum circuits for the discrete Hartley transform and for the fractional Fourier transforms. The sound mathematical basis of this design method allows to give meaningful and natural interpretations of the resulting circuits. We demonstrate this aspect by giving a natural interpretation of known teleportation circuits.

scholarly journals Block-based quantum-logic synthesis

2011 ◽  
Vol 11 (3&4) ◽  
pp. 262-277
Author(s):  
Mehdi Saeedi ◽  
Mona Arabzadeh ◽  
Morteza Saheb Zamani ◽  
Mehdi Sedighi
Keyword(s):  
Quantum Logic ◽  
Nearest Neighbor ◽  
Synthesis Method ◽  
Logic Synthesis ◽  
Quantum Circuit ◽  
Basic Block ◽  
Synthesis Methods ◽  
Quantum Logic Synthesis ◽  
Block Based ◽  
Arbitrary Quantum

In this paper, the problem of constructing an efficient quantum circuit for the implementation of an arbitrary quantum computation is addressed. To this end, a basic block based on the cosine-sine decomposition method is suggested which contains $l$ qubits. In addition, a previously proposed quantum-logic synthesis method based on quantum Shannon decomposition is recursively applied to reach unitary gates over $l$ qubits. Then, the basic block is used and some optimizations are applied to remove redundant gates. It is shown that the exact value of $l$ affects the number of one-qubit and CNOT gates in the proposed method. In comparison to the previous synthesis methods, the value of $l$ is examined consequently to improve either the number of CNOT gates or the total number of gates. The proposed approach is further analyzed by considering the nearest neighbor limitation. According to our evaluation, the number of CNOT gates is increased by at most a factor of $\frac{5}{3}$ if the nearest neighbor interaction is applied.

scholarly journals Nonnegative Combined Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Rafael Bru ◽  
Maria T. Gassó ◽  
Isabel Giménez ◽  
Máximo Santana
Keyword(s):  
Real Matrix ◽  
Sign Pattern ◽  
Doubly Stochastic ◽  
Classes Of Matrices ◽  
Combined Matrix

The combined matrix of a nonsingular real matrixAis the Hadamard (entrywise) productA∘A-1T. It is well known that row (column) sums of combined matrices are constant and equal to one. Recently, some results on combined matrices of different classes of matrices have been done. In this work, we study some classes of matrices such that their combined matrices are nonnegative and obtain the relation with the sign pattern ofA. In this case the combined matrix is doubly stochastic.

scholarly journals Universal control of quantum processes using sector-preserving channels

2021 ◽  
Vol 21 (15&16) ◽  
pp. 1320-1352
Author(s):  
Augustin Vanrietvelde ◽  
Giulio Chiribella
Keyword(s):  
Quantum Channel ◽  
Coherent Control ◽  
Quantum Circuit ◽  
Compact Representation ◽  
Dimensional System ◽  
Quantum Processes ◽  
Information Theoretic ◽  
Higher Dimensional ◽  
Standard Notion ◽  
Arbitrary Quantum

No quantum circuit can turn a completely unknown unitary gate into its coherently controlled version. Yet, coherent control of unknown gates has been realised in experiments, making use of a different type of initial resources. Here, we formalise the task achieved by these experiments, extending it to the control of arbitrary noisy channels, and to more general types of control involving higher dimensional control systems. For the standard notion of coherent control, we identify the information-theoretic resource for controlling an arbitrary quantum channel on a $d$-dimensional system: specifically, the resource is an extended quantum channel acting as the original channel on a $d$-dimensional sector of a $(d+1)$-dimensional system. Using this resource, arbitrary controlled channels can be built with a universal circuit architecture. We then extend the standard notion of control to more general notions, including control of multiple channels with possibly different input and output systems. Finally, we develop a theoretical framework, called supermaps on routed channels, which provides a compact representation of coherent control as an operation performed on the extended channels, and highlights the way the operation acts on different sectors.

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.

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