Now, let us present the method for Bermudan option pricing by Chebyshev interpolation and QAE.
Assumptions
We begin with making some assumptions necessary to execute the proposed method. The first one is as follows.
Assumption 1
We are given the access to the oracle \(O_{{\mathrm{step}},k}\), which generates the state corresponding to the probability distribution of \(\vec{S}_{k+1}\) conditional on \(\vec{S}_{k}\). That is, for every \(k\in [K1]_{0}\) and \(\vec{S}\in \mathcal{S}\),
$$ O_{{\mathrm{step}},k}: \vec{S} \rangle  0 \rangle \mapsto \sum _{\vec{s}\in \widetilde{\mathcal{S}}_{k+1}(\vec{S})}\sqrt{p_{k+1}(\vec{s};\vec{S})}  \vec{S} \rangle  \vec{s} \rangle , $$
(38)
where \(\widetilde{\mathcal{S}}_{k+1}(\vec{S})\) is the set of possible values of \(\vec{S}_{k+1}\) under the condition that \(\vec{S}_{k}=\vec{S}\), and
$$ p_{k+1}(\vec{s};\vec{S}):=\mathbb{P} (\vec{S}_{k+1}= \vec{s} \mid \vec{S}_{k}=\vec{S} ). $$
(39)
We here make comments on how to implement \(O_{{\mathrm{step}},k}\). As mentioned in Sect. 3, usually, following some SDE and some numerical method such as EulerMaruyama, we can generate random sample values of \(\vec{S}_{k+1}\) with the given value of \(\vec{S}_{k}\) as the initial condition. Implementations of such a calculation on quantum circuits have been discussed in the previous papers [7, 9, 13]. That is, we can prepare the states corresponding to some (discretely approximated) random variables (e.g. standard normal) on the other registers, and, using them at discretized time steps, generate the path of \(\vec{S}(t)\) from \(t_{k}\) to \(t_{k+1}\). This yields the state like (38). We should also note that, in Assumption 1, it is assumed that \(\vec{S}_{k+1}\) can take only a finite number of values for the fixed \(\vec{S}_{k}\). This is not the case in the most models of \(\vec{S}(t)\), in which it takes continuous values. However, under the aforementioned implementations for time evolution of \(\vec{S}(t)\), in which both time and random variables are discretely approximated, the number of the possible values of \(\vec{S}_{k}\) necessarily becomes finite.
Hereafter, we are mainly interested in the number of calls to \(O_{{\mathrm{step}},k}\) in calculating the option price as a measure of complexity, since calculation for time evolution of underlying asset prices is typically the most timeconsuming part in option pricing.
The second assumption is as follows. Here, \(\mathcal{I}_{\mathcal{A}}\) denotes the set of all realvalued functions on a given subset \(\mathcal{A}\subseteq \mathbb{R}^{d}\).
Assumption 2
For every \(k\in [K1]\), we are given the following

the hyperrectangle \(\mathcal{D}_{k}:=[L_{1,k},U_{1,k}]\times \cdots\times [L_{d,k},U_{d,k}] \subseteq \mathcal{S}\), with \(L_{1,k},\ldots,L_{d,k},U_{1,k},\ldots,U_{d,k}\in \mathbb{R}\) satisfying \(L_{1,k}< U_{1,k},\ldots , L_{d,k}< U_{d,k}\),

\(V^{\mathrm{OB}}_{k}\in \mathcal{I}_{\mathcal{S}\setminus \mathcal{D}_{k}}\)
such that the following (i) and (ii) are satisfied.

(i)
There exists \(\epsilon ^{\mathrm{OB}}_{k}\in \mathbb{R}_{+}\) such that either
$$ \bigl\vert V^{\mathrm{OB}}_{k}(\vec{s})V_{k}( \vec{s}) \bigr\vert < \epsilon ^{\mathrm{OB}}_{k} $$
(40)
or
$$ \bigl\vert \mathbf{F}_{k}[V_{k}]( \vec{s})V_{k}(\vec{s}) \bigr\vert < \epsilon ^{\mathrm{OB}}_{k} $$
(41)
is satisfied for any \(\vec{s}\in \mathcal{S}\setminus \mathcal{D}_{k}\). Here, \(\mathbf{F}_{k}[\cdot ]\) is the ‘flat extrapolation operator’ defined as
$$\begin{aligned}& \mathbf{F}_{k}[F](\vec{s}) := F\bigl(b_{k}( \vec{s})\bigr), \end{aligned}$$
(42)
$$\begin{aligned}& b_{k}(\vec{s}) := \bigl(\min \bigl\{ U_{1,k},\max \{L_{1,k},s_{1}\}\bigr\} ,\ldots, \min \bigl\{ U_{d,k},\max \{L_{d,k},s_{d}\}\bigr\} \bigr)^{T} \end{aligned}$$
(43)
for any \(F\in \mathcal{I}_{\mathcal{D}_{k}}\) and \(\vec{s}=(s_{1},\ldots,s_{d})^{T}\in \mathcal{S}\).

(ii)
If, for some \(G\in \mathcal{I}_{\mathcal{D}_{k}}\), we have the access to the oracle \(O_{G}\) such that
$$ O_{G} \vec{s} \rangle  0 \rangle = \vec{s} \rangle \bigl G(\vec{s}) \bigr\rangle $$
(44)
for any \(\vec{s}\in \mathcal{D}_{k}\), we also have the access to the oracle \(\widetilde{O}_{G}\), which acts as
$$ \widetilde{O}_{G} \vec{s} \rangle  0 \rangle = \vec{s} \rangle \bigl \mathbf{G}_{k}[G](\vec{s}) \bigr\rangle . $$
(45)
Here, \(\mathbf{G}_{k}[\cdot ]\) is defined as
$$ \mathbf{G}_{k}[H](\vec{s}) := \textstyle\begin{cases} V^{\mathrm{OB}}_{k}(\vec{s}) ;& \text{if } \vec{s}\in \mathcal{A}_{k}, \\ \mathbf{F}_{k}[H](\vec{s}) ;& \text{otherwise} \end{cases} $$
(46)
for any \(H\in \mathcal{I}_{\mathcal{D}_{k}}\) and \(\vec{s}\in \mathcal{S}\), where \(\mathcal{A}_{k}\) is a subset of \(\mathcal{S}\setminus \mathcal{D}_{k}\) such that (40) and (41) hold for any \(\vec{s}\in \mathcal{A}_{k}\) and any \(\vec{s}\in (\mathcal{S}\setminus \mathcal{D}_{k})\setminus \mathcal{A}_{k}\), respectively.
We also define \(\mathbf{G}_{K}[H](\vec{s}) := H(\vec{s})\) for any \(H\in \mathcal{I}_{\mathcal{S}}\) and \(\vec{s}\in \mathcal{S}\).
Roughly speaking, this assumption means that, when some of underlying asset prices are extremely large or small, we can approximate the option value \(V_{k}\) by some known and easily computable function \(V^{\mathrm{OB}}_{k}\) or the flat extrapolation of \(V_{k}\) from moderate underlying asset prices. Postponing explanation on why this assumption is necessary to Sect. 5.2, we here see that it is actually satisfied in some typical settings in option pricing. For example, let us consider a basket put option, whose payoff function is \(f^{\mathrm{pay}}_{k}((s_{1},\ldots,s_{d})^{T})=\max \{\kappa s_{1}\cdotss_{d},0 \}\) with some \(\kappa \in \mathbb{R}\) for every \(k\in [K]\), under some model in which \(S_{1}(t),\ldots, S_{d}(t)\) are unbounded from above but bounded from below, say, by 0, as the BlackScholes model. Then, in each of the following situations, (40) or (41) holds.

If some of \(S_{1,k},\ldots, S_{d,k}\) are extremely large, the option is far outofmoney, and therefore its price is almost 0.

If some of \(S_{1,k},\ldots, S_{d,k}\) are smaller than the sufficiently small thresholds \(L_{1,k},\ldots,L_{d,k}\in \mathbb{R}_{+}\) respectively, but the others are not, setting the former to the thresholds hardly affects the option price.

If all of \(S_{1,k},\ldots, S_{d,k}\) are sufficiently close to 0, the option is exercised at \(t_{k}\), and therefore \(V_{k}(\vec{S}_{k})=f^{\mathrm{pay}}_{k}(\vec{S}_{k})\).
Thirdly, we make the following assumption, which is necessary for bounding the interpolation error in the proposed method.
Assumption 3
For every \(k\in [K1]\), \(Q_{k}(\vec{S})\) has an analytic extension to \(\mathcal{B}_{\mathcal{D}_{k},\rho _{k}}\), where \(\mathcal{D}_{k}\) is given in Assumption 2 and \(\rho _{k}\) is some real number greater than 1, and
$$ \sup_{\vec{s}\in \mathcal{B}_{\mathcal{D}_{k},\rho _{k}}} \bigl\vert Q_{k}( \vec{S}) \bigr\vert \le B_{k} $$
(47)
holds, where \(B_{k}\) is some positive real number.
The proposed method
Under these assumptions, we can construct the procedure for Bermudan option pricing based on QAE and Chebyshev interpolation. This is also a backward calculation similarly to LSM; we sequentially calculate the approximate continuation value \(\widetilde{Q}_{k}\) and option price \(\widetilde{V}_{k}\) at \(t_{k}\), going from the final maturity to the present. Roughly, the outline is as follows. As preparation, for every \(k\in [K1]\), we set \(m_{k}\in \mathbb{N}\), the degree of Chebyshev polynomials used for the approximation, and the hyperrectangle \(\mathcal{D}_{k}=[L_{1,k},U_{1,k}]\times \cdots \times [L_{d,k},U_{d,k}] \subseteq \mathcal{S}\). We begin the iterative calculation by setting \(\widetilde{V}_{K}(\vec{S}):=f^{\mathrm{pay}}_{K}(\vec{S})\) for every \(\vec{S}\in \mathcal{S}\). Then, for \(k\in [K1]\), given \(\widetilde{V}_{k+1}\), we estimate the expected value of \(\widetilde{V}_{k+1}(\vec{S}_{k+1})\) under the condition that \(\vec{S}_{k}=\vec{S}^{\mathcal{D}_{k},m_{k}}_{\vec{j}}\) for every Chebyshev node \(\vec{S}^{\mathcal{D}_{k},m_{k}}_{\vec{j}}\) by QAE, and denote the estimation as \(\widehat{Q}_{k,\vec{j}}^{\mathrm{QAE}}\). Using these, we construct \(\widetilde{Q}_{k}\), the Chebyshev interpolation of the approximate continuation value, and set \(\widetilde{V}_{k}(\vec{S})=\max \{\widetilde{Q}_{k}(\vec{S}),f^{\mathrm{pay}}_{k}( \vec{S})\}\) for every \(\vec{S}\in \mathcal{S}\). We repeat these steps until we reach \(k=1\). Finally, we estimate the expected value of \(\widetilde{V}_{1}(\vec{S}_{1})\) by QAE again, and let the result be an approximation of \(V_{0}\).
The fully detailed procedure is shown in Algorithm 1.
Some additional explanations should be made. The first one is on \( \Psi _{k,\vec{j}} \rangle \) in Step 4. For every \(k\in [K1]\) and \(\vec{j}\in \mathcal{J}_{k}\), given the approximation \(\widetilde{V}_{k+1}\in \mathcal{I}_{\mathcal{D}_{k+1}}\) of \(V_{k+1}\), we generate the state \( \Psi _{k,\vec{l}} \rangle \) on the appropriate multiregister system with the last one being singlequbit, by the following operation:
$$\begin{aligned}&  0 \rangle  0 \rangle  0 \rangle  0 \rangle \\& \quad \rightarrow \bigl \vec{S}^{\mathcal{D}_{k},m_{k}}_{\vec{j}} \bigr\rangle  0 \rangle  0 \rangle  0 \rangle \\& \quad \rightarrow \bigl \vec{S}^{\mathcal{D}_{k},m_{k}}_{\vec{j}} \bigr\rangle \sum _{ \vec{s}\in \widetilde{\mathcal{S}}_{k+1} (\vec{S}^{\mathcal{D}_{k},m_{k}}_{ \vec{j}} )} \sqrt{p_{k+1} \bigl(\vec{s}; \vec{S}^{\mathcal{D}_{k},m_{k}}_{ \vec{j}} \bigr)} \vec{s} \rangle  0 \rangle  0 \rangle \\& \quad \rightarrow \bigl \vec{S}^{\mathcal{D}_{k},m_{k}}_{\vec{j}} \bigr\rangle \sum _{ \vec{s}\in \widetilde{\mathcal{S}}_{k+1} (\vec{S}^{\mathcal{D}_{k},m_{k}}_{ \vec{j}} )} \sqrt{p_{k+1} \bigl(\vec{s}; \vec{S}^{\mathcal{D}_{k},m_{k}}_{ \vec{j}} \bigr)} \vec{s} \rangle \bigl \mathbf{G}_{k+1}[\widetilde{V}_{k+1}](\vec{s}) \bigr\rangle  0 \rangle \\& \quad \rightarrow \bigl \vec{S}^{\mathcal{D}_{k},m_{k}}_{\vec{j}} \bigr\rangle \sum _{ \vec{s}\in \widetilde{\mathcal{S}}_{k+1} (\vec{S}^{\mathcal{D}_{k},m_{k}}_{ \vec{j}} )} \sqrt{p_{k+1} \bigl(\vec{s}; \vec{S}^{\mathcal{D}_{k},m_{k}}_{ \vec{j}} \bigr)} \vec{s} \rangle \bigl \mathbf{G}_{k+1}[\widetilde{V}_{k+1}](\vec{s}) \bigr\rangle \\& \qquad {} \otimes \biggl(\sqrt{\frac{1}{2}+ \frac{\mathbf{G}_{k+1}[\widetilde{V}_{k+1}](\vec{s})}{2\widetilde{V}_{k+1}^{\mathrm{max}}}}  1 \rangle + \sqrt{\frac{1}{2} \frac{\mathbf{G}_{k+1}[\widetilde{V}_{k+1}](\vec{s})}{2\widetilde{V}_{k+1}^{\mathrm{max}}}}  0 \rangle \biggr) \\& \quad =:  \Psi _{k,\vec{j}} \rangle , \end{aligned}$$
(50)
where \(O_{{\mathrm{step}},k}\) in Assumption 1 and \(\widetilde{O}_{\widetilde{V}_{k+1}}\) in Assumption 2 are used at the second and third arrows, respectively. Note that the probability to obtain 1 on the last qubit in measuring \( \Psi _{k,\vec{l}} \rangle \) is
$$ P_{k,\vec{j}}=\frac{1}{2}+ \frac{\widehat{Q}_{k} (\vec{S}^{\mathcal{D}_{k},m_{k}}_{\vec{j}} )}{2\widetilde{V}_{k+1}^{\mathrm{max}}}, $$
(51)
where
$$\begin{aligned} \widehat{Q}_{k} (\vec{S} ) :=&\mathbb{E} \bigl[ \mathbf{G}_{k+1}[ \widetilde{V}_{k+1}]( \vec{S}_{k+1}) \mid \vec{S}_{k}=\vec{S} \bigr] \\ =&\sum_{\vec{s}\in \widetilde{\mathcal{S}}_{k+1} (\vec{S} )} p_{k+1} (\vec{s}; \vec{S} )\mathbf{G}_{k+1}[ \widetilde{V}_{k+1}](\vec{s}). \end{aligned}$$
(52)
Therefore, as long as \(\mathbf{G}_{k+1}[\widetilde{V}_{k+1}]\) is close to \(V_{k+1}\), \((2P_{k,\vec{j}}1)\widetilde{V}^{\mathrm{max}}_{k+1}=\widehat{Q}_{k} (\vec{S}^{\mathcal{D}_{k},m_{k}}_{\vec{j}} )\) is close to \(Q_{k} (\vec{S}^{\mathcal{D}_{k},m_{k}}_{\vec{j}} )\). This is why we can obtain approximations of the continuation values at Chebyshev nodes by Step 4, with the errors from QAEs being also small.
Second, let us explain the state \( \Psi _{0} \rangle \) in Step 9. Given \(\widetilde{V}_{1}\), we can generate \( \Psi _{0} \rangle \) similarly to \( \Psi _{k,\vec{j}} \rangle \) as
$$\begin{aligned}&  0 \rangle  0 \rangle  0 \rangle  0 \rangle \\& \quad \rightarrow  \vec{S}_{0} \rangle  0 \rangle  0 \rangle  0 \rangle \\& \quad \rightarrow  \vec{S}_{0} \rangle \sum_{\vec{s}\in \widetilde{\mathcal{S}}_{1}(\vec{S}_{0})} \sqrt{p_{1}(\vec{s};\vec{S}_{0})}  \vec{s} \rangle  0 \rangle  0 \rangle \\& \quad \rightarrow  \vec{S}_{0} \rangle \sum_{\vec{s}\in \widetilde{\mathcal{S}}_{1}(\vec{S}_{0})} \sqrt{p_{1}(\vec{s};\vec{S}_{0})}  \vec{s} \rangle \bigl \mathbf{G}_{1}[\widetilde{V}_{1}](\vec{s}) \bigr\rangle  0 \rangle \\& \quad \rightarrow  \vec{S}_{0} \rangle \sum_{\vec{s}\in \widetilde{\mathcal{S}}_{1}(\vec{S}_{0})} \sqrt{p_{1}(\vec{s};\vec{S}_{0})}  \vec{s} \rangle \bigl \mathbf{G}_{1}[\widetilde{V}_{1}](\vec{s}) \bigr\rangle \\& \qquad {} \otimes \biggl(\sqrt{\frac{1}{2}+ \frac{\mathbf{G}_{1}[\widetilde{V}_{1}](\vec{s})}{2\widetilde{V}_{1}^{\mathrm{max}}}}  1 \rangle + \sqrt{\frac{1}{2} \frac{\mathbf{G}_{1}[\widetilde{V}_{1}](\vec{s})}{2\widetilde{V}_{1}^{\mathrm{max}}}}  0 \rangle \biggr) \\& \quad =:  \Psi _{0} \rangle , \end{aligned}$$
(53)
where the last ket corresponds to a singlequbit register. Since the probability \(P_{0}\) to obtain 1 on the last qubit in measuring \( \Psi _{0} \rangle \) satisfies
$$ (2P_{0}1)\widetilde{V}^{\mathrm{max}}_{1}= \widehat{V}_{0}, $$
(54)
where
$$ \widehat{V}_{0}:=\mathbb{E} \bigl[\mathbf{G}_{1}[ \widetilde{V}_{1}]( \vec{S}_{1}) \bigr]=\sum _{\vec{s}\in \widetilde{\mathcal{S}}_{1}( \vec{S}_{0})} p_{1}(\vec{s};\vec{S}_{0}) \mathbf{G}_{1}[\widetilde{V}_{1}]( \vec{s}), $$
(55)
we can obtain an approximation of \(V_{0}\) by Step 9, as long as \(\mathbf{G}_{1}[\widetilde{V}_{1}]\) is close to \(V_{1}\) and the QAE error is small.
Lastly, let us comment on the reason why Assumption 2 is necessary. This is because we have to handle underlying asset prices out of \(\mathcal{D}_{k+1}\) in Steps 4 and 9, or, more specifically, in generating \( \Psi _{k,\vec{j}} \rangle \) and \( \Psi _{0} \rangle \). In fact, when we generate \( \Psi _{k,\vec{j}} \rangle \), \(\vec{S}_{k+1}\) can be out of \(\mathcal{D}_{k+1}\) with some probability. In particular, when \( \Psi _{k,\vec{j}} \rangle \) corresponds to a Chebyshev node \(\vec{S}^{\mathcal{D}_{k},m_{k}}_{\vec{j}}\) close to the boundary of \(\mathcal{D}_{k}\), or, in other words, the condition that \(\vec{S}_{k}\) is close to the boundary of \(\mathcal{D}_{k}\) is imposed, such a probability becomes nonnegligible.
Evaluation of the error
Then, let us consider the error on the present option price in the proposed method. First, we have the following theorem.
Theorem 3
Under Assumptions 1to 3, consider Algorithm 1. Suppose that, for every \(k\in [K1]\) and \(\vec{j}\in \mathcal{J}_{k}\),
$$ \bigl\vert \widehat{Q}_{k} \bigl(\vec{S}^{\mathcal{D}_{k},m_{k}}_{\vec{j}} \bigr)\widehat{Q}^{\mathrm{QAE}}_{k,\vec{j}} \bigr\vert \le \epsilon ^{\mathrm{QAE}}_{k} $$
(56)
is satisfied, where \(\epsilon ^{\mathrm{QAE}}_{k}\) is some positive real number. Moreover, suppose that
$$ \vert \widehat{V}_{0}\widetilde{V}_{0} \vert \le \epsilon ^{\mathrm{QAE}}_{0} $$
(57)
is satisfied for some \(\epsilon ^{\mathrm{QAE}}_{0}\in \mathbb{R}_{+}\). Then,
$$ \vert V_{0}\widetilde{V}_{0} \vert \le \sum _{k=1}^{K1} \widetilde{\Lambda }_{1,k1} \epsilon ^{\mathrm{int}}_{k} + \sum _{k=1}^{K1} \widetilde{\Lambda }_{1,k1} \epsilon ^{\mathrm{OB}}_{k}+ \sum _{k=0}^{K1} \widetilde{\Lambda }_{1,k} \epsilon ^{\mathrm{QAE}}_{k} $$
(58)
holds, where, for \(k\in [K1]\) and \(k^{\prime }\in [K1]_{0}\),
$$ \epsilon ^{\mathrm{int}}_{k} := \epsilon _{\mathrm{int}}(\rho _{k},d,m_{k},B_{k}) $$
(59)
and
$$\begin{aligned}& \widetilde{\Lambda }_{k,k^{\prime }}:= \textstyle\begin{cases} \prod_{i=k}^{k^{\prime }}\Lambda _{i} ;& \textit{if } k\le k^{\prime }\\ 1 ;& \textit{otherwise} \end{cases}\displaystyle \end{aligned}$$
(60)
$$\begin{aligned}& \Lambda _{k} := \biggl(\frac{2}{\pi }\log (m_{k}+1)+1 \biggr)^{d}. \end{aligned}$$
(61)
The proof is presented in Appendix A.1.
Complexity
Based on Theorem 3, we can evaluate the complexity, that is, the number of calls to \(O_{{\mathrm{step}},k}\) sufficient to achieve the desired level of the error on the present option price.
Corollary 1
Let ϵ be a real number in \((0,0.1)\). Under Assumptions 1to 3, consider Algorithm 1 with the following parameters:

(i)
\(m_{k},k\in [K1]\) satisfying \(m_{k}\ge m^{\mathrm{th}}_{k}\) with
$$ \begin{aligned} &m^{\mathrm{th}}_{1} = \biggl\lceil \frac{1}{\log \rho _{1}}\log \biggl( \frac{2^{d/2+2}\sqrt{d}(K1)(1\rho _{1}^{2})^{d/2}B_{1}}{\epsilon \widetilde{V}^{\mathrm{max}}_{1}} \biggr) \biggr\rceil \\ &m^{\mathrm{th}}_{k} = \biggl\lceil \frac{1}{\log \rho _{k}}\log \biggl( \frac{2^{d/2+2}\sqrt{d}(K1)(1\rho _{k}^{2})^{d/2}\widetilde{\Lambda }^{\mathrm{th}}_{1,k1}B_{k}}{\epsilon \widetilde{V}^{\mathrm{max}}_{1}} \biggr) \biggr\rceil \\ & \quad \textit{for } k=2,\ldots,K1, \end{aligned} $$
(62)
where \(\widetilde{\Lambda }^{\mathrm{th}}_{1,k1}\) is determined as \(\widetilde{\Lambda }_{1,k1}\) in (60) with \(m_{1}=m^{\mathrm{th}}_{1},\ldots , m_{k1}=m^{\mathrm{th}}_{k1}\).

(ii)
\(N^{\mathrm{QAE}}_{k},k\in [K1]_{0}\) set as
$$ N^{\mathrm{QAE}}_{k} = \biggl\lceil \frac{7}{\bar{\epsilon }_{k}} \biggr\rceil . $$
(63)
Here, \(\bar{\epsilon }_{0},\ldots, \bar{\epsilon }_{K1}\) are given by
$$ \begin{aligned} &\bar{\epsilon }_{0} = \frac{1}{1+\sum_{k^{\prime }=1}^{K1}\sqrt{(m_{k^{\prime }}+1)^{d}\widetilde{\Lambda }_{1,k^{\prime }}}} \cdot \frac{\epsilon }{4} \\ &\bar{\epsilon }_{k} = \frac{\sqrt{(m_{k}+1)^{d} /\widetilde{\Lambda }_{1,k}}}{1+\sum_{k^{\prime }=1}^{K1}\sqrt{(m_{k^{\prime }}+1)^{d}\widetilde{\Lambda }_{1,k^{\prime }}}} \cdot \frac{\widetilde{V}^{\mathrm{max}}_{1} \epsilon }{4\widetilde{V}^{\mathrm{max}}_{k}} \quad \textit{for } k=1,\ldots,K1, \end{aligned} $$
(64)
where \(m_{0},\ldots, m_{K1}\) are set as (i) and \(\widetilde{\Lambda }_{1,1},\ldots, \widetilde{\Lambda }_{1,K1}\) are given as (60) with such \(m_{0},\ldots, m_{K1}\).

(iii)
\(N^{\mathrm{rep}}_{k}\) set to
$$ N_{\mathrm{rep}} := 12 \biggl\lceil \log \biggl(\frac{N_{\mathrm{est}}}{0.01} \biggr) \biggr\rceil + 1, $$
(65)
for every \(k\in [K1]_{0}\). Here, \(N_{\mathrm{est}}:=1+\sum_{k^{\prime }=1}^{K1}(m_{k^{\prime }}+1)^{d}\) with \(\{m_{k}\}\) set as (i).
Moreover, suppose that \(\epsilon ^{\mathrm{OB}}_{1},\ldots, \epsilon ^{\mathrm{OB}}_{K1}\) are 0. Then, Algorithm 1 outputs \(\widetilde{V}_{0}\) satisfying \(V_{0}\widetilde{V}_{0}\le \epsilon \widetilde{V}^{\mathrm{max}}_{1}\) with probability higher than 0.99.
The proof is presented in Appendix A.2.
We here explain why the parameters are set as above. As we see in the proof in Appendix A.2, \(m_{1},\ldots, m_{K1}\) satisfying (62) make the first term in the RHS in (58) smaller than \(\epsilon \widetilde{V}^{\mathrm{max}}_{1}/2\). Then, for such \(\{m_{k}\}_{k=1,\ldots,K1}\), \(\{N^{\mathrm{QAE}}_{k}\}_{k=0,\ldots,K1}\) are determined as (63) so that
$$ \frac{N_{\mathrm{tot}}}{N_{\mathrm{rep}}}=N^{\mathrm{QAE}}_{0}+\sum _{k=1}^{K1}(m_{k}+1)^{d}N^{ \mathrm{QAE}}_{k}, $$
(66)
that is, the total number \(N_{\mathrm{tot}}\) of calls to \(\{O_{{\mathrm{step}},k}\}_{k=0,\ldots,K1}\) divided by the QAE repetition number \(N^{\mathrm{rep}}\), is minimized under the constraint that, if all the QAEs in Algorithm 1 succeed, the third term in the RHS in (58) is smaller than \(\epsilon \widetilde{V}^{\mathrm{max}}_{1}/2\). Finally, \(\{N^{\mathrm{rep}}_{k}\}_{k=0,\ldots,K1}\) are determined so that the probability that these QAEs all succeed becomes higher than \(0.99=10.01\). In total, Algorithm 1 with the setting in Corollary 1 gives an approximation of \(V_{0}\) with an error at most \(\epsilon \widetilde{V}^{\mathrm{max}}_{1}\) with probability higher than 0.99.
Note that, in reality, it is difficult to set \(m_{k}\) to \(m^{\mathrm{th}}_{k}\), since \(\rho _{k}\) and \(B_{k}\) are usually unknown. In practice, we might set them to some conservatively large values, based on, for example, the calculation results of some benchmark pricing problems for various \(\{m_{k}\}_{k=1,\ldots,K1}\). Besides, note that, in the above setting, the half of the error tolerance ϵ is assigned to the interpolation error and another half is assigned to the QAE error. Although we can of course change this assignment ratio, it affects the complexity only logarithmically, since the sufficient levels of \(\{m_{k}\}_{k=1,\ldots,K1}\) are logarithmically affected by such a change and so are \(\{N^{\mathrm{rep}}_{k}\}\) compensating the change of \(\{m_{k}\}\).
Let us consider the dependency of the total complexity on the error tolerance ϵ. We see that
$$\begin{aligned}& m_{k} = O \bigl(\log \bigl(\epsilon ^{1}\bigr){ \mathrm{polyloglog}}\bigl(\epsilon ^{1} \bigr) \bigr), \end{aligned}$$
(67)
$$\begin{aligned}& N^{\mathrm{QAE}}_{k} = O \bigl(\epsilon ^{1}\times { \mathrm{polyloglog}}\bigl( \epsilon ^{1}\bigr) \bigr) \end{aligned}$$
(68)
for every \(k\in [K1]\), and that
$$ N^{\mathrm{QAE}}_{0}= O \bigl(\epsilon ^{1}\log ^{d/2}\bigl(\epsilon ^{1}\bigr)\operatorname{polyloglog} \bigl(\epsilon ^{1}\bigr) \bigr), $$
(69)
where \(\operatorname{polyloglog}(\cdot )\) means \(\operatorname{polylog} (\log (\cdot ) )\). Combining these with (66), we obtain
$$ N_{\mathrm{tot}} = O \bigl(\epsilon ^{1}\log ^{d} \bigl(\epsilon ^{1}\bigr)\operatorname{polyloglog}\bigl( \epsilon ^{1}\bigr) \bigr), $$
(70)
which eventually beats LSM’s complexity \(\widetilde{O}(\epsilon ^{2})\) for small ϵ.