# Quantum linear system solver based on time-optimal adiabatic quantum computing and quantum approximate optimization algorithm

We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can solve a quantum linear system problem (QLSP) with O(κ/ϵ) runtime, where κ is the condition number, and ϵ is the target accuracy. This achieves the optimal time complexity with respect to κ. The success of the time-optimal AQC implies that the quantum approximate optimization algorithm (QAOA) can also achieve the O(κ) complexity with respect to κ. Our method is applicable to general non-Hermitian matrices (possibly dense), but the efficiency can be improved when restricted to Hermitian matrices, and further to Hermitian positive definite matrices. Numerical results indicate that QAOA can yield the lowest runtime compared to the time-optimal AQC, vanilla AQC, and the recently proposed randomization method. The runtime of QAOA is observed numerically to be only O(κpoly(log(1/ϵ))).

## Authors

• 9 publications
• 36 publications
• ### Solving quantum linear system problem with near-optimal complexity

We present a simple algorithm to solve the quantum linear system problem...
10/31/2019 ∙ by Lin Lin, et al. ∙ 0

• ### Quantum Data Fitting Algorithm for Non-sparse Matrices

We propose a quantum data fitting algorithm for non-sparse matrices, whi...
07/16/2019 ∙ by Guangxi Li, et al. ∙ 0

• ### A Quantum IP Predictor-Corrector Algorithm for Linear Programming

We introduce a new quantum optimization algorithm for Linear Programming...
02/18/2019 ∙ by P. A. M. Casares, et al. ∙ 0

• ### A Quantum Interior Point Method for LPs and SDPs

We present a quantum interior point method with worst case running time ...
08/28/2018 ∙ by Iordanis Kerenidis, et al. ∙ 0

• ### Quantum Algorithm for Optimization and Polynomial System Solving over Finite Field and Application to Cryptanalysis

In this paper, we give quantum algorithms for two fundamental computatio...
02/12/2018 ∙ by Yu-Ao Chen, et al. ∙ 0

• ### Quantum message-passing algorithm for optimal and efficient decoding

Recently, one of us proposed a quantum algorithm called belief propagati...
09/16/2021 ∙ by Christophe Piveteau, et al. ∙ 0

• ### Quantum algorithms for spectral sums

We propose and analyze new quantum algorithms for estimating the most co...
11/12/2020 ∙ by Alessandro Luongo, et al. ∙ 0

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## I The gap of H(f(s)) for the Hermitian positive definite A

The Hamiltonian can be written in the block matrix form as

 H(f)=(0((1−f)I+fA)QbQb((1−f)I+fA)0). (S1)

Let be an eigenvalue of , then

 0 =det(λI−((1−f)I+fA)Qb−Qb((1−f)I+fA)λI) =det(λ2I−((1−f)I+fA)Q2b((1−f)I+fA))

where the second equality holds because the bottom two blocks are commutable. Thus is an eigenvalue of , and equals the smallest non-zero eigenvalue of . Applying a proposition of matrices that and have the same non-zero eigenvalues, also equals the smallest non-zero eigenvalue of .

Now we focus on the matrix . Note that is the unique eigenstate corresponding to the eigenvalue 0, all eigenstates corresponding to non-zero eigenvalues must be orthogonal with . Therefore

 Δ2(f) =inf⟨b|φ⟩=0,⟨φ|φ⟩=1⟨φ∣∣Qb((1−f)I+fA)2Qb∣∣φ⟩ =inf⟨b|φ⟩=0,⟨φ|φ⟩=1⟨φ∣∣((1−f)I+fA)2∣∣φ⟩ ≥inf⟨φ|φ⟩=1⟨φ∣∣((1−f)I+fA)2∣∣φ⟩ =(1−f+f/κ)2,

and .

## Ii Relations among Different Measurements of Accuracy

The quantum adiabatic theorem (Jansen et al., 2007, Theorem 3) states that for any ,

We will show that also serves as an error bound for the density distance and bounds the fidelity from below.

Note that is the eigenstate for both and corresponding the 0 eigenvalue, we have , and thus . Together with the initial condition , the overlap of and remains to be 0 for the whole time period, i.e.  Since is a rank-2 projector, we have . Therefore the error used in the adiabatic theorem becomes

Since is exactly the fidelity , the fidelity can be bounded from below by .

Furthermore, by using , the distance between and can be bounded by the error of the fidelity as

 ∥|ψT(s)⟩⟨ψT(s)|−|˜x(s)⟩⟨˜x(s)|∥22 ≤ ∥|ψT(s)⟩⟨ψT(s)|−|˜x(s)⟩⟨˜x(s)|∥2F =

which implies

## Iii Proof of Theorem 1 and Theorem 2

The proof of Theorem 1 and Theorem 2 can be completed by carefully analyzing the -dependence of each term in given in Eq. (3). Note that in both cases , and we introduce a constant with for the proof of Theorem 1 and for the proof of Theorem 2 due to the different scaling parameter of . We first compute the derivatives of

by chain rule as

 H(1)(s)=ddsH(f(s))=dH(f(s))dfdf(s)ds=(H1−H0)cpΔp∗(f(s)),

and

 H(2)(s) =ddsH(1)(s)=dds((H1−H0)cpΔp∗(f(s))) =(H1−H0)cppΔp−1∗(f(s))dΔ∗(f(s))dfdf(s)ds =c′(−1+1/κ)(H1−H0)c2ppΔ2p−1∗(f(s)).

Then the first two terms of can be rewritten as

 ∥H(1)(0)∥2TΔ2(0)+∥H(1)(s)∥2TΔ2(f(s))≤∥H(1)(0)∥2TΔ2∗(0)+∥H(1)(s)∥2TΔ2∗(f(s)) = ∥(H1−H0)cpΔp∗(f(0))∥2TΔ2∗(0)+∥(H1−H0)cpΔp∗(f(s))∥2TΔ2∗(f(s)) ≤ CT(cpΔp−2∗(0)+cpΔp−2∗(f(s))) ≤ CT(cpΔp−2∗(0)+cpΔp−2∗(1))

Here stands for a general positive constant independent of . To compute the remaining two terms of , we use the following change of variable

 u=f(s′),du=dds′f(s′)ds′=cpΔp∗(f(s′))ds′,

and the last two terms of become

 1T∫s0∥H(2)∥2Δ2ds′≤1T∫s0∥H(2)∥2Δ2∗ds′ = 1T∫s0∥c′(−1+1/κ)(H1−H0)c2ppΔ2p−1∗(f(s′))∥2Δ2∗(f(s′))ds′ = 1T∫f(s)0∥c′(−1+1/κ)(H1−H0)c2ppΔ2p−1∗(u)∥2Δ2∗(u)ducpΔp∗(u) ≤ CT((1−1/κ)cp∫f(s)0Δp−3∗(u)du) ≤ CT((1−1/κ)cp∫10Δp−3∗(u)du),

and similarly

 1T∫s0∥H(1)∥22Δ3ds′≤1T∫s0∥H(1)∥22Δ3∗ds′ = 1T∫s0∥(H1−H0)cpΔp∗(f(s′))∥22Δ3∗(f(s′))ds′ = 1T∫f(s)0∥(H1−H0)cpΔp∗(u)∥22Δ3∗(u)ducpΔp∗(u) ≤ CT(cp∫f(s)0Δp−3∗(u)du) ≤ CT(cp∫10Δp−3∗(u)du).

Summarize all terms above, an upper bound of is

 η(s) ≤CT{(cpΔp−2∗(0)+cpΔp−2∗(1))+((1−1/κ)cp∫10Δp−3∗(u)du)+(cp∫10Δp−3∗(u)du)} =CT{c′p−2(cp+cpκ2−p)+((1−1/κ)cp∫10Δp−3∗(u)du)+(cp∫10Δp−3∗(u)du)}.

Finally, since for

 cp=∫10Δ−p∗(u)du=c′−pp−1κκ−1(κp−1−1),

and

 ∫10Δp−3∗(u)du=c′p−32−pκκ−1(κ2−p−1),

we have

 η(s)≤ CT{κκ−1(κp−1−1)+κκ−1(κ−κ2−p) +κκ−1(κp−1−1)(κ2−p−1)+(κκ−1)2(κp−1−1)(κ2−p−1)}.

The leading term of the bound is when .

Now we consider the limiting case when . Note that the bound for can still be written as

 η(s) ≤CT{(cpΔp−2∗(0)+cpΔp−2∗(1))+((1−1/κ)cp∫10Δp−3∗(u)du)+(cp∫10Δp−3∗(u)du)} =CT{c′p−2(cp+cpκ2−p)+(1−1/κ)cpc3−p+cpc3−p}.

Straightforward computation shows that

 c1=∫10Δ−1∗(u)du=1c′κκ−1log(κ)

and

 c2=∫10Δ−2∗(u)du=1c′2κκ−1(κ−1).

Hence when ,

 η(s)≤CT{c′p−2(cp+cpκ2−p)+(1−1/κ)c1c2+c1c2}≤Cκlog(κ)T.

This completes the proof of Theorem 1 and Theorem 2.

## Iv Details of the numerical examples

For concreteness, for the Hermitian positive definite example, we choose . Here

is an orthogonal matrix obtained by Gram-Schmidt orthogonalization (implemented via a QR factorization) of the discretized periodic Laplacian operator given by

 L=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1−0.5−0.5−0.51−0.5−0.51−0.5⋱⋱⋱−0.51−0.5−0.5−0.51⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (S2)

is chosen to be a diagonal matrix with diagonals uniformly distributed in

. More precisely, with . Such construction ensures to be a Hermitian positive definite matrix which satisfies and the condition number of is . We choose where is the set of the column vectors of . Here .

For the non-Hermitian positive definite example, we choose . Here and are the same as those in the Hermitian positive definite case, except that the dimension is reduced to . is an orthogonal matrix obtained by Gram-Schmidt orthogonalization of the matrix

 K=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝2−0.5−0.5−0.52−0.5−0.52−0.5⋱⋱⋱−0.52−0.5−0.5−0.52⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (S3)

Such construction ensures to be non-Hermitian, satisfying and the condition number of is . We choose the same as that in the Hermitian positive definite example.