Compact quantum circuit of variational quantum eigensolver
for quantum impurity models
Rihito Sakurai (Saitama Univ.), Wataru Mizukami (Osaka Univ.), Hiroshi Shinaoka (Saitama Univ.)

Take home message“Sparse” ansatz

New⚠ Solving impurity models is the biggest bottleneck of DMFT.
⚠ VQA suffers from an increasing # of parameters with the # of orbitals.

Assumption: Most two-particle excitation operators between
bath sites do not contribute to the ground state significantly.
k-uCJuCCGSD

💡 Compact quantum circuit for quantum impurity models

Introduction■ Dynamical mean-field theory (DMFT)[1]
Lattice model

Bottleneck

Impurity model
SolverLocal Green’s functionGbath† †c2↑
c2↓c3↓c3↑† †c1↑
c5↓c6↓c5↑† †c1↑
c6↓c5↓c6↑

We remove operators with morethan 3 bath orbitals.

Update bath

We remove operators between
different bath site.

Discretize continuous bath“Sparse” uCCGSD■ Imaginary-time formalism# of impurity orbitals: nimp

fewer bath sites# of bath orbitals: nbathnbath ∝ 10 × nimp
e.g. Fe-based superconductor(LaFeAsO)[2]norb = nimp + nbath ∼ 40 + 332- Fault-tolerant QC[4]- QC with limited hardware resources (e.g. NISQ devices)
Variational quantum algorithms (VQA) for computing imaginary-time orfrequency Green’s functions G(τ)[5], G(iω)[6]in ansatz grows rapidly with the number of orbitals.

Unitary coupled cluster
generalized singles anddoubles (uCCGSD)[7]

k-unitary cluster jastrow(k-uCJ)[8]O ((nimp + nbath ) )O ((nimp + nbath ) )U(θ ⃗ ) =Ĵ=n orbe
∏ {a>i,b>jkK̂i Jî −K̂i∑[e e ei=1

θ a †a †a a
∑ jl j l l jjln orbK̂ =θ a† a∑ ∑ jl jσ lσjl σ =↑,↓]

norb

NP10000††n2orb
n4orb

5000

∑ ∑

k=2 σ =↑,↓

k=2 σ =↑,↓

Sparse k-uCJ (original)

272 (1752)

228 (288)

H=

∑
i=1−

̂ ni↓
̂ −μU ni↑∑∑ ∑j=1,2 k σ =↑,↓∑

i=1,2,σ =↑,↓

̂ −niσ̂nkσ

40

̂ c1σ̂ )t c†̂ c ̂ + c†2σ
∑ ( 1σ 2σσ =↑,↓̂ ckσ
̂ + c†kσ
̂ cjσ̂ ) + ϵbV (c†jσ∑ ∑k=3 σ =↑,↓̂nbσU = 4.0, μ = U / 2, t = 1.0, V =0.5, ϵb = [−1,0,1]- Sparse k-ucj works for cluster- Sparse k-ucj outperforms
sparse uCCGSD.

Number of orbitals (sites) : norb

References[1] G. Kotliar et al., Rev. Mod. Phys. 78, 865 (2006)[2] H. Shinaoka et al., Phys. Rev. B 103, 045120 (2021)[3] F. Alexander Wolf et al., Phys. Rev. X 5, 041032 (2015)[4] B. Bauer et al., Phys. Rev. X 6, 031045 (2016)[5] R. Sakurai et al., Phys. Rev. Research 4, 023219 (2022)[6] H. Chen et al., Phys. Rev. A 104, 032405 (2021)[7] J. Lee et al., J. Chem. Theory. Comput. 15, 1 (2019)[8] Y. Kurashige et al., J. Chem. Theory. Comput. 16, 2 (2020)[9] https://github.com/qulacs/qulacs∑ ∑Sparse uCCGSD (original)

impurity model.

2500
20

̂ ckσ
̂ + c†kσ
̂ c1σ
̂ ) + ϵkV (c†1σ■Two-site
impurity with six bath sites
2a,i7500σ =↑,↓

̂ −n1σΔ = | Eorigi(V ) − Esparse(V ) | < O(10−4 ) ← Small influence on ground state!

θi (ca ci−ci ca )} ∏ {e}a∑¢EG = |Eexact ° Ekucj|k-uCJ

U(θ ⃗ ) =

θijab (ca†cb†cj ci−ci†cj†cb ca )■ One impurity with five bath sites

U = 4.0, μ = U / 2, t=1.0, V ∈ [0.0,1.0] , ϵk = [−2, − 1,0,1,2]

2

Number of variational params : NPuCCGSD

2
O (nimp), (nbath ∼ O(nimp)- Ground state calculation: Variational quantum eigensolver- Qulacs[9] wrapped by julia- Optimizer: BFGS
̂ n1↓
̂ −μH = U n1↑

Common problem: The number of variational parameters NPnorb

O ((nimp)4)■ Numerical details■ Quantum algorithms for impurity solver

4NPNumerical results

State-of-the-art “classical” simulation
Two-site impurity with three orbitals by MPS[3]NP“Sparse” k-uCJ

10°1

sparse uCCGSD
sparse k-ucj
k-ucj (origial)

10°2
10

k=1

k=2
k=3°3

k=4500Future plan
- Computing spectral functions with moments expansion- Effect of noise1000

Np1500