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Quantum belief function.

Qianli Zhou, Guojing Tian, Yong Deng

Published Jul 8, 2021• 1 citation
OpenAlex

Abstract

The belief function in Dempster Shafer evidence theory can express more information than the traditional Bayesian distribution. It is widely used in approximate reasoning, decision-making and information fusion. However, its power exponential explosion characteristics leads to the extremely high computational complexity when handling large amounts of elements in classic computers. In order to solve the problem, we encode the basic belief assignment (BBA) into quantum states, which makes each qubit correspond to control an element. Besides the high efficiency, this quantum expression is very conducive to measure the similarity between two BBAs, and the measuring quantum algorithm we come up with has exponential acceleration theoretically compared to the corresponding classical algorithm. In addition, we simulate our quantum version of BBA on Qiskit platform, which ensures the rationality of our algorithm experimentally. We believe our results will shed some light on utilizing the characteristic of quantum computation to handle belief function more conveniently.

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The transferable belief model (TBM), developed within Dempster-Shafer theory, represents ambiguity and partial ignorance through set-valued belief masses. Its classical operations, however, can grow combinatorially over the power set. We formulate TBM reasoning on quantum circuits using the mass function quantum state (MFQS). The resulting framework covers belief representation, credal-level fusion and correction, product-space operations, and the generalized Bayesian theorem. For prepared MFQS inputs, CCR and the entire $α$-junction use element-wise gate modules whose logical counts are linear in the number of frame elements. These circuits avoid explicit operation-stage updates over all $2^n$ focal-set coordinates. A noisy multi-domain classifier-fusion example further demonstrates interpretable source fusion and improved robustness to gate-control noise. The results establish Dempster-Shafer structures as a semantic layer for quantum information processing beyond singleton-probability representations.

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Quantum supremacy: some fundamental concepts

Man‐Hong Yung

Quantum computation was envisioned by Feynman as a valuable means of solving quantum problems [1]. The question is, how do we prove the superiority of quantum computing over classical devices? A common misconception in describing the power of a quantum computer over a classical computer is that quantum bits can be prepared in a superposition of an exponential number of states, which cannot be achieved with classical bits. This statement is certainly true, but it does not tell us, specifically, what problems quantum computers can efficiently solve while classical computers fail to do so. In fact, based exactly on the technique of Feynman's path integral, the output (i.e. transition amplitude) of any quantum circuit can be calculated classically with a polynomial amount of memory (although the temporal cost would be exponential). Therefore any quantum computation can be simulated classically; the key question is about the efficiency. Much effort has been made trying to pin down the power of quantum computers based on the machinery of the computational complexity theory. In particular, the class of computational (decision) problems that can be solved efficiently by a quantum computer is called BQP (bounded-error quantum polynomial time); the counterpart for classical computers is called P (polynomial time). Of course, BQP cannot be less than P; in principle quantum computers can simulate classical computers efficiently. The point is that the foundation of quantum computation cannot be established without a proof showing that BQP is strictly larger than P (i.e. BQP ≠ P). In this sense, this question is as interesting as the famous challenge of proving P ≠ NP (non-deterministic polynomial time). A well known instance of an NP problem is the factoring problem; Shor's quantum algorithm is capable of solving the factoring problem in polynomial time, which is not achievable with the best classical algorithm discovered so far. However, we still cannot rigorously exclude the possibility of the existence of an efficient classical algorithm for factoring. Although again without a proof, it is commonly believed that quantum computers are unable to solve some of the NP problems. In fact, the proof of this statement (if true) provides a possible path to prove P ≠ NP. A potentially less demanding question would be, when do we expect a quantum computer to be able to perform some well defined tasks (not necessarily related to any practical problem) that cannot be simulated with any currently available classical device, within a reasonable time? Once achieved, the status is called ‘quantum supremacy' [2]. One may ask, what about Grover's search algorithm, which provides a quadratic speedup over the classical search? Can we say that we can already achieve quantum supremacy with Grover's algorithm? The problem is that, instead of taking the large-N limit, quantum supremacy requires us to determine the actual number of qubits and gates that can no longer be simulated by classical computers within a reasonable time. To elaborate further, we remark that there can be two notions of classical simulation, namely ‘strong simulation' versus ‘weak simulation'. Strong simulation refers to the cases of calculating the transition probabilities, or expectation values of observables, to a high accuracy in polynomial time. Weak simulation requires the probability distributions to be accurately reproduced, which involves ‘sampling' the different outcomes of the quantum devices. Some quantum circuits that cannot be strongly simulated with an efficient classical means may have efficient classical methods for weak simulation [3]. However, we should be careful about the accuracy requirement in the simulation tasks. For example, for many decision problems, it may be sufficient to determine the transition amplitudes to within an additive error, instead of multiplicative error. In other words, the classical simulability of quantum computational problems depends on the accuracy requirement. Currently, there are three popular approaches for achieving quantum supremacy (see Fig. 1), namely (i) boson sampling [4], (ii) sampling IQP (instantaneous quantum polynomial) circuits [5], and (iii) sampling chaotic quantum circuits [6]. In all of these approaches, distributions of different bit strings or photon numbers are sampled from the quantum devices. Furthermore, they all involve the assumption that classical computers are unable to efficiently determine the transition amplitudes, and/or reproduce (or approximate) the distributions of quantum devices performing these tasks, in the sense of both strong and weak simulations. Three different approaches for achieving quantum supremacy: (a) boson sampling; (b) IQP circuits; (c) chaotic quantum circuits. In boson sampling, single photons are injected into different modes of a linear optical network. The task is to determine the photon distributions at the output ports. The key feature of boson sampling is that the transition amplitudes are related to the permanents of complex matrices, which are in general very difficult to calculate exactly, or approximate to within an multiplicative error; these problems belong to the #P-hard complexity class. Moreover, efficient weak simulation of boson sampling is believed to be impossible, unless the polynomial hierarchy collapses to its third level. However, a recent numerical study suggests that, in order to achieve quantum supremacy with boson sampling, one needs to simultaneously generate at least 50 or more single photons, which is still highly technologically challenging. An interesting question is whether linear optics, in the setting of boson sampling, can be applied to solve decision problems. In this case, the transition amplitudes may need to be determined to an additive error. However, we found that a large class of decision problems associated with boson sampling can be simulated by a classical computer [7], settling an open problem [8] by Aaronson and Hance. IQP circuits represent a simplified quantum-circuit model. The initial state starts from the all-zero state, |000…0〉, followed by applying Hadamard gates to each qubit. Then, diagonal gates are applied to the qubits, followed by applying Hadamard gates to each qubit again. The argument of showing the complexity in simulating IQP circuits is similar to that of boson sampling. In fact, the complexity of boson sampling was inspired by the complexity results of IQP circuits. However, when subject to noise, IQP circuits may become classically simulable [9]. Lastly, chaotic quantum circuits refer to the class of quantum circuits where two-qubit gates are applied regularly, while single-qubit gates are applied randomly from a gate set. The output distributions of these circuits approach the so-called Porter–Thomas distribution, which is a characteristic feature of quantum chaos. Recently, there have been several numerical investigations aiming to explore the ultimate limit of classical computing in simulating low-depth quantum circuits. For the ideal cases, one needs to consider both the qubit number and circuit depth for benchmarking quantum supremacy. However, in the presence of noise, these chaotic circuits may also become classically simulable for high-depth circuits [10]. Finally, other than these three approaches, one should expect that quantum supremacy can be achieved for many practical applications, e.g. simulation of quantum chemistry or quantum machine learning. When will it happen? Only time will tell!

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Zichang He, Jiang Wen

The sure thing principle and the law of total probability are basic laws in classic probability theory. A disjunction fallacy leads to the violation of these two classical probability laws. In this paper, a new quantum dynamic belief decision making model based on quantum dynamic modelling and Dempster-Shafer (D-S) evidence theory is proposed to address this issue and model the real human decision-making process. Some mathematical techniques are borrowed from quantum mathematics. Generally, belief and action are two parts in a decision making process. The uncertainty in belief part is represented by a superposition of certain states. The uncertainty in actions is represented as an extra uncertainty state. The interference effect is produced due to the entanglement between beliefs and actions. Basic probability assignment (BPA) of decisions is generated by quantum dynamic modelling. Then BPA of the extra uncertain state and an entanglement degree defined by an entropy function named Deng entropy are used to measure the interference effect. Compared the existing model, the number of free parameters is less in our model. Finally, a classical categorization decision-making experiment is illustrated to show the effectiveness of our model.

arXiv:1703.02386•Mar 6, 2017•0 citations
The belief function in Dempster Shafer evidence theory can express more information than the traditional Bayesian distribution. It is widely used in approximate reasoning, decision-making and information fusion. However, its power exponential explosion characteristics leads to the extremely high computational complexity when handling large amounts of elements in classic computers. In order to solve the problem, we encode the basic belief assignment (BBA) into quantum states, which makes each qubit correspond to control an element. Besides the high efficiency, this quantum expression is very conducive to measure the similarity between two BBAs, and the measuring quantum algorithm we come up with has exponential acceleration theoretically compared to the corresponding classical algorithm. In addition, we simulate our quantum version of BBA on Qiskit platform, which ensures the rationality of our algorithm experimentally. We believe our results will shed some light on utilizing the characteristic of quantum computation to handle belief function more conveniently.
The transferable belief model (TBM), developed within Dempster-Shafer theory, represents ambiguity and partial ignorance through set-valued belief masses. Its classical operations, however, can grow combinatorially over the power set. We formulate TBM reasoning on quantum circuits using the mass function quantum state (MFQS). The resulting framework covers belief representation, credal-level fusion and correction, product-space operations, and the generalized Bayesian theorem. For prepared MFQS inputs, CCR and the entire ααα-junction use element-wise gate modules whose logical counts are linear in the number of frame elements. These circuits avoid explicit operation-stage updates over all 2n2^n2n focal-set coordinates. A noisy multi-domain classifier-fusion example further demonstrates interpretable source fusion and improved robustness to gate-control noise. The results establish Dempster-Shafer structures as a semantic layer for quantum information processing beyond singleton-probability representations.
Quantum computation was envisioned by Feynman as a valuable means of solving quantum problems [1]. The question is, how do we prove the superiority of quantum computing over classical devices? A common misconception in describing the power of a quantum computer over a classical computer is that quantum bits can be prepared in a superposition of an exponential number of states, which cannot be achieved with classical bits. This statement is certainly true, but it does not tell us, specifically, what problems quantum computers can efficiently solve while classical computers fail to do so. In fact, based exactly on the technique of Feynman's path integral, the output (i.e. transition amplitude) of any quantum circuit can be calculated classically with a polynomial amount of memory (although the temporal cost would be exponential). Therefore any quantum computation can be simulated classically; the key question is about the efficiency. Much effort has been made trying to pin down the power of quantum computers based on the machinery of the computational complexity theory. In particular, the class of computational (decision) problems that can be solved efficiently by a quantum computer is called BQP (bounded-error quantum polynomial time); the counterpart for classical computers is called P (polynomial time). Of course, BQP cannot be less than P; in principle quantum computers can simulate classical computers efficiently. The point is that the foundation of quantum computation cannot be established without a proof showing that BQP is strictly larger than P (i.e. BQP ≠ P). In this sense, this question is as interesting as the famous challenge of proving P ≠ NP (non-deterministic polynomial time). A well known instance of an NP problem is the factoring problem; Shor's quantum algorithm is capable of solving the factoring problem in polynomial time, which is not achievable with the best classical algorithm discovered so far. However, we still cannot rigorously exclude the possibility of the existence of an efficient classical algorithm for factoring. Although again without a proof, it is commonly believed that quantum computers are unable to solve some of the NP problems. In fact, the proof of this statement (if true) provides a possible path to prove P ≠ NP. A potentially less demanding question would be, when do we expect a quantum computer to be able to perform some well defined tasks (not necessarily related to any practical problem) that cannot be simulated with any currently available classical device, within a reasonable time? Once achieved, the status is called ‘quantum supremacy' [2]. One may ask, what about Grover's search algorithm, which provides a quadratic speedup over the classical search? Can we say that we can already achieve quantum supremacy with Grover's algorithm? The problem is that, instead of taking the large-N limit, quantum supremacy requires us to determine the actual number of qubits and gates that can no longer be simulated by classical computers within a reasonable time. To elaborate further, we remark that there can be two notions of classical simulation, namely ‘strong simulation' versus ‘weak simulation'. Strong simulation refers to the cases of calculating the transition probabilities, or expectation values of observables, to a high accuracy in polynomial time. Weak simulation requires the probability distributions to be accurately reproduced, which involves ‘sampling' the different outcomes of the quantum devices. Some quantum circuits that cannot be strongly simulated with an efficient classical means may have efficient classical methods for weak simulation [3]. However, we should be careful about the accuracy requirement in the simulation tasks. For example, for many decision problems, it may be sufficient to determine the transition amplitudes to within an additive error, instead of multiplicative error. In other words, the classical simulability of quantum computational problems depends on the accuracy requirement. Currently, there are three popular approaches for achieving quantum supremacy (see Fig. 1), namely (i) boson sampling [4], (ii) sampling IQP (instantaneous quantum polynomial) circuits [5], and (iii) sampling chaotic quantum circuits [6]. In all of these approaches, distributions of different bit strings or photon numbers are sampled from the quantum devices. Furthermore, they all involve the assumption that classical computers are unable to efficiently determine the transition amplitudes, and/or reproduce (or approximate) the distributions of quantum devices performing these tasks, in the sense of both strong and weak simulations. Three different approaches for achieving quantum supremacy: (a) boson sampling; (b) IQP circuits; (c) chaotic quantum circuits. In boson sampling, single photons are injected into different modes of a linear optical network. The task is to determine the photon distributions at the output ports. The key feature of boson sampling is that the transition amplitudes are related to the permanents of complex matrices, which are in general very difficult to calculate exactly, or approximate to within an multiplicative error; these problems belong to the #P-hard complexity class. Moreover, efficient weak simulation of boson sampling is believed to be impossible, unless the polynomial hierarchy collapses to its third level. However, a recent numerical study suggests that, in order to achieve quantum supremacy with boson sampling, one needs to simultaneously generate at least 50 or more single photons, which is still highly technologically challenging. An interesting question is whether linear optics, in the setting of boson sampling, can be applied to solve decision problems. In this case, the transition amplitudes may need to be determined to an additive error. However, we found that a large class of decision problems associated with boson sampling can be simulated by a classical computer [7], settling an open problem [8] by Aaronson and Hance. IQP circuits represent a simplified quantum-circuit model. The initial state starts from the all-zero state, |000…0〉, followed by applying Hadamard gates to each qubit. Then, diagonal gates are applied to the qubits, followed by applying Hadamard gates to each qubit again. The argument of showing the complexity in simulating IQP circuits is similar to that of boson sampling. In fact, the complexity of boson sampling was inspired by the complexity results of IQP circuits. However, when subject to noise, IQP circuits may become classically simulable [9]. Lastly, chaotic quantum circuits refer to the class of quantum circuits where two-qubit gates are applied regularly, while single-qubit gates are applied randomly from a gate set. The output distributions of these circuits approach the so-called Porter–Thomas distribution, which is a characteristic feature of quantum chaos. Recently, there have been several numerical investigations aiming to explore the ultimate limit of classical computing in simulating low-depth quantum circuits. For the ideal cases, one needs to consider both the qubit number and circuit depth for benchmarking quantum supremacy. However, in the presence of noise, these chaotic circuits may also become classically simulable for high-depth circuits [10]. Finally, other than these three approaches, one should expect that quantum supremacy can be achieved for many practical applications, e.g. simulation of quantum chemistry or quantum machine learning. When will it happen? Only time will tell!
This thesis explores the intersection of Bayesian inference and quantum computing, with a particular focus on developing and analyzing quantum approaches to probabilistic modeling. The motivation arises from the limitations of classical inference methods, which often become intractable when applied to high-dimensional distributions and complex Bayesian networks. By leveraging quantum algorithms, the work investigates how probabilistic reasoning can benefit from computational speedups, scalability, and more efficient handling of uncertainty.The implementation targets canonical benchmark problems—the Sprinkler Network and the Asia Network—where both classical and quantum Bayesian inference techniques are applied and compared. Using Qiskit, the thesis demonstrates how quantum circuits can encode probabilistic models through parameterized rotations and controlled operations, and how posterior distributions can be computed by sampling and amplitude amplification. Comparative experiments systematically evaluate the accuracy, variance, and scalability of quantum Bayesian inference against classical baselines, highlighting trade-offs between precision and shot counts while exploring the potential for quadratic speedups.The results show that although present-day hardware constraints, such as limited qubit counts and noise, restrict large-scale deployment, quantum implementations already display strong promise for complex probabilistic reasoning tasks. Importantly, the conclusions underscore the role of quantum Bayesian networks as a pathway toward practical quantum-enhanced machine learning.
In the Bayesian approach to probability theory, probability quantifies a degree of belief for a single trial, without any a priori connection to limiting frequencies. Despite being prescribed by a fundamental law, probabilities for individual quantum systems can be understood within the Bayesian approach. We argue that the distinction between classical and quantum probabilities lies not in their definition, but in the nature of the information they encode. In the classical world, maximal information about a physical system is complete in the sense of providing definite predictions for all possible questions that can be asked of the system. In the quantum world, maximal information is not complete and cannot be completed. Using this distinction, we show that any Bayesian probability assignment in quantum mechanics must have the form of the quantum probability rule, that maximal information about a quantum system leads to a unique quantum-state assignment, and that quantum theory provides a stronger connection between probability and measured frequency than can be justified classically. Finally we give a Bayesian formulation of quantum-state tomography. There are excellent reasons for interpreting quantum states as states of knowledge. A classic
This paper proposes a novel Bayesian inference algorithm termed "Quantum-Inspired Bayesian Inference" (QIBI), drawing inspiration from quantum mechanics, particularly superposition and entanglement. Classical Bayesian inference struggles with representing and processing highly complex and uncertain data, often relying on a single, definitive probability distribution. QIBI addresses this limitation by introducing a mechanism analogous to quantum superposition, allowing the algorithm to simultaneously consider multiple hypotheses, each weighted by its "quantum coherence." Quantum coherence represents the degree to which a hypothesis consistently explores the solution space. The algorithm's core equation is: P(Hi | D) = (1/Z) * [Ψ*(D) * P(Hi)] Where: * P(Hi | D) is the posterior probability of hypothesis *Hi* given data *D*. * Ψ(D) is the quantum coherence vector representing the superposition state. * P(Hi) is the prior probability of hypothesis *Hi*. * Z is the normalization constant. The implementation of QIBI offers potential advantages over traditional Bayesian methods, particularly in scenarios with high dimensionality, complex dependencies, and significant uncertainty. This work outlines the theoretical framework, the core equation, and the rationale behind QIBI's approach, suggesting a pathway towards more robust and efficient Bayesian inference by leveraging principles from the quantum world. The algorithm's performance is theoretically assessed, and future research directions are discussed, focusing on practical implementation and comparative analysis with established Bayesian techniques.
The sure thing principle and the law of total probability are basic laws in classic probability theory. A disjunction fallacy leads to the violation of these two classical probability laws. In this paper, a new quantum dynamic belief decision making model based on quantum dynamic modelling and Dempster-Shafer (D-S) evidence theory is proposed to address this issue and model the real human decision-making process. Some mathematical techniques are borrowed from quantum mathematics. Generally, belief and action are two parts in a decision making process. The uncertainty in belief part is represented by a superposition of certain states. The uncertainty in actions is represented as an extra uncertainty state. The interference effect is produced due to the entanglement between beliefs and actions. Basic probability assignment (BPA) of decisions is generated by quantum dynamic modelling. Then BPA of the extra uncertain state and an entanglement degree defined by an entropy function named Deng entropy are used to measure the interference effect. Compared the existing model, the number of free parameters is less in our model. Finally, a classical categorization decision-making experiment is illustrated to show the effectiveness of our model.