dorsal/arxiv
View SchemaOptimal Control-Based Efficient Synthesis of Building Blocks of Quantum Algorithms Seen in Perspective from Network Complexity towards Time Complexity
| Authors | T. Schulte-Herbrueggen, A. K. Spoerl, N. Khaneja, S. J. Glaser |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0502104 |
| URL | https://arxiv.org/abs/quant-ph/0502104 |
| DOI | 10.1103/PhysRevA.72.042331 |
| Journal | Phys. Rev. A 72 (2005) 042331 |
Abstract
In this paper, we demonstrate that optimal control algorithms can be used to speed up the implementation of modules of quantum algorithms or quantum simulations in networks of coupled qubits. The gain is most prominent in realistic cases, where the qubits are not all mutually coupled. Thus the shortest times obtained depend on the coupling topology as well as on the characteristicratio of the time scales for local controls {\em vs} non-local ({\em i.e.} coupling) evolutions in the specific experimental setting. Relating these minimal times to the number of qubits gives the tightest known upper bounds to the actual time complexity of the quantum modules. As will be shown, time complexity is a more realistic measure of the experimental cost than the usual gate complexity. In the limit of fast local controls (as {\em e.g.} in NMR), time-optimised realisations are shown for the quantum Fourier transform (QFT) and the multiply controlled {\sc not}-gate ({\sc c$^{n-1}$not}) in various coupling topologies of $n$ qubits. The speed-ups are substantial: in a chain of six qubits the quantum Fourier transform so far obtained by optimal control is more than eight times faster than the standard decomposition into controlled phase, Hadamard and {\sc swap} gates, while the {\sc c$^{n-1}$not}-gate for completely coupled network of six qubits is nearly seven times faster.
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"abstract": "In this paper, we demonstrate that optimal control algorithms can be used to\nspeed up the implementation of modules of quantum algorithms or quantum\nsimulations in networks of coupled qubits. The gain is most prominent in\nrealistic cases, where the qubits are not all mutually coupled. Thus the\nshortest times obtained depend on the coupling topology as well as on the\ncharacteristicratio of the time scales for local controls {\\em vs} non-local\n({\\em i.e.} coupling) evolutions in the specific experimental setting. Relating\nthese minimal times to the number of qubits gives the tightest known upper\nbounds to the actual time complexity of the quantum modules. As will be shown,\ntime complexity is a more realistic measure of the experimental cost than the\nusual gate complexity.\n In the limit of fast local controls (as {\\em e.g.} in NMR), time-optimised\nrealisations are shown for the quantum Fourier transform (QFT) and the multiply\ncontrolled {\\sc not}-gate ({\\sc c$^{n-1}$not}) in various coupling topologies\nof $n$ qubits. The speed-ups are substantial: in a chain of six qubits the\nquantum Fourier transform so far obtained by optimal control is more than eight\ntimes faster than the standard decomposition into controlled phase, Hadamard\nand {\\sc swap} gates, while the {\\sc c$^{n-1}$not}-gate for completely coupled\nnetwork of six qubits is nearly seven times faster.",
"arxiv_id": "quant-ph/0502104",
"authors": [
"T. Schulte-Herbrueggen",
"A. K. Spoerl",
"N. Khaneja",
"S. J. Glaser"
],
"categories": [
"quant-ph"
],
"doi": "10.1103/PhysRevA.72.042331",
"journal_ref": "Phys. Rev. A 72 (2005) 042331",
"title": "Optimal Control-Based Efficient Synthesis of Building Blocks of Quantum Algorithms Seen in Perspective from Network Complexity towards Time Complexity",
"url": "https://arxiv.org/abs/quant-ph/0502104"
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