Stochastic emulation of quantum algorithms
Quantum algorithms profit from the interference of quantum states in an exponentially large Hilbert space and the fact that unitary transformations on that Hilbert space can be broken down to universal gates that act only on one or two qubits at the same time. The former aspect renders the direct cl...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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IOP Publishing
2022-01-01
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| Series: | New Journal of Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/1367-2630/ac4b0f |
| _version_ | 1797748570042925056 |
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| author | Daniel Braun Ronny Müller |
| author_facet | Daniel Braun Ronny Müller |
| author_sort | Daniel Braun |
| collection | DOAJ |
| description | Quantum algorithms profit from the interference of quantum states in an exponentially large Hilbert space and the fact that unitary transformations on that Hilbert space can be broken down to universal gates that act only on one or two qubits at the same time. The former aspect renders the direct classical simulation of quantum algorithms difficult. Here we introduce higher-order partial derivatives of a probability distribution of particle positions as a new object that shares these basic properties of quantum mechanical states needed for a quantum algorithm. Discretization of the positions allows one to represent the quantum mechanical state of n _bit qubits by 2( n _bit + 1) classical stochastic bits. Based on this, we demonstrate many-particle interference and representation of pure entangled quantum states via derivatives of probability distributions and find the universal set of stochastic maps that correspond to the quantum gates in a universal gate set. We prove that the propagation via the stochastic map built from those universal stochastic maps reproduces up to a prefactor exactly the evolution of the quantum mechanical state with the corresponding quantum algorithm, leading to an automated translation of a quantum algorithm to a stochastic classical algorithm. 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 the emulation. |
| first_indexed | 2024-03-12T16:07:41Z |
| format | Article |
| id | doaj.art-8823d2700b50468e93e756ca5c975a68 |
| institution | Directory Open Access Journal |
| issn | 1367-2630 |
| language | English |
| last_indexed | 2024-03-12T16:07:41Z |
| publishDate | 2022-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | New Journal of Physics |
| spelling | doaj.art-8823d2700b50468e93e756ca5c975a682023-08-09T14:17:46ZengIOP PublishingNew Journal of Physics1367-26302022-01-0124202302810.1088/1367-2630/ac4b0fStochastic emulation of quantum algorithmsDaniel Braun0https://orcid.org/0000-0001-8598-2039Ronny Müller1https://orcid.org/0000-0002-4276-9657Eberhard-Karls-Universität Tübingen, Institut für Theoretische Physik , 72076 Tübingen, GermanyEberhard-Karls-Universität Tübingen, Institut für Theoretische Physik , 72076 Tübingen, GermanyQuantum algorithms profit from the interference of quantum states in an exponentially large Hilbert space and the fact that unitary transformations on that Hilbert space can be broken down to universal gates that act only on one or two qubits at the same time. The former aspect renders the direct classical simulation of quantum algorithms difficult. Here we introduce higher-order partial derivatives of a probability distribution of particle positions as a new object that shares these basic properties of quantum mechanical states needed for a quantum algorithm. Discretization of the positions allows one to represent the quantum mechanical state of n _bit qubits by 2( n _bit + 1) classical stochastic bits. Based on this, we demonstrate many-particle interference and representation of pure entangled quantum states via derivatives of probability distributions and find the universal set of stochastic maps that correspond to the quantum gates in a universal gate set. We prove that the propagation via the stochastic map built from those universal stochastic maps reproduces up to a prefactor exactly the evolution of the quantum mechanical state with the corresponding quantum algorithm, leading to an automated translation of a quantum algorithm to a stochastic classical algorithm. 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 the emulation.https://doi.org/10.1088/1367-2630/ac4b0fquantum algorithmstochastic matricesclassical entanglementquantum interference |
| spellingShingle | Daniel Braun Ronny Müller Stochastic emulation of quantum algorithms New Journal of Physics quantum algorithm stochastic matrices classical entanglement quantum interference |
| title | Stochastic emulation of quantum algorithms |
| title_full | Stochastic emulation of quantum algorithms |
| title_fullStr | Stochastic emulation of quantum algorithms |
| title_full_unstemmed | Stochastic emulation of quantum algorithms |
| title_short | Stochastic emulation of quantum algorithms |
| title_sort | stochastic emulation of quantum algorithms |
| topic | quantum algorithm stochastic matrices classical entanglement quantum interference |
| url | https://doi.org/10.1088/1367-2630/ac4b0f |
| work_keys_str_mv | AT danielbraun stochasticemulationofquantumalgorithms AT ronnymuller stochasticemulationofquantumalgorithms |