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arXiv:2301.06572

Mapping of Quantum Systems to the Probability Simplex

Deniz D. Yavuz, Anirudh Yadav

Published Jan 16, 2023• 0 citations
arXivHTMLPDF

Abstract

We start with the simplest quantum system (a two-level system, i.e., a qubit) and discuss a one-to-one mapping of the quantum state in a two-dimensional Hilbert space to a vector in an eight dimensional probability space (probability simplex). We then show how the usual transformations of the quantum state, specifically the Hadamard gate and the single-qubit phase gate, can be accomplished with appropriate transformations of the mapped vector in the probability simplex. One key defining feature of both the mapping to the simplex and the transformations in the simplex is that they are not linear. These results show that both the initial state and the time evolution of a qubit can be fully captured in an eight dimensional probability simplex (or equivalently using three classical probabilistic bits). We then discuss multi-partite quantum systems and their mapping to the probability simplex. Here, the key tool is the identical tensor product structure of combining multiple quantum systems as well as multiple probability spaces. Specifically, we explicitly show how to implement an analog of the two-qubit controlled-not (CNOT) gate in the simplex. We leave it an open problem how much the quantum dynamics of $N$ qubits can be captured in a probability simplex with $3N$ classical probabilistic bits. Finally, we also discuss the equivalent of the Schrodinger's equation for the wavefunction (in a Hilbert space of arbitrary dimension), which dictates the time evolution of the vectors in the simplex.

DOI
10.48550/arxiv.2301.06572

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We start with the simplest quantum system (a two-level system, i.e., a qubit) and discuss a one-to-one mapping of the quantum state in a two-dimensional Hilbert space to a vector in an eight dimensional probability space (probability simplex). We then show how the usual transformations of the quantum state, specifically the Hadamard gate and the single-qubit phase gate, can be accomplished with appropriate transformations of the mapped vector in the probability simplex. One key defining feature of both the mapping to the simplex and the transformations in the simplex is that they are not linear. These results show that both the initial state and the time evolution of a qubit can be fully captured in an eight dimensional probability simplex (or equivalently using three classical probabilistic bits). We then discuss multi-partite quantum systems and their mapping to the probability simplex. Here, the key tool is the identical tensor product structure of combining multiple quantum systems as well as multiple probability spaces. Specifically, we explicitly show how to implement an analog of the two-qubit controlled-not (CNOT) gate in the simplex. We leave it an open problem how much the quantum dynamics of NNN qubits can be captured in a probability simplex with 3N3N3N classical probabilistic bits. Finally, we also discuss the equivalent of the Schrodinger's equation for the wavefunction (in a Hilbert space of arbitrary dimension), which dictates the time evolution of the vectors in the simplex.
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 presentation provides a comprehensive overview of the foundational principles of quantum computation. It begins by contrasting classical and quantum information, highlighting how quantum bits (qubits) leverage superposition, entanglement, and interference to process information in ways that surpass classical limits. The document explains the four postulates of quantum mechanics: state space (qubits as unit vectors in a Hilbert space), time evolution (via unitary operators), measurement (probabilistic collapse of the state), and composite systems (tensor products of subsystems). Key concepts such as the Bloch sphere representation, quantum gates (e.g., Hadamard, CNOT, Pauli gates), and quantum circuits are discussed, illustrating how quantum operations manipulate qubit states. The presentation also covers quantum evolution, the gate model for computation, and the universality of certain gate sets, emphasizing the potential for quantum parallelism and the transformative impact of quantum computing on fields like cryptography and optimization.
Classical computing architectures encode information into discrete binary digits residing in isolated regions of phase space. Quantum computing replaces this paradigm by leveraging complex state vectors defined within continuous, multi-dimensional Hilbert spaces. This paper provides a unified mathematical framework examining finite-dimensional Hilbert spaces in quantum information processing. It traces the structural scaling of quantum state spaces from single-qubit systems up through composite tensor product structures and multi-qubit registers. By detailing computational basis representations, state transformations, probability amplitudes, and multipartite entanglement classes, such as Bell, GHZ, and W states, this paper demonstrates how exponential dimensional scaling enables quantum parallelism while presenting computational barriers to classical simulation.
This work discusses simple examples how quantum systems are obtained as subsystems of classical statistical systems. For a single qubit with arbitrary Hamiltonian and for the quantum particle in a harmonic potential we provide explicitly all steps how these quantum systems follow from an overall ''classical" probability distribution for events at all times. This overall probability distribution is the analogue of Feynman's functional integral for quantum mechanics or for the functional integral defining a quantum field theory. In our case the action and associated weight factor are real, however, defining a classical probabilistic system. Nevertheless, a unitary time-evolution of wave functions can be realized for suitable systems, in particular probabilistic automata. Based on these insights we discuss novel aspects for correlated computing not requiring the extreme isolation of quantum computers. A simple neuromorphic computer based on neurons in an active or quiet state within a probabilistic environment can learn the unitary transformations of an entangled two-qubit system. Our explicit constructions constitute a proof that no-go theorems for the embedding of quantum mechanics in classical statistics are circumvented. We show in detail how subsystems of classical statistical systems can explain various ``quantum mysteries". Conceptually our approach is a straightforward derivation starting from an overall probability distribution without invoking non-locality, acausality, contextuality, many worlds or other additional concepts. All quantum laws follow directly from the standard properties of classical probabilities.
Complex Hilbert spaces play an important role in the description of quantum systems. In fact, with every quantum system there is associated an infinite dimensional or a finite dimensional separable complex Hilbert space ℍ that consists of the states of the quantum system. In physics terminology, the Hilbert space ℍ is usually referred to as the space of (pure) states. Throughout this monograph, the mathematical description of a quantum system shall be based on a certain complex (separable) Hilbert space ℍ, and therefore the quantum system will simply be denoted by ℍ. The quantum system ℍ is said to be a finite-dimensional system if ℍ is a finite-dimensional complex Hilbert space. Otherwise, the quantum system ℍ is said to be an infinite-dimensional system. The Hilbert space ℍ representing a composite quantum system that consists of n subsystems is a tensor product of the Hilbert spaces of n component systems described by ℍ 1 , … ℍ n , i.e., ℍ = ℍ 1 ⊗ … ⊗ ℍ n . In the theory of Hilbert spaces, a tensor product of Hilbert spaces is a way to create a new Hilbert space out of 2 (or more) Hilbert spaces. In the following, we illustrate the concept and construction of tensor product ℍ ⊗ 핂 of only 2 Hilbert spaces ℍ and 핂. The concepts and constructions can be easily extended to more than 2 Hilbert spaces such as ℍ 1 ⊗ ℍ 2 ⊗ … ⊗ ℍ n . Recall that the tensor product of ℍ with 핂 denoted by ℍ ⊗ 핂 is a new Hilbert space that consists of elements ϕ ⊗ φ ( ϕ ∈ ℍ and φ ∈ 핂) and is equipped with the Hilbertian inner product 〈·, ·〉 ℍ⊗핂 :(ℍ ⊗ 핂) × (ℍ ⊗ 핂) → C defined by 〈 ϕ 1 ⊗ φ 1 , ϕ 2 ⊗ φ 2 〉 ℍ⊗핂 = 〈 ϕ 1 , ϕ 2 〉 ℍ 〈 φ 1 , φ 2 〉 핂 , ϕ i ∈ ℍ and φ i ∈ 핂 for i = 1, 2.