Training Optimization for Gate-Model Quantum Neural Networks

Gate-based quantum computations represent an essential to realize near-term quantum computer architectures. A gate-model quantum neural network (QNN) is a QNN implemented on a gate-model quantum computer, realized via a set of unitaries with associated gate parameters. Here, we define a training optimization procedure for gate-model QNNs. By deriving the environmental attributes of the gate-model quantum network, we prove the constraint-based learning models. We show that the optimal learning procedures are different if side information is available in different directions, and if side information is accessible about the previous running sequences of the gate-model QNN. The results are particularly convenient for gate-model quantum computer implementations.


A Appendix
A.1 Abbreviations AI Artificial Intelligence DAG Directed Acyclic Graph QG Quantum Gate structure of a gate-model quantum computer

A.2 Notations
The notations of the manuscript are summarized in Table A.1.
where P is a generalized Pauli operator formulated by a tensor product of Pauli operators {X,Y, Z}, while θ i is referred to as the gate parameter associated to U i (θ i ).
U j (θ i j ) Selection of θ j for the unitary U j to realize the operation U i (θ i )U j (θ j ), i.e., the application of U j (θ j ) on the output of U i (θ i ) at a particular gate parameter θ j .
|ψ, ϕ Input system, where |ψ = |z is a computational basis state, where z is an n-length string, while the (n + 1)-th quantum state initialized as |ϕ = |1 , and is referred to as the readout quantum state.

|Y
An (n + 1)-length output quantum system of the gate-model quantum neural network.
z An n-length string, z = z 1 z 2 . . . z n , where z i represents a classical bit,

|Y (r)
An output system realization, r = 1, . . . , R, where R is the total number of output instances.
S T Training set, formulated via N input strings and labels, A directed acyclic graph (DAG), with a set V of vertexes, and a set S of arcs.
V Set of vertexes in the G environmental graph.

Set of arcs in the
x An identifier.
X Perceptual space.
Z Mapped space.
x An element (vector) of the perceptual space X ⊂ C d .

♦
Symbol of missing features.
v 0 A vertex associated to the input in the environmental graph.
θ i j Gate parameter, associated to the directed arch s i j between v U i and v U j .
x U i (θ i ) An element of X associated to unitary U i (θ i ).
x 0 An element of X associated to the input, x 0 = |z, 1 .
Topological ordering function on the environmental graph.
H Hessian matrix.
|γ v System state associated to state variable γ v .
ζ v Constraint on f T QNN QG for a QNN QG .
H t Unit vector for a unitary U t (θ t ), t = 1, . . . , L − 1, H t = x t + iy t , where x t and y t are real values.

Z t+1
System state, Z t+1 = U( θ )H t + Ex t+1 , where E is a basis vector matrix. f * Compact function subject to be determined.

S L(QNN)
Non-empty supervised learning set.
Differential operator, = P † P, where P † is the adjoint of P.
H (x), Φ, χ κ Parameters used in the calculation of compact function f * (x).

A C (QNNQG)
Learning method for C QNN QG .

A D(RQNN QG)
Learning method for D RQNN QG .

Ξ (k)
Parents of k ∈ V in the environmental graph. P (r) (G QNN QG ) Post-processing associated to the r-th measurement on G QNN QG .
δ U i (θ i ) Error associated to unitary U i (θ i ) in the environmental graph.
ν U L (θ L ) Vertex associated to U L (θ L ) in the environmental graph. (δ Q U l (θ l ),U i (θ i ) ) 2 Square error between unitaries U l (θ l ) and U i (θ i ) at a particular output |Q of RQNN QG .