Apparatus and methods for implementing arbitrary unitary transformations on optical modes via a rectangular architecture

1. An apparatus, comprising:

a first optical circuit including a network of interconnected interferometers and configured to perform an M-mode universal transformation on N input optical modes that are divided into (M−1) groups of pulses, the first optical circuit including:

M input ports, each input port of a first (M−1) input ports from the M input ports configured to receive a corresponding group of pulses in the (M−1) groups of pulses;

M output ports; and

a first delay line configured to couple an Mth output port in the M output ports with an Mth input port in the M input ports; and

a second optical circuit in optical communication with the first optical circuit and configured to perform an (2M−3)-mode residual transformation on the N input optical modes, the second optical circuit including a network of beamsplitters and swap gates,

the first optical circuit and the second optical circuit being configured to perform an arbitrary N-mode unitary transformation to the N input optical modes via a rectangular architecture.

2. The apparatus of claim 1 , wherein N−2M+2 input optical modes in the N input optical modes are configured to transmit through about a common number of optical elements in the apparatus.

3. The apparatus of claim 1 , wherein:

each group of pulses in a first (M−2) groups of pulses from the (M−1) groups of pulses includes k pulses and a (M−1)th group includes k+1 pulses, adjacent pulses in each group of the (M−1) groups being separated by a delay τ, N, M, k being positive integers, N being greater than M, and k=(N−1)/(M−1), and

the first delay line is configured to introduce the delay τ between the Mth output port and the Mth input port.

4. The apparatus of claim 1 , wherein the second optical circuit includes (M−1)(M−2)/2 beamsplitters and (M−1)(M−2) swap gates.

5. The apparatus of claim 1 , wherein the first optical circuit is configured to perform the M-mode universal transformation by:

receiving a single pulse in the (M−1)th group of pulses via one of the first (M−1) input ports;

transmitting the single pulse to the Mth output port; and

sending the single pulse from the Mth output port to the Mth input port in concurrence with receiving one pulse via each input port from the first (M−1) input ports.

6. The apparatus of claim 1 , wherein M is about 3 to about 100.

7. The apparatus of claim 1 , wherein N is about 10 to about 10000.

8. The apparatus of claim 3 , wherein the first optical circuit and the second optical circuit are configured to perform an arbitrary transformation to the N input optical modes by:

transmitting the N input optical modes through the first optical circuit for k(k+1)/2 rounds; and

transmitting the N input optical modes through the second optical circuit for k(k−1)/2 rounds.

9. The apparatus of claim 3 , wherein the second optical circuit includes:

a second delay line configured to couple a last output port of the second optical circuit with a last input port of the second optical circuit and introduce the delay τ.

10. The apparatus of claim 1 , wherein the first optical circuit further includes:

M bypass waveguides, each bypass waveguide in the M bypass waveguides configured to transmit an optical mode from an input port in the M input ports to a corresponding output port in the M output ports without transmitting the optical mode through the network of interconnected interferometers.

11. An apparatus, comprising:

a first optical circuit including a network of interconnected interferometers, the first optical circuit configured to perform an M-mode universal transformation on N input optical modes that are divided into M groups of pulses, each group of pulses in the M groups of pulses including/pulses and adjacent pulses being separated by a delay τ, N, M, and/being positive integers, N being greater than M, and l=N/M; and

a second optical circuit in optical communication with the first optical circuit and configured to perform an (2M)-mode cosine-sine transformation on the N input optical modes, the second optical circuit including:

a first plurality of M input ports in optical communication with M output ports of the first optical circuit;

a second plurality of M input ports in optical communication with the M output ports of the first optical circuit via M delay lines, each delay line from the M delay lines configured to couple an output port in the M output ports of the first optical circuit with a corresponding input port in the second plurality of M input ports; and

M beamsplitters, each beamsplitter in the M beamsplitters being in optical communication with one input port in the first plurality of M input ports and one input port in the second plurality of M input ports.

12. The apparatus of claim 11 , wherein the second optical circuit is configured to:

receive, via the first plurality of M input ports, a first plurality of M output optical modes from the M output ports of the first optical circuit, the first plurality of M output optical modes being generated by the first optical circuit based on a first plurality of M input optical modes in the N input optical modes, each input optical mode from the first plurality of M input optical modes being from a corresponding group in the M groups of pulses; and

receive, via the second plurality of M input ports, a second plurality of M output optical modes from the M output ports of the first optical circuit, the second plurality of M output optical modes being generated by the first optical circuit based on a second plurality of M input optical modes in the N input optical modes, each input optical mode from the second plurality of M input optical modes being from a corresponding group in the M groups of pulses.

13. The apparatus of claim 11 , wherein M is about 3 to about 100.

14. The apparatus of claim 11 , wherein N is about 10 to about 10000.

15. The apparatus of claim 11 , wherein the first optical circuit and the second optical circuit are configured to perform an arbitrary transformation to the N input optical modes, and each optical mode in the N optical modes is configured to transmit through about a common number of optical elements in the apparatus.

16. The apparatus of claim 11 , wherein the first optical circuit and the second optical circuit are configured to perform an arbitrary transformation to the N input optical modes by:

transmitting the N input optical modes through the first optical circuit for l 2 rounds; and

transmitting the N input optical modes through the second optical circuit for l(l−1)/2 rounds.

17. The apparatus of claim 11 , wherein the first optical circuit further includes:

M bypass waveguides, each bypass waveguide in the M bypass waveguides configured to transmit an optical mode from an input port in the M input ports to a corresponding output port in the M output ports without transmitting the optical mode through the network of interconnected interferometers.

18. The apparatus of claim 11 , further comprising:

M switchable devices in optical communication with and between the first optical circuit and the second optical circuit, each switchable device in the M switchable devices being switchable between a first state and a second state,

during the first state, the M switchable devices configured to direct M output optical modes from the first optical circuit to the first plurality of M input ports of the second optical circuit,

during the second state, the M switchable devices configured to direct M output optical modes from the first optical circuit to the second plurality of M input ports of the second optical circuit.