Revision #3 Authors: Adam Bouland, Scott Aaronson

Accepted on: 13th June 2014 20:51

Downloads: 708

Keywords:

In 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.

Edited Lemma 3.3 and updated references. Results are unchanged.

Revision #2 Authors: Scott Aaronson, Adam Bouland

Accepted on: 15th May 2014 21:51

Downloads: 661

Keywords:

In 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.

Added references and material to address subgroups of SU(3) which were missing in the previous version. Results are unchanged.

Revision #1 Authors: Adam Bouland, Scott Aaronson

Accepted on: 10th December 2013 04:32

Downloads: 672

Keywords:

In 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.

Changed title, edited introduction, added references.

TR13-147 Authors: Adam Bouland, Scott Aaronson

Publication: 25th October 2013 01:10

Downloads: 1412

Keywords:

In 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.