Mapping of Quantum Systems to the Probability Simplex

• URL: http://arxiv.org/abs/2301.06572v1
• Date: Mon, 16 Jan 2023 19:06:09 GMT
• Title: Mapping of Quantum Systems to the Probability Simplex
• Authors: D. D. Yavuz and A. Yadav
• Abstract summary: We show a one-to-one mapping of the quantum state in a two-dimensional Hilbert space to a vector in an eight dimensional probability space. We then discuss multi-partite quantum systems and their mapping to the probability simplex.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Variational method for learning Quantum Channels via Stinespring Dilation on neutral atom systems [0.0]
  Quantum systems interact with their environment, resulting in non-reversible evolutions. For many quantum experiments, the time until which measurements can be done might be limited. We introduce a method to approximate a given target quantum channel by means of variationally approximating equivalent unitaries on an extended system.
  arXiv  Detail & Related papers  (2023-09-19T13:06:44Z)
• Simulation of quantum algorithms using classical probabilistic bits and circuits [0.0]
  We discuss a new approach to simulate quantum algorithms using classical probabilistic bits and circuits. 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. We show how to implement the analogs of single-qubit and two-qubit gates in the probability space using correlation-inducing operations.
  arXiv  Detail & Related papers  (2023-07-26T18:49:42Z)
• A vertical gate-defined double quantum dot in a strained germanium double quantum well [48.7576911714538]
  Gate-defined quantum dots in silicon-germanium heterostructures have become a compelling platform for quantum computation and simulation. We demonstrate the operation of a gate-defined vertical double quantum dot in a strained germanium double quantum well. We discuss challenges and opportunities and outline potential applications in quantum computing and quantum simulation.
  arXiv  Detail & Related papers  (2023-05-23T13:42:36Z)
• Quantum Circuit Completeness: Extensions and Simplifications [44.99833362998488]
  The first complete equational theory for quantum circuits has only recently been introduced. We simplify the equational theory by proving that several rules can be derived from the remaining ones. The complete equational theory can be extended to quantum circuits with ancillae or qubit discarding.
  arXiv  Detail & Related papers  (2023-03-06T13:31:27Z)
• Simple Tests of Quantumness Also Certify Qubits [69.96668065491183]
  A test of quantumness is a protocol that allows a classical verifier to certify (only) that a prover is not classical. We show that tests of quantumness that follow a certain template, which captures recent proposals such as (Kalai et al., 2022) can in fact do much more. Namely, the same protocols can be used for certifying a qubit, a building-block that stands at the heart of applications such as certifiable randomness and classical delegation of quantum computation.
  arXiv  Detail & Related papers  (2023-03-02T14:18:17Z)
• A Quantum Algorithm for Computing All Diagnoses of a Switching Circuit [73.70667578066775]
  Faults are by nature while most man-made systems, and especially computers, work deterministically. This paper provides such a connecting via quantum information theory which is an intuitive approach as quantum physics obeys probability laws.
  arXiv  Detail & Related papers  (2022-09-08T17:55:30Z)
• Why we should interpret density matrices as moment matrices: the case of (in)distinguishable particles and the emergence of classical reality [69.62715388742298]
  We introduce a formulation of quantum theory (QT) as a general probabilistic theory but expressed via quasi-expectation operators (QEOs) We will show that QT for both distinguishable and indistinguishable particles can be formulated in this way. We will show that finitely exchangeable probabilities for a classical dice are as weird as QT.
  arXiv  Detail & Related papers  (2022-03-08T14:47:39Z)
• Stochastic emulation of quantum algorithms [0.0]
  We introduce higher-order partial derivatives of a probability distribution of particle positions as a new object that shares basic properties of quantum mechanical states needed for a quantum algorithm. We prove that the propagation via the map built from those universal maps reproduces up to a prefactor exactly the evolution of the quantum mechanical state. We implement several well-known quantum algorithms, analyse the scaling of the needed number of realizations with the number of qubits, and highlight the role of destructive interference for the cost of emulation.
  arXiv  Detail & Related papers  (2021-09-16T07:54:31Z)
• Tossing Quantum Coins and Dice [0.0]
  This case is an important example of a quantum procedure because it presents a typical framework employed in quantum information processing and quantum computing. The emphasis is on the clarification of the difference between quantum and classical conditional probabilities.
  arXiv  Detail & Related papers  (2021-03-31T11:39:56Z)
• Quantum Probability's Algebraic Origin [0.0]
  We show that quantum probabilities and classical probabilities have very different origins. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy. It provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.
  arXiv  Detail & Related papers  (2020-09-17T18:19:41Z)
• Quantum Gram-Schmidt Processes and Their Application to Efficient State Read-out for Quantum Algorithms [87.04438831673063]
  We present an efficient read-out protocol that yields the classical vector form of the generated state. Our protocol suits the case that the output state lies in the row space of the input matrix. One of our technical tools is an efficient quantum algorithm for performing the Gram-Schmidt orthonormal procedure.
  arXiv  Detail & Related papers  (2020-04-14T11:05:26Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.