Equation 1 · How Post-CMOS, Neuromorphic, Photonic, and Quantum AI Compute Actually Works
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Symbol M
M is part of the quantity the equation computes from the expression on the right.
Symbol U
U is one factor in the product that computes the quantity on the left.
Symbol Sigma
Sigma is one factor in the product that computes the quantity on the left.
Symbol V^*
is one factor in the product that computes the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
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A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
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What the article says around this equation
An MZI mesh built this way implements a unitary matrix: a transformation that preserves the total optical power passing through it. Most matrices a neural-network layer actually needs are not unitary. The standard fix is a singular value decomposition, writing the desired matrix M as . where U and V* are unitary matrices, each realized by its own MZI mesh, and Σ is a diagonal matrix of non-negative singular values, realized by a bank of tunable optical attenuators placed between the two meshes. Light entering the chip passes through the first mesh, has each of its modes independently attenuated by the corresponding singular value, then passes through the second mesh — end…
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An MZI mesh built this way implements a unitary matrix: a transformation that preserves the total optical power passing through it. Most matrices a neural-network layer actually needs are not unitary. The standard fix is a singular value decomposition, writing the desired matrix M as . where U and V* are unitary matrices, each realized by its own MZI mesh, and Σ is a diagonal matrix of non-negative singular values, realized by a bank of tunable optical attenuators placed between the two meshes. Light entering the chip passes through the first mesh, has each of its modes independently attenuated by the corresponding singular value, then passes through the second mesh — end to end, the optical power distribution leaving the chip is the matrix-vector product of M and the input vector encoded in the light entering it [ 1 ] .
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