Three kinds of "quantum" finance research — what "quantum-inspired" means, and what it does not

Search for “quantum finance” and you will find quantum computers, new algorithms and physics equations applied to share prices, all listed side by side. The term “quantum-inspired” also refers to different things in different contexts.

These kinds of research make different claims and should be judged by different standards. Reading them as one leads to mistaking the results of one kind for those of another. This note sorts finance research that carries the word “quantum” into three kinds.

1. Computing on a quantum computer

The first kind solves financial computing problems on a quantum computer. Its claims take the form: “the same problem can be solved with less computation than on a classical computer”.

The best-known example is the Monte Carlo method. Pricing derivatives or measuring risk often means generating many random paths and averaging over them. With classical Monte Carlo, the number of samples needed for an error ε grows roughly as 1/ε² (for a fixed spread of the values) (halving the error takes four times as many samples). In 2015 Montanaro (1) showed that, for a random computation whose spread (standard deviation) is bounded by σ, quantum amplitude estimation reduces the number of calls needed to estimate its mean to within ε to roughly σ/ε, up to logarithmic factors (classically it grows as σ²/ε²). One “call” here runs the whole quantum circuit that prepares the probability distribution as a quantum state and then computes the quantity being averaged, such as a payoff (or its inverse). This is a “quadratic speed-up” in the sense that an accuracy requiring N classical samples can be reached with about √N calls. It does not mean that the total computational effort falls to its square root across the board. It assumes the circuit can be built efficiently, and it has been pointed out that, depending on how the distribution is prepared, the speed-up can be lost — for example when a log-concave distribution is prepared with the Grover–Rudolph method (2) (3). In 2018 Rebentrost and colleagues (4) gave a concrete construction applying this to option pricing.

Applications to optimisation problems such as portfolio optimisation, and to machine learning, are also being studied. The 2022 survey by Herman and colleagues (3) (the same authors published a review in Nature Reviews Physics in 2023 (5)) covers three areas — stochastic modelling, optimisation and machine learning — setting out the proposed algorithms and whether they can run on current and near-term hardware.

The standards for judging this kind of research are clear.

• Faster than what? The comparison should be with the best classical algorithm for the problem.
• On what hardware? Many theoretical speed-ups assume large, error-corrected quantum computers. A study estimating the resources needed for derivative pricing also puts the required scale far beyond today’s devices (6). The same results cannot necessarily be achieved on today’s devices.

2. Quantum-inspired classical algorithms

The second kind consists of algorithms that run on ordinary computers but were devised from ideas in research on quantum algorithms or quantum systems. The term “quantum-inspired”, however, is used in at least three different contexts, and the nature of the claim differs in each.

In complexity theory: dequantisation

In complexity theory, “quantum-inspired” is used in a narrower sense: in work that asks whether the speed of a quantum algorithm really comes from quantum effects.

In 2017 Kerenidis and Prakash (7) gave a quantum algorithm for a recommendation-system problem — selecting one product to recommend to a given user, on the premise that the matrix of user–product ratings is well approximated by a few components (is low-rank). Assuming the matrix is stored in a special data structure that can be read quantumly, it runs in time polylogarithmic in the size of the matrix. In 2019 Ewin Tang (8) observed that the same data structure also lets an ordinary computer read entries and sample rows and entries with probability proportional to their squared magnitude, and showed that with these operations a classical algorithm can produce the same kind of output in time logarithmic in the size of the matrix. This line of work is known as “dequantisation”.

The scope of the result is narrow. It concerns linear-algebra problems involving low-rank matrices, and the comparison holds when the classical side is given data access equivalent to the quantum side’s. As the 2022 survey by Herman and colleagues points out (3), dequantised algorithms scale steeply with the rank of the matrix and the required precision, so room remains for quantum algorithms to keep a polynomial advantage. Tang herself acknowledges that the dependence on rank and precision is much worse than on the quantum side (8). It is not a result showing that quantum algorithms in general can be replaced by classical ones.

As optimisation methods

In combinatorial optimisation, heuristic methods that implement ideas from research on quantum computers as numerical computation on classical hardware are also called “quantum-inspired”. One example, presented by Goto and colleagues in 2019 (9), takes its cue from research on quantum adiabatic optimisation, proposed as a kind of quantum computer, and numerically solves the corresponding classical equations of motion on an ordinary computer. Such methods carry no complexity guarantee; they are judged by how good a solution they find on real problems, and how quickly.

As numerical methods

Tensor networks, a numerical technique developed for the study of quantum many-body systems (10) (and developed independently in numerical analysis as the tensor-train decomposition (11)), are also sometimes called “quantum-inspired” when carried over to approximate computation in fields outside physics; in finance they have been applied to portfolio optimisation (12). These too are approximation methods that run on ordinary computers.

The only thing the three senses share is that no quantum computer is used. The first is a theorem about computational cost under restricted assumptions; the second and third are methods judged by their performance on real problems. When something is described as “quantum-inspired”, the first step is to establish which sense is meant.

3. Models that describe markets using the mathematics of quantum mechanics

The third kind uses the mathematical formalism of quantum mechanics as a model of markets or of investors’ judgements. The computation is done on ordinary computers; no quantum computer is involved. An overview of research applying quantum mathematics to the social sciences is given in (13). Within this kind, however, there are two strands with fundamentally different starting points.

3a. Carrying physics equations over to markets

One strand brings the equations of physics into the description of prices. In 2010 Zhang and Huang (14), reasoning by analogy with wave–particle duality, treated share prices as a kind of quantum system, defined a wave function and operators for them, and set up a Schrödinger equation. For models of this kind, the premise that a market can be treated as a quantum system is the first thing to be questioned.

3b. Describing judgements with quantum probability

The other strand uses the Hilbert-space formalism of quantum mechanics not as a description of a physical system but as a set of rules for calculating probabilities. People’s judgements are known to change with the order in which questions are asked and with context. To express such effects in classical probability theory, a separate assumption has to be added for each order. The starting point of this strand is that the probability rules of quantum mechanics — descriptions by non-commuting operators, under which the order of measurements can change the outcome — express them with fewer assumptions; it is studied in the psychology of decision-making as “quantum cognition” (15). An a priori prediction about question-order effects derived from these rules has been tested against a large set of survey data (16). A study testing another prediction derived from the same model on similar data, however, found the simplest form of the model unsupported in most cases (17). In 2012 Aerts and colleagues (18), using a framework that rebuilds quantum mechanics at the operational level (the SCoP formalism) and a model in which quantum probabilities arise from fluctuations in the context of measurement, argued that a share has no definite price until it is traded, and that its price is actualised in the context of trading. The core of the claim is that the trading context gives the probabilities of prices a structure that does not fit within classical probability theory. Quantum mechanics here is not a claim that markets are physically quantum systems, but mathematics for describing judgements that change with context.

Where quantum random walk models fit

The quantum random walk models discussed in the previous note sit between the two strands. De Backer and colleagues (19) use the quantum random walk phenomenologically, as a mathematical mechanism for producing return distributions, and do not claim that markets are quantum systems. Orrell (20) gives the quantum random walk a quantum-cognition interpretation. De Backer and colleagues, incidentally, call their own approach “quantum-inspired”. The term is used not only in the senses of section 2 (complexity theorems and computational methods) but also for models of this kind. Here it means borrowing the mathematics of quantum mechanics for a model of probability distributions, not a claim that computation becomes faster.

Standards for judging

For both strands, the standard is not computational speed but quality as a model.

• How well does it explain actual data?
• Compared with existing models (GARCH-type models (21) or stable distributions (22), for example), what can it now explain?
• Is that worth the additional parameters?
• Can the parameters be given a meaning on the market side?

In addition, for 3a the validity of treating markets as quantum systems is in question, and for 3b whether the context effects in judgement are actually observed.

The three side by side

	1. Quantum computing	2. Quantum-inspired	3. Models using quantum mathematics
Computer	Quantum computer	Ordinary computer	Ordinary computer
Form of the claim	Solves the same problem with fewer samples or less computation	A complexity theorem under restricted assumptions, or practical performance	Describes the phenomenon or the judgement better
Compared with	Best classical algorithm	Quantum algorithm, or existing methods	Existing statistical models (for 3b, classical-probability models of judgement)
What to check	Hardware scale and errors, assumptions about preparing the data	Which sense; for a theorem, assumptions and dependence on rank and precision; for a method, the conditions of comparison	Fit to real data, number of parameters, validity of the premise (3a), observed context effects (3b)

Three questions to ask when reading

When reading about finance research or products that describe themselves as “quantum”, three questions help bring out what is actually being claimed.

1. Which sense of “quantum”? Is a quantum computer being used, is it classical computation inspired by quantum algorithms, or is the mathematics of quantum mechanics being used as a model?
2. Compared with what? Is the comparison with the standard method for the problem?
3. Tested where? Is this theoretical potential, results on simulated data, or testing on real data?

Each of the three kinds is legitimate research on its own terms. Confusion arises when the claims of one kind (for example, “a quantum computer makes this faster”) are read onto another (for example, a model that runs on an ordinary computer).

This series covers published research in quantum finance and econophysics broadly. It does not assess the performance of any particular quantum computer or offer a view on its prospects.