Quantum analog-digital conversion

Many quantum algorithms, such as the Harrow-Hassidim-Lloyd (HHL) algorithm, depend on oracles that efficiently encode classical data into a quantum state. The encoding of the data can be categorized into two types: analog encoding, where the data are stored as amplitudes of a state, and digital encoding, where they are stored as qubit strings. The former has been utilized to process classical data in an exponentially large space of a quantum system, whereas the latter is required to perform arithmetics on a quantum computer. Quantum algorithms such as HHL achieve quantum speedups with a sophisticated use of these two encodings. In this work, we present algorithms that convert these two encodings to one another. While quantum digital-to-analog conversions have implicitly been used in existing quantum algorithms, we reformulate it and give a generalized protocol that works probabilistically. On the other hand, we propose a deterministic algorithm that performs a quantum analog-to-digital conversion. These algorithms can be utilized to realize high-level quantum algorithms such as a nonlinear transformation of amplitudes of a quantum state. As an example, we construct a “quantum amplitude perceptron,” a quantum version of the neural network that hence has a possible application in the area of quantum machine learning.

Figure 1

(a) Schematic sketch of analog encoding and digital encoding. QDAC and QADC mediate these two encodings. (b) A brief flowchart of the HHL algorithm [4]. $\left\{|a_j\rangle \right\}$ denote eigenvectors of a Hermitian matrix $A$, each corresponding to eigenvalues $\left\{{\lambda }_j\right\}$. ${\chi }_j$ are complex numbers such that ${\sum }_{j=1}^N{x}_j|j\rangle ={\sum }_{j=1}^N{\chi }_j|a_j\rangle$.

Figure 2

Quantum circuit of the swap test [23].

Figure 3

Quantum circuit through steps (i) to (iv) of absolute QADC in the main text.

Figure 4

Definition of gate $G$ in absolute QADC.

Figure 5

Step (vi) of the absolute-QADC algorithm. The phase estimation is performed to encode the analog-encoded value $x_j$ into qubit bit strings. IQFT: inverse quantum Fourier transformation [25].

Figure 6

A quantum circuit through steps (i) to (iii) of real QADC in the main text.

Figure 7

A quantum circuit element for imaginary QADC. The imaginary QADC can be performed by replacing $W$ in the real QADC by this circuit.