量子計算と場の量子論
Masazumi Honda(本多正純)

25th Mar, 2024

KEK 素核研・物構研 連携研究会2024

お題：

量子コンピュータでの場の量子論のダイナミクス解明へ向けた話
特に、これまでの例だけでなく・より将来へ向けた展望・そのための乗り越えるべき壁
・新たに参入しようとする人へ向けた挑戦状

本講演：・(場の量子論に関して) 何ができるようになりそうか
・そのためにどうするべきか

できるようになりそうなこと (期待)etc…

量子計算機でできるようになりそうなこと： (詳細は後述)
従来の数値的アプローチ(=モンテカルロ法)では問題(=符号問題)がある状況のシミュレーション
符号問題が起きやすい物理的状況：・トポロジカル項をもつ系 (ex. Chern-Simons項, theta項)
・フェルミオン系に化学ポテンシャルを入れた場合
これが原因でQCD(量子色力学)の相図が未完成
・実時間系

言うまでもなく重要課題・ 一定以上の性能をもつ量子計算機が必要(qubit数, fidelity, etc...)・ 量子計算機で効率的に解ける問題の開拓

課題・ 一定以上の性能をもつ量子計算機が必要(qubit数, fidelity, etc...)・ 量子計算機で効率的に解ける問題の開拓
やるべきこと：(ハードの発展を待つ以外に)・ベンチマーク・必要な性能の見積り・手法の開発・改善 → 必要な性能を下げる
演算子形式での格子上の場の理論の整備
特にゲージ理論の適切な正則化方法の模索
・場の量子論 → 量子計算？
新たな誤り訂正方法の提案など

Contents
1. Introduction
2. QC for QFT
3. QFT for QC (?)
4. Summary

Sign problem in Monte Carlo simulation
Conventional approach to simulate QFT:
① Discretize Euclidean spacetime by lattice:& make path integral finite dimensional:
probability②Sign problem in Monte Carlo simulation
Conventional approach to simulate QFT:
① Discretize Euclidean spacetime by lattice:& make path integral finite dimensional:
probability② Numerically Evaluate it by (Markov Chain) Monte Carlo method

regarding the Boltzmann factor as a probability:Sign problem in Monte Carlo simulation
Markov Chain Monte Carlo:
probability

can’t directly apply when Boltzmann factor isn’t R≧0Sign problem in Monte Carlo simulation
Markov Chain Monte Carlo:
probability

can’t directly apply when Boltzmann factor isn’t R≧0Naïve way to avoid = reweighting:𝒪 𝜙=∫ 𝐷𝜙 𝒪 𝜙 𝑒 −𝑆 𝜙

∫ 𝐷𝜙 𝑒 −𝑆[𝜙]=∫ 𝐷𝜙 𝒪 𝜙 𝑒 −𝑆 𝜙 ∫ 𝐷𝜙 𝑒 −𝑆 𝜙=

∫ 𝐷𝜙 𝑒 −𝑆 𝜙

∫ 𝐷𝜙 𝑒 −𝑆[𝜙]𝒪 𝜙 ⋅phase(𝑒 −𝑆 ) no−phase
phase(𝑒 −𝑆 ) no−phaseSign problem in Monte Carlo simulation
Markov Chain Monte Carlo:
probability

can’t directly apply when Boltzmann factor isn’t R≧0Naïve way to avoid = reweighting:𝒪 𝜙=∫ 𝐷𝜙 𝒪 𝜙 𝑒 −𝑆 𝜙

∫ 𝐷𝜙 𝑒 −𝑆[𝜙]=∫ 𝐷𝜙 𝒪 𝜙 𝑒 −𝑆 𝜙 ∫ 𝐷𝜙 𝑒 −𝑆 𝜙=
For highly oscillating integral, ∼∫ 𝐷𝜙 𝑒 −𝑆 𝜙∫ 𝐷𝜙 𝑒 −𝑆[𝜙]𝒪 𝜙 ⋅phase(𝑒 −𝑆 ) no−phasephase(𝑒 −𝑆 ) no−phase00

needs huge statistics“sign problem”

Sign problem in Monte Carlo simulation (Cont’d)Markov Chain Monte Carlo:
probability

problematic when Boltzmann factor isn’t R≧0 & is highly oscillating
Examples w/ sign problem:
・topological termcomplex action

・chemical potential

indefinite sign of fermion determinant・real time

“ 𝑒 𝑖𝑆(𝜙) ” much worse

In operator formalism,
sign problem is absent from the beginning(∃various approaches within framework of path integral formalism but I’ll skip it )

Cost of operator formalism
We have to play with huge vector spacesince QFT typically has ∞-dim. Hilbert space
regularization needed!

Technically, computers have tomemorize huge vector & multiply huge matrices

Quantum computers do this job?

“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 2285 at this moment)(1+1)d free Dirac fermion
Continuum:

𝛾 0 = 𝜎3,ത 1 𝜕1 𝜓 + 𝑚𝜓𝜓ത
𝐻 = ∫ 𝑑𝑥 −𝑖𝜓𝛾𝛾 1 = 𝑖𝜎2

= ∫ 𝑑𝑥 −𝑖(𝜓𝑢† 𝜕1 𝜓𝑑 + 𝜓𝑑† 𝜕1 𝜓𝑢 ) + 𝑚(𝜓𝑢† 𝜓𝑢 − 𝜓𝑑† 𝜓𝑑 )Lattice (w/ 𝑁 sites and spacing 𝑎):𝜒𝑛𝑎1/2“Staggered fermion”

𝜓(𝑥) =𝜓𝑢𝜓𝑑[Susskind, Kogut-Susskind ’75]

odd siteeven site

𝑁−1𝑁

𝑛=1𝑛=1𝑖†
𝐻=−෍ 𝜒𝑛† 𝜒𝑛+1 − 𝜒𝑛+1
𝜒𝑛 + 𝑚 ෍ −1 𝑛 𝜒𝑛† 𝜒𝑛2𝑎{𝜒𝑚 , 𝜒𝑛† } = 𝛿mn , {𝜒𝑚 , 𝜒𝑛 } = 0

Jordan-Wigner transformation†{𝜒𝑚 , 𝜒𝑛 } = 𝛿mn , {𝜒𝑚 , 𝜒𝑛 } = 0

This is satisfied by the operator:[Jordan-Wigner’28](𝑋𝑛 , 𝑌𝑛 , 𝑍𝑛 : 𝜎1,2,3 at site 𝑛)

Then the system is mapped to the spin system:

Now we can apply quantum algorithms to QFT!

Scalar field theory
Continuum Hamiltonian:
1 2 1
𝐻 = ∫ 𝑑 𝒙 Π + 𝜕𝑖 𝜙 2 + 𝑉 𝜙22𝑑∫ 𝑑 𝑑 𝑥 → 𝑎𝑑 ෍ ,𝑛𝜕𝜇 𝜙 𝑥 → Δ𝜇 𝜙 𝑥𝑛 ≡𝜙 𝑥𝑛 +𝑎𝑒𝜇 −𝜙(𝑥𝑛 )Lattice Hamiltonian (simplest):
𝐻 = 𝑎𝑑 ෍𝑛

1 2 1
Πn + ෍ Δ𝑖 𝜙𝑛 2 + 𝑉(𝜙𝑛 )22𝜙 𝒙𝒎 , Π 𝒙𝒏𝑖= 𝑖𝛿𝒎,𝒏technically the same as multi-particle QM𝑎

Regularization for single particle QM
1 2 𝜔2 2෡ = 𝑝Ƹ +𝐻𝑥ො + 𝑉int (𝑥)ො22Most naïve approach = truncation in harmonic osc. basis:∞𝑎ො =𝜔𝑖𝑥ො +𝑝Ƹ = ෍ 𝑛 + 1 |𝑛⟩⟨𝑛 + 1|
22𝜔
𝑛=0
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Regularization for 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 20Then,′𝑛 𝑛 + 1 = ⊗𝐾−1(|𝑏ℓ=0ℓ ⟩⟨𝑏ℓ |)either one of𝐾 ≡ log 2 Λ(binary representation)

Pure Maxwell theoryContinuum:

1 2 1 2
ℋ = 𝐸𝑖 + 𝐵𝑖
22

𝜕𝑖 𝐸 𝑖 = 0Lattice:𝑎𝑑ℋ=෍ 𝐿2𝒏,𝑖 + Re
2𝒏,𝑖෍

෍

ෑ

plaquette 𝑖<𝑗 𝑃∈plaquette[𝑈𝒎,𝑖 , 𝐿𝒏,𝑗 ] = 𝑖𝛿𝑖𝑗 𝛿𝒎,𝒏Gauss law:෍(𝐿𝒏+𝒆𝒊 ,𝑖 −𝐿𝒏,𝑖 ) = 0𝑖𝑈𝑃

Ex. (1+1)d pure Maxwell theory w/ 𝜃Continuum:
1 2𝜃
ℒ = 2 𝐹01 +𝐹012𝑔2𝜋

Lattice:

Gauss law:

Π=1𝜃ሶ+𝐴
g22𝜋1𝜃
ℋ=
Π−22𝜋𝑔2 𝑎𝜃𝐻=෍ 𝐿𝑛 +22𝜋22Π 𝑥
𝐿𝑛 ↔ −𝑔𝑛

𝐿𝑛+1 − 𝐿𝑛 = 0・open b.c.
𝐿𝑛 = 𝐿𝑛−1 = 𝐿𝑛−2 = ⋯ = 𝐿1 = (𝑏. 𝑐. )・p.b.c.
𝐿𝑛 = 𝐿𝑛−1 = ⋯ = 𝐿1 = ⋯ = 𝐿𝑛+1 = 𝐿𝑛
one d.o.f. remains

分野の大体の研究の流れ (?)
物理Spin chain

基底状態 時間発展 散乱/崩壊 非平衡 ・・・

Schwinger model
2+1d abelian
2+1d non-abelian

3+1d abelian
3+1d non-abelian

理論

Charge-𝑞 Schwinger model
gauge/electric field
𝑈0 , 𝐿0

𝑈2 , 𝐿2

𝑈1 , 𝐿1xxx

𝜒0𝜒1𝜒2x𝑎𝑈𝑁−2 , 𝐿𝑁−2・・・𝜒3fermionxx

𝜒𝑁−2

𝜒𝑁−1

Physical states are subject to Gauss law:
𝐿𝑛 − 𝐿𝑛−1 |phys⟩"∇ ⋅ 𝐸 (𝑥)"=𝑞1−†
𝜒𝑛 𝜒𝑛 −−1 𝑛|phys⟩2"𝜌(𝑥) "Schwinger model as qubits
1. Take open b.c. & solve Gauss law:w/ 𝐿−1 = 0

2. Take the gauge 𝑈𝑛 = 1
3. Map to spin system:(𝑋𝑛 , 𝑌𝑛 , 𝑍𝑛 : 𝜎1,2,3 at site 𝑛)“Jordan-Wigner transformation”[Jordan-Wigner’28]

Schwinger model as qubits
1. Take open b.c. & solve Gauss law:w/ 𝐿−1 = 0

2. Take the gauge 𝑈𝑛 = 1
3. Map to spin system:(𝑋𝑛 , 𝑌𝑛 , 𝑍𝑛 : 𝜎1,2,3 at site 𝑛)“Jordan-Wigner transformation”[Jordan-Wigner’28]

Qubit description of the Schwinger model !!

Ground state expectation value in massless case
[Chakraborty-MH-Kikuchi-Izubuchi-Tomiya ’20]

(after continuum limit)

exact result

Screening versus Confinement
Let’s consider
potential between 2 heavy charged particles+𝑞𝑝−𝑞𝑝

Classical picture:

𝑉 𝑥 =2 𝑔2
𝑞𝑝2Coulomb law in 1+1d𝑥?

confinement

too naive in the presence of dynamical fermionsExpectations from previous analyzes
Potential between probe charges ±𝑞𝑝 has been analytically computed[Iso-Murayama ’88, Gross-Klebanov-Matytsin-Smilga ’95 ]・massless case:𝑉 𝑥 =・massive case:𝑞𝑝2 𝑔22𝜇𝜇 ≡ 𝑔/ 𝜋(1 − 𝑒 −𝑞𝜇𝑥 )screeningExpectations from previous analyzes
Potential between probe charges ±𝑞𝑝 has been analytically computed[Iso-Murayama ’88, Gross-Klebanov-Matytsin-Smilga ’95 ]・massless case:𝑉 𝑥 =・massive case:𝑞𝑝2 𝑔22𝜇𝜇 ≡ 𝑔/ 𝜋

(1 − 𝑒 −𝑞𝜇𝑥 )screeningΣ ≡ 𝑔𝑒 𝛾 /2𝜋 3/2[cf. Misumi-Tanizaki-Unsal ’19 ]𝜃 + 2𝜋𝑞𝑝𝜃
𝑉 𝑥 ∼ 𝑚𝑞Σ cos− cos𝑞𝑞𝑥(m ≪ 𝑔, 𝑥 ≫ 1/𝑔 )screening= Const.

for qp /q = 𝐙∝𝑥

for qp /q ≠ 𝒁 confinement?
but sometimes negative slope!

That is, as changing the parameters…

Let’s explore this aspect by quantum simulation!

Positive / negative string tension[MH-Itou-Kikuchi-Tanizaki ’21]
[cf. MH-Itou-Kikuchi-Nagano-Okuda ’21]

Parameters: 𝑔 = 1, 𝑎 = 0.4, 𝑁 = 25, 𝑇 = 99, 𝑞𝑝 /𝑞 = −1/3, 𝑚 = 0.15

Sign(tension) changes as changing 𝜃-angle!!

100 qubit simulation of Schwinger model(127-qubit device: ibm_cusco w/ error mitigation)

Ground state exp. of local chiral condensate :[Farrel-Illa-Ciavarella-Savage ’23]

Other simulations of Schwinger model・decay of massive vacuum under time evolution[cf. Martinez etal. Nature 534 (2016) 516-519]

・quenched dynamics of 𝜃 [Nagano-Bapat-Bauer ’23]

・Schwinger model in open quantum system[De Jong-Metcalf-Mulligan-Ploskon-Ringer-Yao ’20, de Jong-Lee-Mulligan-Ploskon-Ringer-Yao ’21,
Lee-Mulligan-Ringer-Yao ’23]

・112 qubit simulation of meson propagation[Farrell-Illa-Ciavarella-Savage ’24]

・finding energy spectrum [MH-Ghim, work in progress]
・finite temperature [Itou-Sun-Pedersen-Yunoki ’23]

etc…

Scattering in Thirring model
Thirring model on lattice:[Chai-Crippa-Jansen-Kuhn-Pascuzzi-Tacchino-Tavernelli ’23]

Scattering in Thirring model
Thirring model on lattice:[Chai-Crippa-Jansen-Kuhn-Pascuzzi-Tacchino-Tavernelli ’23]

Particle density of two wave packets:(12-qubit device: ibm_peekskill
w/ error mitigation )On higher dimensional fermion
Go to higher dimensions!

[MH, work in progress]

1st step: find a nice way to map 2d fermion to spins
Problem in naïve approach:

・1d
†
𝜒𝑛+1
𝜒𝑛

・2d

Jordan-Wigner∃𝑋𝑛+1 𝑋𝑛 ,𝑌𝑛+1 𝑌𝑛 , 𝑋𝑛+1 𝑌𝑛 , 𝑌𝑛+1 𝑋𝑛

local

On higher dimensional fermion
Go to higher dimensions!

[MH, work in progress]

1st step: find a nice way to map 2d fermion to spins
Problem in naïve approach:・1d†𝜒𝑛+1𝜒𝑛Jordan-Wigner∃𝑋

𝑛+1 𝑋𝑛 ,𝑌𝑛+1 𝑌𝑛 , 𝑋𝑛+1 𝑌𝑛 , 𝑌𝑛+1 𝑋𝑛local

・2d (𝑁 × 𝑁 square lattice)
Relabeling site (𝑖, 𝑗) like 1d label (say 𝑛 = 𝑖 + 𝑁𝑗),††
𝜒(𝑖,𝑗+1) 𝜒(𝑖,𝑗) = 𝜒𝐼+𝑁 𝜒𝐼

JW

𝐼+𝑁−1ς𝑋
𝐼+𝑁 𝐼 𝑖=𝐼+1 𝑍𝑖 , etc…

∃𝑋

(cf. 𝒪(log𝑁) for Bravyi-Kitaev trans.)

non-local

Application of a new map to field theory[Chen-Kapustin-Radicevic ’17]

2 Majorana fermions on faceSpin op. on edge

where

“Gauss law” constraint at site 𝑣:𝐿(𝑒)𝑒 𝑅(𝑒)𝑟(𝑒)𝑒𝑒𝑟(𝑒)

ex.)local

Some other applications
・Efficient simulation of (2+1)d U(1) gauge th.
[Kane-Grabowska-Nachman-Bauer ’22]

・Inflation (scalar in curved spacetime) [Liu-Li ’20]
・Chiral fermion [Hayata-Nakayama-Yamamoto ’23]
・Quantum group approach to Non-abelian gauge th.
[Zache-Gonzalez-Cuadra-Zoller ’23, Hayata-Hidaka ’23]

・Conformal bootstrap [Bao-Liu ’18]
・Dark sector showers[Chigusa-Yamazaki ’22, Bauer-Chigusa-Yamazaki ’23]

・Measurement-based quantum computation[Okuda-Sukeno ’22]

・quantum machine learning [Nagano-Miessen-OnoderaTavernelli-Tacchino-Terashi ’23, etc…]
・String/M-theory[Gharibyan-Hanada-MH-Liu ’20]

etc…

Contents
1. Introduction
2. QC for QFT
3. QFT for QC (?)
4. Summaryrelations between QEC & gauge theoryMotivations[Spirit may be similar to Rajput-Roggaro-Wiebe ’21, Gustafson-Lamm ’23, etc...]1. ∃explicit examplesex.) Toric code = 𝒁𝟐 lattice gauge theory [Kitaev ’97]
2.

3.
4.

relations between QEC & gauge theory
Motivations[Spirit may be similar to Rajput-Roggaro-Wiebe ’21, Gustafson-Lamm ’23, etc...]1. ∃explicit examplesex.) Toric code = 𝒁𝟐 lattice gauge theory [Kitaev ’97]
2. Conceptual similarities:
QEC= redundant description of logical qubits
Gauge theory = redundant description of physical states

3.
4.

relations between QEC & gauge theory
Motivations[Spirit may be similar to Rajput-Roggaro-Wiebe ’21, Gustafson-Lamm ’23, etc...]1. ∃explicit examplesex.) Toric code = 𝒁𝟐 lattice gauge theory [Kitaev ’97]
2. Conceptual similarities:
QEC= redundant description of logical qubits
Gauge theory = redundant description of physical states

3. Nature = Gauge theory & Nature = Quantum computer
Gauge theory may know something on QEC?
4.

relations between QEC & gauge theory
Motivations[Spirit may be similar to Rajput-Roggaro-Wiebe ’21, Gustafson-Lamm ’23, etc...]1. ∃explicit examplesex.) Toric code = 𝒁𝟐 lattice gauge theory [Kitaev ’97]
2. Conceptual similarities:
QEC= redundant description of logical qubits
Gauge theory = redundant description of physical states

3. Nature = Gauge theory & Nature = Quantum computer
Gauge theory may know something on QEC?
4. ∃proposals on relations among QEC & concepts in HEPex.) Holography, Black hole, CFT, Renormalization group[Almheiri-Dong-Harlow ’14, Hayden-Preskill ’07, Dymarsky-Shapere ’20, Kawabata-Nishioka-Okuda ’22,
Furuya-Lashkari-Moosa ’21, etc...]Gauge theory on QC w/ error correction
When we don’t solve Gauss law before simulation…

|phys⟩

Gauge theory on QC w/ error correction
When we don’t solve Gauss law before simulation…

|phys⟩

unconstrained(logical qubits)

Gauge theory on QC w/ error correction
When we don’t solve Gauss law before simulation…

|phys⟩

unconstrained(logical qubits)

encoded2

redundancy ‼

What I’m doing…

[MH, work in progress]

to make dictionary for classes of codes/gauge theories:

QEC

Gauge theoryerrors

unphysical op. (& excitation)

logical qubits

physical states (w/ low energy)“no error conditions”

Gauss law (& min[energy])(stabilizer)

logical op.

gauge invariant op.

ancilla for recovery

additional matter⋮⋮QECGauge

QECorGauge

QECGaugeor???

QECGauge

QFT as a generator of error correcting code?
Toric code
・Lattice model interpreted as QEC
・Low energy effective theory = QFT (BF theory)

QFT ↔ Lattice model ↔ QEC
Idea：if we get something new in one of them,
then try to fill the other partsex.) “Dipolar” generalization of Toric code

[Pace-Wen ’22]

corresponds to a “layer” of BF theory w/ some rule
[Ebisu-MH-Nakanishi ’23]

Summary

できるようになりそうなこと (期待)

etc…

課題
・ 一定以上の性能をもつ量子計算機が必要(qubit数, fidelity, etc...)・ 量子計算機で効率的に解ける問題の開拓
やるべきこと：(ハードの発展を待つ以外に)・ベンチマーク・必要な性能の見積り・手法の開発・改善 → 必要な性能を下げる
演算子形式での格子上の場の理論の整備
特にゲージ理論の適切な正則化方法の模索
・場の量子論 → 量子計算？
新たな誤り訂正方法の提案など

Thanks!