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Robert R. Tucci

Publications and source records attributed to Robert R. Tucci.

At least 37 records · Page 2Linked to original sources
arXiv2010

An Adaptive, Fixed-Point Version of Grover's Algorithm

We give an adaptive, fixed-point version of Grover's algorithm. By this we mean that our algorithm performs an infinite sequence of gradually diminishing steps (so we say it's adaptive) that drives the starting state to the target state with absolute certainty (so we say it's a fixed-point algorithm). Our algorithm is motivated by Bloch sphere geometry. We include with the ArXiv distribution of this paper some simple software (Octave/Matlab m-files) that implements, tests and illustrates some of the results of this paper.

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arXiv2010

Quantum Gibbs Sampling Using Szegedy Operators

We present an algorithm for doing Gibbs sampling on a quantum computer. The algorithm combines phase estimation for a Szegedy operator, and Grover's algorithm. For any $ε>0$, the algorithm will sample a probability distribution in ${\cal O}(\frac{1}{\sqrtδ})$ steps with precision ${\cal O}(ε)$. Here $δ$ is the distance between the two largest eigenvalue magnitudes of the transition matrix of the Gibbs Markov chain used in the algorithm. It takes ${\cal O}(\frac{1}δ)$ steps to achieve the same precision if one does Gibbs sampling on a classical computer.

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arXiv2009

Code Generator for Quantum Simulated Annealing

This paper introduces QuSAnn v1.2 and Multiplexor Expander v1.2, two Java applications available for free. (Source code included in the distribution.) QuSAnn is a "code generator" for quantum simulated annealing: after the user inputs some parameters, it outputs a quantum circuit for performing simulated annealing on a quantum computer. The quantum circuit implements the algorithm of Wocjan et al. (arXiv:0804.4259), which improves on the original algorithm of Somma et al. (arXiv:0712.1008). The quantum circuit generated by QuSAnn includes some quantum multiplexors. The application Multiplexor Expander allows the user to replace each of those multiplexors by a sequence of more elementary gates such as multiply controlled NOTs and qubit rotations.

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arXiv2008

Use of a Quantum Computer to do Importance and Metropolis-Hastings Sampling of a Classical Bayesian Network

Importance sampling and Metropolis-Hastings sampling (of which Gibbs sampling is a special case) are two methods commonly used to sample multi-variate probability distributions (that is, Bayesian networks). Heretofore, the sampling of Bayesian networks has been done on a conventional "classical computer". In this paper, we propose methods for doing importance sampling and Metropolis-Hastings sampling of a classical Bayesian network on a quantum computer.

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arXiv2008

Use of a Quantum Computer and the Quick Medical Reference To Give an Approximate Diagnosis

The Quick Medical Reference (QMR) is a compendium of statistical knowledge connecting diseases to findings (symptoms). The information in QMR can be represented as a Bayesian network. The inference problem (or, in more medical language, giving a diagnosis) for the QMR is to, given some findings, find the probability of each disease. Rejection sampling and likelihood weighted sampling (a.k.a. likelihood weighting) are two simple algorithms for making approximate inferences from an arbitrary Bayesian net (and from the QMR Bayesian net in particular). Heretofore, the samples for these two algorithms have been obtained with a conventional "classical computer". In this paper, we will show that two analogous algorithms exist for the QMR Bayesian net, where the samples are obtained with a quantum computer. We expect that these two algorithms, implemented on a quantum computer, can also be used to make inferences (and predictions) with other Bayesian nets.

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arXiv2008

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.

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arXiv2008

QuanFou, QuanGlue, QuanOracle and QuanShi, Four Special Purpose Quantum Compilers

This paper introduces QuanFou v1.1, QuanGlue v1.1, QuanOracle v1.1, QuanShi v1.1, four Java applications available for free. (Source code included in the distribution.) Each application compiles a different kind of input quantum evolution operator. The applications output a quantum circuit that equals the input evolution operator.

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arXiv2008

Java Application that Outputs Quantum Circuit for Some NAND Formula Evaluators

This paper introduces QuanFruit v1.1, a Java application available for free. (Source code included in the distribution.) Recently, Farhi-Goldstone-Gutmann (FGG) wrote a paper arXiv:quant-ph/0702144 that proposes a quantum algorithm for evaluating NAND formulas. QuanFruit outputs a quantum circuit for the FFG algorithm.

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arXiv2007

How to Compile Some NAND Formula Evaluators

We say a unitary operator acting on a set of qubits has been compiled if it has been expressed as a SEO (sequence of elementary operations, like CNOTs and single-qubit operations). SEO's are often represented as quantum circuits. arXiv:quant-ph/0702144 by Farhi-Goldstone-Gutmann has inspired a recent flurry of papers, that propose quantum algorithms for evaluating NAND formulas via quantum walks over tree graphs. These algorithms use two types of unitary evolution: oracle and non-oracle. Non-oracle evolutions are independent of the NAND formula input, whereas oracle evolutions depend on this input. In this paper we compile (i.e., give explicit SEOs and their associated quantum circuits for) the oracle and non-oracle evolution operators used in some of these NAND formula evaluators. We consider here only the case of balanced binary NAND trees. Our compilation methods are based on the CSD (Cosine Sine Decomposition), a matrix decomposition from Linear Algebra. The CS decomposition has been used very successfully in the past to compile unstructured unitary matrices exactly.

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arXiv2007

Factorization of Quantum Density Matrices According to Bayesian and Markov Networks

We show that any quantum density matrix can be represented by a Bayesian network (a directed acyclic graph), and also by a Markov network (an undirected graph). We show that any Bayesian or Markov net that represents a density matrix, is logically equivalent to a set of conditional independencies (symmetries) satisfied by the density matrix. We show that the d-separation theorems of classical Bayesian and Markov networks generalize in a simple and natural way to quantum physics. The quantum d-separation theorems are shown to be closely connected to quantum entanglement. We show that the graphical rules for d-separation can be used to detect pairs of nodes (or of node sets) in a graph that are unentangled. CMI entanglement (a.k.a. squashed entanglement), a measure of entanglement originally discovered by analyzing Bayesian networks, is an important part of the theory of this paper.

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arXiv2006

Simplifying Quantum Circuits via Circuit Invariants and Dressed CNOTs

Quantum Compiling Algorithms decompose (exactly, without approximations) an arbitrary $2^\nb$ unitary matrix acting on $\nb$ qubits, into a sequence of elementary operations (SEO). There are many possible ways of decomposing a unitary matrix into a SEO, and some of these decompositions have shorter length (are more efficient) than others. Finding an optimum (shortest) decomposition is a very hard task, and is not our intention here. A less ambitious, more doable task is to find methods for optimizing small segments of a SEO. Call these methods piecewise optimizations. Piecewise optimizations involve replacing a small quantum circuit by an equivalent one with fewer CNOTs. Two circuits are said to be equivalent if one of them multiplied by some external local operations equals the other. This equivalence relation between circuits has its own class functions, which we call circuit invariants. Dressed CNOTs are a simple yet very useful generalization of standard CNOTs. After discussing circuit invariants and dressed CNOTs, we give some methods for simplifying 2-qubit and 3-qubit circuits. We include with this paper software (written in Octave/Matlab) that checks many of the algorithms proposed in the paper.

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arXiv2005

Replacing Two Controlled-U's with Two CNOTs

We give a simple, closed-form formula, what we call the Deflation Identity, for converting any 2-qubit circuit with exactly two controlled-U's (and some 1-qubit rotations) into an equivalent circuit with just two CNOTs (and some 1-qubit rotations). We also give two interesting applications of the Deflation Identity; one to "opening and closing a breach" in a quantum circuit, the other to the CS decomposition of a 2-qubit operator.

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arXiv2005

An Introduction to Cartan's KAK Decomposition for QC Programmers

This paper presents no new results; its goals are purely pedagogical. A special case of the Cartan Decomposition has found much utility in the field of quantum computing, especially in its sub-field of quantum compiling. This special case allows one to factor a general 2-qubit operation (i.e., an element of U(4)) into local operations applied before and after a three parameter, non-local operation. In this paper, we give a complete and rigorous proof of this special case of Cartan's Decomposition. From the point of view of QC programmers who might not be familiar with the subtleties of Lie Group Theory, the proof given here has the virtues, that it is constructive in nature, and that it uses only Linear Algebra. The constructive proof presented in this paper is implemented in some Octave/Matlab m-files that are included with the paper. Thus, this paper serves as documentation for the attached m-files.

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arXiv2005

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.

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arXiv2004

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).

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arXiv2004

Qubiter Algorithm Modification, Expressing Unstructured Unitary Matrices with Fewer CNOTs

A quantum compiler is a software program for decomposing ("compiling") an arbitrary unitary matrix into a sequence of elementary operations (SEO). The author of this paper is also the author of a quantum compiler called Qubiter. Qubiter uses a matrix decomposition called the Cosine-Sine Decomposition (CSD) that is well known in the field of Computational Linear Algebra. One way of measuring the efficiency of a quantum compiler is to measure the number of CNOTs it uses to express an unstructured unitary matrix (a unitary matrix with no special symmetries). We will henceforth refer to this number as $ε$. In this paper, we show how to improve $ε$ for Qubiter so that it matches the current world record for $ε$, which is held by another quantum compiling algorithm based on CSD.

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arXiv2004

QC Paulinesia

My goal in producing this document was to create a collection of qubit circuit identities that are used in Quantum Computing. Mathematicians and Physicists may consider it as being analogous to a Table of Integrals or a Mathematical Handbook such as Gradshteyn-Ryzhik or Abramowitz-Stegun. Computer Programmers may think of it as a scrapbook of code snippets that are elegant, instructive, well documented, and useful. Electronics experts may view it as a compendium of circuits for performing a large assortment of tasks. The vast majority of the circuit identities collected in this work were not discovered for the first time by me, and I take no credit for discovering them. In producing this document, I am acting as a collector, not as a discoverer.

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