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A quantum unstructured search algorithm for discrete optimisation: the use case of portfolio optimisation

Titos Matsakos, Adrian Lomas

TL;DR

This work introduces QSERA, a quantum unstructured search algorithm for finding extrema or roots of discrete functions by mapping the problem to a Grover oracle via a rescaled function g(x) in [0,1] and a raised power n, enabling a quadratic speedup ($O(\sqrt{N})$) over classical search. It extends the QUSA framework to incorporate higher-order objective terms by expressing g(x)^n as a polynomial in binary variables and constructing a gate-based oracle from a hierarchical set of QSERA gates, including Q_*0, Q_*1, Q_*2, and beyond. The authors validate the approach with an end-to-end portfolio-optimisation example, encoding 16 portfolios on 4 qubits and demonstrating that the optimal portfolio can be identified with high probability after a small number of Grover iterations, albeit affected by approximate bounds for f_min and f_max. Overall, QSERA broadens quantum optimisation beyond QUBO by accommodating higher-order terms and enabling discretized treatment of continuous problems, offering a pathway to quantum-accelerated discrete optimisation with caveats related to bound accuracy and circuit depth.

Abstract

We propose a quantum unstructured search algorithm to find the extrema or roots of discrete functions, $f(\mathbf{x})$, such as the objective functions in combinatorial and other discrete optimisation problems. The first step of the Quantum Search for Extrema and Roots Algorithm (QSERA) is to translate conditions of the form $f(\mathbf{x}_*) \simeq f_*$, where $f_*$ is the extremum or zero, to an unstructured search problem for $\mathbf{x}_*$. This is achieved by mapping $f(\mathbf{x})$ to a function $u(z)$ to create a quantum oracle, such that $u(z_*) = 1$ and $u(z \neq z_*) = 0$. The next step is to employ Grover's algorithm to find $z_*$, which offers a quadratic speed-up over classical algorithms. The number of operations needed to map $f(\mathbf{x})$ to $u(z)$ determines the accuracy of the result and the circuit depth. We describe the implementation of QSERA by assembling a quantum circuit for portfolio optimisation, which can be formulated as a combinatorial problem. QSERA can handle objective functions with higher order terms than the commonly-used Quadratic Unconstrained Binary Optimisation (QUBO) framework. Moreover, while QSERA requires some a priori knowledge of the extrema of $f(\mathbf{x})$, it can still find approximate solutions even if the conditions are not exactly satisfied.

A quantum unstructured search algorithm for discrete optimisation: the use case of portfolio optimisation

TL;DR

This work introduces QSERA, a quantum unstructured search algorithm for finding extrema or roots of discrete functions by mapping the problem to a Grover oracle via a rescaled function g(x) in [0,1] and a raised power n, enabling a quadratic speedup () over classical search. It extends the QUSA framework to incorporate higher-order objective terms by expressing g(x)^n as a polynomial in binary variables and constructing a gate-based oracle from a hierarchical set of QSERA gates, including Q_*0, Q_*1, Q_*2, and beyond. The authors validate the approach with an end-to-end portfolio-optimisation example, encoding 16 portfolios on 4 qubits and demonstrating that the optimal portfolio can be identified with high probability after a small number of Grover iterations, albeit affected by approximate bounds for f_min and f_max. Overall, QSERA broadens quantum optimisation beyond QUBO by accommodating higher-order terms and enabling discretized treatment of continuous problems, offering a pathway to quantum-accelerated discrete optimisation with caveats related to bound accuracy and circuit depth.

Abstract

We propose a quantum unstructured search algorithm to find the extrema or roots of discrete functions, , such as the objective functions in combinatorial and other discrete optimisation problems. The first step of the Quantum Search for Extrema and Roots Algorithm (QSERA) is to translate conditions of the form , where is the extremum or zero, to an unstructured search problem for . This is achieved by mapping to a function to create a quantum oracle, such that and . The next step is to employ Grover's algorithm to find , which offers a quadratic speed-up over classical algorithms. The number of operations needed to map to determines the accuracy of the result and the circuit depth. We describe the implementation of QSERA by assembling a quantum circuit for portfolio optimisation, which can be formulated as a combinatorial problem. QSERA can handle objective functions with higher order terms than the commonly-used Quadratic Unconstrained Binary Optimisation (QUBO) framework. Moreover, while QSERA requires some a priori knowledge of the extrema of , it can still find approximate solutions even if the conditions are not exactly satisfied.

Paper Structure

This paper contains 24 sections, 58 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quantum optimisation
  3. Portfolio optimisation
  4. Quantum Search for Extrema and Roots Algorithm (QSERA)
  5. Overview of QUSA
  6. Description of QSERA
  7. Implementation of the quantum oracle $u_n(z)$ with quantum gates
  8. Remarks
  9. Application of QSERA to portfolio optimisation
  10. Summary and conclusions
  11. Quantum optimisation algorithms
  12. Quantum Adiabatic Algorithm (QAA)
  13. Variational Quantum Eigensolvers (VQE)
  14. Quantum Approximate Optimisation Algorithm (QAOA)
  15. QUSA worked example
  16. ...and 9 more sections

Figures (6)

  • Figure 1: Visualisation of the first step of QSERA, i.e. the transformation $f(\mathbf{x}) \to u(z)$ for maximisation (left column), minimisation (middle column), and root finding (right column) problems. The example discrete function $f(\mathbf{x})$ is the same in all three cases and depends on four binary variables, $x_i\in\{0,1\}$, that can be mapped to integers with $z \equiv \mathrm{b}_3\mathrm{b}_2\mathrm{b}_1\mathrm{b}_0$, where the right hand side has the elements of the vector $\mathbf{x}$ written as a binary number. The first row shows $f(\mathbf{x})$ with blue points, whereas the maximum (left), minimum (middle), and root (right) are shown with red points. The second row shows the transformation $f(\mathbf{x}) \to g(\mathbf{x})$ based on Eq. (\ref{['eq:g_max']}) (left), Eq. (\ref{['eq:g_min']}) (middle), and Eq. (\ref{['eq:g_root']}) (right), respectively. The third and fourth rows show the transformation $g(\mathbf{x}) \to u_n(z)$ of Eq. (\ref{['eq:u_n']}) for $n = 8$ and $n = 64$, respectively.
  • Figure 2: The returns and volatility of the $2^{N_\mathrm{a}} = 16$ portfolios that can be constructed from the combinations of the $N_\mathrm{a} = 4$ available assets. The 0000 portfolio and the portfolios consisting of one asset only are shown with open blue points, those with two or more assets with filled blue points, the benchmark with a red cross, and the portfolio that minimises $f(\mathbf{x})$ --- which QSERA is expected to find --- with a red point.
  • Figure 3: The functions $f(z)$ (Eq. \ref{['eq:f']}, top), $g(z)$ (Eq. \ref{['eq:g_min']} with $f_\mathrm{min}$ set to $0$, middle), and $u_{24}(z)$ (Eq. \ref{['eq:u_n']}, bottom), which show how the quantum oracle is created.
  • Figure 4: The probabilities of measuring the different states, each one representing a portfolio. QSERA identifies the portfolio 1001 which minimises the objective function of Eq. (\ref{['eq:f']}). The portfolio 0101, which has the second lowest value, also has a high probability to be measured relative to all other ones.
  • Figure 5: The probability of measuring the portfolio that minimises $f(\mathbf{x})$ as a function of the power $n$ used to create the oracle $u_n(z)$.
  • ...and 1 more figures