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arxiv: 2301.09241 · v6 · submitted 2023-01-23 · 🪐 quant-ph · cs.NA· math.NA· q-fin.CP· q-fin.MF

Quantum Monte Carlo algorithm for option pricing and its complexity analysis

Jianjun Chen , Yongming Li , Ariel Neufeld This is my paper

Pith reviewed 2026-05-24 10:21 UTC · model grok-4.3

classification 🪐 quant-ph cs.NAmath.NAq-fin.CPq-fin.MF
keywords quantum Monte Carlooption pricingBlack-Scholes PDEquantum algorithmcomplexity analysismultidimensional pricingpiecewise affine payoff
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The pith

A quantum Monte Carlo algorithm solves multidimensional Black-Scholes PDEs for option pricing with polynomial complexity in dimension and accuracy.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a quantum Monte Carlo algorithm for pricing options under multidimensional Black-Scholes models that incorporate asset correlations. It requires the payoff to be continuous and piecewise affine and supplies a full error analysis along with a complexity bound. The key result is that the number of quantum operations scales polynomially with the dimension d and with 1/ε. For bounded payoffs the method is shown to be faster than classical Monte Carlo sampling. Numerical tests in two dimensions confirm the approach works in practice.

Core claim

We provide a quantum Monte Carlo algorithm to solve multidimensional Black-Scholes PDEs with correlation for option pricing. The payoff function is continuous and piecewise affine. We prove that the computational complexity is bounded polynomially in the space dimension d and the reciprocal of the accuracy ε. For bounded payoffs the algorithm has a speed-up compared to classical Monte Carlo methods.

What carries the argument

The quantum Monte Carlo algorithm based on quantum state preparation for the Black-Scholes dynamics and amplitude estimation to evaluate the expectation under the payoff.

Load-bearing premise

The payoff function must be continuous and piecewise affine to allow quantum state preparation and error analysis to succeed.

What would settle it

An explicit payoff that is continuous and piecewise affine yet requires superpolynomial resources in d, or a bounded-payoff case where the quantum resource count exceeds classical Monte Carlo.

Figures

Figures reproduced from arXiv: 2301.09241 by Ariel Neufeld, Jianjun Chen, Yongming Li.

Figure 1
Figure 1. Figure 1: Implementation of the Toffoli gate as a quantum circuit using single-qubit gates and [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Flowchart of Algorithm 1. (Top left) (1) Construction of the operator [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the expected payoff estimates from the algorithm for the vanilla call option across a range of strike [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Expected payoff estimates from our proposed algorithm for the basket call option across a range of strike prices, [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Expected payoff estimates from the algorithm for the spread call option across a range of strike prices, compared [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Expected payoff estimates from the algorithm for the call-on-max option across a range of strike prices, compared [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Expected payoff estimates from the algorithm for the call-on-min option across a range of strike prices, compared [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Expected payoff estimates from the algorithm for the best-of-call option across a range of strike prices, compared [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Description of the circuit to compare three variables. The first [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: An example for the quantum circuit for permutation in Lemma 4.8, with [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Circuit diagram for Qe(+) in Corollary 4.10. where k is the number of cycles, and a1, . . . , amk ∈ {1, . . . , n} are distinct numbers. Note by convention that each of these cycles has length ml ≥ 2. For each of these cycles Cl = (aml−1+1 · · · aml ), l = 1 . . . , k, with m0 := 0 we construct the quantum circuits TCl , l = 1, . . . , k, via TCl = mYl−1 i=ml−1 Tai↔ai+1 = Taml−1↔aml · · · Taml−1+1↔aml−1+2… view at source ↗
Figure 12
Figure 12. Figure 12: Circuit diagram for Qe(×) in Corollary 4.12. Moreover, by Lemma 4.8, there is a quantum circuit Tπ2 such that |a ⊞ b⟩n1+m1+1 |b⟩n2+m2 7→ |b⟩n2+m2 |a ⊞ b⟩n1+m1+1 , (107) which uses at most 2(n + m + 1)2 swap gates. Define the quantum circuit Qe (+) := Tπ2Q(+)Tπ1 . Observe that (104), (105), and (107) shows that Qe (+) satisfies (103), and that the total number of elementary gates required to construct Qe (… view at source ↗
Figure 13
Figure 13. Figure 13: Circuit diagram for Qe(comp) in Corollary 4.14. and that the number of elementary gates required to construct Q(×) is at most ( 1 2 (5(n1 + m1) 2 + (n1 + m1)) + 4(n2 + m2) 2 + 4(n1 + m1)(n2 + m2) + 6(n2 + m2) + 7). (113) By another application of Lemma 4.8, there is a quantum circuit Tπ′ with at most 2(2n + 2m + n2 + m2 + 3)2 swap gates satisfying Tπ′ : |anc⟩ |a  b⟩n+m |a⟩n1+m1 |b⟩n2+m2 |anc⟩n2+m2+2 7→ |… view at source ↗
Figure 14
Figure 14. Figure 14: Circuit diagram for CRy(θ) in Lemma 4.15. Lemma 4.15 (Controlled Y -rotations) For any θ ∈ (0, 4π), there is a controlled Y -rotation gate acting on two qubits that performs the following operation CRy(θ) : |c⟩ |0⟩ 7→ |c⟩(Ry(θ))c |0⟩ = ( |c⟩ |0⟩, if c = 0, |c⟩(cos(θ/2)|0⟩ + sin(θ/2)|1⟩), if c = 1. (125) The quantum circuit to construct CRy(θ) requires two Ry(θ/2) gates (see Example 2.9) and two CNOT gates… view at source ↗
Figure 15
Figure 15. Figure 15: Circuit diagram for Q d,n,m + in Lemma 4.19. Remark 4.17 In case one uses the quantum circuit constructed in [15, Appendix E] to upload the discretized multivariate log-normal distribution, the corresponding constant C3 defined in Assumption 4.16 can be chosen to be C3 := max{2, L}, where L ∈ N is the depth of each variational quantum circuit involved in [15, Appendix E] for approximating the involved Gau… view at source ↗
Figure 16
Figure 16. Figure 16: Circuit diagram for Q n,m (max) in Lemma 4.20. 6. We consolidate the ancillary qubits by combining the qubits (labeled |a2  i2⟩, . . . , |ad  id⟩, |b⟩) under ancilla qubits |anc⟩⋆ . The permutation circuit Tπ from Lemma 4.8 performs the following operation Tπ : |i1⟩n1+m1 · · · |id⟩n1+m1 |a2  i2⟩n+m · · · |ad  id⟩n+m |b⟩n2+m2 [PITH_FULL_IMAGE:figures/full_fig_p032_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Circuit diagram for Q I,n,m (max) in Corollary 4.21. Thus, we have |a⟩n+m |b⟩n+m |c1⟩ |c2⟩ |c3⟩ |anc⟩ 2   nO−1 j=−m Xc3bjXc2bjXc1aj |0⟩n+m   =    |a⟩n+m |b⟩n+m |c1⟩ |c2⟩ |c3⟩ |anc⟩ 2 Nn−1 j=−m Xaj |0⟩n+m  , if Dn,m(a) > Dn,m(b), |a⟩n+m |b⟩n+m |c1⟩ |c2⟩ |c3⟩ |anc⟩ 2 Nn−1 j=−m Xbj |0⟩n+m  , if Dn,m(a) ≤ Dn,m(b), = ( |a⟩n+m |b⟩n+m |c1⟩ |c2⟩ |c3⟩ |anc⟩ 2 |a⟩n+m , Dn,m(a) > Dn,m(b), |a⟩n+m |b⟩n+m |c… view at source ↗
Figure 18
Figure 18. Figure 18: Circuit diagram for Qh in Proposition 4.22. Proposition 4.22 (Quantum circuit for loading CPWA component functions) Let I, d, n1, n2, m1, m2 ∈ N. Define n := n1 + n2, m := m1 + m2, and p := d(2n2 + 2m2 + 3) + (n2 + m2) + (d − 1)(n + m). Let {al,j}l=1,...,I;j=1,...,d, {bl}l=1,...,I ⊂ Fn2,m2 . Let hl : F d n1,m1 → Fn+d,m, l = 1, . . . , I be functions defined by hl(i1, . . . , id) = d ⊞ j=1 (al,j  ij ) ⊞ b… view at source ↗
Figure 19
Figure 19. Figure 19: Circuit diagram for Rf in Lemma 4.23. Thus, the total number of elementary gates used is at most I[2N 2 + 563d 3 (n + m + 1)2 ] + 1189I 2 (n + m + d + 1)3 + 2N 2 = (I + 1)2N 2 + 563Id3 (n + m + 1)2 + 1189I 2 (n + m + d + 1)3 ≤ 4IN2 + 563Id3 (n + m + 1)2 + 1189I 2 (2d) 3 (n + m + 1)3 ≤ 4I(12Id(n + m + 1))2 + 563Id3 (n + m + 1)2 + 1189I 2 (2d) 3 (n + m + 1)3 ≤ (4 · 122 + 563 + 1189 · 2 3 )I 3 d 3 (n + m + 1… view at source ↗
Figure 20
Figure 20. Figure 20: Circuit diagram for Rh in Proposition 4.24 (Steps 1–3). Proof. The construction of the quantum circuit Rh involves the following steps: 1. We first prepare the K component functions hk using Proposition 4.22. For k = 1, . . . , K, we apply the quantum circuits (Qhk )k=1,...,K of Proposition 4.22 (with I, d, n1, m1, n2, m2 ← Ik, d, n1, m1, n2, m2, al,j ← ak,l,j , and bl ← bk,l in the notation of Propositio… view at source ↗
Figure 21
Figure 21. Figure 21: Circuit diagram for Rh in Proposition 4.24 (Steps 4–6). 2.6) |i1⟩n1+m1 · · · |id⟩n1+m1 |anc⟩ q1+···+qK |h1(i)⟩n+m+d |0⟩ 2 |0⟩n+m+d+2 |0⟩ 5 · |h2(i)⟩n+m+d |0⟩ 2 |0⟩n+m+d+2 |0⟩ 5 · · · |hK(i)⟩n+m+d |0⟩ 2 |0⟩n+m+d+2 |0⟩ 5 |0⟩K 7−→ |i1⟩n1+m1 · · · |id⟩n1+m1 |anc⟩ q1+···+qK |h1(i)⟩n+m+d [PITH_FULL_IMAGE:figures/full_fig_p041_21.png] view at source ↗
read the original abstract

In this paper we provide a quantum Monte Carlo algorithm to solve multidimensional Black-Scholes PDEs with correlation for option pricing. The payoff function of the option is of general form and is only required to be continuous and piecewise affine, which covers most of the relevant payoff functions used in finance. We provide a rigorous error analysis and complexity analysis of our algorithm. In particular, we prove that the computational complexity of our algorithm is bounded polynomially in the space dimension $d$ of the PDE and the reciprocal of the prescribed accuracy $\varepsilon$. Moreover, we show that for payoff functions which are bounded, our algorithm indeed has a speed-up compared to classical Monte Carlo methods. Furthermore, we provide numerical simulations in two dimensions using our developed package within the Qiskit framework tailored to price continuous piecewise affine options with respect to the Black-Scholes model, as well as discuss the potential extension of the numerical simulations to arbitrary space dimension.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript presents a quantum Monte Carlo algorithm for solving multidimensional Black-Scholes PDEs with correlations to price options. The payoff functions are assumed to be continuous and piecewise affine. A rigorous error analysis is provided, along with a complexity analysis showing that the algorithm's complexity is polynomial in the space dimension d and the reciprocal of the accuracy ε. For bounded payoffs, a speedup over classical Monte Carlo is claimed. Numerical simulations in two dimensions are performed using a custom Qiskit package.

Significance. If the analysis holds, the work provides a concrete quantum algorithm for a practical finance problem with an explicit polynomial complexity bound in d and 1/ε, plus a claimed speedup for bounded payoffs. The rigorous error/complexity analysis and the open-source Qiskit implementation are strengths that support reproducibility and verifiability.

minor comments (3)
  1. [§1] §1 and abstract: the statement that the continuous piecewise affine assumption 'covers most of the relevant payoff functions used in finance' would benefit from one or two concrete examples (e.g., European calls vs. certain exotics) to clarify scope.
  2. [Numerical simulations] Numerical section: the 2D Qiskit implementation is described at a high level; adding pseudocode or a brief description of the state-preparation circuit for the piecewise-affine payoff would improve clarity and reproducibility.
  3. [Complexity analysis] The complexity proof sketch in the main text could explicitly flag where the piecewise-affine property is used to bound the state-preparation cost, even if the full derivation is in an appendix.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the recognition of its strengths in rigorous analysis and open-source implementation, and the recommendation for minor revision. No specific major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper states a quantum Monte Carlo algorithm for multidimensional Black-Scholes PDEs whose complexity is proven polynomial in dimension d and 1/ε under the explicit modeling assumption that payoffs are continuous and piecewise affine (with boundedness for the speedup claim). This is presented as a rigorous proof against external classical Monte Carlo benchmarks rather than any fitted parameter, self-definition, or self-citation chain. The abstract and reader's summary give no indication that any load-bearing step reduces by construction to the inputs; the derivation is therefore treated as self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard assumptions of quantum computing (unitary evolution, amplitude estimation) and the Black-Scholes model (log-normal asset dynamics, constant coefficients). No free parameters or invented entities are introduced in the abstract. The piecewise-affine payoff restriction is a domain modeling choice rather than an ad-hoc invention.

axioms (2)
  • standard math Quantum amplitude estimation provides quadratic speedup over classical Monte Carlo sampling
    Invoked implicitly when claiming speedup for bounded payoffs; this is a standard result in quantum algorithms literature.
  • domain assumption The Black-Scholes PDE with correlation admits a Feynman-Kac representation that can be sampled quantum-mechanically
    Required to map the PDE solution to a quantum expectation value; standard in stochastic finance but not proved in the abstract.

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