In the current noisy intermediate-scale quantum (NISQ)era, quantum machine learning is emerging as a dominantparadigm to program gate-based quantum computers. Inquantum machine learning, the gates of a quantum circuitare parametrized, and the parameters are tuned via classical optimization based on data and on measurements ofthe outputs of the circuit. Parametrized quantum circuits(PQCs) can efficiently address combinatorial optimizationproblems, implement probabilistic generative models, andcarry out inference (classification and regression). This monograph provides a self-contained introduction to quantummachine learning for an audience of engineers with a background in probability and linear algebra. It first describesthe necessary background, concepts, and tools necessary todescribe quantum operations and measurements. Then, itcovers parametrized quantum circuits, the variational quantum eigensolver, as well as unsupervised and supervisedquantum machine learning formulations.
4.3.1 Shift Operators Given a binary string a = a0, ..., am1 {0, 1}m, the shift operator X a is a separable quantum gate that applies either the identity gate I or the Pauli X gate to each k-th qubit, with k {0, 1, ..., m 1}. Specically, the Pauli X gate is applied to qubit k if the corresponding cbit ak equals ak = 1, and no operation is applied to qubit k if ak = 0. Mathematically, as illustrated in Fig. 4.2, the shift operator X a with shift cbit string a can be expressed as the Kronecker product X a = X a0 X a1 X am1. Note that the shift operator X a is a Pauli string as dened in Sec. 2.6.1. As an example, with m = 2 qubits and the shift bit string a = 01, the shift operator is given as X 01 = I X. Therefore, the operator X 01 applies a Pauli X gate to the second qubit only. 113 (4.1) (4.2) 114 Quantum Computing X a0 X a1 ...
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X am1 Figure 4.2: The shift operator X a on m qubits, dened in (4.1) or equivalently in (4.4), is a Pauli string that applies single-qubit Pauli X gates or the identity I in a manner controlled by the cbit string a = a0, ..., am1 {0, 1}m as in (4.1). The name shift operator arises from the change-of-basis form of the unitary (4.1). To describe this interpretation, for two binary strings a = a0, ..., am1 {0, 1}m and b = b0, ..., bm1 {0, 1}m, we write as a b the m-cbit string obtained as the bit-wise XOR of the two cbit strings, i.e., a b = a0 b0, ..., am1 bm1. Following the characterization of unitary matrices in Sec. 2.6.2, we can then express the shift operator (4.1) as (4.3) X a = 2m1 X |y aihy|. (4.4) y=0 Accordingly, the shift operator maps each vector |yi in the computational basis to the shifted version |y ai. This can be readily veried by applying (4.1) to a vector |yi in the computational basis, which yields
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X a|yi = |y ai. (4.5) As an example, the shift operator (4.2) can be expressed in the change-of-basis form X 01 = |01ih00| + |00ih01| + |11ih10| + |10ih11|. (4.6) 4.3.2 Controlled Shift Operators Consider now a partition of the set of qubits into two subsystems, with n qubits in the rst subsystem and m in the second subsystem. 4.3. Computing Binary Functions and Quantum RAM |xi n n |xi |yi m X f (x) m
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|y f (x)i Figure 4.3: The controlled shift operator CX f is a controlled qubit gate in which the rst m qubits control the shift operator X f (x) applied to last n qubits. In this subsection, we will introduce the controlled shift operator, in which the rst set of qubits determines, in a sense to be specied below, the shift to be applied to the second set of qubits. We will see in the following subsections that this operator allows the computation of binary functions on a quantum computer, as well as the implementation of a form of quantum memory. To start, let us study the eect of applying the shift operator X a to the second subsystem of m qubits within the larger system encompassing also the rst n qubits. This operation amounts to the application of the unitary I X a, with the identity matrix I being of size 2n 2n, i.e., to the transformation (I X a)|x, yi = |x, y ai,
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