UNITARY MATRIX CODE

Implementing such matrix in Q# was one way to solve this problem. Here are matrices of these gates: $$$ X = egin $$$ We will use X gate and rotation gates ( R1, Ry, Rz). What operations does Q# support?įirstly, Q# has a variety of 1-qubit gates.

UNITARY MATRIX CODE

This algorithm was inspired by problem B2 from Microsoft Q# Coding Contest - Winter 2019, to solve which I had to write Q# code implementing a particular 4x4 unitary matrix. Please note that Q and Q -1 represent the conjugate. If Q is a complex square matrix and if it satisfies Q Q -1 then such matrix is termed as unitary.

In other words, I am going to describe an algorithm, which for arbitrary unitary matrix produces Q# source code implementing action of that matrix on a register of qubits. When the conjugate transpose of a complex square matrix is equal to the inverse of itself, then such matrix is called as unitary matrix. In this post we will consider different universal set of gates: set of gates which you can describe using Q# language.

One famous result is that you can implement any unitary operation by using only one-qubit gates and CNOT gate (which means these gates form a universal set). Therefore, any sequence of quantum gates can be described by a unitary matrix.īut what about going the other way: given a unitary matrix, represent it as a sequence of quantum gates? Turns out it's possible if we can use gates from universal set. Applying multiple gates multiplies state by a matrix which is matrix product of matrices for individual gates. Any quantum gate can be described as a unitary matrix acting on a vector representing the current state of a quantum computer. IntroductionĪlgorithms for a quantum computer are usually described as a sequence of quantum gates (such as Pauli X, Hadamard or CNOT) acting on register of qubits, followed by a sequence of measurements. In this post I am going to tell you about an algorithm which implements arbitrary unitary matrix as a sequence of elementary quantum gates in Q# programming language. Check it out for more great posts about quantum computing and Q#! Ma.This post is part of the Q# Advent Calendar 2020.

'Show that the eigenvalues of a unitary matrix have modulus 1'.

(2001), 'Unitary matrix', Encyclopedia of Mathematics, Springer Science+Business Media B.V. 'Restrictions on realizable unitary operations imposed by symmetry and locality' (in en). In fact, there are some similarities between orthogonal matrices and unitary matrices. For real matrices, unitary is the same as orthogonal.

Unitary matrices leave the length of a complex vector unchanged. Physical Review A (American Physical Society (APS)) 52 (5): 3457–3467. where denotes the conjugate transpose and is the matrix inverse. 'Elementary gates for quantum computation'. Margolus, Norman Shor, Peter Sleator, Tycho Smolin, John A.

↑ Barenco, Adriano Bennett, Charles H.

Quantum Computation and Quantum Information. Linear Algebra and Its Applications 547: 32–44. ↑ Führ, Hartmut Rzeszotnik, Ziemowit (2018).Linear and Multilinear Algebra 50 (4): 321–326. 'Additive decomposition of real matrices'. Hermitian matrix and Skew-Hermitian matrix.Many other factorizations of a unitary matrix in basic matrices are possible. This expression highlights the relation between 2 × 2 unitary matrices and 2 × 2 orthogonal matrices of angle θ. Given two complex vectors x and y, multiplication by U preserves their inner product that is, ⟨ U x, U y⟩ = ⟨ x, y⟩.For any unitary matrix U of finite size, the following hold: