This article presents a systematic survey of six prominent quantum computing applications in finance, unified under the paradigm of optimization as the foundational use case from which derivative applications are constructed. We formalize the transition from the classical Markowitz portfolio optimization framework to a quantum implementation via the Quantum Approximate Optimization Algorithm (QAOA), including explicit mathematical derivations, theoretical performance bounds, and convergence guarantees. Beyond algorithmic formalism, we critically assess prevailing hardware limitations, focusing on noise thresholds and coherence constraints that currently preclude a demonstrable quantum advantage over classical counterparts. Furthermore, we address the underexplored institutional prerequisites for financial deployment, including regulatory compliance, model validation protocols, and structural barriers to adoption. We conclude that despite ongoing hardware maturation, proactive engagement with quantum algorithm development is imperative for financial institutions to preempt technological obsolescence upon the achievement of hardware parity.
We start by applying quantum techniques to solve one of the most common problems in finance. Six related applications derived from this general solution are discussed in
Section 2. Modern portfolio theory, originating from Markowitzβs seminal work [
1], establishes the mathematical foundation for optimal asset allocation. The classical mean-variance optimization problem can be formulated as
where
πββπ is the portfolio weight vector,
Ξ£ββπΓπ is the covariance matrix of asset returns,
πββπ is the expected return vector, and
π>0 is the risk aversion parameter (which is commonly written as its reciprocal in classical economics literature).
For
n assets, solving this quadratic programming problem classically requires
πͺ(π3) operations for matrix inversion when using interior-point methods such as that of Karmarkar [
2]. In practice, financial institutions often deal with
πβ₯500, such as the S&P 500 Index components, making real-time optimization challenging. Moreover, once integer constraints are introduced (e.g., minimum investment lots), the problem becomes an integer programming problem, which is NP-hard.
For
n assets, solving this quadratic programming problem classically requires
πͺ(π3.5πΏ) operations using Karmarkarβs original interior-point method [
2], where
L is the input bit length. Subsequent improvements, including Renegarβs path-following method [
3] and modern primal-dual interior-point methods [
4,
5], achieve
πͺ(π3πΏ) complexity.
While this article provides a survey of several financial applications of quantum computing, the principal technical contribution lies in the treatment of portfolio optimization. The remaining application domains are surveyed solely to establish contextual background and to delineate the broader research landscape, and are explicitly not afforded mathematical treatment of equivalent rigor.
For a broader overview of the field, the reader is referred to recent comprehensive surveys of quantum computing applications in finance [
6,
7].
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Paper Structure: The remainder of
Section 1 develops the mathematical foundations of quantum optimization, focusing on QAOA and its application to portfolio construction (readers less familiar with quantum formalism may focus on the conceptual summaries).
Section 2 surveys six quantum financial applications, with an emphasis on practical deployment considerations.
Section 3 examines institutional challenges, including compliance, validation, and hardware limitations, culminating in strategic recommendations for financial institutions.
Section 4 concludes.
Quantum Computing Fundamentals
Definition 1
(Quantum State Space [
8])
. For a system of n qubits, the state space is the Hilbert space β=(β2)βπβ
β2π. A pure quantum state is represented by a unit vector |πβͺββ satisfying β©π|πβͺ=1.
Definition 2
(Quantum Gates and Circuits)
. A quantum gate on k qubits is a unitary operator πβπ(2π). Common single-qubit gates include the Pauli-X, Pauli-Y, Pauli-Z, Hadamard, and Rotation-Z gates:
This entry is adapted from the peer-reviewed paper 10.3390/encyclopedia6050095
