ECCC-Report TR13-147https://eccc.weizmann.ac.il/report/2013/147Comments and Revisions published for TR13-147en-usFri, 13 Jun 2014 20:51:21 +0300
Revision 3
| Generation of Universal Linear Optics by Any Beamsplitter |
Adam Bouland,
Scott Aaronson
https://eccc.weizmann.ac.il/report/2013/147#revision3In 1994, Reck et al. showed how to realize any linear-optical unitary transformation using a product of beamsplitters and phaseshifters. Here we show that any single beamsplitter that nontrivially mixes two modes, also densely generates the set of m by m unitary transformations (or orthogonal transformations, in the real case) on m>=3 modes. (We prove the same result for any two-mode real optical gate, and for any two-mode optical gate combined with a generic phaseshifter.) Experimentally, this means that one does not need tunable beamsplitters or phaseshifters for universality: any nontrivial beamsplitter is universal. Theoretically, it means that one cannot produce "intermediate" models of quantum-optical computation (analogous to the Clifford group for qubits) by restricting the allowed beamsplitters and phaseshifters: there is a dichotomy; one either gets a trivial set or else a universal set. No similar classification theorem for gates acting on qubits is currently known. We leave open the problem of classifying optical gates that act on three or more modes.Fri, 13 Jun 2014 20:51:21 +0300https://eccc.weizmann.ac.il/report/2013/147#revision3
Revision 2
| Generation of Universal Linear Optics by Any Beamsplitter |
Adam Bouland,
Scott Aaronson
https://eccc.weizmann.ac.il/report/2013/147#revision2In 1994, Reck et al. showed how to realize any linear-optical unitary transformation using a product of beamsplitters and phaseshifters. Here we show that any single beamsplitter that nontrivially mixes two modes, also densely generates the set of m by m unitary transformations (or orthogonal transformations, in the real case) on m>=3 modes. (We prove the same result for any two-mode real optical gate, and for any two-mode optical gate combined with a generic phaseshifter.) Experimentally, this means that one does not need tunable beamsplitters or phaseshifters for universality: any nontrivial beamsplitter is universal. Theoretically, it means that one cannot produce "intermediate" models of quantum-optical computation (analogous to the Clifford group for qubits) by restricting the allowed beamsplitters and phaseshifters: there is a dichotomy; one either gets a trivial set or else a universal set. No similar classification theorem for gates acting on qubits is currently known. We leave open the problem of classifying optical gates that act on three or more modes.Thu, 15 May 2014 21:51:38 +0300https://eccc.weizmann.ac.il/report/2013/147#revision2
Revision 1
| Generation of Universal Linear Optics by Any Beamsplitter |
Adam Bouland,
Scott Aaronson
https://eccc.weizmann.ac.il/report/2013/147#revision1In 1994, Reck et al. showed how to realize any unitary transformation on a single photon using a product of beamsplitters and phaseshifters. Here we show that any single beamsplitter that nontrivially mixes two modes, also densely generates the set of unitary transformations (or orthogonal transformations, in the real case) on the single-photon subspace with m>=3 modes. (We prove the same result for any 2-mode real optical gate, and for any 2-mode optical gate combined with a generic phaseshifter.) Experimentally, this means that one does not need tunable beamsplitters or phaseshifters for universality: any nontrivial beamsplitter is universal for linear optics. Theoretically, it means that one cannot produce "intermediate" models of linear optical computation (analogous to the Clifford group for qubits) by restricting the allowed beamsplitters and phaseshifters: there is a dichotomy; one either gets a trivial set or else a universal set. No similar classification theorem for gates acting on qubits is currently known. We leave open the problem of classifying optical gates that act on 3 or more modes.Tue, 10 Dec 2013 04:32:32 +0200https://eccc.weizmann.ac.il/report/2013/147#revision1
Paper TR13-147
| Any Beamsplitter Generates Universal Quantum Linear Optics |
Adam Bouland,
Scott Aaronson
https://eccc.weizmann.ac.il/report/2013/147In 1994, Reck et al. showed how to realize any linear-optical unitary transformation using a product of beamsplitters and phaseshifters. Here we show that any single beamsplitter that nontrivially mixes two modes, also densely generates the set of m by m unitary transformations (or orthogonal transformations, in the real case) on m>=3 modes. (We prove the same result for any 2-mode real optical gate, and for any 2-mode optical gate combined with a generic phaseshifter.) Experimentally, this means that one does not need tunable beamsplitters or phaseshifters for universality: any nontrivial beamsplitter is universal. Theoretically, it means that one cannot produce "intermediate" models of quantum-optical computation (analogous to the Clifford group for qubits) by restricting the allowed beamsplitters and phaseshifters: there is a dichotomy; one either gets a trivial set or else a universal set. No similar classification theorem for gates acting on qubits is currently known. We leave open the problem of classifying optical gates that act on 3 or more modes.Fri, 25 Oct 2013 01:10:45 +0300https://eccc.weizmann.ac.il/report/2013/147