Towards Quantum Computational Mechanics

TL;DR

This work demonstrates how quantum computing can indeed be used to solve representative volume element (RVE) problems in computational homogenisation with polylogarithmic complexity of $\mathcal{O}((\log N)^c)$ in classical computing and attains exponential acceleration with respect to classical solvers, bringing concurrent multiscale computing closer to practicality.

Abstract

The advent of quantum computers, operating on entirely different physical principles and abstractions from those of classical digital computers, sets forth a new computing paradigm that can potentially result in game-changing efficiencies and computational performance. Specifically, the ability to simultaneously evolve the state of an entire quantum system leads to quantum parallelism and interference. Despite these prospects, opportunities to bring quantum computing to bear on problems of computational mechanics remain largely unexplored. In this work, we demonstrate how quantum computing can indeed be used to solve representative volume element (RVE) problems in computational homogenisation with polylogarithmic complexity of $\mathcal{O}((\log N)^c)$, compared to $\mathcal{O}(N^c)$ in classical computing. Thus, our quantum RVE solver attains exponential acceleration with respect to classical solvers, bringing concurrent multiscale computing closer to practicality. The proposed quantum RVE solver combines conventional algorithms such as a fixed-point iteration for a homogeneous reference material and the Fast Fourier Transform (FFT). However, the quantum computing reformulation of these algorithms requires a fundamental paradigm shift and a complete rethinking and overhaul of the classical implementation. We employ or develop several techniques, including the Quantum Fourier Transform (QFT), quantum encoding of polynomials, classical piecewise Chebyshev approximation of functions and an auxiliary algorithm for implementing the fixed-point iteration and show that, indeed, an efficient implementation of RVE solvers on quantum computers is possible. We additionally provide theoretical proofs and numerical evidence confirming the anticipated $\mathcal{O} \left ((\log N)^c \right)$ complexity of the proposed solver.

Figures (17)

• Figure 1: A basic quantum circuit with two qubits $k_0$ and $k_1$. As indicated in (b) the identity gate $I$ is usually omitted in circuit diagrams so that circuits (a) and (b) are equivalent. The states of both qubits are initialised with $\ket {k_0} = \ket 0$ and $\ket {k_1} = \ket 0$ so that $\ket q = \ket {k_0} \otimes \ket {k_1} = \ket 0 \otimes \ket 0$ which is abbreviated as $\ket q = \ket{0}\ket{0}$ or $\ket q = \ket{00}$. The output of the circuit is $(I \otimes X) (\ket 0 \otimes \ket 0) = I \ket 0 \otimes X \ket 0= \ket 0 \otimes \ket 1 \equiv \ket 0 \ket 1 \equiv \ket {01}$.
• Figure 2: Quantum circuits with two qubits $k_0$ and $k_1$. The circuit (a) depicts the $CNOT$ gate where $k_0$ is the control and $k_1$ the target wire. The respective unitary matrix is given in \ref{['eq:cnotMatrix']}. The state of the target $k_1$ is flipped only when the state of the control $k_0$ is $\ket 1$, otherwise it remains unchanged. The output of the circuit (b) is the entangled state $\frac{1}{\sqrt 2} (\ket 0 \otimes \ket 0 + \ket 1 \otimes \ket 1) \equiv \frac{1}{\sqrt 2} (\ket {0} \ket{0} + \ket {1} \ket{1}) \equiv \frac{1}{\sqrt 2} (\ket {00} + \ket {11})$.
• Figure 3: Multi-controlled $C_m U$ gates. (a) A multi-controlled $C_3 U$ gate with three control qubits $\ket{c_0c_1c_2}$ and one target qubit $\ket{k}$. (b) The V-chain implementation of the $C_3 U$ gate using the three-qubit Toffoli gates and the $CU$ gate. In addition, two ancillary (auxiliary) qubits $\ket{a_0}$ and $\ket{a_1}$ are introduced to store temporary information.
• Figure 4: Quantum circuit for implementing the mapping $\ket k \ket 0 \mapsto \cos (f(k)) \ket k \ket 0 + \sin ( f(k)) \ket k \ket 1$. Here, $\ket k= \ket{k_0 k_1 k_2}$ is represented with three qubits and the ancillary qubit is initially in state $\ket 0$. For instance, providing the circuit with the input $\ket{k} = \ket 1 \ket 1 \ket 0$, i. e., $k=1 \cdot 2^2+ 1 \cdot 2^1 + 0 \cdot 2^0= 6$, it will yield the output $\cos (f(6)) \ket 1 \ket 1 \ket 0 \ket 0 + \sin ( f(6)) \ket 1 \ket 1 \ket 0 \ket 1$.
• Figure 5: Quantum state preparation via function encoding. The four-qubit gate in (a) represents the unitary $U_P(f(k))$ for encoding $f(k)$ and is composed as the circuit depicted in Figure \ref{['fig:polyEncode']}. Prepending the circuit with three Hadamard gates as shown in (b) yields a uniform superposition state, which becomes after multiplication by $U_P(f(k))$ an entangled state.