Quantum Compiling with Approximation of Multiplexors

A quantum compiling algorithm is an algorithm for decomposing ("compiling") an arbitrary unitary matrix into a sequence of elementary operations (SEO). Suppose $U_{in}$ is an \nb-bit unstructured unitary matrix (a unitary matrix with no special symmetries) that we wish to compile. For \nb>10, expressing $U_{in}$ as a SEO requires more than a million CNOTs. This calls for a method for finding a unitary matrix that: (1)approximates $U_{in}$ well, and (2) is expressible with fewer CNOTs than $U_{in}$. The purpose of this paper is to propose one such approximation method. Various quantum compiling algorithms have been proposed in the literature that decompose an arbitrary unitary matrix into a sequence of U(2)-multiplexors, each of which is then decomposed into a SEO. Our strategy for approximating $U_{in}$ is to approximate these intermediate U(2)-multiplexors. In this paper, we will show how one can approximate a U(2)-multiplexor by another U(2)-multiplexor that is expressible with fewer CNOTs.Comment: Ver1:18 pages (files: 1 .tex, 1 .sty, 7 .eps); Ver2:26 pages (files: 1 .tex, 1 .sty, 7 .eps, 7 .m) Ver2 = Ver1 + new material, including 7 Octave/Matlab m-file

Entanglement of Formation and Conditional Information Transmission

We show that the separability of states in quantum mechanics has a close counterpart in classical physics, and that conditional mutual information (a.k.a. conditional information transmission) is a very useful quantity in the study of both quantum and classical separabilities. We also show how to define entanglement of formation in terms of conditional mutual information. This paper lays the theoretical foundations for a sequel paper which will present a computer program that can calculate a decomposition of any separable quantum or classical state.Comment: 14 pages (files: 1 .tex, 1 .eps, 2 .sty

A Rudimentary Quantum Compiler

We present a new algorithm for reducing an arbitrary unitary matrix into a sequence of elementary operations (operations such as controlled-nots and qubit rotations). Such a sequence of operations can be used to manipulate an array of quantum bits (i.e., a quantum computer). We report on a C++ program called "Qubiter" that implements our algorithm. Qubiter source code is publicly available.Comment: 1 LaTeX file (25 pages), 9 eps file

QuanTree and QuanLin, Two Special Purpose Quantum Compilers

This paper introduces QuanTree v1.1 and QuanLin v1.1, two Java applications available for free. (Source code included in the distribution.) Each application compiles a different type of input quantum evolution operator. The applications output a quantum circuit that is approximately equal to the input evolution operator. QuanTree compiles an input evolution operator whose Hamiltonian is proportional to the incidence matrix of a balanced, binary tree graph. QuanLin compiles an input evolution operator whose Hamiltonian is proportional to the incidence matrix of a line (open string) graph. Both applications also output an error, defined as the distance in the Frobenius norm between the input evolution operator and the output quantum circuit.Comment: 14 pages (files: 1 .tex, 1 .sty, 10 .pdf).Ver2 of paper, for software ver. 1.1 instead of 1.

Quantum Fast Fourier Transform Viewed as a Special Case of Recursive Application of Cosine-Sine Decomposition

A quantum compiler is a software program for decomposing ("compiling") an arbitrary unitary matrix into a sequence of elementary operations (SEO). Coppersmith showed that the \nb-bit Discrete Fourier Transform matrix $U_{FT}$ can be decomposed in a very efficient way, as a sequence of order(\nb^2) elementary operations. Can a quantum compiler that doesn't know a priori about Coppersmith's decomposition nevertheless decompose $U_{FT}$ as a sequence of order(\nb^2) elementary operations? In other words, can it rediscover Coppersmith's decomposition by following a much more general algorithm? Yes it can, if that more general algorithm is the recursive application of the Cosine-Sine Decomposition (CSD).Comment: 14 pages (files: 1 .tex, 2 .sty, 5 .eps

An Introduction to Quantum Bayesian Networks for Mixed States

This paper is intended to be a pedagogical introduction to quantum Bayesian networks (QB nets), as I personally use them to represent mixed states (i.e., density matrices, and open quantum systems). A special effort is made to make contact with notions used in textbooks on quantum Shannon Information Theory (quantum SIT), such as the one by Mark Wilde (arXiv:1106.1445)Comment: 20 pages (3 files: 1 .tex, 2 .sty

Counterexamples to the Theory of Thermodynamics With Feedback Proposed By Sagawa and Ueda

We give some counterexamples to the theory of thermodynamics with feedback that has been proposed by Sagawa and Ueda. Our counterexamples are evidence that their theory, in its present form, is flawed.Comment: 4 pages (2 files: 1 .tex, 1 .sty

Entanglement of Distillation and Conditional Mutual Information

In previous papers, we expressed the Entanglement of Formation in terms of Conditional Mutual Information (CMI). In this brief paper, we express the Entanglement of Distillation in terms of CMI.Comment: 10 pages (files: 1 .tex, 2 .sty, 3 .eps

Introduction to Judea Pearl's Do-Calculus

This is a purely pedagogical paper with no new results. The goal of the paper is to give a fairly self-contained introduction to Judea Pearl's do-calculus, including proofs of his 3 rules.Comment: 16 pages (11 files: 1 .tex, 1 .sty, 9 .jpg

All Moments of the Uniform Ensemble of Quantum Density Matrices

Given a uniform ensemble of quantum density matrices $\rho$, it is useful to calculate the mean value over this ensemble of a product of entries of $\rho$. We show how to calculate such moments in this paper. The answer involves well known results from Group Representation Theory and Random Matrix Theory. This quantum problem has a well known classical counterpart: given a uniform ensemble of probability distributions $P=(P_1, P_2, ..., P_N)$ where the $P_j$ are non-negative reals that sum to one, calculate the mean value over this probability simplex of products of $P$ components. The answer to the classical problem follows from an integral formula due to Dirichlet.Comment: 17 pages (files: 1 .tex, 3 .sty