Gausss law and Hilbert space constructions for U(1) lattice gauge - PowerPoint PPT Presentation

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 0

Outline Context of U(1) study 1 Conventional Hamiltonian LGT set-up 2 Reformulation with Gauss’s law solved 3 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 Context of U(1) study 1 Conventional Hamiltonian LGT set-up 2 Reformulation with Gauss’s law solved 3 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 Context of U(1) study 1 Conventional Hamiltonian LGT set-up 2 Reformulation with Gauss’s law solved 3 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 Periodic Boundary Link operators raise or lower electric P P e e r r i i o o field: .. d d i i c c .. B B o o u u n .. .. n d d a a .. .. r r .. y y .. .. eigenbasis eigenbasis .. 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 Periodic Boundary Link operators raise or lower electric P P e e r r i i o o field: .. d d i i c c .. B B o o u u n .. .. n d d a a .. .. r r .. y y .. .. eigenbasis eigenbasis .. Kogut-Susskind Hamiltonian: « ¯ff 1 1 1 ´ ÿ g 2 t ˆ E 2 ÿ 2 ´ ˆ P p ´ ˆ P : H E “ H B “ ˜ ℓ , p g 2 2 a s 2 a s ˜ s p ℓ Ñ H “ 1 ż a s Ñ 0 d D x p E 2 ` B 2 q H E ` H B Ý 2 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 Must impose Gauss’s law on kets [Kogut and Susskind 1975; 1 Zohar et al. 2017] Most directions in H unphysical. Danger of leaving H phys due to errors, noise If truncating states (by e.g. | E ℓ | ď Λ in U p 1 q ), 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 Must impose Gauss’s law on kets [Kogut and Susskind 1975; 1 Zohar et al. 2017] Most directions in H unphysical. Danger of leaving H phys due to errors, noise If truncating states (by e.g. | E ℓ | ď Λ in U p 1 q ), makes awkward constraints around cutoff. Electric fluctuations large at weak coupling 2 Expect large E fluctuations as a s Ñ 0 in D “ 2 gauge theories and in asymptotically-free theories in D “ 3 Rate of convergence as a s 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 Context of U(1) study 1 Conventional Hamiltonian LGT set-up 2 Reformulation with Gauss’s law solved 3 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 E ` ˆ ˆ “ H H B , « ¯ff 1 1 ´ ˆ ÿ 2 ´ ˆ P p ´ ˆ P : “ H B , p g 2 2 a s ˜ s p « ¯ff g 2 1 ˜ ´ Q : ˆ t ÿ 2 ´ ˆ Q ℓ ´ ˆ “ H E . ℓ ξ 2 2 a s ℓ Hilbert space H and ˆ H B are conventional 1 Different, 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 E ` ˆ ˆ “ H H B , « ¯ff 1 1 ´ ˆ ÿ 2 ´ ˆ P p ´ ˆ P : “ H B , p g 2 2 a s ˜ s p « ¯ff g 2 1 ˜ ´ Q : ˆ t ÿ 2 ´ ˆ Q ℓ ´ ˆ “ H E . ℓ ξ 2 2 a s ℓ Hilbert space H and ˆ H B are conventional But we exponentiated E : Q ℓ ” e iξ ˆ ˆ E ℓ ξ ! 1 1 Different, 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? H E Ą ˆ ˆ Q ℓ : just apply phases H B Ą ˆ ˆ P p : 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? H E Ą ˆ ˆ Q ℓ : just apply phases H B Ą ˆ ˆ P p : excite electric flux loops Simplest physical state: | Ω y ” b ℓ | 0 y ℓ 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? H E Ą ˆ ˆ Q ℓ : just apply phases H B Ą ˆ ˆ P p : excite electric flux loops Simplest physical state: | Ω y ” b ℓ | 0 y ℓ Basis for H phys 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 Plaquette powers, A p : Encoding for valid ¯ A p | Ω y ´ ź ˆ E ℓ configurations. . . or new quantum num- | A L y ” P p bers p 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 Plaquette powers, A p : Encoding for valid ¯ A p | Ω y ´ ź ˆ E ℓ configurations. . . or new quantum num- | A L y ” P p bers p L Notice: Plaquettes p „ dual sites n ‹ . ñ A p is scalar field A n ‹ on L ‹ . E ℓ on a link „ difference ∆ A n ‹ along a dual link L * D.B. Kaplan & J.R. Stryker (INT@UW) Gauss’s law & Hilbert spaces: U(1) 2018-09-12 Next steps in QS 11