# Quantum Circuit Implementation of Multi-Dimensional Non-Linear Lattice Models
## 3 Quantum Computing Principles
### Data Encoding in Quantum Algorithms  
The majority of existing quantum algorithms were designed such that data are encoded in the amplitudes defining the quantum state of the qubit register. This is commonly referred to as amplitude-based encoding or encoding within the amplitudes of the quantum state. An important feature of this encoding technique is that it facilitates the quantum algorithm to benefit from quantum parallelism. Using \(n\) qubits, \(2^{n}\) degrees of freedom are created, with the potential of an exponential saving in terms of memory when comparing the number of qubits and classical bits. Quantum parallelism also enables the simultaneous manipulation of \(2^{n}\) amplitudes, defining the quantum state by performing a single unitary operator on an \(n\)-qubit coherent quantum register. An alternative data-encoding approach is usually referred to as quantum computational basis encoding or encoding within the computational basis of the quantum state. In this approach, the \(n\) coherent qubits in the register are used to represent data using a fixed-point representation. A quantum algorithm using this encoding as a first step initializes the quantum state by applying an oracle that transforms the qubit register from the initial zero-state to the fixed-point representation of the required input state, such that in contrast to amplitude-based encoding, only a single of the \(2^{n}\) possible states has non-zero amplitude. Successive gate operations then effectively change which of the quantum states has the non-zero amplitude. Using this type of encoding, it is less clear than in amplitude-based encoding how quantum algorithms can take advantage of quantum parallelism. However, crucially, it facilitates non-linear operations in terms of arithmetic operations, as previously demonstrated in existing quantum algorithms for arithmetic operations. Encoding within the computational basis has not been widely used in applications related to scientific computing. A notable example is the quantum algorithm for computing the Fourier transform in the computational basis (termed QFTC) by [PERSON] et al. [PERSON] _et al._ (2019). Research work covering efficient transformations between the two representation types has, so far, been very limited, e.g., the conversion from amplitude-encoding to computational basis encoding was investigated by [PERSON] et al. [PERSON] _et al._ (2019), while the reverse transformation was investigated by [PERSON] _et al._ (2019). From previous and ongoing work, it is clear that important and meaningful quantum algorithms using computational-basis encoding can be obtained despite quantum parallelism not being exploited directly as in an algorithm using amplitude-based encoding. However, when algorithms using  
Figure 1: Illustration of the Variational Quantum Computing (VQC) approach, shown here for four layers, each with 6 input qubits and depth 5 for the operators \(U(\tilde{\lambda})\) with 13 variational parameters defining two-qubit gates. The Quantum Non-linear Processing Unit (\(QNPU\)) is designed to efficiently create non-linear products using problem-specific quantum circuits. The figure is adapted from [PERSON] et al. [PERSON] _et al._ (2019).  
quantum-basis encoding are used as part of a larger application it is expected that quantum speed-up can still be achieved, particularly if further improvements to the algorithms for conversion between the two different encoding techniques can be made. The availability of efficient, updatable quantum memory further supports the potential of algorithms using computational-basis encoding.