digital quantum simulation for screening and confinement in

Digital quantum simulation for screening and confinement in gauge

theory with a topological termEtsuko Itou

(RIKEN/ RCNP/Keio U.)

arXiv: 2105.03276 collaboration with M.Honda (YITP, Kyoto U.), Y.Kikuchi(BNL), L.Nagano, T.Okuda(U. of Tokyo) Work in progress with M.Honda (YITP, Kyoto U.), Y.Kikuchi(BNL), Y.Tanizaki (YITP, Kyoto U.)

RIKEN-Vancouver Joint Workshop on Quantum Computing, Online, 2021/08/25

Introduction (my research interests)

• I am a high energy theorist. Mainly working on numerical simulations using supercomputer.

• In high energy theory, we would like to study nonperturbative property of quantum field theories. (ex. Phase transition, thermodynamics, correlation of observables)

• In particular, theory of elementary particles is described by gauge theory. (For instance, quantum chromodynamics (QCD) is given by SU(3) gauge theory)

• Lattice QCD simulation based on Markov-Chain-Monte-Carlo method has been a standard method in this subject, but it suffers from the sign problem if we consider some parameter regime.

• Implementation of quantum computing for QCD with such a regime is my long goal.

Today's talk• We consider the simple gauge theory 1+1 dimensional U(1) gauge theory + topological term coupled with fermion, Schwinger model (e.g. QCD is 3+1 dimensional SU(3) gauge theory coupled with fermions)

• We use a simulator (not real quantum device) to see if our strategy works well even in the parameter regime where the sign problem appears in conventional Monte Carlo

• State preparation here is adiabatic state preparation

• See the systematic error from adiabatic state preparation and adiabatic schedule function

θ

Schwinger model

Schwinger model• Schwinger model = 1+1 dim. U(1) gauge theory

• term induces the sign problem in Monte Carlo simulation

• In this work, we numerically obtain the potential between two probe charges with the distance

• Analytical calculation predicts the potential between two probe depends on and .

• We compare the results with analytic results in both infinite vol. and finite vol. using the mass perturbation

ℒ = −14

FμνFμν +gθ0

4πϵμνFμν + iψγμ(∂μ + igAμ)ψ − mψ ψ

θ

ℓ

m q

integer q in fractional q in

m = 0, m ≠ 0

m = 0fractional q in m ≠ 0

theoretical predictions in infinite vol. limit

screening potential

linear potential

V(ℓ)

ℓ

V(ℓ)

ℓ

S. Iso and H. Murayama Prog.Theor.Phys.84(1990)142 D. J. Gross et al., NPB461 (1996)

M.Honda, E.I., Y.Kikuchi, L.Nagano, T.Okuda, arXiv: 2105.03276

• Lagrangian in continuum

• Hamiltonian in continuum

• Hamiltonian on lattice (staggered fermion, link variable)

• Remove gauge d.o.f. (OBC and Gauss law constraint)

• Spin Hamiltonian using Pauli matrices(Jordan-Wigner trans.)

From to for Quantum computerℒ ℋEx) Schwinger model with open b.c.

ℒ = −14

FμνFμν +gθ0

4πϵμνFμν + iψγμ(∂μ + igAμ)ψ − mψψ

Hcon = ∫ dx [ 12 (Π −

gθ0

2π )2

− iψγ1(∂1 + igA1)ψ + mψ ψ]

0 = ∂1Π + gψ†ψ → Ln − Ln−1 = χ†n χn −

1 − (−1)n

2

χn

a↔ {ψu(x) n : even

ψd(x) n : oddH = J

N−2

∑n=0

(Ln +ϑn

2π )2

− iwN−2

∑n=0

(χ†nUn χn+1 − χ†

n+1U†n χn) + m

N−1

∑n=0

(−1)n χ†n χn

H = JN−2

∑n=0 (ϵ−1 +

n

∑i=0 (χ†

i χi −1 − (−1)i

2 ) +ϑn

2π )2

− iwN−2

∑n=0

(χ†n χn+1 − χ†

n+1χn) + mN−1

∑n=0

(−1)n χ†n χn

H = JN−2

∑n=0 [

n

∑i=0

Zi + (−1)i

2+

ϑn

2π ]2

+w2

N−2

∑n=0

[XnXn+1 + YnYn+1] +m2

N−1

∑n=0

(−1)nZn

PBC: Shaw et al. Quantum 4, 306 (2020) arXiv:2002.11146

Apply quantum algorithms to this spin hamiltonian.

Canonical momentum: Π = ∂0A1 +gθ2π

Link variable: Staggered fermion:

Ln ↔ − Π(x)/g, Un ↔ e−iagA1(x),

Gauss law:

χn =Xn − Yn

2

n−1

∏i=0

(−iZi)Jordan-Wigner trans.:

Map the Lagrangian to the Spin Hamiltonian

• Lagrangian in continuum

• Hamiltonian in continuum

• Hamiltonian on lattice (staggered fermion, link variable)

• Remove gauge d.o.f. (OBC and Gauss law constraint)

• Spin Hamiltonian using Pauli matrices(Jordan-Wigner trans.)

From to for Quantum computerℒ ℋEx) Schwinger model with open b.c.

ℒ = −14

FμνFμν +gθ0

4πϵμνFμν + iψγμ(∂μ + igAμ)ψ − mψψ

Hcon = ∫ dx [ 12 (Π −

gθ0

2π )2

− iψγ1(∂1 + igA1)ψ + mψ ψ]

0 = ∂1Π + gψ†ψ → Ln − Ln−1 = χ†n χn −

1 − (−1)n

2

χn

a↔ {ψu(x) n : even

ψd(x) n : oddH = J

N−2

∑n=0

(Ln +ϑn

2π )2

− iwN−2

∑n=0

(χ†nUn χn+1 − χ†

n+1U†n χn) + m

N−1

∑n=0

(−1)n χ†n χn

H = JN−2

∑n=0 (ϵ−1 +

n

∑i=0 (χ†

i χi −1 − (−1)i

2 ) +ϑn

2π )2

− iwN−2

∑n=0

(χ†n χn+1 − χ†

n+1χn) + mN−1

∑n=0

(−1)n χ†n χn

H = JN−2

∑n=0 [

n

∑i=0

Zi + (−1)i

2+

ϑn

2π ]2

+w2

N−2

∑n=0

[XnXn+1 + YnYn+1] +m2

N−1

∑n=0

(−1)nZn

PBC: Shaw et al. Quantum 4, 306 (2020) arXiv:2002.11146

Apply quantum algorithms to this spin hamiltonian.

Canonical momentum: Π = ∂0A1 +gθ2π

Link variable: Staggered fermion:

Ln ↔ − Π(x)/g, Un ↔ e−iagA1(x),

Gauss law:

χn =Xn − Yn

2

n−1

∏i=0

(−iZi)Jordan-Wigner trans.:

• Spin Hamiltonian:

• Theta-term can be expressed as a background electric field (link variable) in Hamiltonian formalism. Probe charge shift the value of theta

• Generate the ground state ( ) using adiabatic state preparation with 2nd order Suzuki-Trotter decomposition

• Directly measure the energy , and Cf.) In Lattice QCD, we calculate the Wilson loop and extract the potential from its exponent.

with

H = JN−2

∑n=0 [

n

∑i=0

Zi + (−1)i

2+

ϑn

2π ]2

+w2

N−2

∑n=0

[XnXn+1 + YnYn+1] +m2

N−1

∑n=0

(−1)nZn

|Ω⟩

E(ℓ) = ⟨Ω |H(ℓ) |Ω⟩ V(ℓ) = E(ℓ) − E(0)

⟨W(R, T )⟩ ∼ exp(−V(R)/T ) V(R) ∼ σR + α/R

Summary of simulation strategyℓ0

ℓ + ℓ00 N − 1

qp −qp

ℓℓ0 ℓ0

ϑn

2πqp + θ0

θ0

n

Simulation results

• IBM Qiskit library, Simulator (not real quantum device)

• lattice size: N=15,21, open b.c.(remove d.o.f. of gauge field)

• # of shots = 100,000 - 400,000

• a=0.4, g=1.0, mass= 0.00 - 0.25 cf) 1st order phase transition emerges at with in infinite limitmc/g ≈ 0.33 θ0 = π, q = 0

Simulation Setup

0.0 2.5 5.0 7.5g`

0.0

0.5

1.0

1.5

2.0

2.5

Vf(`

)/g

q = 1, m = 0

V (0)(`)/g

V (0)f (`)/g, N = 15

V (0)f (`)/g, N = 21

N = 15

N = 21

0.0 2.5 5.0 7.5g`

0

1

2

3

4

Vf(`

)/g

q = 1, m/g = 0.2

V (0)(`)/g

N = 15

N = 21

N = 101

N = 15

N = 21

Result(1): integer charge ( )θ0 = 0• screening potential

- - - - denotes

• In and finite N is given by

• const. term remains in

-・-・(roughly) denotes

• Our results are consistent with the theoretical predictions in the finite N We expect the discrepancy comes from finite a effect

a → 0, N → ∞

V(0)

a → 0

V(0)f (ℓ) =

q2g2

2μ(1 − e−μℓ)(1 + e−μ(L−ℓ))

1 + e−μL,

𝒪(m)

V0 + V(1)

Result(2): fractional charge ( )θ0 = 0

• If Q is fractional, in is screening for massless case is confinement in large for

nonzero masses

• In and finite N, potential shape gradually changes (see black curve)

• Around , black curve depicts linear behavior

a → 0,N → ∞

V(ℓ)

V(ℓ) ℓ

a → 0

0.20 ≲ m/g ≲ 0.25

0 1 2 3 4 5 6gl0

0.05

0.1

0.15

0.2

V(l)

/g

0 1 2 3 4 5 6gl0

0.05

0.1

0.15

0.2

0 1 2 3 4 5 6gl

0

0.05

0.1

0.15

0.2Vf

(0)(l)+Vf(1)(l), (N=15)

qiskit (snapshot)

0 1 2 3 4 5 6gl

0

0.05

0.1

0.15

0.2

V(l)

/g

0 1 2 3 4 5 6gl

0

0.05

0.1

0.15

0.2

0 1 2 3 4 5 6gl

0

0.05

0.1

0.15

0.2

m=0.00 m/g=0.05

m/g=0.15 m/g=0.20

m/g=0.10

m/g=0.25

Potential shape as a function of mass

Result(2): fractional charge ( )θ0 = 0

0.0 0.2 0.4q

0.00

0.02

0.04

0.06

0.08

0.10

æf/g

2

q2

2 (Coulomb)

mß(1 ° cos(2ºq))

analytic

m/g = 0.2

0 2 4g`

0

0.05

0.1

0.15

Vf(`

)/g

V (0)(`)/g

V (0)f (`)/g

[V (0)f (`) + V (1)

f (`)]/g

m = 0

m/g = 0.2

• string tension is calculated by fitting long data

• It shows the large discrepancy from Coulomb potential (pure U(1) gauge theory)

• Our data are consistent with the analytical prediction of 1st order mass perturbation in finite vol.

• Around , string tension in finite vol. is quite different from result in infinite volume:

because of open boundary condition (We will show the restoration of the periodicity Work in progress w/ Honda, Kikuchi and Tanizaki)

ℓ

q = 0.5 (θ = π)

V(1)(ℓ) = mΣ(1 − cos(2πq))ℓ

N=15, q=0.25String tension ( )σ

V(ℓ) = σℓ + c0

Potential

Result(3): non-zero caseθ0

°1.0 °0.5 0.0 0.5 1.0µ0/2º

0.0

0.2

0.4

0.6

0.8

1.0

1.2

[E(µ

0,q

,`)°

E(0

,0,0

)]/g

2L mß(1 ° cos µ0)/g2

finite interval

q = 0, `/a = 0

q = 1, `/a = 6

q = 1, `/a = 10

q = 1, `/a = 14

• non-zero simulation is doable

• periodicity is broken because of open b.c. Infinite volume limit: )

• , is a shift of boundary charge ( )

for ,

θ0

2π

E(θ) = mΣ(1 − cos θ)ℓ

q = 1 ℓ/a = 14 2π θ0

q = 0 ℓ/a = 0

Energy density

Systematic error

Systematic errors from adiabatic state preparation

• Systematic errors: Adiabatic error (T) Trotter error ( ) (initial mass ( ))

δt

minit. ≈ m + 0.5

w → wtT

, θ0 → θ0tT

, q → qtT

, m → m0 (1 −tT ) + m

tT

|Ω⟩ = limT→∞

𝒯 exp (−i∫T

0dt HA(t)) |Ω⟩0

Generate (approximate) ground state based on adiabatic theoremWork in progress with M.Honda, Y.Kikuchi, Y.Tanizaki

In finite T, an approximate ground state is obtained. In the adiabatic Hamiltonian, the parameters of model depend on the time (t/T), ( e.g. where with ) If we take a linear fn,

θ0 f(t/T ) f(x) f(0) = 0 , f(1) = 1

HA(0) = H0 HA(T ) = HHere, is adiabatic HamiltonianHA(t)

Adiabatic error and adiabatic schedule

Lecture note by A. Childs

Adiabatic error ( ) scales if the first derivatives of the Hamiltonian is zero at the beginning and end of the evolution.

ϵ 𝒪(1/Tm+1) m

denotes the first derivative of adiabatic Hamiltonian at t=0 and t=T denotes the gap energy

·H(t/T )

Δ(t)

Rezakhani A T, Pimachev A K and Lidar D A 2010 Phys. Rev. A 82 052305 Lidar D A, Rezakhani A T and Hamma A 2009 J. Math. Phys. 50 102106

Thus, naively if the adiabatic schedule fn. has a mild slope at both edges of time evolution, then we expect that adiabatic error becomes small.

Adiabatic schedule function• It is difficult to estimate where the gap energy is small in quantum field theory

0 0.2 0.4 0.6 0.8 1x

0

0.2

0.4

0.6

0.8

1

f(x)

xx2

tanh(x)/tanh(1)

tanh2(2x)/tanh2(2)

0 1 2 3 4 5 6 7gl

0

1

2

3

4

V(l)

/g

x; T=99

x2, T=99tanh(x)/tanh(1), T=99

tanh2(2x)/tanh2(2), T=99x, T= 792

We practically find that is good schedule fn if we take a parameter set of this model.

Why ?? Around x=0, and 1, or has more gentle slop.

tanh(x)/tanh(1)

tanh2(2x)/tanh2(2) x2

Work in progress with M.Honda, Y.Kikuchi, Y.Tanizaki

Adiabatic error

(True value)

Adiabatic schedule function• It is difficult to estimate where the gap energy is small in quantum field theory

0.0 0.2 0.4 0.6 0.8 1.0adiabatic time

0.5

1.0

1.5

2.0

2.5

(E1

°E

0)/

g

N = 17

l = 4, q = 2

l = 12, q = 2

l = 4, q = °1

l = 12, q = °1

Gap energy between ground and 1st excited states in adiabatic process (exact diagonalization calculation)

Orange data has a rapid decreasing of gap energy around t/T =0.5. It explains why is the best schedule fn, since has the most gentle slope around t/T = 0.5.

tanh(x)/tanh(1)

tanh(x)/tanh(1)

Work in progress with M.Honda, Y.Kikuchi, Y.Tanizaki

Summary and outlook• We carried out the numerical simulation for the screening-type and confinement potentials of Schwinger model using quantum algorithm

• Thanks to the Hamiltonian formalism, even in regime, we can perform the simulations

• The potential can be directly obtained from the vev of Hamiltonian.

• If we consider limits, some improvements are necessary.

• Choice of adiabatic schedule is nontrivial in some QFT.

• Using quantum algorithm, we can investigate various QFTs which suffer from the sign problem in conventional Monte Carlo approach.

θ0 ≠ 0

a → 0,N → ∞

Confinement potential in Lattice QCD

• Lattice QCD simulations started the derivation of the confinement potential.

• Now, QC for QFT (or QCD) have just started!

M.Creutz, PRD21 (1980) 2308