Step-by-Step HHL Algorithm Walkthrough to Enhance the Understanding of Critical Quantum Computing Concepts

Abstract

After learning basic quantum computing concepts, it is desirable to reinforce the learning using an important and relatively complex algorithm through which the students can observe and appreciate how the qubits evolve and interact with each other. Harrow-Hassidim-Lloyd (HHL) quantum algorithm, which can solve Linear System Problems with exponential speed-up over classical method and is the basic of many important quantum computing algorithms, is used to serve this purpose. The HHL algorithm is explained analytically followed by a 4-qubit numerical example in bra-ket notation. Matlab code corresponding to the numerical example is available for students to gain a deeper understanding of the HHL algorithm from a pure matrix point of view.A quantum circuit programmed using qiskit is also provided which can be used for real hardware execution in IBM quantum computers. After going through the material, students are expected to have a better appreciation of the concepts such as basis transformation, bra-ket and matrix rep-

Application of Quantum Computers in Foreign Exchange Reserves Management

“…The interested reader can find technical details in the above-mentioned papers. Examples of how to build a quantum circuit implementing the HHL algorithm for a particular linear system are provided in [32], [33] (suitable for complete beginners), and [9].…”

“…The latter vector is in fact a quantum state |i − 1 which can be easily prepared with X gates only. 33 Note that the first two members of state |x opt are in fact Lagrange multipliers λ and µ, therefore the first weight of an asset is the third member of |x opt .…”

“…This is still better than the best known classical algorithm introduced in [35], which has complexity O(N 2.376 ), and the more often used algorithm designed in [36], which has complexity O(N 2.8074 ). 34 33 Assume, for example, that our optimization involves 16 variables (i.e., four qubits are employed to represent the optimization results) and we are interested in the seventh one. This means that we need to calculate the inner product xopt|6 (the index runs from zero, hence the seventh position is represented by the number 6).…”

“…Details are provided in[34] 29. To get an idea of how to set the rotation angles of Ry gates in practice, consult[32] and[33]. It is obvious that in practice we need to estimate the eigenvalues, because we do not know them in advance as they are calculated in the phase estimation process.…”