The Quantum Fourier Transform (QFT) is one of the most important quantum operations for numerous quantum computing applications. The major obstacle to the construction of QFT is the large number of elementary gates required to build the circuit. Of these gates, CNOT gate is the most expensive resource required for NISQ implementation. We have been applying our proprietary quantum Karnaugh map technology [1] to optimized design of QFT and quantum amplitude estimation algorithm (QAEA) [2,3]

[1] J. H. Bae et al., “Quantum circuit optimization using quantum Karnaugh map,” Scientific Reports 10, 15651 (2020).

[2] B. Park & D. Ahn, “Optimization of T-count and T-depth in quantum Fourier transform,” arXiv:2203.07739 (2022).

[3] B. Park & D. Ahn,”Optimizing the number of CNOT gates in one-dimensional nearest neighbor quantum Fourier transfom circuit,” arXiv:2208.14249 (2022).

Figure 1. Circuit depth optimization of partial quantum Fourier transform (QFT) circuit. This is the first step, and the circuit depth is reduced from 29 to 20. We have shown that the number of CNOT gates is reduced to 26 from 50 by using quantum Karnaugh map.

Quantum computing (QC) is a promising candidate to provide new disruptive means to perform computations for a range of computational problems with increased complexity. However, little attention is given to quantum computation of a classical nonlinear systems such as a viscous fluid even though this too is hard for classical computers. Such fluid is governed by the Navier-Stokes (NS) nonlinear partial differential equations (PDEs), whose solution is essential to the aerospace industry, weather forecasting, and astrophysics, to name a few. The alternative to the NS PDEs, the direct simulation Monte Carlo (DSMC) method has also evolved into a primary computational workhorse and is routinely being applied to various flow problems of scientific and technological interest including rarefied hypersonic gas flows. Therefore, a construction of quantum algorithms for solving the NS PDEs and DSMC will be a pivotal task considering the significance of both scientific and economic importance of these applications.

At First Quantum, we are developing quantum amplitude estimation algorithm (QAEA) for solving both NS PDE and DSMC, optimized with our proprietary QKM technology. The principal beneficiaries will be scientists and engineers working on hypersonic flow, turbulence, hydrodynamics, magnetohydrodynamics and atmospheric science, to name a few. The successful development of quantum computing algorithms for the nonlinear fluidics would be extremely important both from scientific and technological viewpoints.

Quantum computing algorithms to simulate larger, more complex molecules that might help cure diseases could be game changing and could have disruptive impacts on pharmaceutical industries. Pharma as an industry is a natural candidate for quantum computing considering its focus on molecular formation. The molecules are actually quantum systems. Quantum computing is expected to be able to predict and simulate the structure, properties, and behavior of these molecules more effectively than conventional computing can. Exact methods are computationally intractable for standard computers, and approximate methods are often not sufficiently accurate when interactions on the atomic level are critical, as is the case for many compounds. Theoretically, quantum computers have the capacity to efficiently simulate the complete problem, including interactions on the atomic level. As these quantum computers become more powerful, the tremendous value will be at stake [1]. Currently, we are investigating the possibility of applying quantum Karnaugh map to variational quantum simulation (VQS) of molecules [2]. Schematics of VQS is as follows:

Where (a) and (b) are the circuit representations of the single and double excitiation operators in exp[*iHt*].

[1] Pharma’s digital Rx: Quantum computing in drug research and development, McKinsey & Company, June 2021.

[2] X. Yuan et al., Theory of variational quantum simulation. arXiv:1812.08767

[3] M. Motta and J. E. Rice, Emerging quantum computing algorithms for quantum chemistry. WIRES computational molecular science 12, e1580 (2022)

Quantum-computing use cases in finance may be a bit further in the future, and the advantages of possible short-term uses are speculative. However, according to McKinsey & Company, the most promising use cases of quantum computing in finance are in portfolio and risk management [1]. Recently, it was shown that risk management model can be built upon using quantum amplitude estimation algorithm (QAEA) [2]. The standard quantum amplitude estimation (QAE) circuit is given by figure 1 [3]

Figure 1. quantum amplitude estimation (QAE) circuit. Here QFT-dagger is the inverse quantum Fourier transform (QFT), H is the Hadamard gate and Q is the Grover operator.

Figure 2. Implementation of five-qubit quantum amplitude estimation (QAE) circuit.

Figure 2 shows our preliminary results for the implementation of QAE circuit. The peaks are the value in (±0.87) of 70 in which 70 is the right value reproducing the result of reference 2. Currently, we are collaborating with Professor Patrick Rebentrost and his colleagues of Center for Quantum Technologies, National University of Singapore in quantum computing application to finance and financial engineering.

[1] Quantum computing: An emerging eco systems and use cases, McKinsey & Company, Dec. 2021.

[2] S. Woerner and D. J. Egger, Quantum risk analysis, npj Quantum Information 5, 1 (2019).

[3] G. Brassard, P. Hoyer, M. Mosca, and A. Tapp, Quantum Amplitude Amplification and Estimation, Contemporary Mathematics 305, 53 (2002).

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