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IEEE National Aerospace and Electronics Conference (NAECON)

A Quantum Probabilistic Comparator: Circuit Implementation

Linh Nguyen, Zhenhua Jiang

Published Aug 28, 2023• 2 citations
Publisher

Abstract

Decision making in stochastic autonomous systems, which considers uncertainty of many factors, is always a challenge. The problem can typically be addressed by probabilistic methods such as approximate stochastic optimization or Monte Carlo (MC) simulation. The potential solution may usually be constrained by the system's computing power, due to the size of the probabilistic problem. This paper aims to present and implement a quantum probabilistic comparison (QPC) algorithm that can help to find a desired solution or make decision between two random variables under uncertainty. This approach takes advantage of quantum computing to perform parallel comparison and fast calculation of the probability of matched state vectors. In the presented scheme, two random variables have different values (e.g., 2, 4, 8, or more values) and different probability distributions. The data of the variables are encoded to the basis states of the corresponding qubits. The quantum probabilistic comparator is used to make up the quantum output matrix that stores the comparison solutions as well as their probability probabilities. Moreover, unlike the existing quantum comparator algorithm, which uses bit string comparison approaches, the proposed probabilistic comparator is not only comparing the basis state, but also accumulating the outcome's probabilities. The calculation results of the quantum probabilistic comparator circuit are compared with those from classical computation and quantum circuit simulation as well as on an IBM quantum computer, for analysis and verification.

DOI
10.1109/naecon58068.2023.10365860

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Decision making in stochastic autonomous systems, which considers uncertainty of many factors, is always a challenge. The problem can typically be addressed by probabilistic methods such as approximate stochastic optimization or Monte Carlo (MC) simulation. The potential solution may usually be constrained by the system's computing power, due to the size of the probabilistic problem. This paper aims to present and implement a quantum probabilistic comparison (QPC) algorithm that can help to find a desired solution or make decision between two random variables under uncertainty. This approach takes advantage of quantum computing to perform parallel comparison and fast calculation of the probability of matched state vectors. In the presented scheme, two random variables have different values (e.g., 2, 4, 8, or more values) and different probability distributions. The data of the variables are encoded to the basis states of the corresponding qubits. The quantum probabilistic comparator is used to make up the quantum output matrix that stores the comparison solutions as well as their probability probabilities. Moreover, unlike the existing quantum comparator algorithm, which uses bit string comparison approaches, the proposed probabilistic comparator is not only comparing the basis state, but also accumulating the outcome's probabilities. The calculation results of the quantum probabilistic comparator circuit are compared with those from classical computation and quantum circuit simulation as well as on an IBM quantum computer, for analysis and verification.
Quantum circuit optimization remains a critical challenge in the development of quantum computing. Traditional dynamic programming (DP) methods, while effective for deterministic circuit design, fail to adequately address the inherent probabilistic nature of quantum operations. This paper proposes a novel approach that extends DP algorithms by explicitly incorporating probabilistic models representing uncertainties in quantum gate operations and qubit measurements. The core idea is to leverage Monte Carlo simulations to estimate expected costs and rewards within the DP framework. This allows for a more realistic and potentially efficient solution to the circuit optimization problem, considering the statistical fluctuations that are fundamental to quantum computing. We demonstrate the feasibility of this method conceptually, outlining the key components and algorithmic steps. Further research and implementation will be necessary to fully realize the potential of this approach.
We discuss a new approach to simulate quantum algorithms using classical probabilistic bits and circuits. Each qubit (a two-level quantum system) is initially mapped to a vector in an eight dimensional probability space (equivalently, to a classical random variable with eight probabilistic outcomes). The key idea in this mapping is to store both the amplitude and phase information of the complex coefficients that describe the qubit state in the probabilities. Due to the identical tensor product structure of combining multiple quantum systems as well as multiple probability spaces, nnn qubits are then mapped to a tensor product of nnn 8-dimensional probabilistic vectors (i.e., the Hilbert space of dimension 2n2^n2n is mapped to a probability space of dimension 8n8^n8n). After this initial mapping, we show how to implement the analogs of single-qubit and two-qubit gates in the probability space using correlation-inducing operations on these classical random variables. The key defining feature of both the mapping to the probability space and the transformations in this space (i.e., operations on the random variables) is that they are not linear, but instead affine. Using this architecture, the evolution of the 2n2^n2n complex coefficients of the quantum system can be tracked in the joint fully-correlated probabilities of the polynomial number of random variables. We then give specific procedures for implementing (1) the Deutsch-Jozsa algorithm, and (2) the Quantum Fourier Transform in the probability space. Identical to the Quantum case, simulating the Quantum Fourier Transform in the probability space requires O(n)O(n)O(n) probabilistic bits and O(n2)O(n^2)O(n2) (i.e., quadratic in the number of quantum bits) operations.
This paper presents a novel approach to simulating quantum computing algorithms using Monte Carlo methods, specifically designed to account for the inherent probabilistic nature of quantum systems. Traditional simulation methods often rely on deterministic models, which fail to capture the fundamental uncertainty associated with quantum mechanics. Our method employs a stochastic simulation engine that accurately models the probabilistic behavior of quantum gates and measurements. The core of the approach involves generating a large number of samples, each representing a possible outcome of a quantum computation. By analyzing the statistical distribution of these samples, we can approximate the expected results of the algorithm. This technique offers a practical way to evaluate the performance of quantum algorithms, especially those with high levels of uncertainty, and provides a valuable tool for algorithm design and optimization. The method avoids the limitations of purely deterministic simulations and offers a more realistic representation of quantum computation. Key aspects include the development of a robust sampling strategy and the efficient calculation of probabilities. The approach is demonstrated through a detailed explanation and illustrative examples, focusing on the conceptual framework and the underlying mathematical principles.
This paper introduces a novel hybrid algorithm framework combining the strengths of probability logic and quantum computing to address complex optimization problems. We propose a system architecture that leverages the inherent scalability and parallel processing capabilities of quantum computing to enhance the robustness and efficiency of probabilistic optimization algorithms. The core mechanism involves integrating probabilistic logic modules with quantum computation layers, allowing for the efficient handling of large and complex problem instances. This framework aims to achieve significantly improved accuracy and scalability compared to traditional optimization methods, particularly when tackling problems with inherent uncertainties and combinatorial nature. The study explores the potential benefits of this integration, highlighting the advantages of combining these two powerful computational paradigms. We present a preliminary implementation demonstrating the framework's efficacy in a specific optimization scenario.
Quantum-mechanical phenomena are playing an increasing role in information processing as transistor sizes approach the nanometer level, while the securest forms of communication rely on quantum data encoding. When they involve a finite number of basis states, these phenomena can be modeled as quantum circuits, the quantum analogue of conventional or classical logic circuits. Simulation of quantum circuits can therefore be used as a tool to evaluate issues in the design of quantum information processors. Unfortunately, simulating such phenomena efficiently is exceedingly difficult. The matrices representing quantum operators (gates) and vectors modeling quantum states grow exponentially with the number of quantum bits. The information represented by quantum states and operators often exhibits structure that can be exploited when simulating certain classes of quantum circuits. We study the development of simulation methods that run on classical computers and take advantage of such repetitions and redundancies. In particular, we define a new data structure for simulating quantum circuits called the quantum information decision diagram (QuIDD). A QuIDD is a compressed graph representation of a vector or matrix and permits computations to be performed directly on the compressed data. We develop a comprehensive set of algorithms for operating on QuIDDs in both the state-vector and density-matrix formats, and evaluate their complexity. These algorithms have been implemented in a general-purpose simulator program for quantum-mechanical applications called QuIDDPro. Through extensive experiments conducted on representative quantum simulation applications, including Grover's search algorithm, error characterization, and reversible circuits, we demonstrate that QuIDDPro is faster than other existing quantum-mechanical simulators such as the National Institute of Standards and Technology's QCSim program, and is far more memory-efficient. Using QuIDDPro, we explore the advantages of quantum computation over classical computation, simulate quantum errors and error correction, and study the impact of numerical precision on the fidelity of simulations. We also develop several novel algorithms for testing quantum circuit equivalence and compare them empirically. The QuIDDPro software is equipped with a user-friendly interface and is distributed with numerous example scripts. It has been used as a laboratory supplement for quantum computing courses at several universities.
Abstract This laboratory work provides a practical exploration of random quantum circuits and their statistical behavior using the Qiskit framework. The material is based on educational resources developed by the International Quantum Center at CERN (Switzerland) and aims to introduce participants to probabilistic outputs in quantum computation and the comparison between experimental and theoretical distributions. In Laboratory Work No. 8, participants generate samples from randomly constructed quantum circuits, which consist of qubits initialized in superposition and subjected to sequences of randomly chosen quantum gates. Measurement operations are applied to obtain classical outcomes, producing a distribution of bitstrings. The resulting empirical distribution is then compared to the theoretical probability distribution expected from the ideal quantum state. This exercise demonstrates the intrinsic probabilistic nature of quantum measurement, the impact of circuit depth and gate selection on outcome distributions, and the role of randomness in quantum circuits. Participants learn to implement, execute, and analyze random quantum circuits in Qiskit, including statistical validation against theoretical models. The laboratory guide emphasizes hands-on experience with quantum circuit sampling, data collection, histogram visualization, and statistical comparison, bridging the gap between theoretical quantum mechanics and practical quantum computation on simulators and cloud-accessible hardware.