Potential Applications of Quantum Computing for the Insurance Industry Michael Adam

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Potential Applications of Quantum Computing for the
Insurance Industry
Michael Adam∗
AXA Konzern AG†
October 10, 2022
Abstract
This paper is the documentation of a pre-study performed by AXA Konzern AG in collabo-
ration with Fraunhofer ITWM to assess the relevance of quantum computing for the insurance
industry. Beside a general overview of the status quo of quantum computing technologies, we
investigate its applicability for the valuation of insurance contracts as a concrete use case. This
valuation is a computationally intensive problem because the lack of closed pricing formulas
requires the use of Monte Carlo methods. Therefore current technical capabilities force insurers
to apply approximation methods for many subsequent tasks like economic capital calculation or
optimization of strategic asset allocations. The business-criticality of these tasks combined with
the existence of a quantum algorithm called Amplitude Estimation which promises a quadratic
speed-up of Monte Carlo simulation makes this use case obvious. We provide a detailed explana-
tion of Amplitude Estimation and present two quantum circuits which describe insurance-related
payoff features in a quantum circuit model. An exemplary circuit that encodes dynamic lapse
is evaluated both on a simulator and on real quantum hardware.
∗michael.adam@axa.de
†in collaboration with Fraunhofer ITWM
1
arXiv:2210.06172v1 [quant-ph] 10 Oct 2022
CONTENTS
Contents
1 Introduction 3
2 Quantum Computing Basics 5
2.1 AQuantumBit....................................... 5
2.2 Evolution of a Quantum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Measurement ........................................ 8
2.4 CompositeSystems..................................... 9
3 Building Blocks of Payoff Valuation on a Quantum Computer 10
3.1 DistributionLoading.................................... 10
3.2 PayoffImplementation................................... 11
3.3 Calculation of the Expected Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Amplitude Estimation 14
4.1 Amplitude Estimation based on Phase Estimation . . . . . . . . . . . . . . . . . . . 14
4.2 Amplitude Estimation without Phase Estimation . . . . . . . . . . . . . . . . . . . . 26
4.3 Excursus: Grover’s Quantum Search Algorithm . . . . . . . . . . . . . . . . . . . . . 26
5 Insurance-related Payoffs 27
5.1 GeneralPayoff ....................................... 27
5.2 Specifications ........................................ 28
6 Insurance-related Quantum Circuits 29
6.1 Wholelifeinsurance .................................... 29
6.2 DynamicLapse....................................... 30
7 Quantum Hardware Results 33
7.1 Simulator .......................................... 34
7.2 RealHardware ....................................... 36
8 An Overview of current Quantum Computing Technology 38
8.1 Software........................................... 39
8.2 Hardware .......................................... 40
9 Conclusion 41
2
1 INTRODUCTION
1 Introduction
Quantum computers work fundamentally different than classical computers. By leveraging the
laws of quantum mechanics, they promise significant speed-ups for certain problems. The areas of
potential applications are versatile and include i.a. cryptography, machine learning and simulation.
This opens up several possibilities for the insurance industry to benefit from the new technology.
Broadly speaking, quantum computers can help solving computationally intensive problems. When
asking an actuary for a common task with high computational requirements, the valuation of in-
surance contracts is an obvious answer. The most expensive simulation concerns the liabilities of
life and health companies with very long time horizons (40 years and beyond), complex contractual
and legal frameworks, sophisticated customer behavior and interactions with assets (e.g. through
profit participation). Due to the complexity of these ”instruments”, closed-form valuation is not
available and hence Monte Carlo (MC) methods are widely used. But the fact is, that the valuation
of liabilities is essential for many subsequent tasks like calculation of economic capital, stress test
exercises or optimization of strategic asset allocations. Unfortunately, in many cases the current
technical capabilities do not allow full nested simulations. If we for example consider economic
capital, the full probability distribution of profits and losses is needed and hence MC simulations
”within” MC simulations are necessary. Actually that often means that at least 10’000 so called
outer scenarios and additionally not less than 1’000 inner scenarios for each outer are needed. Hence
we easily reach millions of simulations with long time horizons.
In practice there are different approaches to reduce the total number of inner scenarios, in particular
replicating portfolios [2] and least square Monte Carlo [1]. In both approaches a reduced number
of simulations is processes and an ”easy-to-valuate-proxy” is fit to the calculated information. The
obvious drawbacks are (1) potential inaccuracies especially in extreme scenarios (which are often
the most interesting ones), (2) additional work to generate the proxy and (3) additional effort to
analyze the discrepancies. If quantum computing would speed-up the simulation of inner scenarios
so that these proxies become obsolete, the impact on life and health insurers would be enormous:
Besides reducing the effort due to the above-mentioned drawbacks, increasing calculation frequency
could open the door to much more dynamic steering strategies. Import indicators like the Solvency
2 coverage ratio are for example currently calculated only a few times a year and consequently it
is hard to use it for dynamic hedging or optimization purposes.
Recently published quantum algorithms promise a quadratic speed-up for Monte Carlo valuations
of plain vanilla European options [21], basket options [23] as well as path-depended options like
Asian [21] or Barrier options [23]. I.e. the convergence rate increases from 1/√Mto 1/M where
Mis the number of samples. The underlying idea is Amplitude Estimation [3], where the target
quantity is first encoded to a quantum circuit which is then manipulated so that a subsequent
measurement delivers the desired result with high probability. Despite these promising results,
the implementation of a realistic insurance model on a real hardware quantum computer cannot
be expected in the upcoming years. Both hardware and software development are still in their
infancy and many theoretical results can only be executed in their most basic version. On the
other hand the development progresses and with the launch of IBM’s Q System One in Germany
another milestone was reached in June 2021.
In this documentation, we concentrate on quantum software development in a circuit model. In
particular we implement a payoff representing a whole life insurance and a second one simulating
3
1 INTRODUCTION
stochastic customer behavior linked to interest rates. This is a contribution to the growing library of
quantum circuits that can be used to model financial instruments or specifically insurance contracts.
Before we start the description of the insurance-related circuits, we give a brief introduction to
quantum computing in section 2 followed by a general description of the building blocks needed for
calculating expected values in section 3. Since amplitude estimation (AE) leads to the quadratic
speed-up which is the main motivation for our work, we dedicate an own section to the derivation
of AE (section 4). Then, after a mathematical formulation of the considered payoffs in section 5
we describe the new insurance-related quantum circuits in section 6. In section 7 we show results
from running parts of the dynamics lapse circuit on a simulator and on real quantum hardware.
The paper finishes with a short overview of current quantum computing technology, both for hard-
and software in section 8.
Notation
|ψi,|ϕi,|ρi,|χi,|υiQubit register in arbitrary superposition
m, n, r, s Size of qubit registers. Corresponding number of basis states is denoted
by capital letter, e.g. M= 2m
|ψimIndex outside ket gives size of the qubit register in number of qubits.
If size is 1 or not relevant in the current context, the index is usually
omitted.
|ψiiIndex inside ket indicates ith qubit in register |ψim. If omitted entire
register is meant.
|ψiimiIndex in- and outside ket indicates that |ψimshould be interpreted as
register of registers, i.e. |ψim=|ψ1im1⊗ ··· ⊗ |ψnimnwith Pmi=m.
Then |ψiimidenotes ith qubit register of |ψim.
|kim,|limQubit in kth or lth basis state. k, l ∈ {0,...,2m−1}
αk,√akQubit Amplitudes
i, tiUsually used as time associated iterators
A,B, . . . Linear operators
H, H⊗mSingle and m-qubit Hadamard operator
ϑ, θ, φ, γ Angles
H(χ, φ) Hyperplane spanned by |χiand its orthogonal space χ⊥with φdenot-
ing the rotation angle in the complex dimension
λkEigenvalues
MC Monte Carlo
PE, AE, AA Phase Estimation, Amplitude Estimation, Amplitude Amplification
QF T, QF T −1Quantum Fourier Transformation and its inverse
VHilbert Space
P V Present Value
⊕,∧xor,and
4
2 QUANTUM COMPUTING BASICS
2 Quantum Computing Basics
We will use the Quantum Circuit Model of Computation. The model is described in a finite-
dimensional complex Hilbert space V. A quantum circuit defines a sequence of operations which is
applied to qubit registers that are initialized to a certain state. A qubit register (also called qubit
system) is a list of qubits.
Before getting in the basics of quantum computing we introduce the useful Dirac Notation. Vectors
are written inside a ”ket” |ki, their dual is denoted with a ”bra” hk|. The scalar product of two
vectors |kiand |liis written as hl|ki. This is why the notation is also called ”bra-ket notation”.
Since the Hilbert space is finite-dimensional, we can choose a fixed basis and enumerate the basis
vectors instead of writing potentially large column vectors. The fixed basis is called computational
basis and can usually be associated with the canonical basis. Let for example C4be our vector
space V. The computational basis may then be defined as
1
0
0
0
,
0
1
0
0
,
0
0
1
0
,
0
0
0
1
.
In Dirac notation we would enumerate the basis vectors, either in decimal or in binary notation,
i.e.
{|0i,|1i,|2i,|3i} or {|00i,|01i,|10i,|11i},
which saves us a lot of writing in case of higher dimensions. The binary notation has the advantage,
that the vectors can also be interpreted as tensor products, for example
|01i=|0i⊗|1i=
0
1
0
0
.
We will often use a subscript to indicate the vector’s size in terms of binary digits, i.e. the dimen-
sion of the vector space spanned by |kim, k = 0,...,2m−1,is 2m.
In the following sub-sections, we will briefly describe the four postulates of quantum mechanics:
(2.1) The state space postulate, (2.2) the evolution postulate, (2.3) the measurement postulate and
(2.4) the composite systems postulate. We closely follow chapter 3 of [16].
2.1 A Quantum Bit
State Space Postulate
The state of a qubit system is described by a unit vector in a Hilbert space V.
As the smallest unit of computation, the Quantum Bit or Qubit is the quantum analogue of a
bit on a classical computer. While the classical bit always is either in state 0 or 1, the qubit can
be both at the same time. Mathematically, a qubit is a 2-dimensional Hilbert space and all unit
vectors in this space are potential states of the qubit. The state of a qubit |ψican hence be written
as linear combination of the basis vectors, i.e.
|ψi=α0|0i+α1|1i,
5
摘要：

PotentialApplicationsofQuantumComputingfortheInsuranceIndustryMichaelAdam*AXAKonzernAG„October10,2022AbstractThispaperisthedocumentationofapre-studyperformedbyAXAKonzernAGincollabo-rationwithFraunhoferITWMtoassesstherelevanceofquantumcomputingfortheinsuranceindustry.Besideageneraloverviewofthestatus...

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