dorsal/arxiv
View SchemaDecoherence on Grover's quantum algorithm: perturbative approach
| Authors | Hiroo Azuma |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0110101 |
| URL | https://arxiv.org/abs/quant-ph/0110101 |
| DOI | 10.1103/PhysRevA.65.042311 10.1103/PhysRevA.66.019903 |
| Journal | Phys. Rev. A 65, 042311 (2002); Phys. Rev. A 66, 019903(E) (2002) |
Abstract
In this paper, we study decoherence on Grover's quantum searching algorithm using a perturbative method. We assume that each two-state system (qubit) suffers \sigma_{z} error with probability p (0\leq p\leq 1) independently at every step in the algorithm. Considering an n-qubit density operator to which Grover's operation is applied M times, we expand it in powers of 2Mnp and derive its matrix element order by order under the n\to \infty limit. (In this large n limit, we assume p is small enough, so that 2Mnp(\geq 0) can take any real positive value or 0.) This approach gives us an interpretation about creation of new modes caused by \sigma_{z} error and an asymptotic form of an arbitrary order correction. Calculating the matrix element up to the fifth order term numerically, we investigate a region of 2Mnp (perturbative parameter) where the algorithm finds the correct item with a threshold of probability P_{th} or more. It satisfies 2Mnp<(8/5)(1-P_{th}) around 2Mnp\simeq 0 and P_{th}\simeq 1, and this linear relation is applied to a wide range of P_{th} approximately. This observation is similar to a result obtained by E. Bernstein and U. Vazirani concerning accuracy of quantum gates for general algorithms. We cannot investigate a quantum to classical phase transition of the algorithm, because it is outside the reliable domain of our perturbation theory.
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"abstract": "In this paper, we study decoherence on Grover\u0027s quantum searching algorithm\nusing a perturbative method. We assume that each two-state system (qubit)\nsuffers \\sigma_{z} error with probability p (0\\leq p\\leq 1) independently at\nevery step in the algorithm. Considering an n-qubit density operator to which\nGrover\u0027s operation is applied M times, we expand it in powers of 2Mnp and\nderive its matrix element order by order under the n\\to \\infty limit. (In this\nlarge n limit, we assume p is small enough, so that 2Mnp(\\geq 0) can take any\nreal positive value or 0.) This approach gives us an interpretation about\ncreation of new modes caused by \\sigma_{z} error and an asymptotic form of an\narbitrary order correction. Calculating the matrix element up to the fifth\norder term numerically, we investigate a region of 2Mnp (perturbative\nparameter) where the algorithm finds the correct item with a threshold of\nprobability P_{th} or more. It satisfies 2Mnp\u003c(8/5)(1-P_{th}) around 2Mnp\\simeq\n0 and P_{th}\\simeq 1, and this linear relation is applied to a wide range of\nP_{th} approximately. This observation is similar to a result obtained by E.\nBernstein and U. Vazirani concerning accuracy of quantum gates for general\nalgorithms. We cannot investigate a quantum to classical phase transition of\nthe algorithm, because it is outside the reliable domain of our perturbation\ntheory.",
"arxiv_id": "quant-ph/0110101",
"authors": [
"Hiroo Azuma"
],
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"quant-ph"
],
"doi": "10.1103/PhysRevA.65.042311 10.1103/PhysRevA.66.019903",
"journal_ref": "Phys. Rev. A 65, 042311 (2002); Phys. Rev. A 66, 019903(E) (2002)",
"title": "Decoherence on Grover\u0027s quantum algorithm: perturbative approach",
"url": "https://arxiv.org/abs/quant-ph/0110101"
},
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