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Gauss’s law and Hilbert space constructions for U(1) lattice gauge theories

David B. Kaplan & Jesse R. Stryker

Institute for Nuclear Theory University of Washington

Next steps in Quantum Science for HEP Work based on arXiv:1806.08797

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS

Outline

1

Context of U(1) study

2

Conventional Hamiltonian LGT set-up

3

Reformulation with Gauss’s law solved Original theory set-up Closer look at physical Hilbert space Formulation in the dual

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 1

Context of U(1) study

Roadmap

1

Context of U(1) study

2

Conventional Hamiltonian LGT set-up

3

Reformulation with Gauss’s law solved Original theory set-up Closer look at physical Hilbert space Formulation in the dual

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 2

Context of U(1) study

Unitary evolution on a quantum computer

Digital quantum computers (QC): Unitary gates „ e´it ˆ

H of

some ˆ H. Want to simulate a lattice gauge theory (LGT) How to map its ˆ H and its Hilbert space H on to QC?

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 3

Context of U(1) study

Unitary evolution on a quantum computer

Digital quantum computers (QC): Unitary gates „ e´it ˆ

H of

some ˆ H. Want to simulate a lattice gauge theory (LGT) How to map its ˆ H and its Hilbert space H on to QC? Near-term QC architectures will have very limited capabilities How to most wisely spend those qubits?

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 3

Context of U(1) study

Previous work

Arena for these questions is the Hamiltonian formalism of LGT

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 4

Context of U(1) study

Previous work

Arena for these questions is the Hamiltonian formalism of LGT We seek most economical construction for pure U(1) LGT

Small step toward more interesting gauge theories Can serve as benchmark for near-term quantum simulations

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 4

Context of U(1) study

Previous work

Arena for these questions is the Hamiltonian formalism of LGT We seek most economical construction for pure U(1) LGT

Small step toward more interesting gauge theories Can serve as benchmark for near-term quantum simulations

Construction leads directly to dual theory

Dualities also extensively studied in LGTs and many other areas See, e.g., [Anishetty and Sharatchandra 1990; Mathur 2006; Anishetty and Sreeraj 2018]

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 4

Conventional Hamiltonian LGT set-up

Roadmap

1

Context of U(1) study

2

Conventional Hamiltonian LGT set-up

3

Reformulation with Gauss’s law solved Original theory set-up Closer look at physical Hilbert space Formulation in the dual

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 5

Conventional Hamiltonian LGT set-up

Conventional construction

Periodic Boundary eigenbasis P e r i

• d

i c B

• u

n d a r y Periodic Boundary eigenbasis

.. .. .. .. .. .. .. .. .. ..

P e r i

• d

i c B

• u

n d a r y

Link operators raise

• r

lower electric field:

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 6

Conventional Hamiltonian LGT set-up

Conventional construction

Periodic Boundary eigenbasis P e r i

• d

i c B

• u

n d a r y Periodic Boundary eigenbasis

.. .. .. .. .. .. .. .. .. ..

P e r i

• d

i c B

• u

n d a r y

Link operators raise

• r

lower electric field: Kogut-Susskind Hamiltonian: HE “ 1 2as ÿ

ℓ

˜ g2

t ˆ

E2

ℓ ,

HB “ 1 2as « 1 ˜ g2

s

ÿ

p

´ 2 ´ ˆ Pp ´ ˆ P :

p

¯ff HE ` HB

asÑ0

Ý Ñ H “ 1 2 ż dDx pE2 ` B2q

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 6

Conventional Hamiltonian LGT set-up

Issues with standard formulation

1

Must impose Gauss’s law on kets [Kogut and Susskind 1975; Zohar et al. 2017]

Most directions in H unphysical. Danger of leaving Hphys due to errors, noise If truncating states (by e.g. |Eℓ| ď Λ in Up1q), makes awkward constraints around cutoff.

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 7

Conventional Hamiltonian LGT set-up

Issues with standard formulation

1

Must impose Gauss’s law on kets [Kogut and Susskind 1975; Zohar et al. 2017]

Most directions in H unphysical. Danger of leaving Hphys due to errors, noise If truncating states (by e.g. |Eℓ| ď Λ in Up1q), makes awkward constraints around cutoff.

2

Electric fluctuations large at weak coupling

Expect large E fluctuations as as Ñ 0 in D “ 2 gauge theories and in asymptotically-free theories in D “ 3 Rate of convergence as as Ñ 0 unclear when truncating on E

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 7

Reformulation with Gauss’s law solved

Roadmap

1

Context of U(1) study

2

Conventional Hamiltonian LGT set-up

3

Reformulation with Gauss’s law solved Original theory set-up Closer look at physical Hilbert space Formulation in the dual

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 8

Reformulation with Gauss’s law solved Original theory set-up

Starting point for original theory

We start with a symmetric Hamiltonian,1 ˆ H “ ˆ HE ` ˆ HB , ˆ HB “ 1 2as « 1 ˜ g2

s

ÿ

p

´ 2 ´ ˆ Pp ´ ˆ P :

p

¯ff , ˆ HE “ 1 2as « ˜ g2

t

ξ2 ÿ

ℓ

´ 2 ´ ˆ Qℓ ´ ˆ Q:

ℓ

¯ff . Hilbert space H and ˆ HB are conventional

1Different, but similar to [Horn, Weinstein, and Yankielowicz 1979]. D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 9

Reformulation with Gauss’s law solved Original theory set-up

Starting point for original theory

We start with a symmetric Hamiltonian,1 ˆ H “ ˆ HE ` ˆ HB , ˆ HB “ 1 2as « 1 ˜ g2

s

ÿ

p

´ 2 ´ ˆ Pp ´ ˆ P :

p

¯ff , ˆ HE “ 1 2as « ˜ g2

t

ξ2 ÿ

ℓ

´ 2 ´ ˆ Qℓ ´ ˆ Q:

ℓ

¯ff . Hilbert space H and ˆ HB are conventional But we exponentiated E: ˆ Qℓ ” eiξ ˆ

Eℓ

ξ ! 1

1Different, but similar to [Horn, Weinstein, and Yankielowicz 1979]. D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 9

Reformulation with Gauss’s law solved Closer look at physical Hilbert space

Physical Hilbert space generation

Practical question: What does ˆ H do in electric basis? ˆ HE Ą ˆ Qℓ: just apply phases ˆ HB Ą ˆ Pp: excite electric flux loops

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 10

Reformulation with Gauss’s law solved Closer look at physical Hilbert space

Physical Hilbert space generation

Practical question: What does ˆ H do in electric basis? ˆ HE Ą ˆ Qℓ: just apply phases ˆ HB Ą ˆ Pp: excite electric flux loops Simplest physical state: |Ωy ” bℓ |0yℓ

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 10

Reformulation with Gauss’s law solved Closer look at physical Hilbert space

Physical Hilbert space generation

Practical question: What does ˆ H do in electric basis? ˆ HE Ą ˆ Qℓ: just apply phases ˆ HB Ą ˆ Pp: excite electric flux loops Simplest physical state: |Ωy ” bℓ |0yℓ Basis for Hphys is generated by acting with plaquettes on trivial state!

eigenbasis Powers of plaquettes

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 10

Reformulation with Gauss’s law solved Closer look at physical Hilbert space

Hilbert space transcription

|ALy ” ź

p

´ ˆ Pp ¯Ap |Ωy Plaquette powers, Ap: Encoding for valid Eℓ configurations. . . or new quantum num- bers

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 11

Reformulation with Gauss’s law solved Closer look at physical Hilbert space

Hilbert space transcription

|ALy ” ź

p

´ ˆ Pp ¯Ap |Ωy Plaquette powers, Ap: Encoding for valid Eℓ configurations. . . or new quantum num- bers

L L *

Notice: Plaquettes p „ dual sites n‹. ñ Ap is scalar field An‹ on L‹. Eℓ on a link „ difference ∆An‹ along a dual link

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 11

Reformulation with Gauss’s law solved Closer look at physical Hilbert space

Hilbert space transcription

|ALy ” ź

p

´ ˆ Pp ¯Ap |Ωy Plaquette powers, Ap: Encoding for valid Eℓ configurations. . . or new quantum num- bers

L L *

Notice: Plaquettes p „ dual sites n‹. ñ Ap is scalar field An‹ on L‹. Eℓ on a link „ difference ∆An‹ along a dual link Identify ź

p

´ ˆ Pp ¯Ap ˇ ˇ ˇ ˇ ˇ

Ap“An‹ppq

|Ωy Ð Ñ bn‹ |An‹y

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 11

Reformulation with Gauss’s law solved Closer look at physical Hilbert space

Hilbert space transcription

1.) Define identical local orthonor- mal bases, t|An‹yu, which diago- nalize ˆ Un‹ ”

8

ÿ

An‹“´8

|An‹y eiξAn‹ xAn‹| . 2.) Global basis states: |AL‹y ” bn‹ |An‹y 3.) (Local) raising operators: ˆ Qn‹ ”

8

ÿ

An‹“´8

|An‹ ` 1y xAn‹| .

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 12

Reformulation with Gauss’s law solved Closer look at physical Hilbert space

Hilbert space transcription

1.) Define identical local orthonor- mal bases, t|An‹yu, which diago- nalize ˆ Un‹ ”

8

ÿ

An‹“´8

|An‹y eiξAn‹ xAn‹| . 2.) Global basis states: |AL‹y ” bn‹ |An‹y 3.) (Local) raising operators: ˆ Qn‹ ”

8

ÿ

An‹“´8

|An‹ ` 1y xAn‹| . Redundancy:

… Same electric fields!

Since ś

p

´ ˆ Pp ¯ “ ˆ 1, must impose ź

n‹

ˆ Qn‹ |AL‹y “ |AL‹y

• n H‹.

This is magnetic Gauss law.

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 12

Reformulation with Gauss’s law solved Formulation in the dual

Reformulation in dual variables

Original Dual plaquette, p Ø site, n‹ plaquette operator, ˆ Pp Ø site raising operator, ˆ Qn‹ link, ℓ Ø (perpendicular) link, ℓ‹ field square, E2

ℓ

Ø field laplacian, ˆ U :

n‹B` i B´ i

ˆ Un‹

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 13

Reformulation with Gauss’s law solved Formulation in the dual

Reformulation in dual variables

Original Dual plaquette, p Ø site, n‹ plaquette operator, ˆ Pp Ø site raising operator, ˆ Qn‹ link, ℓ Ø (perpendicular) link, ℓ‹ field square, E2

ℓ

Ø field laplacian, ˆ U :

n‹B` i B´ i

ˆ Un‹ We have xA 1

L| ˆ

H |ALy “ xA 1

L‹| H |AL‹y for the dual Hamiltonian

ˆ H “ 1 2as ÿ

n‹

„ 1 ˜ g2

s

´ 2 ´ ˆ Qn‹ ´ ˆ Q:

n‹

¯ ´ ˜ g2

t

ξ2 a2

s ˆ

U :

n‹B` i B´ i

ˆ Un‹  , pD “ 2q (subject to magnetic Gauss).

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 13

Reformulation with Gauss’s law solved Formulation in the dual

Solving the dual Gauss law

1

Fix one An‹ “ 0 and solve its ˆ Qn‹ ë Break translational symmetries ë ˆ H becomes nonlocal

Truncation possible via |An‹| ď Amax (a modified E truncation)

L *

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 14

Reformulation with Gauss’s law solved Formulation in the dual

Solving the dual Gauss law

1

Fix one An‹ “ 0 and solve its ˆ Qn‹ ë Break translational symmetries ë ˆ H becomes nonlocal

Truncation possible via |An‹| ď Amax (a modified E truncation)

2

Restrict states to subspace on which ś

n‹ ˆ

Qn‹ “ 1

Truncation can be done on argument of Qn‹ phases (equivalent to regulating B in

• riginal theory)

L *

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 14

Summary

Summary

1

Building ∇ ¨ E “ 0 into 2+1 U(1) theory leads to dual, with fewer variables

2

Formulating and truncating dual theory potentially preferable for weak coupling

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 15

Summary

Summary

1

Building ∇ ¨ E “ 0 into 2+1 U(1) theory leads to dual, with fewer variables

2

Formulating and truncating dual theory potentially preferable for weak coupling Future directions Generalizing to matter & non-Abelian

Want: Local Hilbert spaces, ˆ H built from local operators How much redundancy?

Target calculations for near-term QCs. . .

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Summary

A next step: Benchmarks

Near-term QCs need simple benchmarks to compare against Tailor analytical/numerical calculations to what QCs can extract E.g.,

Low-energy spectrum Operator vacuum expectation values

Notice: Many extrapolations to control

as, volume, Trotter step, Hilbert cutoff

Issue of theoretical and practical interest: Quantitative comparison of E vs B truncation (current)

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Acknowledgments

Acknowledgments

Thank you to the Fermilab QIS/Theory workshop organizers. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant

• No. DGE-1256082.

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Appendix Further details

E fluctuations at weak coupling

Analogy to SHO: (electric field is momentum, gauge field is coordinate) HE “ 1 2as ÿ

ℓ

˜ g2

t ˆ

E2

ℓ

„ 1 2m ˆ p2 HB “ 1 2as « 1 ˜ g2

s

ÿ

p

´ 2 ´ ˆ Pp ´ ˆ P :

p

¯ff „ k 2 ˆ x2 Read off m „ 1{˜ g2

t ,

k „ 1{˜ g2

s

By dimensional analysis, xˆ p2y 9 ? mk „ 1 ˜ gt˜ gs , xˆ x2y 9 1 ? mk „ ˜ gt˜ gs

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 1

Appendix Further details

Topological sectors

Original formulation (on periodic lattice) has many gauge-invariant states decoupled from |Ωy Topological Polyakov loops are gauge-invariant Define class representatives, |νy ”

d

ź

i“1

´ ˆ WpCiq ¯νi |0y , νi P Z . with ˆ WpCiq the product of oriented ˆ Uℓ’s along a closed loop Ci wrapping direction i. An ˆ H containing only elementary Wilson loops cannot cause transitions Fully general state: |A yν “ ź

p

´ ˆ Pp ¯Ap |νy , Ap P Z

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Appendix Further details

Dual Hamiltonian with topology

Since ν’s don’t talk to each other, we fix ν. We must adapt H to get the right matrix elements: H Ñ H ν “ HB ` H ν

E ,

(HB unchanged) H ν

E

“ 1 2as ÿ

n‹

„ ´˜ g2

t

ξ2 a2

s ˆ

U :

n‹∆ ˆ

Un‹  , pD “ 2q Here we have generalized to a covariant Laplacian ∆ “ Σ2

i“1D` i D´ i ,

D`

1 Fn‹

“ pWtn‹,n‹´e1uFn‹´e1 ´ Fn‹q{as , D`

2 Fn‹

“ pWtn‹,n‹`e2uFn‹`e2 ´ Fn‹q{as , involving the (dual lattice) connection Wℓ‹ “ # eiξνi, if ℓ P Ci ; 1,

• therwise

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Appendix Further details

Dual Hamiltonian in d “ 3 ` 1

For D “ 3 spatial dimensions, p Ø ℓ‹ (rather than p Ø n‹). We define ˆ Qℓ‹’s and ˆ Uℓ‹’s on local dual link Hilbert spaces by direct analogy. Then ˆ Hν “ 1 2as „ÿ

ℓ‹

1 ˜ g2

s

´ 2 ´ ˆ Qℓ‹ ´ ˆ Q:

ℓ‹

¯ ` ˜ g2

t

ξ2 ÿ

p‹

´ 2 ´ ´ Wp‹ ˆ Pp‹ ` h.c. ¯¯ pD “ 3q.

Dual plaquettes ˆ Pp‹ are usual products of ˆ Uℓ‹’s, and Wp‹ “ # eiξνi, if ℓ P Ci ; 1,

• therwise .

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Appendix References

References I

Anishetty, R. and H. S. Sharatchandra (1990). “Duality transformation for non-Abelian lattice gauge theories”. In: Phys. Rev. Lett. 65,

• pp. 813–815. DOI: 10.1103/PhysRevLett.65.813.

Anishetty, Ramesh and T. P . Sreeraj (2018). “Mass gap in the weak coupling limit of (2+1)-dimensional SU(2) lattice gauge theory”. In:

• Phys. Rev. D97.7, p. 074511. DOI:

10.1103/PhysRevD.97.074511. arXiv: 1802.06198 [hep-lat]. Horn, D., M. Weinstein, and S. Yankielowicz (1979). “Hamiltonian Approach to Z(N) Lattice Gauge Theories”. In: Phys. Rev. D19,

• p. 3715. DOI: 10.1103/PhysRevD.19.3715.

Kogut, John B. and Leonard Susskind (1975). “Hamiltonian Formulation of Wilson’s Lattice Gauge Theories”. In: Phys. Rev. D11, pp. 395–408. DOI: 10.1103/PhysRevD.11.395.

D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 5

Appendix References

References II

Mathur, Manu (2006). “The loop states in lattice gauge theories”. In:

• Phys. Lett. B640, pp. 292–296. DOI:

10.1016/j.physletb.2006.08.022. arXiv: hep-lat/0510101 [hep-lat]. Zohar, Erez et al. (2017). “Digital lattice gauge theories”. In: Phys. Rev. A95.2, p. 023604. DOI: 10.1103/PhysRevA.95.023604. arXiv: 1607.08121 [quant-ph].

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