Abstract
This chapter provides an overview of quantum algorithms that have relevance for econometricians and computational economists. It is divided into two subsections. The first covers theoretical developments in the construction of quantum algorithms and related applications on quantum hardware. We identify relevant problems within economics and finance, determine whether computational speedups are achievable with existing algorithms, and evaluate whether an algorithm has additional restrictions that do not apply to its classical counterpart. The second part of this chapter describes experimental progress in the development of quantum computing devices and its implications for the implementation of quantum algorithms. We will also discuss the limitations of different quantum computing devices.
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Notes
- 1.
See http://quantumalgorithmzoo.org, which is a regularly updated list of quantum algorithms maintained by Stephen Jordan.
- 2.
- 3.
The large literature on ARCH and GARCH models makes use of numerical gradient and Hessian computation (Bollerslev 1986a; Engle et al. 1987; Bollerslev 1986b; Bollerslev 1987; Danielsson 1994; Zakoian 1994; Engle and Russell 1998; Gray 1996; Engle and Russell 1998; De Santis and Gérard 1998; and Engel 2000). Numerical differentiation is also employed to solve a variety of different models of financial markets (Hsieh 1991; Hiemstra and Jones 1994; Longstaff and Schwartz 1992; Karolyi and Stulz 1996; De Santis and Gérard 1997; Dufour and Engle (2000); Bae et al. 2003).
- 4.
Aguirregabiria and Mira (2002) and Judd and Su (2012) provide algorithms for structural microeconometric models that make use of numerical derivatives. Aguirregabiria and Mira (2010) offer a survey of the literature on dynamic discrete choice models, which makes extensive use of gradient-based methods. Burtless and Hausman (1978), Lancaster (1979), and Heckman and Macurdy (1980) employed gradient-based methods to solve structural microeconomic models.
- 5.
Maximum likelihood estimation (MLE) is used for a variety of different economic and financial problems, including the estimation of structural models (see, e.g., Belsley 1980; Greene 1982; White 1982; Bunch 1988; Rabe-Hesketh et al. 2005; Fernández-Villaverde and Rubio-Ramírez 2007; and Judd and Su 2012). It is often necessary to compute the gradient and Hessian of the likelihood function, which can create a bottleneck in the estimation algorithm for high-dimensional problems.
- 6.
Packages used to solve and estimate DSGE models, such as Dynare, commonly make use of numerical gradients and Hessian matrices.
- 7.
- 8.
See Sect. 2.11 for an overview of computational complexity and related notation.
- 9.
- 10.
A wide variety of computational models in economics and finance employ interpolation in the solution method. For a range of applications, see Keane and Wolpin (1994), Ackerberg (2003), Rust (1997), and Crawford and Shum (2005). For surveys of problem classes that often employ interpolation, see Heckman and Navarro (2007), Aguirregabiria and Mira (2010), and Keane (2011).
- 11.
See Judd (1998) for an overview of interpolation methods.
- 12.
Heer and Maussner (2009) compare run times and Euler equation residuals for an infinite horizon Ramsey model under several different solution methods, including value function iteration with and without interpolation. When the state space contains 5,000 nodes, they find that cubic polynomial interpolation is 32 times faster than value function iteration and also generates small Euler equation residuals.
- 13.
In the case where A is not Hermitian, the authors point out that we may instead use \(C=\begin {bmatrix} 0 & A \\ A^{\dagger } & 0 \\ \end {bmatrix}.\) Furthermore, we may replace x with \(\begin {bmatrix} 0 \\ x \end {bmatrix}\) and b with \(\begin {bmatrix} b \\ 0 \end {bmatrix}\).
- 14.
See Hughes (2000) for an introduction to linear finite element analysis.
- 15.
- 16.
- 17.
A density matrix is an alternative way to express a quantum state. It is often used when there is classical uncertainty about the true underlying state. In such cases, we express the density matrix as a mixture of pure states, \(\rho = \sum _{i} p_{i} {\left\vert \phi _{j}\right\rangle} {\left\langle \phi _{j}\right\vert}\). Note that the density matrix for each pure state is given by the outer product of its state in ket or vector form.
- 18.
- 19.
Incomplete markets models with many heterogeneous agents were introduced by Bewley (1977), Huggett (1993), and Aiyagari (1994). Krusell and Smith (1998) provided a tractable solution method for incomplete market models with aggregate uncertainty. Kaplan et al. (2018) showed how monetary policy could be included in such models. A large and growing literature has made use of such models to study the distributional impact of policy. See, e.g., Heaton and Lucas (1996), Gourinchas and Parker (2003), Castañeda et al. (2003), Kreuger and Perri (2006), Carroll and Samwick (1998), Chatterjee et al. (2007), Blundell et al. (2008), Heathcote et al. (2010), Hornstein et al. (2011), Kaplan and Violante (2014), and Guerrieri and Guido (2017).
- 20.
- 21.
- 22.
- 23.
- 24.
See Adame and McMahon (2020) for a discussion of inhomogeneous annealing.
- 25.
In November of 2020, ID Quantique sold QRNG devices for roughly $1000.
- 26.
- 27.
Formally, this could be done in polynomial time in the size of the puzzle.
- 28.
“Nature isn’t classical, dammit, and if you want to make a simulation of nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem, because it doesn’t look so easy.”
- 29.
- 30.
GPUs, which were originally developed to render graphics, have since been exploited to perform massively parallel computation of basic floating point operations. TPUs were developed to perform the computational function of a GPU, but without the capacity to render graphics. Had there not been substantial progress in the development of GPUs over the last decade, it is unlikely that machine learning would have experienced as much success as it has as a field. Similarly, it is possible that quantum computing could generate similar transformations by unlocking the solution and estimation of otherwise intractable models.
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Hull, I., Sattath, O., Diamanti, E., Wendin, G. (2024). Quantum Algorithms. In: Quantum Technology for Economists. Contributions to Economics. Springer, Cham. https://doi.org/10.1007/978-3-031-50780-9_3
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