素粒子物理学のための量子計算入門～場の量子論編～

Masazumi Honda(本多正純)

11th June, 2021

第11回HPC-Phys勉強会※zoomにて録画予定

Plan
1. QFT as qubits (mapping to spin system)
2. Schwinger model as qubits
3. Time evolution operator
4. Without probe charges
5. With probe charges
6. Summary & Outlook[Chakraborty-MH-Kikuchi-Izubuchi-Tomiya ’20]

(if time is allowed)[MH-Itou-Kikuchi-Nagano-Okuda ’21]

“Regularization” of Hilbert space
Hilbert space of QFT is typically ∞ dimensional
Make it finite dimensional!
・Fermion is easiest (up to doubling problem)
Putting on spatial lattice, Hilbert sp. is finite dimensional

・scalar
Hilbert sp. at each site is ∞ dimensional(need truncation or additional regularization)

・gauge field (w/ kinetic term)no physical d.o.f. in 0+1D/1+1D (w/ open bdy. condition)∞ dimensional Hilbert sp. in higher dimensions

Citation history of “Hamiltonian Formulation of
Wilson's Lattice Gauge Theories” by Kogut-Susskind(totally 1923 at this moment)

Free Dirac fermion in 1+1D
Continuum:

Lattice (w/ N sites and spacing a):
For staggered fermion:(anti-)commutation relation:

Jordan-Wigner transformation
This is satisfied by the operator:[Jordan-Wigner’28]

Then the system is mapped to the spin system:

We can apply quantum algorithms to QFT!

Bosonic case: illustration by single particle QM
1 2 𝜔2 2
෡ = 𝑝Ƹ +𝐻𝑥ො + 𝑉int (𝑥)ො22Most naïve approach = truncation in harmonic osc. basis:∞

𝜔𝑖𝑥ො +𝑝Ƹ = ෍ 𝑛 + 1 |𝑛⟩⟨𝑛 + 1|22𝜔
𝑛=0𝑎ො =

Λ−2

෍ 𝑛 + 1 |𝑛⟩⟨𝑛 + 1|

regularize! 𝑛=0

Then replace 𝑝Ƹ & 𝑥ො by𝑥ො ቚregularized≡12𝜔(𝑎ො + 𝑎ො † ) ቚregularized

1 𝜔𝑝Ƹ ቚ≡(𝑎ො − 𝑎ො † ) ቚ𝑖 2
regularized
regularizedBosonic case: illustration by single particle QM (Cont’d)Λ−2𝑎ො ቚregularized= ෍ 𝑛 + 1 |𝑛⟩⟨𝑛 + 1|
𝑛=0

We can rewrite the Fock basis in terms of qubits:𝑛 = 𝑏𝐾−1 𝑏𝐾−2 ⋯ |𝑏0 ⟩𝑛 = bK−1 2K−1 + bK−2 2K−2 + ⋯ + 𝑏0 20
Then,′𝑛 𝑛 + 1 = ⊗𝐾−1(|𝑏ℓ=0ℓ ⟩⟨𝑏ℓ |)either one of𝐾 ≡ log 2 Λ(binary representation)

Plan1. QFT as qubits (mapping to spin system)
2. Schwinger model as qubits
3. Time evolution operator
4. Without probe charges
5. With probe charges
6. Summary & Outlook[Chakraborty-MH-Kikuchi-Izubuchi-Tomiya ’20]

(if time is allowed)[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Schwinger model w/ topological term
Continuum ①:(will be used for the case w/ probes)

Continuum ➁:(equivalent via “chiral anomaly”, used here)

Taking temporal gauge 𝐴0 = 0,

Physical states are constrained by Gauss law:

[Fujikawa’79]( Π = 𝐴ሶ1 )Sign problem in path integral formalism
In Minkowski space,
𝑆 = ∫ 𝑑4𝑥

1 2− 𝐹𝜇𝜈 + 𝜓ത 𝑖𝛾 𝜇 𝐷𝜇 − 𝑚 𝜓4∫ 𝐷𝐴𝐷𝜓𝐷 𝜓ത 𝒪 𝑒 𝑖𝑆
𝒪 =∫ 𝐷𝐴𝐷𝜓𝐷 𝜓ത 𝑒 𝑖𝑆𝑔𝜃+ ∫𝐹 ∈ 𝑹
4𝜋

highly oscillating

In Euclidean space,
𝑆 = ∫ 𝑑4𝑥

1 2𝑔𝜃𝜇− 𝐹𝜇𝜈 + 𝜓ത 𝑖𝛾 𝐷𝜇 − 𝑚 𝜓 + 𝑖∫𝐹 ∈ 𝑪
44𝜋∫ 𝐷𝐴𝐷𝜓𝐷 𝜓ത 𝒪 𝑒 −𝑆𝒪 =∫ 𝐷𝐴𝐷𝜓𝐷 𝜓ത 𝑒 −𝑆

highly oscillating for non-small θAccessible region by analytic computation
・Massive limit:

The fermion can be integrated out&the theory becomes effectively pure Maxwell theory w/ 𝜃・Bosonization:[Coleman ’76]

1𝑔2 2𝑒𝛾𝑔2ℒ=𝜕𝜇 𝜙 − 2 𝜙 + 3/2 𝑚 cos(𝜙 + 𝜃)
8𝜋8𝜋2𝜋exactly solvable for 𝑚 = 0&small 𝑚 regime is approximated by perturbation

Map of accessibility/difficulty𝜃
We can makeprediction here

Massperturb.

PureMaxwellsolvable

Monte Carlo𝑚

Put the theory on lattice・Fermion (on site):“Staggered fermion” [Susskind, Kogut-Susskind ’75]

𝜒𝑛𝑎1/2

𝜓(𝑥) =𝜓𝑢𝜓𝑑odd siteeven sitelattice spacing・Gauge field (on link):𝜙𝑛x𝜒1

𝜙1 , 𝐿1x
𝜒2↔ −𝑎𝑔𝐴1 𝑥 ,𝜙2 , 𝐿2x𝜒3𝜙3 , 𝐿3x
𝜒4Π 𝑥
𝐿𝑛 ↔ −𝑔・・・

𝜙𝑁−1 , 𝐿𝑁−1xx

𝜒𝑁−1

𝜒𝑁

Lattice theory w/ staggered fermion
Hamiltonian:Commutation relation:

Gauss law:Eliminate gauge d.o.f.
1. Take open b.c. & solve Gauss law:

2. Redefine fermion to absorb 𝜙𝑛 :Then,This acts on finite dimensional Hilbert space

Going to spin system
This is satisfied by the operator:“Jordan-Wigner transformation”[Jordan-Wigner’28]

Going to spin system
This is satisfied by the operator:“Jordan-Wigner transformation”[Jordan-Wigner’28]

Now the system is purely a spin system:

Qubit description of the Schwinger model !!

Plan
1. QFT as qubits (mapping to spin system)
2. Schwinger model as qubits
3. Time evolution operator
4. Without probe charges
5. With probe charges
6. Summary & Outlook[Chakraborty-MH-Kikuchi-Izubuchi-Tomiya ’20]

(if time is allowed)[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Time evolution operator
Suzuki-Trotter decomposition:(M: large positive integer)

Can we express it in terms of elementary gates?

Time evolution operator (cont’d)
The 1st one is trivial:

The 2nd one appeared in Ising model:

The 3rd one (see next slide):

The 4th one:

Time evolution operator (Cont’d)
Proof:

Thus,

Quantum circuit for time evolution op. (N=4)

Improvement of Suzuki-Trotter decomposition
The leading order decomposition:

The 2nd order improvement:Improvement of Suzuki-Trotter decomposition
The leading order decomposition:

The 2nd order improvement:

cf. Baker-Campbell-Hausdorff formula:This increases the number of gates at each time step
but we can take larger δt (smaller M) to achieve similar accuracy.
Totally we save the number of gates.

Survival probability of massive vacuum[cf. Martinez etal. Nature 534 (2016) 516-519]

The ground state in the large mass limit isSurvival probability:“Schwinger effect”
in massive limit

Result of simulator (10000 shots)

4 sites
6 sites

8 sites

10 sitesPlan
1. QFT as qubits (mapping to spin system)
2. Schwinger model as qubits
3. Time evolution operator
4. Without probe charges
5. With probe charges
6. Summary & Outlook[Chakraborty-MH-Kikuchi-Izubuchi-Tomiya ’20]

(if time is allowed)[MH-Itou-Kikuchi-Nagano-Okuda ’21]

VEV of mass operator (chiral condensation)
Instead of the local op., we analyze the average over the space:

Once we get the vacuum, we can compute the VEV asAdiabatic state preparation of vacuum
Step 1: Choose an initial Hamiltonian 𝐻0 of a simple system
whose ground state |vac0 ⟩ is known and unique
Step 2: Introduce adiabatic Hamiltonian 𝐻𝐴 𝑡 s.t.
・ 𝐻𝐴 0 = 𝐻0 , 𝐻𝐴 𝑇 = 𝐻target・𝑑𝐻𝐴𝑑𝑡≪ 1 for 𝑇 ≫ 1

Step 3: Use the adiabatic theorem
If 𝐻𝐴 (𝑡) has a unique ground state w/ a finite gap for ∀𝑡,then the ground state of 𝐻target is obtained by𝑇

vac = lim 𝒯 exp −𝑖 න 𝑑𝑡 𝐻𝐴 𝑡
𝑇→∞0

|vac0 ⟩

Features of adiabatic state preparation1("systematic error") ∼𝑇 gap 2

Advantage:・guaranteed to be correct for 𝑇 ≫ 1 & 𝛿𝑡 ≪ 1
if 𝐻𝐴 (𝑡) has a unique gapped vacuum・can directly get excited states under some conditions

Disadvantage:
・doesn’t work for degenerate vacua・costly — likely requires many gates
perhaps not so efficient in NISQ era

Adiabatic state preparation in the Schwinger model

Here we choose

m0 can be any positive number in principle
but it is practically chosen to have small systematic error

Massless caseFor massless case,𝜃 is absorbed by chiral rotation

𝜃 = 0 w/o loss of generality
No sign problem

Nevertheless,it’s difficult in conventional approach because computation offermion determinant becomes very heavy∃Exact result:

Can we reproduce it?

[Hetrick-Hosotani ’88]

Estimation of systematic errors
Approximation of vacuum:[Chakraborty-MH-Kikuchi-Izubuchi-Tomiya ’20]

Approximation of VEV:Introduce the quantity

independent of t if
dependent on t if

This quantity describes intrinsic ambiguities in prediction
Useful to estimate systematic errors

Estimation of systematic errors (Cont’d)

Oscillating around the correct value
Define central value & error as&

Thermodynamic & Continuum limit#(measurements)

Thermodynamic limit:

Continuum limit:

Result for massless case (after continuum limit)[Chakraborty-MH-Kikuchi-Izubuchi-Tomiya ’20]

exact result

Massive case
Result of mass perturbation theory:[Adam ’98]

However,∃Subtlety in comparison: this quantity is UV divergent

Use a regularization scheme to have the same finite part
Here we subtract free theory result before taking continuum limit:

Result for massive case at 𝑔 = 1[Chakraborty-MH-Kikuchi-Izubuchi-Tomiya ’20]

mass perturbation

𝜃 dependence at 𝑚 = 0.1 & 𝑔 = 1ത⟨𝜓𝜓⟩mass perturbation

Plan
1. QFT as qubits (mapping to spin system)
2. Schwinger model as qubits
3. Time evolution operator
4. Without probe charges
5. With probe charges
6. Summary & Outlook[Chakraborty-MH-Kikuchi-Izubuchi-Tomiya ’20]

(if time is allowed)[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Screening versus Confinement
Let’s consider
potential between 2 heavy charged particles−𝑞Classical picture:+𝑞Screening versus Confinement
Let’s consider
potential between 2 heavy charged particles−𝑞+𝑞Classical picture:𝑉 𝑥 =𝑞 2 𝑔22

Coulomb law in 1+1d𝑥?

confinement

too naive in the presence of dynamical fermionsExpectations from previous analyzes
Potential between probe charges ±𝑞 has been analytically[Iso-Murayama ’88, Gross-Klebanov-Matytsin-Smilga ’95 ]computed for ∞ volume.
・massless case:𝑞 2 𝑔2𝑉 𝑥 =(1 − 𝑒 −𝜇𝑥 )2𝜇・massive case:𝜇 ≡ 𝑔/ 𝜋screening

Expectations from previous analyzes
Potential between probe charges ±𝑞 has been analytically[Iso-Murayama ’88, Gross-Klebanov-Matytsin-Smilga ’95 ]computed for ∞ volume.
・massless case:𝜇 ≡ 𝑔/ 𝜋𝑞 2 𝑔2𝑉 𝑥 =(1 − 𝑒 −𝜇𝑥 )2𝜇screening

・massive case:Σ ≡ 𝑔𝑒 𝛾 /2𝜋 3/2𝑉 𝑥 ∼ 𝑚Σ 1 − cos 2𝜋𝑞𝑥

(m ≪ 𝑔, 𝑥 ≫ 1/𝑔 )= Const.

for q ∈ 𝒁

screening∝𝑥

for q ∉ 𝒁

confinement

Let’s explore this aspect by quantum simulation!

Our strategy

𝑡 = +∞① Introduce the probe charges ±q:

𝑒𝑖𝑞𝑔 ∫𝐶 𝐴𝐶ℓ𝑡 = −∞𝑒➁③𝑖𝑞𝑔 ∫𝑆,𝜕𝑆=𝐶 𝐹

local 𝜃-term w/ 𝜃 = 2𝜋𝑞!!

Our strategy

𝑡 = +∞① Introduce the probe charges ±q:

𝑒𝑖𝑞𝑔 ∫𝐶 𝐴

𝐶ℓ𝑡 = −∞

𝑒𝑖𝑞𝑔 ∫𝑆,𝜕𝑆=𝐶 𝐹

local 𝜃-term w/ 𝜃 = 2𝜋𝑞!!

➁ Include it to the action & switch to Hamilton formalism𝜃 = 𝜃0+𝑞𝜃 = 𝜃0 + 2𝜋𝑞ℓ③−𝑞 𝜃 = 𝜃0𝑥Our strategy

𝑡 = +∞① Introduce the probe charges ±q:𝑒𝑖𝑞𝑔 ∫𝐶 𝐴𝐶ℓ𝑡 = −∞𝑒𝑖𝑞𝑔 ∫𝑆,𝜕𝑆=𝐶 𝐹

local 𝜃-term w/ 𝜃 = 2𝜋𝑞!!

➁ Include it to the action & switch to Hamilton formalism𝜃 = 𝜃0+𝑞𝜃 = 𝜃0 + 2𝜋𝑞−𝑞 𝜃 = 𝜃0𝑥ℓ③ Compute the ground state energy (in the presence of the probes)

Adiabatic state preparation in the presence of the probes

Here we choose
𝐻0 = 𝐻 ቚ𝑤→0, 𝜗𝑛 →0, 𝑚→𝑚0𝐻𝐴 (𝑡) = 𝐻 ቚ𝑡𝑇vac0 = |1010 ⋯ ⟩𝑤→𝑤 𝑡 ,𝜗𝑛 →𝜗𝑛 𝑡 , 𝑚→𝑚 𝑡𝑡𝑇𝑤 𝑡 = 𝑤, 𝜗𝑛 𝑡 = 𝜗𝑛 , 𝑚 𝑡 = 1 −𝑡𝑇𝑡
𝑇𝑚0 + 𝑚𝑚0 can be any positive number in principle
but it is practically chosen to have small systematic error

Results for massless, 𝜃0 = 0 & 𝑞 ∈ 𝒁[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Parameters: 𝑔 = 1, 𝑎 = 0.4, 𝑁 = 15 & 21, 𝑇 = 99, 𝑞 = 1, 𝑚 = 0Lines: analytical results in the continuum limit (finite & ∞ vols.)(probe distance)

Results for massive, 𝜃0 = 0 & 𝑞 ∈ 𝒁[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Parameters: 𝑔 = 1, 𝑎 = 0.4, 𝑁 = 15 & 21, 𝑇 = 99, 𝑞 = 1, 𝑚 = 0.2
Lines: analytical results in the continuum limit (finite & ∞ vols.)(probe distance)Massless vs massive for 𝜃0 = 0 & 𝑞 ∈ 𝒁[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Parameters: 𝑔 = 1, 𝑎 = 0.4, 𝑁 = 15 & 21, 𝑇 = 99, 𝑞 = 1
Lines: analytical results in the continuum limit (finite & ∞ vols.)

Consistent w/ expected screening behavior

Results for 𝜃0 = 0 & 𝑞 ∉ 𝒁[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Parameters: 𝑔 = 1, 𝑎 = 0.4, 𝑁 = 15, 𝑇 = 99, 𝑞 = 0.25, 𝑚/𝑔 = 0 & 0.2
Lines: analytical results in the continuum limit (finite & ∞ vol.)(probe distance)

Results for 𝜃0 = 0 & 𝑞 ∉ 𝒁[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Parameters: 𝑔 = 1, 𝑎 = 0.4, 𝑁 = 15, 𝑇 = 99, 𝑞 = 0.25, 𝑚/𝑔 = 0 & 0.2
Lines: analytical results in the continuum limit (finite & ∞ vol.)Consistent w/ expected confinement behavior-> interesting to estimate stringtension
for various q?
(probedistance)

“String tension” for 𝜃0 = 0
Parameters: 𝑔 = 1, 𝑎 = 0.4, 𝑁 = 15, 𝑇 = 99, 𝑚/𝑔 = 0.2[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Classical Coulomb
mass pert. (finite V)“string tension”(slope for largedistance)mass pert. (∞-vol.)(probe charge)

“String tension” for 𝜃0 = 0
Parameters: 𝑔 = 1, 𝑎 = 0.4, 𝑁 = 15, 𝑇 = 99, 𝑚/𝑔 = 0.2[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Classical Coulomb“string tension”(slope for largedistance)mass pert. (finite V)

mass pert. (∞-vol.)confinement by nontrivial
dynamics!
(probe charge)

Results for 𝜃0 ≠ 0[MH-Itou-Kikuchi-Nagano-Okuda ’21]

(difficult to explore by the conventional Monte Carlo approach)Parameters: 𝑔 = 1, 𝑎 = 0.4, 𝑁 = 15, 𝑇 = 99, 𝑞 = 1, 𝑚/𝑔 = 0.2

mass pert. (finite V)

mass pert. (∞-vol.)Comment on theta angle periodicity

Absence of the periodicity: 𝜃0 ∼ 𝜃0 + 2𝜋 ?
This is expected because we’re taking open b.c.
To get the periodicity back, we need to take ∞-vol. limit[detailed study: work in progress, MH-Itou-Kikuchi-Tanizaki]Comment: density plots of energy gap(known as “Tuna slice plot” inside the collaboration)[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Parameters: 𝑔 = 1, 𝑎 = 0.4, 𝑁 = 15, 𝑞 = 1, 𝑚/𝑔 = 0.15

smaller gap for larger ℓlarger systematic error for larger ℓSummary & Outlook

Summary
fun & ∃many things to do even now
・Quantum computation is suitable for Hamiltonian formalism
which is free from sign problem

・Instead we have to deal with huge vector space.
Quantum computers in future may do this job.
・We’ve constructed the vacuum of Schwinger model w/
the topological term by adiabatic state preparation

・found agreement in the chiral condensate with the exact
result for 𝑚 = 0 & mass perturbation theory for small 𝑚[Chakraborty-MH-Kikuchi-Izubuchi-Tomiya ’20]

・explored the screening vs confinement problem[MH-Itou-Kikuchi-Nagano-Okuda ’21]

Other progress & Outlook
・Searching critical point at 𝜃 = 𝜋[work in progress, Chakraborty-MH-Kikuchi-Izubuchi-Tomiya]・Other ways to prepare vacuum (e.g. variational method, imaginary time evolution)[work in progress, MH-Kikuchi-Rendon]・Multi charge for dynamical fermion [work in progress, MH-Itou-Kikuchi-Tanizaki]・Matrix QM & (non-)SUSY QFTs/String theory[Buser-Gharibyan-Hanada-MH-Liu ’20]

・Non-Abelian & higher dimensions?

・Scattering amplitude?
・Including quantum error correction/mitigation?
・Simulation of matrix QM

Thanks!

Appendix

𝑇-dependence of the systematic errors
Parameters: 𝑔 = 1, 𝑎 = 1, 𝑁 = 8, 106 shots