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Quantum Spectral Methods for Differential Equations


Recently developed quantum algorithms address computational challenges in numerical analysis by performing linear algebra in Hilbert space. Such algorithms can produce a quantum state proportional to the solution of a d-dimensional system of linear equations or linear differential equations with complexity \({{\,\mathrm{poly}\,}}(\log d)\). While several of these algorithms approximate the solution to within \(\epsilon \) with complexity \({{\,\mathrm{poly}\,}}(\log (1/\epsilon ))\), no such algorithm was previously known for differential equations with time-dependent coefficients. Here we develop a quantum algorithm for linear ordinary differential equations based on so-called spectral methods, an alternative to finite difference methods that approximates the solution globally. Using this approach, we give a quantum algorithm for time-dependent initial and boundary value problems with complexity \({{\,\mathrm{poly}\,}}(\log d, \log (1/\epsilon ))\).

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We thank Stephen Jordan and Aaron Ostrander for valuable discussions of quantum algorithms for time-dependent linear differential equations. This work was supported in part by the Army Research Office (MURI award W911NF-16-1-0349), the Canadian Institute for Advanced Research, the National Science Foundation (Grants 1526380 and 1813814), and the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Quantum Algorithms Teams and Quantum Testbed Pathfinder programs (Grant No. DE-SC0019040).

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Communicated by M. M. Wolf


A Chebyshev Polynomials

This appendix defines the Chebyshev polynomials and presents some of their properties that are useful for our analysis.

For any \(k \in {\mathbb {N}}\), the Chebyshev polynomial of the first kind can be defined as the function

$$\begin{aligned} T_k(x)=\cos (k\arccos x),\quad x\in [-1,1]. \end{aligned}$$

It can be shown that this is a polynomial of degree k in x. For example, we have

$$\begin{aligned} T_0(x)&=1,&T_1(x)&=x,&T_2(x)&=2x^2-1,&T_3(x)&=4x^3-3x,&T_4(x)&=8x^4-8x^2+1. \end{aligned}$$

Using the trigonometric addition formula \(\cos (k+1)\theta +\cos (k-1)\theta =2\cos \theta \cos k\theta \), we have the recurrence

$$\begin{aligned} T_{k+1}(x)=2xT_k(x)-T_{k-1}(x) \end{aligned}$$

(which also provides an alternative definition of the Chebyshev polynomials, starting from the initial conditions \(T_0(x)=1\) and \(T_1(x)=x\)). We also have the bounds

$$\begin{aligned} \begin{aligned} |T_k(x)|&\le 1 \text { for } |x|\le 1,&T_k(\pm 1)&=(\pm 1)^k. \end{aligned} \end{aligned}$$

Chebyshev polynomials are orthogonal polynomials on \([-1,1]\) with the weight function \(w(x):=(1-x^2)^{-1/2}\). More concretely, defining an inner product on \(L_w^2(-1,1)\) by

$$\begin{aligned} (f,g)_w&:=\int _{-1}^1 f(x)g(x)\frac{\mathrm {d}{x}}{\sqrt{1-x^2}}, \end{aligned}$$

we have

$$\begin{aligned} (T_m,T_n)_w&=\int _0^{\pi }\cos m\theta \cos n\theta \, \mathrm {d}{\theta } \end{aligned}$$
$$\begin{aligned}&=\frac{\pi }{2}\sigma _n\delta _{m,n} \end{aligned}$$


$$\begin{aligned} \sigma _n:= {\left\{ \begin{array}{ll} 2 &{} n=0\\ 1 &{} n\ge 1. \end{array}\right. } \end{aligned}$$

It is well known from the approximation theorem of Weierstrass that \(\{T_k(x) : k \in {\mathbb {N}}\}\) is complete on the space \(L_w^2(-1,1)\). In other words, we have the following:

Lemma 7

Any function \(u\in L_w^2(-1,1)\) can be expanded by a unique Chebyshev series as

$$\begin{aligned} u(x)=\sum _{k=0}^{\infty }{\hat{c}}_kT_k(x) \end{aligned}$$

where the coefficients are

$$\begin{aligned} {\hat{c}}_k=\frac{2}{\pi }(u,T_k)_w. \end{aligned}$$

For any \(N \in {\mathbb {N}}\), we introduce the orthogonal projection \(P_N :L_w^2(-1,1)\rightarrow \mathbb {P}_N\) (where \(\mathbb {P}_N\) denotes the set of polynomials of degree at most N) by

$$\begin{aligned} P_N u(x)=\sum _{k=0}^N{\hat{c}}_kT_k(x). \end{aligned}$$

By the completeness of the Chebyshev polynomials, we have

$$\begin{aligned} (P_Nu(x),v(x))_w=(u(x),v(x))_w\quad \forall \, v\in \mathbb {P}_N \end{aligned}$$


$$\begin{aligned} \Vert P_Nu(x)-u(x)\Vert _w\rightarrow 0,\quad N\rightarrow \infty . \end{aligned}$$

Finally, we compute the Chebyshev series of \(u'(x)\) in terms of the Chebyshev series of u(x). Since \(T_k(x)=\cos k\theta \) where \(\theta =\arccos x\), we have

$$\begin{aligned} T'_k(x)=\frac{k\sin k\theta }{\sin \theta }. \end{aligned}$$


$$\begin{aligned} 2\cos k\theta =\frac{\sin (k+1)\theta }{\sin \theta }-\frac{\sin (k-1)\theta }{\sin \theta }, \end{aligned}$$

we obtain

$$\begin{aligned} 2T_k(x)=\frac{T'_{k+1}(x)}{k+1}-\frac{T'_{k-1}(x)}{k-1},\quad k\ge 2 \end{aligned}$$


$$\begin{aligned} T_1(x)=\frac{T'_2(x)}{4}. \end{aligned}$$

Since \(P_N u(x) \in \mathbb {P}_N\), the derivative of this projection should be in \(\mathbb {P}_{N-1}\). Indeed, we have

$$\begin{aligned} \begin{aligned} u'(x)&=\sum _{k=0}^{N-1}{\hat{c}}'_kT_k(x) \\&=\frac{1}{2}\sum _{k=1}^{N-1}{\hat{c}}'_k\frac{T'_{k+1}(x)}{k+1} -\frac{1}{2}\sum _{k=2}^{N-1}{\hat{c}}'_k\frac{T'_{k-1}(x)}{k-1} +{\hat{c}}'_0T_0(x)\\&=\sum _{k=2}^{N-2}({\hat{c}}'_{k-1}-{\hat{c}}'_{k+1}) \frac{T'_k(x)}{2k}-\frac{1}{2}{\hat{c}}'_{2}T'_1(x) +\frac{1}{2}{\hat{c}}'_{N-2}\frac{T'_{N-1}(x)}{N-1} +\frac{1}{2}{\hat{c}}'_{N-1}\frac{T'_n(x)}{N}+{\hat{c}}'_0T_0(x)\\&=\sum _{k=1}^N{\hat{c}}_kT'_k(x). \end{aligned} \end{aligned}$$

Comparing the coefficients of both sides, we find

$$\begin{aligned} \begin{aligned} \sigma _k{\hat{c}}'_k&={\hat{c}}'_{k+2}+2(k+1){\hat{c}}_{k+1},\quad k\in [{N}]_0\\ {\hat{c}}'_N&=0 \\ {\hat{c}}'_{N+1}&=0 \end{aligned} \end{aligned}$$

where \(\sigma _k\) is defined in (A.8).

Since \({\hat{c}}'_k=0\) for \(k\ge N\), we can calculate \({\hat{c}}'_{N-1}\) from \({\hat{c}}_N\) and then successively calculate \({\hat{c}}'_{N-2},\ldots ,{\hat{c}}'_1,{\hat{c}}'_0\). This recurrence gives

$$\begin{aligned} {\hat{c}}'_k=\frac{2}{\sigma _k}\sum _{\begin{array}{c} j=k+1 \\ j+k \text { odd} \end{array}}^Nj{\hat{c}}_j,\quad k\in [{N}]_0. \end{aligned}$$

Since \({\hat{c}}'_k\) only depends on \({\hat{c}}_j\) for \(j>k\), the transformation matrix \(D_N\) between the values \({\hat{c}}'_k\) and \({\hat{c}}_k\) for \(k \in [{N+1}]_0\) is an upper triangular matrix with all zero diagonal elements, namely

$$\begin{aligned}{}[D_N]_{kj}= {\left\{ \begin{array}{ll} \frac{2j}{\sigma _k} &{}j>k,~j+k \text { odd} \\ 0 &{} \text {otherwise}. \end{array}\right. } \end{aligned}$$

B An Example of the Quantum Spectral Method

Section 3 defines a linear system that implements the quantum spectral method for solving a system of d time-dependent differential equations. Here we present a simple example of this system for the case \(d=1\), namely

$$\begin{aligned} \frac{\mathrm {d}{x}}{\mathrm {d}{t}}=A(t)x(t)+f(t) \end{aligned}$$

where \(x(t),A(t),f(t)\in {\mathbb {C}}\), \(t\in [0,T]\), and we have the initial condition

$$\begin{aligned} x(0)=\gamma \in {\mathbb {C}}. \end{aligned}$$

In particular, we choose \(m=3\), \(n=2\), and \(p=1\) in the specification of the linear system. We divide [0, T] into \(m=3\) intervals \([0,\Gamma _1],[\Gamma _1,\Gamma _2],[\Gamma _2,T]\) with \(\Gamma _0=0, \Gamma _m=T\), and map each one onto \([-1,1]\) with the linear mapping \(K_h\) satisfying \(K_h(\Gamma _h)=1\) and \(K_h(\Gamma _{h+1})=-1\). Then we take the finite Chebyshev series of x(t) with \(n=2\) into the differential equation with interpolating nodes \(\{t_l=\cos \frac{l\pi }{n} : l \in [{2}]\} = \{0,-1\}\) to obtain a linear system. Finally, we repeat the final state \(p=1\) time to increase the success probability.

With these choices, the linear system has the form

$$\begin{aligned} L= \begin{pmatrix} L_1+L_2(A_0) &{} &{} &{} &{} \\ L_3 &{} L_1+L_2(A_1) &{} &{} &{} \\ &{} L_3 &{} L_1+L_2(A_2) &{} &{} \\ &{} &{} L_3 &{} L_4 &{} \\ &{} &{} &{} L_5 &{} L_4 \\ \end{pmatrix} \end{aligned}$$


$$\begin{aligned} L_1&=|0\rangle \langle 0|P_n+\sum _{l=1}^n|l\rangle \langle l|P_nD_n= \begin{pmatrix} 1 &{} 1 &{} 1 \\ 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} -4 \\ \end{pmatrix} \end{aligned}$$
$$\begin{aligned} L_2(A_h)&=-\sum _{l=1}^nA_h(t_l)\otimes |l\rangle \langle l|P_n=- \begin{pmatrix} 0 &{} 0 &{} 0 \\ A_h(0) &{} 0 &{} -A_h(0) \\ A_h(-1) &{} -A_h(-1) &{} A_h(-1) \\ \end{pmatrix} \end{aligned}$$
$$\begin{aligned} L_3&=\sum _{i=0}^{d}\sum _{k=0}^n(-1)^k|i0\rangle \langle ik|= \begin{pmatrix} 1 &{} -1 &{} 1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{pmatrix} \end{aligned}$$
$$\begin{aligned} L_4&=-\sum _{i=0}^{d}\sum _{l=1}^n|il\rangle \langle il-1|+\sum _{i=0}^{d}\sum _{l=0}^n|il\rangle \langle il|= \begin{pmatrix} 1 &{} 0 &{} 0 \\ -1 &{} 1 &{} 0 \\ 0 &{} -1 &{} 1 \\ \end{pmatrix} \end{aligned}$$
$$\begin{aligned} L_5&=-\sum _{i=0}^{d}|i0\rangle \langle in|= \begin{pmatrix} 0 &{} 0 &{} -1 \\ 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ \end{pmatrix}. \end{aligned}$$

The vector \(|X\rangle \) has the form

$$\begin{aligned} |X\rangle = \begin{pmatrix} c_0(\Gamma _1) \\ c_1(\Gamma _1) \\ c_2(\Gamma _1) \\ c_0(\Gamma _2) \\ c_1(\Gamma _2) \\ c_2(\Gamma _2) \\ c_0(\Gamma _3) \\ c_1(\Gamma _3) \\ c_2(\Gamma _3) \\ x \\ x \\ x \\ x \\ x \\ x \\ \end{pmatrix} \end{aligned}$$

where \(c_l(\Gamma _{h+1})\) are the Chebyshev series coefficients of \(x(\Gamma _{h+1})\) and x is the final state \(x(\Gamma _m)=x(-1)\).

Finally, the vector \(|B\rangle \) has the form

$$\begin{aligned} |B\rangle = \begin{pmatrix} \gamma \\ f_0(0) \\ f_0(-1) \\ 0 \\ f_1(0) \\ f_1(-1) \\ 0 \\ f_2(0) \\ f_2(-1) \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{pmatrix} \end{aligned}$$

where \(\gamma \) comes from the initial condition and \(f_{h}(\cos \frac{l\pi }{n})\) is the value of \(f_h\) at the interpolation point \(t_l=\cos \frac{l\pi }{n} \in \{0,-1\}\).

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Childs, A.M., Liu, JP. Quantum Spectral Methods for Differential Equations. Commun. Math. Phys. 375, 1427–1457 (2020).

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